MILL Wiki - User contributions [en]

Gas Pycnometer

2025-07-17T15:16:43Z

ObordenMEASURE: I wrote everything on the page except the intro.

{| class="infobox"
!
! style="font-size:135%;text-align:center;" |Micrometrics AccuPyc II 1345
|-
|                          
|[[File:Gas Pycnometer.jpg|center|frameless]]
|-
|'''Chamber Inserts'''
|10 cm<math>^3</math>, 3.5 cm<math>^3</math>, 1 cm<math>^3</math>
|-
|'''Maximum Chamber Pressure'''
|20.5 psig
|-
|'''Precision'''
|<math>\pm</math> 0.02% of nominal sample chamber volume
|-
|'''Accuracy'''
|<math>\pm</math> 0.03% of reading plus, <math>\pm</math> 0.03% of sample capacity
|}
The Micrometrics Gas Pycnometer uses high purity helium gas to determine the true volume of a sample. The sample is stored in in an air tight container, which is filled with helium gas, up to a certain pressure. This chamber is then opened up to a reference chamber with a precise known volume. The expansion of volume lowers the overall pressure, and there ratio between the initial and final pressure is used to calculate the true volume of the sample (the volume excluding contributions from pores). The primary application is measurement of porosity and true density.

== Operating Principle ==
Gas pycnometery works by flooding a chamber of known volume with a non-reactive gas. All other gasses are purged out, so that only the gas to be used for the measurement remains. A sample of unknown volume is placed within the chamber. All gas must then be purged from the chamber to start the measurement process.

Because of this, the sample should be as dry as possible, and the sample chamber should be sealed unless a sample is being loaded or unloaded. Any moisture that enters the chamber will have an associated vapor pressure, which will screw measurements.

Once all gas has been evacuated from the chamber, helium gas is pumped to the sample chamber so that a certain pressure is obtained. Helium gas has several advantages over other non-reactive gasses, such as nitrogen. Helium is the smallest gas, which allows it to fill in pores, cracks, and other defects more effectively. It also exhibits very ideal behavior, which is necessary, as the calculation for sample volume relies on a derivation of the ideal gas law.
[[File:Diagram of the operation of a gas pycnometer.png|center|thumb|371x371px|Diagram of a Gas Pycnometer<ref>Lowell, S. Shields, Joan E. Thomas, Martin A. Thommes, Matthias. ''Characterization of Porous Solids and Powders: Surface Area, Pore Size and Density''. Kluwer Academic Publishers, 2004</ref>]]
After the chamber has been flooded with the set amount of helium, a valve is opened, connecting the sample chamber to a reference chamber of known volume. The reference chamber is kept in a vacuum, so that the moles of gas remain constant. However, the pressure lowers as the helium redistributes itself between the sample and reference chambers.

The initial and final pressures of this process are used to calculate sample volume via this equation: <math>\text{V}_{s} = \text{V}_c + \frac{\text{V}_R}{1-(P_1/P_2)}</math>. V<math>_c</math> is the volume of the sample chamber, and V<math>_R</math> is the volume of the reference chamber, both of which are known. P<math>_1</math> and P<math>_2</math> and the measured pressures before and after the volume expansion.

The insertion of additional sample parameters such as mass and envelope density allow porosity, pore volume, and true density to be calculated with the use of the gas pycnometer. However, the only quantity directly measured by the instrument itself is true sample volume.

== Density ==
Density is the ratio of the mass of an object to its volume. However, many objects are porous, or contain interstitial spaces, such as powders. This leads of our having several different classification of density.

=== True Density ===
True density is the ratio of the mass to the volume occupied by that mass. This means that the volume to pores, cracks, or other internal voids should not be considered in calculating true density. The gas pycnometer is particular useful for measuring this type of density.

=== Apparent Density ===
Apparent density is an approximation of true density, which is measured using a liquid rather than a gas. Depending on the surface energy of the sample being measured, as well as the properties of the liquid used, some pores and cavities will not be filled. Gas displacement is the difference between true and apparent density. To approximate true density with a liquid the contact angle between the liquid and the sample should be minimized. If the contact angle exceeds 90º, then the liquid will not fills pores in the sample, unless forced to do so by some external pressure.

=== Bulk Density ===
Bulk density is the density of a material, including the volume of pores, and interstitial spaces in a powder. In essence it is the volume that would be measured it you were to use a graduated cylinder or a measuring cup for the measurement.

Bulk density is useful as a measure of the autoadhesivenss of a sample. This is because the amount of interstitial space in powders depends on particle size, shape, moisture content, and inter particle forces.

Tap density is a similar measurement to bulk density obtained by lifting and dropping the contained ("tapping") to force the powder to settle more compactly. Tap density is almost always higher than bulk density.

=== Envelope or Geometric Density ===
Envelope or geometric density is measured like bulk density, but with a single particle. It takes into account the volume of the sample, and the volume of internal pores to calculate density. Envelope density is generally obtained by measuring the external dimensions of a sample, and computing the volume geometrically. Envelope density can also be measured using a non-wetting fluid, one whose contact angle is greater than 90º. Without external pressure, a non-wetting fluid will not fill any pores or cavities, which allows the envelope density to be calculated.

If the envelope density of the sample is entered in the gas pycnometer software, this can be used to calculated pore volume and porosity.

=== Effective Density ===
A sample may sometimes contain foreign contaminants, or totally encapsulated pores. In these cases, neither gas, nor liquid displacement techniques will provide the true density due to the alien particles and blind or sealed pores. When this is the case, the density measured via gas displacement is called the effective density.

== Porosity ==
Porosity (ø) is ration of the volume taken up by pores within the material to the bulk volume. <math>\phi = \frac{V_B-V_S}{V_B} = \frac{V_P}{V_B}</math>.<ref>“Porosity.” ''Porosity - an Overview | ScienceDirect Topics'', ScienceDirect, www.sciencedirect.com/topics/engineering/porosity. Accessed 17 July 2025. </ref> The bulk volume (V<math>_B</math>) is the volume used to calculate bulk density, and the solid volume (V<math>_S</math>) is the volume used to calculate true density. The difference between two will be equal to the volume of the pores in the sample (V<math>_P</math>).

Porosity can be analyzed with the gas pycnometer if the correct parameters are provided. If you wish to study porosity, when creating your sample file you must select material parameters. So that porosity and pore volume may be calculated you may then supply either the envelope density, or the bulk density. Total Pore Volume is recorded in the "Pressure and Volume Table" report option. With the total pore volume, and the bulk volume of your sample, the porosity can be calculated manually.

File:Diagram of the operation of a gas pycnometer.png

2025-07-17T13:45:17Z

ObordenMEASURE:

Labeled diagram of the components of a gas pycnometer

File:Gas Pycnometer.jpg

2025-07-17T13:31:38Z

ObordenMEASURE:

A picture of the Gas Pycnometer turned on.

Category:Equipment

2025-07-16T15:35:22Z

ObordenMEASURE: Added Pycnometer page

See our [https://sites.gatech.edu/mse-mill/equipment/list/ equipment status list and SOPs].

Learn more about our [https://mill.mse.gatech.edu/equipment/about/ Equipment Teams].

==Characterization==
*[[Scanning Electron Microscope (SEM)]]
*[[Sputter Coater]]
*[[Optical Microscopy]]
**Digital Microscope
**Stereomicroscope
**Inverted Microscope
*[[Optical Profilometer]]
*[[Fourier-Transform Infrared Spectrometer (FTIR)]]
*[[X-ray Diffractometer (XRD)]]
*[[X-ray Fluorescence Spectrometer (XRF)]]
*[[Infrared Spectroscopy]]
*[[Plasma Sputtering]]

==Measurement==
*[[Mechanical Tester]]
*[[Microhardness Tester]]
*[[Differential Scanning Calorimeter (DSC)]]
*[[Thermogravimetric Analyzer (TGA)]]
*[[Moisture Analyzer]]
*[[Infrared Camera]]
*[[UV-Vis Spectrophotometer]]
*[[Contact Angle Goniometer]]
*[[Analytical Balance]]
*[[Brookfield DV2T Viscometer|Viscometer]]
*[[Gas Pycnometer|Pycnometer]]
*[[Abrasion Tester]]
*[[Squidstat]]

==Processing==

=== 3D Printing ===
* [[Fused Filament Fabrication (FFF)]]/Fused Deposition Modeling (FDM) Printers
** Hyrel System 30M
** Prusa i3 MK3S
* [[Stereolithography (SLA)]] Printers
** Prusa Resin Printer
** Formlabs resin printer
* [[Filament Recycling]]
** Granulator
** Filament Extruder

=== Science of Art ===
* Hot Plates
* [[Furnaces & Kilns]]
* [https://sites.gatech.edu/mse-mill/sop/vinyl-cutter/ Vinyl Cutter]

=== Textiles ===
* [[Sewing Machine]]
* [[Embroidery Machine]]
* [[Screen Printer]]
* [[Heat Press]]

=== Other ===
* Vacuum Former
* [[Liquid Nitrogen Generator]]

=== Ceramics Lab (Love 369) ===
* [[Uniaxial Press]]
* [[Ball Mill]]
* Vibratory Tumbler
* Sieve Shaker
* [[Furnaces & Kilns|Electric Furnaces]]

Gas Pycnometer

2025-07-16T15:33:25Z

ObordenMEASURE: Added summary

[[File:Page-under-construction.png|thumb|278x278px]]
The Micrometrics Gas Pycnometer uses high purity helium gas to determine the true volume of a sample. The sample is stored in in an air tight container, which is filled with helium gas, up to a certain pressure. This chamber is then opened up to a reference chamber with a precise known volume. The expansion of volume lowers the overall pressure, and there ratio between the initial and final pressure is used to calculate the true volume of the sample (the volume excluding contributions from pores). The primary application is measurement of porosity and true density.

Abrasion Tester

2025-07-09T19:05:29Z

ObordenMEASURE: Added Picture

The Taber Abrasion Tester is a single platform rotary abrasion tester. It preforms accelerated abrasion testing primarily on fabrics, but can also test plastics, coatings, laminates, paper, carpets, and ceramic.<ref>Taber Industries. Taber Abraser (Abrader)https://www.taberindustries.com/taber-rotary-abraser</ref> It operates by affixing a sample to the platform via a stud, 8mm in diameter. The platform then rotates. It can revolve with two speeds, 60 RPM, or 72 RPM. Two abrading arms lie above the platform. Different abrasion disks and weights can be attached to the arms, depending on the type of material being tested. The test is then run for a set number of cycles, and the durability of the sample is characterized via the wear and tear. This can be done qualitatively or quantitatively.

{|class="infobox"
!
!style="font-size:135%;text-align:center;"| Taber Rotary Abrasion Tester [https://mill.mse.gatech.edu/sop/abrasion-tester/ <nowiki>[SOP]</nowiki>]
|-
|
|[[File:Taber Rotary Abrasion Tester 1700.jpg|frameless]]
|-
|'''Platform Size'''
|~ 100 mm<math> ^2 </math>
|-
|'''Speeds'''
|60 RPM, 72 RPM
|-
|'''Stud Diameter'''
|8 mm
|}

== Sample Prep ==
For fabric or other flexible samples must have a 8 mm diameter hole cut/drilled in the center so that it can be slid onto the stud. Samples should be large enough in size to be draped over the platform; this implies a diameter of ~110 mm. The center piece should be tightened over it with a nut, and the hold down ring should be tightened down on top, to prevent the sample from moving during the test.

Rigid samples should also have an 8 mm in diameter hole drilled in the center. The sample should be cut to 100 mm square. A diameter of 100 mm is also acceptable. It is secured withe the center piece and nut clamping it down at the stud. The hold down ring is not necessary. Recommended sample height is 6.35 mm, higher samples may not be testing without the necessary accessories to do so.

Choice of abrasion disk is heavily dependent not the material being tested. Refer to the SOP for more details on the use of specific wheels.

== Opperation ==
The two abrading disks are designed so that one will abrade the sample towards the center, and the other will scrape it outwards towards the edge of the sample. Thus, as the sample is spun under the two abrading wheels, it will be worn in all directions relative to any potential grain or weave in the material. The total area of the sample to be abraded will cover approximately 30 cm<math>^2</math>.

== Data Analysis ==
There is not a standard unit or method of measuring abrasive wear. However, there are still many methods of measuring the abrasive resistance of a material. Some rely partially on qualitative judgements, while others are purely quantitative.

'''Cycles until End Point'''

The simplest method of quantifying abrasive resistance is in terms of the number of cycles required to reach a specified level of degradation. Reaching this level may be determined via visual qualitative clues, or through quantitive criteria, such as loss of mass, loss of breaking strength, yarn breakage, ect.

'''Mass Loss'''

Another simple method of determining how much a material has been worn down is by measuring the loss in mass. Because the vacuum removes all debris, this measures how much of the material as been removed from the whole by the abrasion disks. However, you must ensure that the sample is well cleaned, and precisely weighted before starting. Mass loss is calculated simply as the change in mass of the sample from before testing til after testing. Test settings must remain unchanged for this to be comparable to other samples.

'''Taber Wear Index'''

This measures the mass loss, per cycle. It can be calculated with this formula: <math> I = \frac{1000\times(M_f-M_i)}{C}</math>, where I is the Taber Wear Index, C is the number of Cycles, and <math> M_f \text{ and } M_i</math> are the final and initial masses, in milligrams, of the sample.

'''Volume Loss'''

This method is used to evaluate the results of materials with different specific gravities. Specific gravity is a measure of the density of a material compared to the density of a known reference usually water. To compare two such materials with varying specific gravities, the Taber Wear Index would be calculated for each material, and the index number would then be divided by the respective specific gravity. This would provide changes in volume, which could then be used to compare the two materials.

'''Residual Breaking Force'''

This evaluation technique is applicable mainly to textiles for fabrics. Tensile testing can be done on some samples before abrasion testing, and on others after abrasion testing. The difference in breaking force required for these tests shows how much weaker the sample has become as a result of the abrasion.

This can also be given as a percent loss in breaking force. Determine the pre and post test breaking loads, and use this to calculate the percent loss of breaking strength. This should be done for the fabric both lengthwise and widthwise. The average is then calculated according to this formula: <math> \text{AR} = \frac{100\times(X-Y)}{X}</math>, where AR is the abrasion resistance, X is the percent loss of breaking strength in lengthwise, and Y is the percent loss of breaking strength widthwise.

'''Depth of Wear'''

Abrasion can also be characterized via the change in depth of the material which has been abraded. When testing coatings, this is sometimes reported as the number of cycles required to wear through a coating of known depth.

File:Taber Rotary Abrasion Tester 1700.jpg

2025-07-09T19:04:36Z

ObordenMEASURE:

Overview picture of the MILL Abrasion Tester, July 2025

Abrasion Tester

2025-07-09T18:11:50Z

ObordenMEASURE: Added overview, Sample Prep, Operation, and Data Analysis section, as well as the info box. Pictures are still needed.

The Taber Abrasion Tester is a single platform rotary abrasion tester. It preforms accelerated abrasion testing primarily on fabrics, but can also test plastics, coatings, laminates, paper, carpets, and ceramic.<ref>Taber Industries. Taber Abraser (Abrader)https://www.taberindustries.com/taber-rotary-abraser</ref> It operates by affixing a sample to the platform via a stud, 8mm in diameter. The platform then rotates. It can revolve with two speeds, 60 RPM, or 72 RPM. Two abrading arms lie above the platform. Different abrasion disks and weights can be attached to the arms, depending on the type of material being tested. The test is then run for a set number of cycles, and the durability of the sample is characterized via the wear and tear. This can be done qualitatively or quantitatively.

{|class="infobox"
!
!style="font-size:135%;text-align:center;"| Taber Rotary Abrasion Tester [https://mill.mse.gatech.edu/sop/abrasion-tester/ <nowiki>[SOP]</nowiki>]
|-
|Place Picture Here
|
|-
|'''Platform Size'''
|~ 100 mm<math> ^2 </math>
|-
|'''Speeds'''
|60 RPM, 72 RPM
|-
|'''Stud Diameter'''
|8 mm
|}

== Sample Prep ==
For fabric or other flexible samples must have a 8 mm diameter hole cut/drilled in the center so that it can be slid onto the stud. Samples should be large enough in size to be draped over the platform; this implies a diameter of ~110 mm. The center piece should be tightened over it with a nut, and the hold down ring should be tightened down on top, to prevent the sample from moving during the test.

Rigid samples should also have an 8 mm in diameter hole drilled in the center. The sample should be cut to 100 mm square. A diameter of 100 mm is also acceptable. It is secured withe the center piece and nut clamping it down at the stud. The hold down ring is not necessary. Recommended sample height is 6.35 mm, higher samples may not be testing without the necessary accessories to do so.

Choice of abrasion disk is heavily dependent not the material being tested. Refer to the SOP for more details on the use of specific wheels.

== Opperation ==
The two abrading disks are designed so that one will abrade the sample towards the center, and the other will scrape it outwards towards the edge of the sample. Thus, as the sample is spun under the two abrading wheels, it will be worn in all directions relative to any potential grain or weave in the material. The total area of the sample to be abraded will cover approximately 30 cm<math>^2</math>.

== Data Analysis ==
There is not a standard unit or method of measuring abrasive wear. However, there are still many methods of measuring the abrasive resistance of a material. Some rely partially on qualitative judgements, while others are purely quantitative.

'''Cycles until End Point'''

The simplest method of quantifying abrasive resistance is in terms of the number of cycles required to reach a specified level of degradation. Reaching this level may be determined via visual qualitative clues, or through quantitive criteria, such as loss of mass, loss of breaking strength, yarn breakage, ect.

'''Mass Loss'''

Another simple method of determining how much a material has been worn down is by measuring the loss in mass. Because the vacuum removes all debris, this measures how much of the material as been removed from the whole by the abrasion disks. However, you must ensure that the sample is well cleaned, and precisely weighted before starting. Mass loss is calculated simply as the change in mass of the sample from before testing til after testing. Test settings must remain unchanged for this to be comparable to other samples.

'''Taber Wear Index'''

This measures the mass loss, per cycle. It can be calculated with this formula: <math> I = \frac{1000\times(M_f-M_i)}{C}</math>, where I is the Taber Wear Index, C is the number of Cycles, and <math> M_f \text{ and } M_i</math> are the final and initial masses, in milligrams, of the sample.

'''Volume Loss'''

This method is used to evaluate the results of materials with different specific gravities. Specific gravity is a measure of the density of a material compared to the density of a known reference usually water. To compare two such materials with varying specific gravities, the Taber Wear Index would be calculated for each material, and the index number would then be divided by the respective specific gravity. This would provide changes in volume, which could then be used to compare the two materials.

'''Residual Breaking Force'''

This evaluation technique is applicable mainly to textiles for fabrics. Tensile testing can be done on some samples before abrasion testing, and on others after abrasion testing. The difference in breaking force required for these tests shows how much weaker the sample has become as a result of the abrasion.

This can also be given as a percent loss in breaking force. Determine the pre and post test breaking loads, and use this to calculate the percent loss of breaking strength. This should be done for the fabric both lengthwise and widthwise. The average is then calculated according to this formula: <math> \text{AR} = \frac{100\times(X-Y)}{X}</math>, where AR is the abrasion resistance, X is the percent loss of breaking strength in lengthwise, and Y is the percent loss of breaking strength widthwise.

'''Depth of Wear'''

Abrasion can also be characterized via the change in depth of the material which has been abraded. When testing coatings, this is sometimes reported as the number of cycles required to wear through a coating of known depth.

Abrasion Tester

2025-07-08T17:14:26Z

ObordenMEASURE: Added Page

The Rotary Abrasion Tester operates by spinning samples under abrasive disks, such that the surface of the sample is roughed up from every possible direction.

Category:Equipment

2025-07-08T17:13:00Z

ObordenMEASURE: Added link to Abrasion Tester page

See our [https://sites.gatech.edu/mse-mill/equipment/list/ equipment status list and SOPs].

Learn more about our [https://mill.mse.gatech.edu/equipment/about/ Equipment Teams].

==Characterization==
*[[Scanning Electron Microscope (SEM)]]
*[[Sputter Coater]]
*[[Optical Microscopy]]
**Digital Microscope
**Stereomicroscope
**Inverted Microscope
*[[Optical Profilometer]]
*[[Fourier-Transform Infrared Spectrometer (FTIR)]]
*[[X-ray Diffractometer (XRD)]]
*[[X-ray Fluorescence Spectrometer (XRF)]]
*[[Infrared Spectroscopy]]
*[[Plasma Sputtering]]

==Measurement==
*[[Mechanical Tester]]
*[[Microhardness Tester]]
*[[Differential Scanning Calorimeter (DSC)]]
*[[Thermogravimetric Analyzer (TGA)]]
*[[Moisture Analyzer]]
*[[Infrared Camera]]
*[[UV-Vis Spectrophotometer]]
*[[Contact Angle Goniometer]]
*[[Analytical Balance]]
*[[Brookfield DV2T Viscometer|Viscometer]]
*Pycnometer
*[[Abrasion Tester]]
*[[Squidstat]]

==Processing==

=== 3D Printing ===
* [[Fused Filament Fabrication (FFF)]]/Fused Deposition Modeling (FDM) Printers
** Hyrel System 30M
** Prusa i3 MK3S
* [[Stereolithography (SLA)]] Printers
** Prusa Resin Printer
** Formlabs resin printer
* [[Filament Recycling]]
** Granulator
** Filament Extruder

=== Science of Art ===
* Hot Plates
* [[Furnaces & Kilns]]
* [https://sites.gatech.edu/mse-mill/sop/vinyl-cutter/ Vinyl Cutter]

=== Textiles ===
* [[Sewing Machine]]
* [[Embroidery Machine]]
* [[Screen Printer]]
* [[Heat Press]]

=== Other ===
* Vacuum Former
* [[Liquid Nitrogen Generator]]

=== Ceramics Lab (Love 369) ===
* [[Uniaxial Press]]
* [[Ball Mill]]
* Vibratory Tumbler
* Sieve Shaker
* [[Furnaces & Kilns|Electric Furnaces]]

Squidstat

2025-07-07T18:57:58Z

ObordenMEASURE: Finalized edits, page is now official :)

{|class="infobox"
!
!style="font-size:135%;text-align:center;"| Admiral Instruments Squidstat Plus [https://mill.mse.gatech.edu/equipment/list/ <nowiki>[SOP]</nowiki>]
|-
|
|[[File:Squidstat picture.jpg|frameless]]
|-
|'''Voltage Scan Range'''
|±10 V
|-
|'''Measured Voltage Accuracy'''
|0.1% of measured, 1 mV min
|-
|'''Measured Voltage Resolution'''
|0.003% of range, 15 μV min
|-
|'''Current Range'''
|± 1 A
|-
|'''Measured Current Accuracy'''
|0.1% of range, 100 pA min
|-
|'''Measured Current Resolution'''
|0.003% of range, 3 pA min
|-
|'''EIS Frequency Range'''
|10 µHz - 2 MHz
|-
|'''Time Resolution'''
|10 ns
|}
The Squidstat Plus is a single channel combined potentiostat galvanostat with FRA/EIS capabilities. It is capable of applying ±10V of potential to a sample or ±1A of current. EIS measurements can be made within the 10 µHz - 2 MHz frequency range. It has 5 alligator clip cables: 2 are lead used to apply potential and current, one attaches to the working electrode and the other to counter electrode; 2 are sense leads used for measuring potential and current at the working and counter electrodes; the fifth is a reference electrode, used to determine what potential will be assigned to 0 V.

== Galvanostat/Potentiostat ==
[[File:Basic Diagram of Potentio-Galvanostat.png|center|thumb|Diagram of Potentiostat/Galvanostat<ref>Metrohm Autolab. (Jul 2018) https://www.metrohm.com/en_us/applications/application-notes/autolab-applikationen-anautolab/an-ec-008.html</ref>]]
A galvanostat applies a current to an electrical circuit or cell and measures a response in the potential difference between electrodes, whereas a potentiostat applies a potential to an electrical cell or circuit and measures the resulting response in current. They both operate in a similar manner.

A working electrode (WE) and counter electrode (CE) must be attached to the sample to be tested, as well as a reference electrode (RE), and a sense (S) connection. The Squidstat employs two sense connections and uses the reference electrode only to set the zero point. The measurements are made between the counter and working senses, which are often stacked on top of the working and counter electrodes. The reference electrode is by convention often placed on the counter electrode to set it as the zero point.

A Galvanostat uses a feedback loop to control the current between the working and counter electrode, while recording the potential difference between the two. The current amplifier (CA) forces a current to run through the electrical cell into the working electrode, where it is then measured by either a Current Follower (LowCF) or a Shunt (HighCR). This provides feedback so that the instrument can maintain a specific current while monitoring the voltage.

A Potentiostat will carefully control the potential difference between the counter and working electrode. This while elicit a current response from the cell under study, which is recorded.

== Experiment Set Up ==
[[File:2 Electrode Set Up for Squidstat Test.jpg|thumb|Set Up for a 2 Electrode Test with a Capacitor<ref>Borden Orrin. (Jul 7, 2025)</ref>]]
The most common set up for the Squidstat is a 2 electrode set up. In this configuration the different leads are stacked together so that only one clamp must be attached to each electrode or to each end of the electrical circuit. The two counter electrode cables (the black ones), are stacked together, and then stacked into the reference lead (the green one). The two working electrodes (the red ones) are also stacked. The working electrode side would then be attached to the cathode, and the counter electrode side to the anode.

This configuration is sufficient for most simple experiments. Another popular set up is the 3 electrode set up. In this set up the reference lead is disconnected from the counter leads and attached to the reference electrode. This allows for very precise control of the potential at the working electrode. The reference lead can also be ignored if the experiment is run with respect to the open circuit potential, as this replaces the reference from the reference lead. Additionally, the counter electrode sense lead is not strictly necessary when testing electrical cells with very low impedance.

== Frequency Response Analysis (FRA) ==
Frequency response analysis (FRA) is a measurement technique used in a variety of applications, including electrochemistry. This technique allows signals to be analyzed based on frequency instead of time. FRA relies on Fourier Transforms to process data. This assumes that all periodic functions can be represented as a sum of sinusoidal functions. Used this analysis the frequency of as signal can be obtained.
[[File:FRA Diagram.png|center|thumb|458x458px|FRA Procedure in a Potentiostat/Galvanostat<ref>BioLogic. (November 15 2024) https://www.biologic.net/topics/what-is-a-frequency-response-analyzer-fra/</ref>]]
A signal, X(t), is imputed to the measurement device. This then records the resulting signal, and outputs it back to the analyzer. In the case of electrochemistry the output signal consists of two parts, a current wave, and a potential wave. Using Fourier transforms these are both represented sinusoidal, and can then be analyzed more effectively by the user.

== Electrochemical Impedance Spectrophotometry (EIS) ==
Electrochemical impedance spectrophotometry (EIS) is the application of a sinusoidal potential or current wave to an electrochemical cell in order to determine the impedance of the circuit. It is a branch of frequency response analysis, which is used by the Squidstat. The application of a potential wave causes a corresponding current wave in the cell, and vis versa with the application of a current wave. The response wave is always separated from the applied wave by some phase shift φ. Measuring this phase shift allows the impedance of the cell to be characterized.

Impedance is essentially the generalized form of resistance. Ohms Law, <math> \text{R} = \frac{\text{V}}{\text{I}} </math>. Ohms Law is accurate only when applied to an ideal resistor, which simplifies the relationship<ref>https://www.gamry.com/application-notes/EIS/basics-of-electrochemical-impedance-spectroscopy/</ref>. In any other circuit or electrical device, Ohms Law becomes less precise, necessitating the use of impedance. Impedance, like resistance, measures the resistance of a circuit to electrical flow. It differs from resistance in that it is dependent on frequency, and the current and potential waves may not necessarily be in phase with each other. When Ohms law is used to calculate resistance, independence of frequency and in phase waves are assumed.

Impedance has 2 components, the phase shift (<math> \phi </math>) in the wave caused by the applied signal, and the amplitude (<math> Z_0 </math>) of the resulting wave. Impedance is represented by the letter '''Z''' and consists of a real and imaginary part.

=== Nyquist Plots ===
Nyquist plots are often used to represent impedance. These are generated through EIS testing across a range of frequencies. The negative imaginary component of impedance is plotted on the y axis and the real component on the x axis. A parametric curve, from the frequencies tested, is plotted. Nyquist plots are popular because they allow the components of the system to be easily identified, and it is easy to see key data points. Different types of circuits produce very distinct shapes. To identify these shapes, Nyquist plots generally follow the convention of being orthonormal. However, neither the phase shift, nor the frequency is not shown on Nyquist plots. Generally the lower left portion is a higher frequency, and the right side is corresponds to the lower frequencies.

In a Nyquist plot, an ideal resistor is represented by a dot on the x axis at the resistance of that resistor. An ideal capacitor appears as a vertical line. However, most real capacitors appear as a slanted line, which starts at some x intercept, and then moves away from the origin. When both a capacitor and a ristorante are connected in parallel, the result is a half circle. This is because at low frequency the impedance of the capacitor increases, and most of the current flows through the resistor. But at high frequency the current through the capacitor dominates, and impedance of the circuit is determined primarily by the capacitor.
[[File:Example Nyquist Plots.png|center|thumb|339x339px|Example Nyquist Plots<ref name=":0">Alex Peroff. (Oct 2, 2024) EIS Data Plotting. https://pineresearch.com/support-article/eis-data-plotting/</ref>]]

=== Bode Plots ===
Bode plots are another common method of graphing EIS data. It plots frequency on the x axis, and uses two y axis. The primary axis shows the magnitude of the impedance, and the secondary axis shows the phase shift. Bode plots are useful because they allow the frequency and phase shift to be read easily. The phase shift also allows some determination of the type of circuit. An ideal resistor has a phase shift of 0º, and an ideal capacitor has a phase shift of -90º, and +90º corresponds to an ideal inductor.
[[File:Example Bode Plots With Circuits.webp|center|thumb|383x383px|Example Bode Plots<ref name=":0" />]]

File:2 Electrode Set Up for Squidstat Test.jpg

2025-07-07T18:30:12Z

ObordenMEASURE:

Picture of the set up for a Squidstat test where all the leads are connected to two different electrodes or wires of the sample.

Squidstat

2025-07-07T16:00:01Z

ObordenMEASURE: Finalized Bode and Nyquist Plot sections, and added final sections to be added.

[[File:Page-under-construction.png|frameless]]
{|class="infobox"
!
!style="font-size:135%;text-align:center;"| Admiral Instruments Squidstat Plus [https://mill.mse.gatech.edu/equipment/list/ <nowiki>[SOP]</nowiki>]
|-
|
|[[File:Squidstat picture.jpg|frameless]]
|-
|'''Voltage Scan Range'''
|±10 V
|-
|'''Measured Voltage Accuracy'''
|0.1% of measured, 1 mV min
|-
|'''Measured Voltage Resolution'''
|0.003% of range, 15 μV min
|-
|'''Current Range'''
|± 1 A
|-
|'''Measured Current Accuracy'''
|0.1% of range, 100 pA min
|-
|'''Measured Current Resolution'''
|0.003% of range, 3 pA min
|-
|'''EIS Frequency Range'''
|10 µHz - 2 MHz
|-
|'''Time Resolution'''
|10 ns
|}
The Squidstat Plus is a single channel combined potentiostat galvanostat with FRA/EIS capabilities. It is capable of applying ±10V of potential to a sample or ±1A of current. EIS measurements can be made within the 10 µHz - 2 MHz frequency range. It has 5 alligator clip cables: 2 are lead used to apply potential and current, one attaches to the working electrode and the other to counter electrode; 2 are sense leads used for measuring potential and current at the working and counter electrodes; the fifth is a reference electrode, used to determine what potential will be assigned to 0 V.

== Galvanostat/Potentiostat ==
[[File:Basic Diagram of Potentio-Galvanostat.png|center|thumb|Diagram of Potentiostat/Galvanostat<ref>Metrohm Autolab. (Jul 2018) https://www.metrohm.com/en_us/applications/application-notes/autolab-applikationen-anautolab/an-ec-008.html</ref>]]
A galvanostat applies a current to an electrical circuit or cell and measures a response in the potential difference between electrodes, whereas a potentiostat applies a potential to an electrical cell or circuit and measures the resulting response in current. They both operate in a similar manner.

A working electrode (WE) and counter electrode (CE) must be attached to the sample to be tested, as well as a reference electrode (RE), and a sense (S) connection. The Squidstat employs two sense connections and uses the reference electrode only to set the zero point. The measurements are made between the counter and working senses, which are often stacked on top of the working and counter electrodes. The reference electrode is by convention often placed on the counter electrode to set it as the zero point.

A Galvanostat uses a feedback loop to control the current between the working and counter electrode, while recording the potential difference between the two. The current amplifier (CA) forces a current to run through the electrical cell into the working electrode, where it is then measured by either a Current Follower (LowCF) or a Shunt (HighCR). This provides feedback so that the instrument can maintain a specific current while monitoring the voltage.

A Potentiostat will carefully control the potential difference between the counter and working electrode. This while elicit a current response from the cell under study, which is recorded.

== Experiment Set Up ==

== Frequency Response Analysis (FRA) ==
Frequency response analysis (FRA) is a measurement technique used in a variety of applications, including electrochemistry. This technique allows signals to be analyzed based on frequency instead of time. FRA relies of Fourier Transforms to process data. This assumes that all periodic functions can be represented as a sum of sinusoidal functions. Used this analysis the frequency of as signal can be obtained.
[[File:FRA Diagram.png|center|thumb|458x458px|FRA Procedure in a Potentiostat/Galvanostat<ref>BioLogic. (November 15 2024) https://www.biologic.net/topics/what-is-a-frequency-response-analyzer-fra/</ref>]]
A signal, X(t), is imputed to the measurement device. This then records the resulting signal, and outputs it back to the analyzer. In the case of electrochemistry the output signal consists of two parts, a current wave, and a potential wave. Using Fourier transforms these are both represented sinusoidal, and can then be analyzed more effectively by the user.

== Electrochemical Impedance Spectrophotometry (EIS) ==
Electrochemical impedance spectrophotometry (EIS) is the application of a sinusoidal potential or current wave to an electrochemical cell in order to determine the impedance of the circuit. It is a branch of frequency response analysis, which is used by the Squidstat. The application of a potential wave causes a corresponding current wave in the cell, and vis versa with the application of a current wave. The response wave is always separated from the applied wave by some phase shift φ. Measuring this phase shift allows the impedance of the cell to be characterized.

Impedance is essentially the generalized form of resistance. Ohms Law, <math> \text{R} = \frac{\text{V}}{\text{I}} </math>. Ohms Law is accurate only when applied to an ideal resistor, which simplifies the relationship<ref>https://www.gamry.com/application-notes/EIS/basics-of-electrochemical-impedance-spectroscopy/</ref>. In any other circuit or electrical device, Ohms Law becomes less precise, necessitating the use of impedance. Impedance, like resistance, measures the resistance of a circuit to electrical flow. It differs from resistance in that it is dependent on frequency, and the current and potential waves may not necessarily be in phase with each other. When Ohms law is used to calculate resistance, independence of frequency and in phase waves are assumed.

Impedance has 2 components, the phase shift (<math> \phi </math>) in the wave caused by the applied signal, and the amplitude (<math> Z_0 </math>) of the resulting wave. Impedance is represented by the letter '''Z''' and consists of a real and imaginary part.

=== Nyquist Plots ===
Nyquist plots are often used to represent impedance. These are generated through EIS testing across a range of frequencies. The negative imaginary component of impedance is plotted on the y axis and the real component on the x axis. A parametric curve, from the frequencies tested, is plotted. Nyquist plots are popular because they allow the components of the system to be easily identified, and it is easy to see key data points. Different types of circuits produce very distinct shapes. To identify these shapes, Nyquist plots generally follow the convention of being orthonormal. However, neither the phase shift, nor the frequency is not shown on Nyquist plots. Generally the lower left portion is a higher frequency, and the right side is corresponds to the lower frequencies.
[[File:Example Nyquist Plots.png|center|thumb|339x339px|Example Nyquist Plots<ref name=":0">Alex Peroff. (Oct 2, 2024) EIS Data Plotting. https://pineresearch.com/support-article/eis-data-plotting/</ref>]]

=== Bode Plots ===
Bode plots are another common method of graphing EIS data. It plots frequency on the x axis, and uses two y axis. The primary axis shows the magnitude of the impedance, and the secondary axis shows the phase shift. Bode plots are useful because they allow the frequency and phase shift to be read easily. The phase shift also allows some determination of the type of circuit. An ideal resistor has a phase shift of 0º, and an ideal capacitor has a phase shift of -90º, and +90º corresponds to an ideal inductor.
[[File:Example Bode Plots With Circuits.webp|center|thumb|383x383px|Example Bode Plots<ref name=":0" />]]

File:Example Nyquist Plots.png

2025-07-07T15:22:34Z

ObordenMEASURE:

6 Example Nyquist Plots with circuits

File:Example Bode Plots With Circuits.webp

2025-07-07T15:18:22Z

ObordenMEASURE:

6 Bode plots from different circuits

Squidstat

2025-07-03T15:31:19Z

ObordenMEASURE: Added diagrams, edited EIS section, added FRA section, and contributed to Potentiostat/Galvanostat section.

[[File:Page-under-construction.png|frameless]]
{|class="infobox"
!
!style="font-size:135%;text-align:center;"| Admiral Instruments Squidstat Plus [https://mill.mse.gatech.edu/equipment/list/ <nowiki>[SOP]</nowiki>]
|-
|
|[[File:Squidstat picture.jpg|frameless]]
|-
|'''Voltage Scan Range'''
|±10 V
|-
|'''Measured Voltage Accuracy'''
|0.1% of measured, 1 mV min
|-
|'''Measured Voltage Resolution'''
|0.003% of range, 15 μV min
|-
|'''Current Range'''
|± 1 A
|-
|'''Measured Current Accuracy'''
|0.1% of range, 100 pA min
|-
|'''Measured Current Resolution'''
|0.003% of range, 3 pA min
|-
|'''EIS Frequency Range'''
|10 µHz - 2 MHz
|-
|'''Time Resolution'''
|10 ns
|}
The Squidstat Plus is a single channel combined potentiostat galvanostat with FRA/EIS capabilities. It is capable of applying ±10V of potential to a sample or ±1A of current. EIS measurements can be made within the 10 µHz - 2 MHz frequency range. It has 5 alligator clip cables: 2 are lead used to apply potential and current, one attaches to the working electrode and the other to counter electrode; 2 are sense leads used for measuring potential and current at the working and counter electrodes; the fifth is a reference electrode, used to determine what potential will be assigned to 0 V.

== Galvanostat/Potentiostat ==
[[File:Basic Diagram of Potentio-Galvanostat.png|center|thumb|Diagram of Potentiostat/Galvanostat<ref>Metrohm Autolab. (Jul 2018) https://www.metrohm.com/en_us/applications/application-notes/autolab-applikationen-anautolab/an-ec-008.html</ref>]]
A galvanostat applies a current to an electrical circuit or cell and measures a response in the potential difference between electrodes, whereas a potentiostat applies a potential to an electrical cell or circuit and measures the resulting response in current. They both operate in a similar manner.

A working electrode (WE) and counter electrode (CE) must be attached to the sample to be tested, as well as a reference electrode (RE), and a sense (S) connection. The Squidstat employs two sense connections and uses the reference electrode only to set the zero point. The measurements are made between the counter and working senses, which are often stacked on top of the working and counter electrodes. The reference electrode is by convention often placed on the counter electrode to set it as the zero point.

A Galvanostat uses a feedback loop to control the current between the working and counter electrode, while recording the potential difference between the two. The current amplifier (CA) forces a current to run through the electrical cell into the working electrode, where it is then measured by either a Current Follower (LowCF) or a Shunt (HighCR). This provides feedback so that the instrument can maintain a specific current while monitoring the voltage.

A Potentiostat will carefully control the potential difference between the counter and working electrode. This while elicit a current response from the cell under study, which is recorded.

== Frequency Response Analysis (FRA) ==
Frequency response analysis (FRA) is a measurement technique used in a variety of applications, including electrochemistry. This technique allows signals to be analyzed based on frequency instead of time. FRA relies of Fourier Transforms to process data. This assumes that all periodic functions can be represented as a sum of sinusoidal functions. Used this analysis the frequency of as signal can be obtained.
[[File:FRA Diagram.png|center|thumb|458x458px|FRA Procedure in a Potentiostat/Galvanostat<ref>BioLogic. (November 15 2024) https://www.biologic.net/topics/what-is-a-frequency-response-analyzer-fra/</ref>]]
A signal, X(t), is imputed to the measurement device. This then records the resulting signal, and outputs it back to the analyzer. In the case of electrochemistry the output signal consists of two parts, a current wave, and a potential wave. Using Fourier transforms these are both represented sinusoidal, and can then be analyzed more effectively by the user.

== Electrochemical Impedance Spectrophotometry (EIS) ==
Electrochemical impedance spectrophotometry (EIS) is the application of a sinusoidal potential or current wave to an electrochemical cell in order to determine the impedance of the circuit. It is a branch of frequency response analysis, which is used by the Squidstat. The application of a potential wave causes a corresponding current wave in the cell, and vis versa with the application of a current wave. The response wave is always separated from the applied wave by some phase shift φ. Measuring this phase shift allows the impedance of the cell to be characterized.

Impedance is essentially the generalized form of resistance. Ohms Law, <math> \text{R} = \frac{\text{V}}{\text{I}} </math>. Ohms Law is accurate only when applied to an ideal resistor, which simplifies the relationship<ref>https://www.gamry.com/application-notes/EIS/basics-of-electrochemical-impedance-spectroscopy/</ref>. In any other circuit or electrical device, Ohms Law becomes less precise, necessitating the use of impedance. Impedance, like resistance, measures the resistance of a circuit to electrical flow. It differs from resistance in that it is dependent on frequency, and the current and potential waves may not necessarily be in phase with each other. When Ohms law is used to calculate resistance, independence of frequency and in phase waves are assumed.

Impedance has 2 components, the phase shift (<math> \phi </math>) in the wave caused by the applied signal, and the amplitude (<math> Z_0 </math>) of the resulting wave. Impedance is represented by the letter '''Z''' and consists of a real and imaginary part.

=== Nyquist Plots ===
Nyquist plots are often used to represent impedance. These are generated through EIS testing across a range of frequencies. The imaginary component of impedance is plotted on the y axis and the real component on the x axis. A parametric curve drawn from the frequencies tested is plotted. Nyquist plots are popular because they allow the components of the system to be easily identified, and it is easy to see key data points. However, the phase shift is not shown on Nyquist plots.

File:FRA Diagram.png

2025-07-03T15:25:29Z

ObordenMEASURE:

Basic Diagram of the function of FRA in a Galvanostat or Potentiostat

File:Basic Diagram of Potentio-Galvanostat.png

2025-07-03T14:37:07Z

ObordenMEASURE:

Basic electrical circuit diagram of a galvanostat or potentiostat.

Mechanical Testing

2025-06-26T19:31:17Z

ObordenMEASURE: Added/adjusted some citations

{|class="infobox"
!
!style="font-size:135%;text-align:center;" | Mark-10 ESM303 [https://sites.gatech.edu/mse-mill/sop/mark-10/ [SOP<nowiki>]</nowiki>]
!<big>'''Mark-10 FS05 [https://mill.mse.gatech.edu/sop/mark-10-fs05/ <nowiki>[SOP]</nowiki>]'''</big>
! style="font-size:135%;text-align:center;" | Instron 5967 [https://sites.gatech.edu/mse-mill/sop/instron/ [SOP<nowiki>]</nowiki>]
|-
|                          ||[[File:Mark10-052621.png|center|250px]]
|[[File:Unknown.jpg|frameless|359x359px]]
|[[File:Instron-052621.png|center|250px]]
|-
|'''Load Cells''':||500 mN, 25 N, 500 N, 1500 N
|0.5 N, 25 N, 250 N, 2.5 kN||30 kN
|-
|'''Load Capacity''':||1.5 kN @ up to 610 mm/min
|2.2 kN||30 kN
|-
|'''Speed range''':||13 - 330 mm/min
|0.5 - 1,100 mm/min||0.001 mm/min - 1000 mm/min
|-
|'''Travel range''':||45.7 cm
|457 mm||1.212 m
|-
|'''Max width''':||n/a
|n/a||0.418 m
|}

The Mark-10 ESM303 Mechanical Tester is a mechanical testing frame capable of testing materials under tensile, cyclic, compressive, and 3-point bending modes. This machine has several load cells: a 1500 N, a 500 N, a 25 N, and a 500 mN load cell.

The Mark-10 FS05 Mechanical Tester is a mechanical testing frame with attachments for tensile, 3 and 4 point bend, compressive, and peel tests. It can also preform cyclic test procedures. It has 0.5 N, 25 N, 250 N, and 2.5 kN load cells.

The Instron Mechanical Tester is a mechanical testing frame capable of testing materials under tensile, compressive, and 3 or 4-point-bending modes. This machine is equipped with a 30kN load cell.

== Loading ==
[[File:Loading.PNG|thumb|199x199px|left|1. Compression 2. Tension 3. Shear 4. Torsion 5. Bending]]

A material in service is subjected to different forces. Depending on the magnitude of these forces, the metal may or may not change its shape. The act of applying the force is known as loading. There are several different ways in which these forces may be applied on a part, including compression, tension, and bending. <ref name="Velling"> Velling, A. (2021, May 5). Stress-Strain Curve: How to Read the Graph? Fractory. https://fractory.com/stress-strain-curve/. </ref> <ref name="Meyers">Meyers, M., & Chawla, K. (2016). Hardness. In Mechanical Behavior of Materials (2nd ed., pp. 214–225). chapter, Cambridge University Press.</ref>

== Stress and Strain ==

=== True Stress/Strain ===
Stress is defined as the ratio of applied force to the cross-sectional area of the material it is applied to. The formula for calculating (true) material stress is:

<math> \sigma_t =\frac{F}{A}, </math> Where ''F'' is force, ''A'' is area, and <math> \sigma </math> is stress.

Stress can be understood as an internal force induced in the material in response to an externally applied force (Newton’s 3rd law). It will try to resist any change in dimension caused by the external force. Strain is defined as the ratio of the change in dimension to the initial dimension of the metal. It does not have a unit (sometimes designated as length/length). The formula for calculating (true) strain is:

<math> \epsilon_t = \int_{l_0} ^l \frac{dl}{l} = ln(\frac{l}{l_0}), </math> Where <math> l_0 </math> is starting or initial length, ''l'' is current length, and <math> \epsilon </math> is strain.

=== Engineering Stress/Strain ===
[[File:SScurve.PNG|thumb|Engineering stress-strain curve of a typical ductile material <ref name="Roylance"> Roylance, D. (2001, August 23). STRESS-STRAIN CURVES.</ref>]]In many applications, a simpler form of strain, commonly called engineering or nominal strain, is used. This type of strain is defined as

<math> \epsilon_e = \frac{(l-l_0)}{l_0} </math>

When the strains are reasonably small, the engineering and true strains are approximately the same. The elastic deformation in metals and ceramics rarely exceeds 0.005, and for this value, the difference between <math> \epsilon_t </math> and <math> \epsilon_e </math> can be neglected. In a likewise fashion, engineering (nominal) stress is defined as:

<math> \sigma_e = \frac{F}{A_0} </math>, Where <math> A_0 </math> is the original area of cross-section.

Normal strain may be positive or negative depending on the external force’s directions and therefore effect on the original length. Tensile stresses and strains are considered to be positive while their compressive counterparts are considered to be negatives. While there are multiple types of stresses and strains, this article focuses on their normal longitudinal variants.

== Tensile Testing & Engineering Stress-Strain Curves ==
[[File:Tensile Test Instron.jpeg|thumb|125x125px|Tensile Test Set Up on Instron tester<ref>Borden, O. 11 June 2025</ref>]]
The most common method to obtain a stress-strain curve is to subject a sample to a tensile test. When setup for tensile testing, the mechanical testers have two claws which hold the two extremes of the sample and pull it at a uniform rate. The force applied and the strain produced is recorded until a fracture occurs. The two parameters are then plotted on an X-Y graph to get a stress-strain curve. Note that the testers produces a force-displacement plot (with the same general shape) that must be manually converted to a stress-strain curve using the aforementioned equations in conjunction with the sample dimensions (or entering the sample dimensions into the software on the Instron). There are various sections on the stress and strain curve (described below) that describe different behavior of a material depending on the amount of stress induced. Stress and strain curves for brittle and ductile materials are different.
[[File:ASTM Dogbone Standard.png|center|thumb|274x274px|ASTM dimensions for 3D printed dog bone standard<ref name=":0">Lmpeeke. (Oct 1, 2018) ASTM D638 Type IV Dog bone Tensile sample. https://www.thingiverse.com/thing:3131427</ref>]]
The ASTM standard for tensile testing is a dog bone shape. This ensures that the break happens near the center of the sample, and not near the grips. This avoids corrupting the data due to force applied by the grips on the material. By clicking [https://drive.google.com/file/d/1GLtQdBHNnlCw7U8GfFs6V392wYZenM0L/view?usp=sharing here]<ref name=":0" /> you can access a .stl file for the ASTM dog bone standard.

A stress-strain curve always has the same 3 sections: elastic deformation, plastic deformation, and fracture. These three section occur differently in different materials and have subcategories as described below.

[[File:BrittleSSCurve.PNG|thumb|300x300px|Typical stress-strain curve of a brittle material <ref name="MechaniCalc> "Mechanical Properties of Materials. MechaniCalc. (n.d.). https://mechanicalc.com/reference/mechanical-properties-of-materials.</ref>]]

== Elastic Deformation ==
In the early (low strain) portion of the curve, most materials obey Hooke’s Law <math> (F_s=kx) </math> to a reasonable approximation, with the constant of proportionality being the modulus of elasticity or Young’s modulus (E), described by the following equation:

<math> \sigma_e = E\epsilon_e \rightarrow E= \frac{\sigma_e}{\epsilon_e} </math>

Also known as the modulus of resilience, it is analogous to the stiffness of a spring (''k''). That’s also why Hooke’s Law includes a spring constant. Materials exhibit elastic flow (ability to return to their original shape upon unloading) up to a certain strain.

== Plastic Deformation ==
As strain is increased, many materials eventually deviate from the linear proportionality of Hooke’s Law, the point of departure being termed the proportional limit. This is usually associated with stress-induced “plastic” flow in the specimen. Here the material is undergoing a rearrangement of its internal molecular or microscopic structure, in which atoms are being moved to new equilibrium positions. This plasticity requires a mechanism for molecular mobility, which in crystalline materials can arise from dislocation motion. Materials lacking this mobility, for instance by having internal microstructures that block dislocation motion, are usually brittle rather than ductile (fail at lower rather than higher strains). The stress-strain curve for brittle materials are typically linear over their full range of strain, eventually terminating in fracture with no to minimal plastic flow (permanent deformation).

=== Yield Stress ===
[[File:0.2Offset.PNG|thumb|Graphical depiction of finding 0.2% offset yield strength (stress)]]
The stress needed to increase the strain beyond the proportional limit in a ductile material continues to rise beyond the proportional limit; the material requires an ever-increasing stress to continue straining, a mechanism termed strain hardening. Microstructural rearrangements associated with plastic flow are usually not reversed. The residual strain induced by a given stress can be determined by drawing an unloading line from the highest point reached on the engineering stress strain curve at that stress back to the strain axis, drawn with a slope equal to that of the initial elastic loading line. This is done because the material unloads elastically, there being no force driving the molecular structure back to its original position. A closely related term is the yield stress <math> (\sigma_y) </math>; this is the stress needed to induce plastic deformation in the specimen. Since it is often difficult to pinpoint the exact stress at which plastic deformation begins, the yield stress is often taken to be the stress needed to induce a specified amount of permanent strain, typically 0.2%. The construction used to find this “offset yield stress” is shown below, where a line of slope E is drawn from the strain axis at <math> \epsilon = 0.2\% </math>; this is the unloading line that would result in the specified permanent strain. The stress at the point of intersection with the curve is the offset yield stress. Engineering calculations typically use a material’s yield strength for determining its ability to resist a load because if the load is greater than the yield strength, the result will be unwanted plastic deformation. See [https://www.youtube.com/watch?v=p0TQ8SV0g2U&t=586s this video] for a detailed explanation of how to find the 0.2% offset yield stress & strain in Excel.

=== Ultimate Strength ===
It appears that the rate of strain hardening decreases up to the ultimate strength (UTS), the greatest level of stress on the curve. Beyond that point, the material appears to strain soften, so that each increment of additional strain requires a smaller stress. The apparent change from strain hardening to strain softening is an artifact of the plotting procedure (using engineering rather than true values), as is the maximum observed in the curve at the UTS. Even though the UTS is perhaps the materials property most commonly reported in tensile tests, it is not a direct measure of the material due to the influence of geometry, and should be used with caution. The yield stress is usually preferred to the UTS in designing with ductile metals, although the UTS is a valid design criterion for brittle materials that do not exhibit flow-induced reductions in cross-sectional area.

=== Necking ===
[[File:Necking.PNG|thumb|Necking in a tensile specimen]]
The true stress is not quite uniform throughout the specimen, and there will always be some location - perhaps a nick or some other defect at the surface - where the local stress is maximum. Once the maximum in the engineering curve has been reached, the localized flow at this site cannot be compensated by further strain hardening, so the area there is reduced further. This increases the local stress even more, which accelerates the flow further. This localized and increasing flow soon leads to a “neck” in the gage length of the specimen such as that seen below.

Until the neck forms, the deformation is essentially uniform throughout the specimen, but after necking all subsequent deformation takes place in the neck. The neck becomes smaller and smaller, local true stress increases all the time, until the specimen fails. This will be the failure mode for most ductile metals. As the neck shrinks, the nonuniform geometry there alters the uniaxial stress state to a complex one involving shear components as well as normal stresses.

=== Drawing ===
[[File:Drawing.PNG|thumb|Necking and drawing in a stress-strain curve]]
This figure shows the engineering stress-strain curve for a semi-crystalline thermoplastic. The response of this material is similar to that of typical metals, in that it shows a proportional limit followed by a maximum in the curve at which necking takes place. (This is commonly described in plastics as the yield stress, although plastic flow has begun at earlier strains.) The polymer, however, differs dramatically in that the neck does not continue shrinking until the specimen fails. Rather, the material in the neck stretches only to a “natural draw ratio” which is a function of temperature and specimen processing, beyond which the material in the neck stops stretching and new material at the neck shoulders necks down. The neck then propagates until it spans the full gage length of the specimen, a process called drawing. This process can be observed without the need for a testing machine, by stretching a “six-pack holder.” Not all polymers are able to sustain this drawing process. Drawing occurs when the necking process produces a strengthened microstructure whose breaking load is greater than that needed to induce necking in the untransformed material just outside the neck.

== Failure ==
[[File:BrittleDuctile.PNG|thumb|Smooth brittle fracture (left) and "cup and cone" shaped ductile fracture (right)]]
Ductile specimens often fail with a “cup and cone” geometry as seen below, in which the outer regions fail in shear and the interior in tension. When the specimen fractures, the engineering strain at break <math> (\epsilon_f) </math> will include the deformation in the necked region and the un-necked region together. Since the true strain in the neck is larger than that in the un-necked material, the value of <math> (\epsilon_f) </math> will depend on the fraction of the gage length that has necked. Therefore, <math> (\epsilon_f) </math> is a function of the specimen geometry as well as the material, and thus is only a crude measure of material ductility. Brittle specimen failure is characterized by very rapid cracking under stress while little or no evidence of ductility or plastic degradation before the fracture occurs. When a piece of equipment suffers a brittle fracture, there is no thinning or necking down. Rather, this damage mechanism often causes cracking without warning, sometimes fracturing equipment into many pieces. Brittle fracture is often caused by low temperatures. If a metal is at or below its brittle-to-ductile transition temperature, then it will be susceptible to brittle fracture. Critical sized flaws, metallurgical degradation, large grain sizes, and high thickness significantly increase the likelihood of a sample experiencing brittle fracture.

== True Stress-Strain Curve ==
[[File:TrueSSC.PNG|thumb|A true stress-strain curve compared its engineering counterpart <ref name="fracture"> United States Naval Academy. (n.d.). FRACTURE OF MATERIALS. </ref> ]]
During elastic deformation, the change in cross-sectional area is less than 1% for most metals and ceramics; thus <math> \sigma_e \approx \sigma_t </math>. However, during plastic deformation, the differences between the true and the engineering values become progressively larger as the specimen dimensions experience substantial change from their original values. Due to this, the engineering stress-strain curve must be interpreted with caution beyond the elastic limits. Using the true stress <math> \sigma_t = \frac{F}{A} </math> rather than the engineering stress <math> \sigma_e = \frac{F}{A_0} </math> can give a more direct measure of the material’s response in the plastic flow range, although this requires using the following additional equations:

<math> \epsilon_t =ln(1+\epsilon_e) </math>
<math> \sigma_t = \sigma_e(1+\epsilon_e) </math>

Ductile metals often have true stress-strain relations that can be described by a simple power-law relation of the form:

<math> \sigma_t = A\epsilon_tn \rightarrow log(\sigma_t) = log(A) + nlog(\epsilon_t)</math>

The strain hardening parameter (n), is the slope of the log-log plot, and is useful as a measure of the resistance to necking. Ductile metals at room temperature usually exhibit values of n from 0.02 to 0.5.
[[File:Toughness.PNG|thumb|Resilience and toughness shown graphically]]

== Strain Energy, Resilience, & Toughness ==
[[File:EnergyDissipated.PNG|thumb|How to graphically determine energy dissipated in a plastically deformed (but not fractured) specimen ]]
The area under the engineering stress-strain curve up to a given value of strain is the total mechanical energy per unit volume consumed by the material in straining it to that value. This is easily shown as follows:

<math> U^* = \frac {1}{V} \int FdL= \int_0^L \frac{FdL}{A_0L_0}= \int_0^{\epsilon}\sigma d\epsilon </math>

In the absence of molecular slip and other mechanisms for energy dissipation, this mechanical energy is stored reversibly within the material as strain energy. The area up to the yield point is termed the modulus of resilience, and the total area up to fracture is termed the modulus of toughness.

The term “resilience” alludes to the concept that up to the point of yielding, the material is unaffected by the applied stress and upon unloading will return to its original shape. But when the strain exceeds the yield point, the material is deformed irreversibly, so that some residual strain will persist even after unloading. The modulus of resilience is then the quantity of energy the material can absorb without suffering damage. Similarly, the modulus of toughness is the energy needed to completely fracture the material. Materials showing good impact resistance, as well as ductile materials, are generally those with high moduli of toughness. The resilience & toughness integrals can be calculated from stress-strain data via methods such as [https://www.youtube.com/watch?v=R8g3B2iySA4&ab_channel=StephanieGlen trapezoidal Riemann sums in Excel].

During loading, the area under the stress-strain curve is the strain energy per unit volume absorbed by the material. Conversely, the area under the unloading curve is the energy released by the material. In the elastic range, these areas are equal and no net energy is absorbed. But if the material is loaded into the plastic range as shown above, the energy absorbed exceeds the energy released and the difference is dissipated as heat.

== Compression ==
Materials that exhibit high tensile strength tend to (but do not always!) exhibit low compressive strength. Likewise, materials high in compressive strength tend to exhibit low tensile strength. Therefore, compression testing is often used on brittle materials such as concrete, metals, plastics, ceramics, composites, and corrugated materials like cardboard. These materials are often used in a load-bearing capacity where their integrity under compressive forces is critical.

Unlike tensile tests, which are usually conducted to determine the tensile properties of a specific material, compression tests are often performed on finished products. Common items such as tennis balls, golf balls, water bottles, protective cases, plastic pipes, and furniture are all examples of products that need to be evaluated for their compressive strength. For example, an engineer may want to conserve plastic by creating water bottles with thinner walls, but the bottles must still be strong enough to be packed in pallets and stacked on top of each other for transport. Compression testing can help the engineer fine tune the balance between product strength and material conservation.
[[File:Instroncompression.jpg|thumb|285x285px|Compression Test on Instron Instrument<ref>Borden, O. (11 June 2025)</ref>]]
As long as the loads are sufficiently small (stresses less than the proportional limit), in many materials the relations outlined for tensile testing apply equally well in compression. The expression for deformation <math> (\delta) </math> from a given load

<math> \delta = \frac{FL}{AE} </math>,

applies just as in tension, with negative values for δ and F indicating compression. Further, the modulus E is the same in tension and compression to a good approximation, and the stress-strain curve simply extends as a straight line into the third quadrant as shown below.

There are some practical difficulties in performing stress-strain tests in compression. If excessively large loads are mistakenly applied in a tensile test, perhaps by wrong settings on the testing machine, the specimen simply breaks and the test must be repeated with a new specimen. But in compression, a mistake can easily damage the load cell or other sensitive components, since even after specimen failure the loads are not necessarily relieved. This is why when designing a method for the Instron tester, there must always be a second stop condition at 28kN load. The ultimate compressive strength of a material is the value of compressive stress reached when the material fails completely. When brittle materials reach their ultimate compressive strength they are crushed, and the load drops drastically. Materials with higher ductility, (most plastics) do not rupture, but instead continue deforming until the load is no longer being applied to the specimen, but rather between the two compression platens. In these cases, compressive strength can be reported as specific deformations such as 1%, 5%, or 10% of the test specimen's original height. <ref name="Instron"> Instron. (n.d.). What is Compression Testing? https://www.instron.us/en-us/our-company/library/test-types/compression-test. </ref>

== Bend Testing ==
[[File:BendVMDiagram.PNG|thumb|Diagram of a three point bend test, with graphs of bending moment ''M'', shear ''Q'' and deflection ''w'' <ref name="Penn"> Penn Center for Musculoskeletal Disorders. Three Point and Four Point Bending | Penn Center for Musculoskeletal Disorders | Perelman School of Medicine at the University of Pennsylvania. (n.d.). https://www.med.upenn.edu/pcmd/3pt-4pt-bending.html. </ref>]]
In bending tests, standardized, mostly cylindrical specimens are placed centrally on a test fixture. <ref name="GOM"> Bending test: Suitable for both brittle and ductile materials. (n.d.). https://www.gom.com/en/topics/bend-test. </ref> The rounded support rollers (bearings) are arranged parallel to each other at a certain distance (support width). The diameter of the cylindrical specimen is proportional to the support width of the bearings. The test punch, which moves down slowly and at a constant speed, loads the sample with increasing force until it breaks or reaches the previously determined deformation. Flexural modulus (elastic modulus in bending), stress-strain behavior, and failure limits can all be obtained using a bend test. <ref name="Bruker"> 3-Point Bend Testing. Bruker. (n.d.). https://www.bruker.com/en/products-and-solutions/test-and-measurement/tribometers-and-mechanical-testers/3-point-bend-testing.html. </ref> The 3-point bending test is the most frequently performed bending test. However, it has the disadvantage that in addition to the compressive and tensile forces exerted, transverse forces are also effective in the material.

During the test, the values of the bending force and deflection are recorded, which are then used to determine the material characteristics. The test sequence is plotted in a stress-strain curve and can also be recorded with a video camera. Bending tests are performed to obtain information about the bending behavior of the tested material from the single-axis bending stress. In case of brittle materials, the bending strength is determined this way. For ductile materials, the limit yield point, greatest possible bending angle, and Young’s modulus are determined.

Calculation of the flexural stress <math>\sigma_f</math>

:<math>\sigma_f = \frac{3 F L}{2 b d^2}</math> for a rectangular cross section

:<math>\sigma_f = \frac{F L}{\pi R^3}</math> for a circular cross section<ref name=Biomaterials>"Chapter 4 Mechanical Properties of Biomaterials". Biomaterials – The intersection of Biology and Material Science. New Jersey, United States: Pearson Prentice Hall. 2008. p. 152.</ref>

Calculation of the flexural strain <math>\epsilon_f</math>

:<math>\epsilon_f = \frac{6Dd}{L^2}</math>

Calculation of flexural modulus <math>E_f</math><ref name="Zweben"> Zweben, C., W. S. Smith, and M. W. Wardle (1979), "Test methods for fiber tensile strength, composite flexural modulus, and properties of fabric-reinforced laminates", Composite Materials: Testing and Design (Fifth Conference), ASTM International</ref>
[[File:Mark10 3point.jpg|thumb|3 Point Bend Test set up on Mark 10 Machine<ref>Borden, O. (11 June 2025)</ref>]]
:<math>E_f = \frac{L^3 m}{4 b d^3}</math>

in these formulas the following parameters are used:
* <math>\sigma_f</math> = Stress in outer fibers at midpoint, (MPa)
* <math>\epsilon_f</math> = Strain in the outer surface, (mm/mm)
* <math>E_f</math> = flexural Modulus of elasticity,(MPa)
* <math>F</math> = load at a given point on the load deflection curve, (N)
* <math>L</math> = Support span, (mm)
* <math>b</math> = Width of test beam, (mm)
* <math>d</math> = Depth or thickness of tested beam, (mm)
* <math>D</math> = maximum deflection of the center of the beam, (mm)
* <math>m</math> = The gradient (i.e., slope) of the initial straight-line portion of the load deflection curve, (N/mm)

* <math>R</math> = The radius of the beam, (mm)

In a bending test, the bending stress is greatest in the center of the specimen (highest deflection). At this point, the greatest bending moment is always present. From the central pressure point, the bending moment decreases linearly in both directions toward the bearings. The material is subjected to pressure on its inner side and tension on its outer side. In the outer fibers of the specimen, the bending stress (tensile and compressive stress) is greatest and decreases inwards toward the neutral fiber.

If a partially plastically deformed specimen is relieved during the bending test by raising the test punch, only the internal stresses (residual stresses) present in the material and the resulting torque are still effective. This will partially reshape the sample.

=== Bending Behavior of Ductile Materials ===
If the bending stress in the sample made of a ductile material is lower than the limit stress of the plastic deformation, the bending stress is exclusively elastic. As the bending stress increases, the yield strength is exceeded first in the peripheral areas of the specimen. These areas then deform plastically (the so-called material flow). The limit yield point is the limit bending stress up to which easily deformable materials can be loaded by bending without permanent deformation in the marginal area.

The moment this kind of deformation occurs can be determined directly from the deflection-force diagram, in a similar manner to how the yield strength is found in a tensile test. With deflection increasing steadily, more and more internal areas of the specimen are involved in the plastic deformation. This is a result of the stress increase. For steels, for example, the limit yield point is between 10 to 20 percent higher than the yield strength due to the linear stress increase. If the yield strength is exceeded in the edge fibers during the bending test, the inner and exclusively elastically stressed fibers impede the flow movement. Bending tests with ductile materials are different from those performed with brittle materials: Tough materials can be subjected to extreme plastic deformation, but they cannot be broken, no matter how strong the applied force is. Therefore, a bending test with a ductile sample is finished when the yield point is exceeded. The bending strength of ductile materials is determined by the point in time at which the plastic deformation occurs.

=== Bending Behavior of Brittle Materials ===
Specimens made of brittle materials show a different bending behavior during materials testing. They break without clearly visible material flow behavior. Therefore, the determination of the limit yield point is more complicated for brittle materials. In order to be able to determine the bending strength nevertheless, the maximum bending stress at which the sample breaks is determined. However, the bending strength is a fictitious value that is not identical to the bending stress actually occurring in the material. Another characteristic of bending tests with brittle materials is the fracture deflection. This technical term describes the greatest possible deflection of a specimen shortly before fracture.

The fracture deflection depends on the support width: Larger distances between the bearings allow for greater deflections. In order to check the strength of brittle materials, the bending test is often more suitable than the tensile test, because the materials are subjected to bending stress only. If this sample were to be subjected to a tensile test, it would break prematurely and measurement problems would occur. For certain brittle materials, the tensile test is therefore replaced by the bending test. These materials include thermosetting sheets and molding materials, thermoplastic injection molding compounds, and fiber-reinforced plastics.

== Related Theory ==
* [[Cyclic Loading]]
* [[Stress Theory]]

== References ==
<references />

[[Index.php?title=Category:Equipment|meas]]

Mechanical Testing

2025-06-26T19:25:43Z

ObordenMEASURE: Added paragraph about dog bone samples, and link to STL file

{|class="infobox"
!
!style="font-size:135%;text-align:center;" | Mark-10 ESM303 [https://sites.gatech.edu/mse-mill/sop/mark-10/ [SOP<nowiki>]</nowiki>]
!<big>'''Mark-10 FS05 [https://mill.mse.gatech.edu/sop/mark-10-fs05/ <nowiki>[SOP]</nowiki>]'''</big>
! style="font-size:135%;text-align:center;" | Instron 5967 [https://sites.gatech.edu/mse-mill/sop/instron/ [SOP<nowiki>]</nowiki>]
|-
|                          ||[[File:Mark10-052621.png|center|250px]]
|[[File:Unknown.jpg|frameless|359x359px]]
|[[File:Instron-052621.png|center|250px]]
|-
|'''Load Cells''':||500 mN, 25 N, 500 N, 1500 N
|0.5 N, 25 N, 250 N, 2.5 kN||30 kN
|-
|'''Load Capacity''':||1.5 kN @ up to 610 mm/min
|2.2 kN||30 kN
|-
|'''Speed range''':||13 - 330 mm/min
|0.5 - 1,100 mm/min||0.001 mm/min - 1000 mm/min
|-
|'''Travel range''':||45.7 cm
|457 mm||1.212 m
|-
|'''Max width''':||n/a
|n/a||0.418 m
|}

The Mark-10 ESM303 Mechanical Tester is a mechanical testing frame capable of testing materials under tensile, cyclic, compressive, and 3-point bending modes. This machine has several load cells: a 1500 N, a 500 N, a 25 N, and a 500 mN load cell.

The Mark-10 FS05 Mechanical Tester is a mechanical testing frame with attachments for tensile, 3 and 4 point bend, compressive, and peel tests. It can also preform cyclic test procedures. It has 0.5 N, 25 N, 250 N, and 2.5 kN load cells.

The Instron Mechanical Tester is a mechanical testing frame capable of testing materials under tensile, compressive, and 3 or 4-point-bending modes. This machine is equipped with a 30kN load cell.

== Loading ==
[[File:Loading.PNG|thumb|199x199px|left|1. Compression 2. Tension 3. Shear 4. Torsion 5. Bending]]

A material in service is subjected to different forces. Depending on the magnitude of these forces, the metal may or may not change its shape. The act of applying the force is known as loading. There are several different ways in which these forces may be applied on a part, including compression, tension, and bending. <ref name="Velling"> Velling, A. (2021, May 5). Stress-Strain Curve: How to Read the Graph? Fractory. https://fractory.com/stress-strain-curve/. </ref> <ref name="Meyers">Meyers, M., & Chawla, K. (2016). Hardness. In Mechanical Behavior of Materials (2nd ed., pp. 214–225). chapter, Cambridge University Press.</ref>

== Stress and Strain ==

=== True Stress/Strain ===
Stress is defined as the ratio of applied force to the cross-sectional area of the material it is applied to. The formula for calculating (true) material stress is:

<math> \sigma_t =\frac{F}{A}, </math> Where ''F'' is force, ''A'' is area, and <math> \sigma </math> is stress.

Stress can be understood as an internal force induced in the material in response to an externally applied force (Newton’s 3rd law). It will try to resist any change in dimension caused by the external force. Strain is defined as the ratio of the change in dimension to the initial dimension of the metal. It does not have a unit (sometimes designated as length/length). The formula for calculating (true) strain is:

<math> \epsilon_t = \int_{l_0} ^l \frac{dl}{l} = ln(\frac{l}{l_0}), </math> Where <math> l_0 </math> is starting or initial length, ''l'' is current length, and <math> \epsilon </math> is strain.

=== Engineering Stress/Strain ===
[[File:SScurve.PNG|thumb|Engineering stress-strain curve of a typical ductile material <ref name="Roylance"> Roylance, D. (2001, August 23). STRESS-STRAIN CURVES.</ref>]]In many applications, a simpler form of strain, commonly called engineering or nominal strain, is used. This type of strain is defined as

<math> \epsilon_e = \frac{(l-l_0)}{l_0} </math>

When the strains are reasonably small, the engineering and true strains are approximately the same. The elastic deformation in metals and ceramics rarely exceeds 0.005, and for this value, the difference between <math> \epsilon_t </math> and <math> \epsilon_e </math> can be neglected. In a likewise fashion, engineering (nominal) stress is defined as:

<math> \sigma_e = \frac{F}{A_0} </math>, Where <math> A_0 </math> is the original area of cross-section.

Normal strain may be positive or negative depending on the external force’s directions and therefore effect on the original length. Tensile stresses and strains are considered to be positive while their compressive counterparts are considered to be negatives. While there are multiple types of stresses and strains, this article focuses on their normal longitudinal variants.

== Tensile Testing & Engineering Stress-Strain Curves ==
[[File:Tensile Test Instron.jpeg|thumb|125x125px|Tensile Test Set Up on Instron tester<ref>Borden, O. 11 June 2025</ref>]]
The most common method to obtain a stress-strain curve is to subject a sample to a tensile test. When setup for tensile testing, the mechanical testers have two claws which hold the two extremes of the sample and pull it at a uniform rate. The force applied and the strain produced is recorded until a fracture occurs. The two parameters are then plotted on an X-Y graph to get a stress-strain curve. Note that the testers produces a force-displacement plot (with the same general shape) that must be manually converted to a stress-strain curve using the aforementioned equations in conjunction with the sample dimensions (or entering the sample dimensions into the software on the Instron). There are various sections on the stress and strain curve (described below) that describe different behavior of a material depending on the amount of stress induced. Stress and strain curves for brittle and ductile materials are different.
[[File:ASTM Dogbone Standard.png|center|thumb|274x274px|ASTM dimensions for 3D printed dog bone standard]]
The ASTM standard for tensile testing is a dog bone shape. This ensures that the break happens near the center of the sample, and not near the grips. This avoids corrupting the data due to force applied by the grips on the material. By clicking [https://drive.google.com/file/d/1GLtQdBHNnlCw7U8GfFs6V392wYZenM0L/view?usp=sharing here] you can access a .stl file for the ASTM dog bone standard.

A stress-strain curve always has the same 3 sections: elastic deformation, plastic deformation, and fracture. These three section occur differently in different materials and have subcategories as described below.

[[File:BrittleSSCurve.PNG|thumb|300x300px|Typical stress-strain curve of a brittle material <ref name="MechaniCalc> "Mechanical Properties of Materials. MechaniCalc. (n.d.). https://mechanicalc.com/reference/mechanical-properties-of-materials.</ref>]]

== Elastic Deformation ==
In the early (low strain) portion of the curve, most materials obey Hooke’s Law <math> (F_s=kx) </math> to a reasonable approximation, with the constant of proportionality being the modulus of elasticity or Young’s modulus (E), described by the following equation:

<math> \sigma_e = E\epsilon_e \rightarrow E= \frac{\sigma_e}{\epsilon_e} </math>

Also known as the modulus of resilience, it is analogous to the stiffness of a spring (''k''). That’s also why Hooke’s Law includes a spring constant. Materials exhibit elastic flow (ability to return to their original shape upon unloading) up to a certain strain.

== Plastic Deformation ==
As strain is increased, many materials eventually deviate from the linear proportionality of Hooke’s Law, the point of departure being termed the proportional limit. This is usually associated with stress-induced “plastic” flow in the specimen. Here the material is undergoing a rearrangement of its internal molecular or microscopic structure, in which atoms are being moved to new equilibrium positions. This plasticity requires a mechanism for molecular mobility, which in crystalline materials can arise from dislocation motion. Materials lacking this mobility, for instance by having internal microstructures that block dislocation motion, are usually brittle rather than ductile (fail at lower rather than higher strains). The stress-strain curve for brittle materials are typically linear over their full range of strain, eventually terminating in fracture with no to minimal plastic flow (permanent deformation).

=== Yield Stress ===
[[File:0.2Offset.PNG|thumb|Graphical depiction of finding 0.2% offset yield strength (stress)]]
The stress needed to increase the strain beyond the proportional limit in a ductile material continues to rise beyond the proportional limit; the material requires an ever-increasing stress to continue straining, a mechanism termed strain hardening. Microstructural rearrangements associated with plastic flow are usually not reversed. The residual strain induced by a given stress can be determined by drawing an unloading line from the highest point reached on the engineering stress strain curve at that stress back to the strain axis, drawn with a slope equal to that of the initial elastic loading line. This is done because the material unloads elastically, there being no force driving the molecular structure back to its original position. A closely related term is the yield stress <math> (\sigma_y) </math>; this is the stress needed to induce plastic deformation in the specimen. Since it is often difficult to pinpoint the exact stress at which plastic deformation begins, the yield stress is often taken to be the stress needed to induce a specified amount of permanent strain, typically 0.2%. The construction used to find this “offset yield stress” is shown below, where a line of slope E is drawn from the strain axis at <math> \epsilon = 0.2\% </math>; this is the unloading line that would result in the specified permanent strain. The stress at the point of intersection with the curve is the offset yield stress. Engineering calculations typically use a material’s yield strength for determining its ability to resist a load because if the load is greater than the yield strength, the result will be unwanted plastic deformation. See [https://www.youtube.com/watch?v=p0TQ8SV0g2U&t=586s this video] for a detailed explanation of how to find the 0.2% offset yield stress & strain in Excel.

=== Ultimate Strength ===
It appears that the rate of strain hardening decreases up to the ultimate strength (UTS), the greatest level of stress on the curve. Beyond that point, the material appears to strain soften, so that each increment of additional strain requires a smaller stress. The apparent change from strain hardening to strain softening is an artifact of the plotting procedure (using engineering rather than true values), as is the maximum observed in the curve at the UTS. Even though the UTS is perhaps the materials property most commonly reported in tensile tests, it is not a direct measure of the material due to the influence of geometry, and should be used with caution. The yield stress is usually preferred to the UTS in designing with ductile metals, although the UTS is a valid design criterion for brittle materials that do not exhibit flow-induced reductions in cross-sectional area.

=== Necking ===
[[File:Necking.PNG|thumb|Necking in a tensile specimen]]
The true stress is not quite uniform throughout the specimen, and there will always be some location - perhaps a nick or some other defect at the surface - where the local stress is maximum. Once the maximum in the engineering curve has been reached, the localized flow at this site cannot be compensated by further strain hardening, so the area there is reduced further. This increases the local stress even more, which accelerates the flow further. This localized and increasing flow soon leads to a “neck” in the gage length of the specimen such as that seen below.

Until the neck forms, the deformation is essentially uniform throughout the specimen, but after necking all subsequent deformation takes place in the neck. The neck becomes smaller and smaller, local true stress increases all the time, until the specimen fails. This will be the failure mode for most ductile metals. As the neck shrinks, the nonuniform geometry there alters the uniaxial stress state to a complex one involving shear components as well as normal stresses.

=== Drawing ===
[[File:Drawing.PNG|thumb|Necking and drawing in a stress-strain curve]]
This figure shows the engineering stress-strain curve for a semi-crystalline thermoplastic. The response of this material is similar to that of typical metals, in that it shows a proportional limit followed by a maximum in the curve at which necking takes place. (This is commonly described in plastics as the yield stress, although plastic flow has begun at earlier strains.) The polymer, however, differs dramatically in that the neck does not continue shrinking until the specimen fails. Rather, the material in the neck stretches only to a “natural draw ratio” which is a function of temperature and specimen processing, beyond which the material in the neck stops stretching and new material at the neck shoulders necks down. The neck then propagates until it spans the full gage length of the specimen, a process called drawing. This process can be observed without the need for a testing machine, by stretching a “six-pack holder.” Not all polymers are able to sustain this drawing process. Drawing occurs when the necking process produces a strengthened microstructure whose breaking load is greater than that needed to induce necking in the untransformed material just outside the neck.

== Failure ==
[[File:BrittleDuctile.PNG|thumb|Smooth brittle fracture (left) and "cup and cone" shaped ductile fracture (right)]]
Ductile specimens often fail with a “cup and cone” geometry as seen below, in which the outer regions fail in shear and the interior in tension. When the specimen fractures, the engineering strain at break <math> (\epsilon_f) </math> will include the deformation in the necked region and the un-necked region together. Since the true strain in the neck is larger than that in the un-necked material, the value of <math> (\epsilon_f) </math> will depend on the fraction of the gage length that has necked. Therefore, <math> (\epsilon_f) </math> is a function of the specimen geometry as well as the material, and thus is only a crude measure of material ductility. Brittle specimen failure is characterized by very rapid cracking under stress while little or no evidence of ductility or plastic degradation before the fracture occurs. When a piece of equipment suffers a brittle fracture, there is no thinning or necking down. Rather, this damage mechanism often causes cracking without warning, sometimes fracturing equipment into many pieces. Brittle fracture is often caused by low temperatures. If a metal is at or below its brittle-to-ductile transition temperature, then it will be susceptible to brittle fracture. Critical sized flaws, metallurgical degradation, large grain sizes, and high thickness significantly increase the likelihood of a sample experiencing brittle fracture.

== True Stress-Strain Curve ==
[[File:TrueSSC.PNG|thumb|A true stress-strain curve compared its engineering counterpart <ref name="fracture"> United States Naval Academy. (n.d.). FRACTURE OF MATERIALS. </ref> ]]
During elastic deformation, the change in cross-sectional area is less than 1% for most metals and ceramics; thus <math> \sigma_e \approx \sigma_t </math>. However, during plastic deformation, the differences between the true and the engineering values become progressively larger as the specimen dimensions experience substantial change from their original values. Due to this, the engineering stress-strain curve must be interpreted with caution beyond the elastic limits. Using the true stress <math> \sigma_t = \frac{F}{A} </math> rather than the engineering stress <math> \sigma_e = \frac{F}{A_0} </math> can give a more direct measure of the material’s response in the plastic flow range, although this requires using the following additional equations:

<math> \epsilon_t =ln(1+\epsilon_e) </math>
<math> \sigma_t = \sigma_e(1+\epsilon_e) </math>

Ductile metals often have true stress-strain relations that can be described by a simple power-law relation of the form:

<math> \sigma_t = A\epsilon_tn \rightarrow log(\sigma_t) = log(A) + nlog(\epsilon_t)</math>

The strain hardening parameter (n), is the slope of the log-log plot, and is useful as a measure of the resistance to necking. Ductile metals at room temperature usually exhibit values of n from 0.02 to 0.5.
[[File:Toughness.PNG|thumb|Resilience and toughness shown graphically]]

== Strain Energy, Resilience, & Toughness ==
[[File:EnergyDissipated.PNG|thumb|How to graphically determine energy dissipated in a plastically deformed (but not fractured) specimen ]]
The area under the engineering stress-strain curve up to a given value of strain is the total mechanical energy per unit volume consumed by the material in straining it to that value. This is easily shown as follows:

<math> U^* = \frac {1}{V} \int FdL= \int_0^L \frac{FdL}{A_0L_0}= \int_0^{\epsilon}\sigma d\epsilon </math>

In the absence of molecular slip and other mechanisms for energy dissipation, this mechanical energy is stored reversibly within the material as strain energy. The area up to the yield point is termed the modulus of resilience, and the total area up to fracture is termed the modulus of toughness.

The term “resilience” alludes to the concept that up to the point of yielding, the material is unaffected by the applied stress and upon unloading will return to its original shape. But when the strain exceeds the yield point, the material is deformed irreversibly, so that some residual strain will persist even after unloading. The modulus of resilience is then the quantity of energy the material can absorb without suffering damage. Similarly, the modulus of toughness is the energy needed to completely fracture the material. Materials showing good impact resistance, as well as ductile materials, are generally those with high moduli of toughness. The resilience & toughness integrals can be calculated from stress-strain data via methods such as [https://www.youtube.com/watch?v=R8g3B2iySA4&ab_channel=StephanieGlen trapezoidal Riemann sums in Excel].

During loading, the area under the stress-strain curve is the strain energy per unit volume absorbed by the material. Conversely, the area under the unloading curve is the energy released by the material. In the elastic range, these areas are equal and no net energy is absorbed. But if the material is loaded into the plastic range as shown above, the energy absorbed exceeds the energy released and the difference is dissipated as heat.

== Compression ==
Materials that exhibit high tensile strength tend to (but do not always!) exhibit low compressive strength. Likewise, materials high in compressive strength tend to exhibit low tensile strength. Therefore, compression testing is often used on brittle materials such as concrete, metals, plastics, ceramics, composites, and corrugated materials like cardboard. These materials are often used in a load-bearing capacity where their integrity under compressive forces is critical.

Unlike tensile tests, which are usually conducted to determine the tensile properties of a specific material, compression tests are often performed on finished products. Common items such as tennis balls, golf balls, water bottles, protective cases, plastic pipes, and furniture are all examples of products that need to be evaluated for their compressive strength. For example, an engineer may want to conserve plastic by creating water bottles with thinner walls, but the bottles must still be strong enough to be packed in pallets and stacked on top of each other for transport. Compression testing can help the engineer fine tune the balance between product strength and material conservation.
[[File:Instroncompression.jpg|thumb|285x285px|Compression Test on Instron Instrument<ref>Borden, O. 11 June 2025</ref>]]
As long as the loads are sufficiently small (stresses less than the proportional limit), in many materials the relations outlined for tensile testing apply equally well in compression. The expression for deformation <math> (\delta) </math> from a given load

<math> \delta = \frac{FL}{AE} </math>,

applies just as in tension, with negative values for δ and F indicating compression. Further, the modulus E is the same in tension and compression to a good approximation, and the stress-strain curve simply extends as a straight line into the third quadrant as shown below.

There are some practical difficulties in performing stress-strain tests in compression. If excessively large loads are mistakenly applied in a tensile test, perhaps by wrong settings on the testing machine, the specimen simply breaks and the test must be repeated with a new specimen. But in compression, a mistake can easily damage the load cell or other sensitive components, since even after specimen failure the loads are not necessarily relieved. This is why when designing a method for the Instron tester, there must always be a second stop condition at 28kN load. The ultimate compressive strength of a material is the value of compressive stress reached when the material fails completely. When brittle materials reach their ultimate compressive strength they are crushed, and the load drops drastically. Materials with higher ductility, (most plastics) do not rupture, but instead continue deforming until the load is no longer being applied to the specimen, but rather between the two compression platens. In these cases, compressive strength can be reported as specific deformations such as 1%, 5%, or 10% of the test specimen's original height. <ref name="Instron"> Instron. (n.d.). What is Compression Testing? https://www.instron.us/en-us/our-company/library/test-types/compression-test. </ref>

== Bend Testing ==
[[File:BendVMDiagram.PNG|thumb|Diagram of a three point bend test, with graphs of bending moment ''M'', shear ''Q'' and deflection ''w'' <ref name="Penn"> Penn Center for Musculoskeletal Disorders. Three Point and Four Point Bending | Penn Center for Musculoskeletal Disorders | Perelman School of Medicine at the University of Pennsylvania. (n.d.). https://www.med.upenn.edu/pcmd/3pt-4pt-bending.html. </ref>]]
In bending tests, standardized, mostly cylindrical specimens are placed centrally on a test fixture. <ref name="GOM"> Bending test: Suitable for both brittle and ductile materials. (n.d.). https://www.gom.com/en/topics/bend-test. </ref> The rounded support rollers (bearings) are arranged parallel to each other at a certain distance (support width). The diameter of the cylindrical specimen is proportional to the support width of the bearings. The test punch, which moves down slowly and at a constant speed, loads the sample with increasing force until it breaks or reaches the previously determined deformation. Flexural modulus (elastic modulus in bending), stress-strain behavior, and failure limits can all be obtained using a bend test. <ref name="Bruker"> 3-Point Bend Testing. Bruker. (n.d.). https://www.bruker.com/en/products-and-solutions/test-and-measurement/tribometers-and-mechanical-testers/3-point-bend-testing.html. </ref> The 3-point bending test is the most frequently performed bending test. However, it has the disadvantage that in addition to the compressive and tensile forces exerted, transverse forces are also effective in the material.

During the test, the values of the bending force and deflection are recorded, which are then used to determine the material characteristics. The test sequence is plotted in a stress-strain curve and can also be recorded with a video camera. Bending tests are performed to obtain information about the bending behavior of the tested material from the single-axis bending stress. In case of brittle materials, the bending strength is determined this way. For ductile materials, the limit yield point, greatest possible bending angle, and Young’s modulus are determined.

Calculation of the flexural stress <math>\sigma_f</math>

:<math>\sigma_f = \frac{3 F L}{2 b d^2}</math> for a rectangular cross section

:<math>\sigma_f = \frac{F L}{\pi R^3}</math> for a circular cross section<ref name=Biomaterials>"Chapter 4 Mechanical Properties of Biomaterials". Biomaterials – The intersection of Biology and Material Science. New Jersey, United States: Pearson Prentice Hall. 2008. p. 152.</ref>

Calculation of the flexural strain <math>\epsilon_f</math>

:<math>\epsilon_f = \frac{6Dd}{L^2}</math>

Calculation of flexural modulus <math>E_f</math><ref name="Zweben"> Zweben, C., W. S. Smith, and M. W. Wardle (1979), "Test methods for fiber tensile strength, composite flexural modulus, and properties of fabric-reinforced laminates", Composite Materials: Testing and Design (Fifth Conference), ASTM International</ref>
[[File:Mark10 3point.jpg|thumb|3 Point Bend Test set up on Mark 10 Machine<ref>Borden, O. 11 June 2025</ref>]]
:<math>E_f = \frac{L^3 m}{4 b d^3}</math>

in these formulas the following parameters are used:
* <math>\sigma_f</math> = Stress in outer fibers at midpoint, (MPa)
* <math>\epsilon_f</math> = Strain in the outer surface, (mm/mm)
* <math>E_f</math> = flexural Modulus of elasticity,(MPa)
* <math>F</math> = load at a given point on the load deflection curve, (N)
* <math>L</math> = Support span, (mm)
* <math>b</math> = Width of test beam, (mm)
* <math>d</math> = Depth or thickness of tested beam, (mm)
* <math>D</math> = maximum deflection of the center of the beam, (mm)
* <math>m</math> = The gradient (i.e., slope) of the initial straight-line portion of the load deflection curve, (N/mm)

* <math>R</math> = The radius of the beam, (mm)

In a bending test, the bending stress is greatest in the center of the specimen (highest deflection). At this point, the greatest bending moment is always present. From the central pressure point, the bending moment decreases linearly in both directions toward the bearings. The material is subjected to pressure on its inner side and tension on its outer side. In the outer fibers of the specimen, the bending stress (tensile and compressive stress) is greatest and decreases inwards toward the neutral fiber.

If a partially plastically deformed specimen is relieved during the bending test by raising the test punch, only the internal stresses (residual stresses) present in the material and the resulting torque are still effective. This will partially reshape the sample.

=== Bending Behavior of Ductile Materials ===
If the bending stress in the sample made of a ductile material is lower than the limit stress of the plastic deformation, the bending stress is exclusively elastic. As the bending stress increases, the yield strength is exceeded first in the peripheral areas of the specimen. These areas then deform plastically (the so-called material flow). The limit yield point is the limit bending stress up to which easily deformable materials can be loaded by bending without permanent deformation in the marginal area.

The moment this kind of deformation occurs can be determined directly from the deflection-force diagram, in a similar manner to how the yield strength is found in a tensile test. With deflection increasing steadily, more and more internal areas of the specimen are involved in the plastic deformation. This is a result of the stress increase. For steels, for example, the limit yield point is between 10 to 20 percent higher than the yield strength due to the linear stress increase. If the yield strength is exceeded in the edge fibers during the bending test, the inner and exclusively elastically stressed fibers impede the flow movement. Bending tests with ductile materials are different from those performed with brittle materials: Tough materials can be subjected to extreme plastic deformation, but they cannot be broken, no matter how strong the applied force is. Therefore, a bending test with a ductile sample is finished when the yield point is exceeded. The bending strength of ductile materials is determined by the point in time at which the plastic deformation occurs.

=== Bending Behavior of Brittle Materials ===
Specimens made of brittle materials show a different bending behavior during materials testing. They break without clearly visible material flow behavior. Therefore, the determination of the limit yield point is more complicated for brittle materials. In order to be able to determine the bending strength nevertheless, the maximum bending stress at which the sample breaks is determined. However, the bending strength is a fictitious value that is not identical to the bending stress actually occurring in the material. Another characteristic of bending tests with brittle materials is the fracture deflection. This technical term describes the greatest possible deflection of a specimen shortly before fracture.

The fracture deflection depends on the support width: Larger distances between the bearings allow for greater deflections. In order to check the strength of brittle materials, the bending test is often more suitable than the tensile test, because the materials are subjected to bending stress only. If this sample were to be subjected to a tensile test, it would break prematurely and measurement problems would occur. For certain brittle materials, the tensile test is therefore replaced by the bending test. These materials include thermosetting sheets and molding materials, thermoplastic injection molding compounds, and fiber-reinforced plastics.

== Related Theory ==
* [[Cyclic Loading]]
* [[Stress Theory]]

== References ==
<references />

[[Index.php?title=Category:Equipment|meas]]

File:ASTM Dogbone Standard.png

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ASTM IV Dog Bone Standard Dimensions

Squidstat

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ObordenMEASURE: Expanded EIS section, and added Nyquist plot subsection

Squidstat

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ObordenMEASURE: Started section on EIS

Squidstat

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ObordenMEASURE: Edited Table

Squidstat

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ObordenMEASURE: Added Page, outlined summary, and created data facts table

[[File:Page-under-construction.png|frameless]]

The Squidstat Plus is a single channel combined potentiostat galvanostat with FRA and EIS capabilities. It is capable of applying ±10V of potential to a sample or ±1A of current. EIS measurements can be made within the 10 µHz - 2 MHz frequency range. It has 5 alligator clip cables: 2 are lead used to apply potential and current, one attaches to the working electrode and the other to counter electrode; 2 are sense leads used for measuring potential and current at the working and counter electrodes; the fifth is a reference electrode, used to determine what potential will be assigned to 0 V.
{| class="wikitable"
|+
!
!Admiral Instruments Squidstat Plus
|-
|
|[[File:Squidstat picture.jpg|frameless]]
|-
|'''Voltage Scan Range'''
|±10 V
|-
|'''Measured Voltage Accuracy'''
|0.1% of measured, 1 mV min
|-
|'''Measured Voltage Resolution'''
|0.003% of range, 15 μV min
|-
|'''Current Range'''
|± 1 A
|-
|'''Measured Current Accuracy'''
|0.1% of range, 100 pA min
|-
|'''Measured Current Resolution'''
|0.003% of range, 3 pA min
|-
|'''EIS Frequency Range'''
|10 µHz - 2 MHz
|-
|'''Time Resolution'''
|10 ns
|}

==== Galvanostat ====
A galvanostat applies a potential to an electrical circuit or cell and measures a response in the current.

==== Potentiostat ====
A potentiostat applies a current to an electrical cell or circuit and measures the resulting response in potential.

File:Squidstat picture.jpg

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ObordenMEASURE:

Profile picture of the Squidstat in SUMS

Category:Equipment

2025-06-17T15:06:42Z

ObordenMEASURE: /* Measurement */ added squidstat page

See our [https://sites.gatech.edu/mse-mill/equipment/list/ equipment status list and SOPs].

Learn more about our [https://mill.mse.gatech.edu/equipment/about/ Equipment Teams].

==Characterization==
*[[Scanning Electron Microscope (SEM)]]
*[[Sputter Coater]]
*[[Optical Microscopy]]
**Digital Microscope
**Stereomicroscope
**Inverted Microscope
*[[Optical Profilometer]]
*[[Fourier-Transform Infrared Spectrometer (FTIR)]]
*[[X-ray Diffractometer (XRD)]]
*[[X-ray Fluorescence Spectrometer (XRF)]]

==Measurement==
*[[Mechanical Tester]]
*[[Microhardness Tester]]
*[[Differential Scanning Calorimeter (DSC)]]
*[[Thermogravimetric Analyzer (TGA)]]
*[[Moisture Analyzer]]
*[[Infrared Camera]]
*[[UV-Vis Spectrophotometer]]
*[[Contact Angle Goniometer]]
*[[Analytical Balance]]
*[[Brookfield DV2T Viscometer|Viscometer]]
*Pycnometer
*Abrasion Tester
*[[Squidstat]]

==Processing==

=== 3D Printing ===
* [[Fused Filament Fabrication (FFF)]]/Fused Deposition Modeling (FDM) Printers
** Ultimaker 2+
** Hyrel System 30M
** Prusa i3 MK3S
** Dimension SST 1200es
* [[Stereolithography (SLA)]] Printers
** Resin Printer
* [[Filament Recycling]]
** Granulator
** Filament Extruder

=== Science of Art ===
* Hot Plates
* [[Furnaces & Kilns]]
* [https://sites.gatech.edu/mse-mill/sop/vinyl-cutter/ Vinyl Cutter]

=== Textiles ===
* [[Sewing Machine]]
* [[Embroidery Machine]]
* [[Screen Printer]]
* [[Heat Press]]

=== Other ===
* Vacuum Former
* [[Liquid Nitrogen Generator]]

=== Ceramics Lab (Love 369) ===
* [[Uniaxial Press]]
* [[Ball Mill]]
* Vibratory Tumbler
* Sieve Shaker
* [[Furnaces & Kilns|Electric Furnaces]]

Mechanical Testing

2025-06-17T14:53:15Z

ObordenMEASURE: Added details on second Mark-10 (FS05) to overview, and updated the Instron bio to show it now has 4-point-bend capabilities.

{|class="infobox"
!
!style="font-size:135%;text-align:center;" | Mark-10 ESM303 [https://sites.gatech.edu/mse-mill/sop/mark-10/ [SOP<nowiki>]</nowiki>]
!<big>'''Mark-10 FS05 [https://mill.mse.gatech.edu/sop/mark-10-fs05/ <nowiki>[SOP]</nowiki>]'''</big>
! style="font-size:135%;text-align:center;" | Instron 5967 [https://sites.gatech.edu/mse-mill/sop/instron/ [SOP<nowiki>]</nowiki>]
|-
|                          ||[[File:Mark10-052621.png|center|250px]]
|[[File:Unknown.jpg|frameless|359x359px]]
|[[File:Instron-052621.png|center|250px]]
|-
|'''Load Cells''':||500 mN, 25 N, 500 N, 1500 N
|0.5 N, 25 N, 250 N, 2.5 kN||30 kN
|-
|'''Load Capacity''':||1.5 kN @ up to 610 mm/min
|2.2 kN||30 kN
|-
|'''Speed range''':||13 - 330 mm/min
|0.5 - 1,100 mm/min||0.001 mm/min - 1000 mm/min
|-
|'''Travel range''':||45.7 cm
|457 mm||1.212 m
|-
|'''Max width''':||n/a
|n/a||0.418 m
|}

The Mark-10 ESM303 Mechanical Tester is a mechanical testing frame capable of testing materials under tensile, cyclic, compressive, and 3-point bending modes. This machine has several load cells: a 1500 N, a 500 N, a 25 N, and a 500 mN load cell.

The Mark-10 FS05 Mechanical Tester is a mechanical testing frame with attachments for tensile, 3 and 4 point bend, compressive, and peel tests. It can also preform cyclic test procedures. It has 0.5 N, 25 N, 250 N, and 2.5 kN load cells.

The Instron Mechanical Tester is a mechanical testing frame capable of testing materials under tensile, compressive, and 3 or 4-point-bending modes. This machine is equipped with a 30kN load cell.

== Loading ==
[[File:Loading.PNG|thumb|199x199px|left|1. Compression 2. Tension 3. Shear 4. Torsion 5. Bending]]

A material in service is subjected to different forces. Depending on the magnitude of these forces, the metal may or may not change its shape. The act of applying the force is known as loading. There are several different ways in which these forces may be applied on a part, including compression, tension, and bending. <ref name="Velling"> Velling, A. (2021, May 5). Stress-Strain Curve: How to Read the Graph? Fractory. https://fractory.com/stress-strain-curve/. </ref> <ref name="Meyers">Meyers, M., & Chawla, K. (2016). Hardness. In Mechanical Behavior of Materials (2nd ed., pp. 214–225). chapter, Cambridge University Press.</ref>

== Stress and Strain ==

=== True Stress/Strain ===
Stress is defined as the ratio of applied force to the cross-sectional area of the material it is applied to. The formula for calculating (true) material stress is:

<math> \sigma_t =\frac{F}{A}, </math> Where ''F'' is force, ''A'' is area, and <math> \sigma </math> is stress.

Stress can be understood as an internal force induced in the material in response to an externally applied force (Newton’s 3rd law). It will try to resist any change in dimension caused by the external force. Strain is defined as the ratio of the change in dimension to the initial dimension of the metal. It does not have a unit (sometimes designated as length/length). The formula for calculating (true) strain is:

<math> \epsilon_t = \int_{l_0} ^l \frac{dl}{l} = ln(\frac{l}{l_0}), </math> Where <math> l_0 </math> is starting or initial length, ''l'' is current length, and <math> \epsilon </math> is strain.

=== Engineering Stress/Strain ===
[[File:SScurve.PNG|thumb|Engineering stress-strain curve of a typical ductile material <ref name="Roylance"> Roylance, D. (2001, August 23). STRESS-STRAIN CURVES.</ref>]]In many applications, a simpler form of strain, commonly called engineering or nominal strain, is used. This type of strain is defined as

<math> \epsilon_e = \frac{(l-l_0)}{l_0} </math>

When the strains are reasonably small, the engineering and true strains are approximately the same. The elastic deformation in metals and ceramics rarely exceeds 0.005, and for this value, the difference between <math> \epsilon_t </math> and <math> \epsilon_e </math> can be neglected. In a likewise fashion, engineering (nominal) stress is defined as:

<math> \sigma_e = \frac{F}{A_0} </math>, Where <math> A_0 </math> is the original area of cross-section.

Normal strain may be positive or negative depending on the external force’s directions and therefore effect on the original length. Tensile stresses and strains are considered to be positive while their compressive counterparts are considered to be negatives. While there are multiple types of stresses and strains, this article focuses on their normal longitudinal variants.

== Tensile Testing & Engineering Stress-Strain Curves ==
[[File:Tensile Test Instron.jpeg|thumb|207x207px|Tensile Test Set Up on Instron tester<ref>Borden, O. 11 June 2025</ref>]]
The most common method to obtain a stress-strain curve is to subject a sample to a tensile test. When setup for tensile testing, the mechanical testers have two claws which hold the two extremes of the sample and pull it at a uniform rate. The force applied and the strain produced is recorded until a fracture occurs. The two parameters are then plotted on an X-Y graph to get a stress-strain curve. Note that the testers produces a force-displacement plot (with the same general shape) that must be manually converted to a stress-strain curve using the aforementioned equations in conjunction with the sample dimensions (or entering the sample dimensions into the software on the Instron). There are various sections on the stress and strain curve (described below) that describe different behavior of a material depending on the amount of stress induced. Stress and strain curves for brittle and ductile materials are different.

A stress-strain curve always has the same 3 sections: elastic deformation, plastic deformation, and fracture. These three section occur differently in different materials and have subcategories as described below.

[[File:BrittleSSCurve.PNG|thumb|300x300px|Typical stress-strain curve of a brittle material <ref name="MechaniCalc> "Mechanical Properties of Materials. MechaniCalc. (n.d.). https://mechanicalc.com/reference/mechanical-properties-of-materials.</ref>]]

== Elastic Deformation ==
In the early (low strain) portion of the curve, most materials obey Hooke’s Law <math> (F_s=kx) </math> to a reasonable approximation, with the constant of proportionality being the modulus of elasticity or Young’s modulus (E), described by the following equation:

<math> \sigma_e = E\epsilon_e \rightarrow E= \frac{\sigma_e}{\epsilon_e} </math>

Also known as the modulus of resilience, it is analogous to the stiffness of a spring (''k''). That’s also why Hooke’s Law includes a spring constant. Materials exhibit elastic flow (ability to return to their original shape upon unloading) up to a certain strain.

== Plastic Deformation ==
As strain is increased, many materials eventually deviate from the linear proportionality of Hooke’s Law, the point of departure being termed the proportional limit. This is usually associated with stress-induced “plastic” flow in the specimen. Here the material is undergoing a rearrangement of its internal molecular or microscopic structure, in which atoms are being moved to new equilibrium positions. This plasticity requires a mechanism for molecular mobility, which in crystalline materials can arise from dislocation motion. Materials lacking this mobility, for instance by having internal microstructures that block dislocation motion, are usually brittle rather than ductile (fail at lower rather than higher strains). The stress-strain curve for brittle materials are typically linear over their full range of strain, eventually terminating in fracture with no to minimal plastic flow (permanent deformation).

=== Yield Stress ===
[[File:0.2Offset.PNG|thumb|Graphical depiction of finding 0.2% offset yield strength (stress)]]
The stress needed to increase the strain beyond the proportional limit in a ductile material continues to rise beyond the proportional limit; the material requires an ever-increasing stress to continue straining, a mechanism termed strain hardening. Microstructural rearrangements associated with plastic flow are usually not reversed. The residual strain induced by a given stress can be determined by drawing an unloading line from the highest point reached on the engineering stress strain curve at that stress back to the strain axis, drawn with a slope equal to that of the initial elastic loading line. This is done because the material unloads elastically, there being no force driving the molecular structure back to its original position. A closely related term is the yield stress <math> (\sigma_y) </math>; this is the stress needed to induce plastic deformation in the specimen. Since it is often difficult to pinpoint the exact stress at which plastic deformation begins, the yield stress is often taken to be the stress needed to induce a specified amount of permanent strain, typically 0.2%. The construction used to find this “offset yield stress” is shown below, where a line of slope E is drawn from the strain axis at <math> \epsilon = 0.2\% </math>; this is the unloading line that would result in the specified permanent strain. The stress at the point of intersection with the curve is the offset yield stress. Engineering calculations typically use a material’s yield strength for determining its ability to resist a load because if the load is greater than the yield strength, the result will be unwanted plastic deformation. See [https://www.youtube.com/watch?v=p0TQ8SV0g2U&t=586s this video] for a detailed explanation of how to find the 0.2% offset yield stress & strain in Excel.

=== Ultimate Strength ===
It appears that the rate of strain hardening decreases up to the ultimate strength (UTS), the greatest level of stress on the curve. Beyond that point, the material appears to strain soften, so that each increment of additional strain requires a smaller stress. The apparent change from strain hardening to strain softening is an artifact of the plotting procedure (using engineering rather than true values), as is the maximum observed in the curve at the UTS. Even though the UTS is perhaps the materials property most commonly reported in tensile tests, it is not a direct measure of the material due to the influence of geometry, and should be used with caution. The yield stress is usually preferred to the UTS in designing with ductile metals, although the UTS is a valid design criterion for brittle materials that do not exhibit flow-induced reductions in cross-sectional area.

=== Necking ===
[[File:Necking.PNG|thumb|Necking in a tensile specimen]]
The true stress is not quite uniform throughout the specimen, and there will always be some location - perhaps a nick or some other defect at the surface - where the local stress is maximum. Once the maximum in the engineering curve has been reached, the localized flow at this site cannot be compensated by further strain hardening, so the area there is reduced further. This increases the local stress even more, which accelerates the flow further. This localized and increasing flow soon leads to a “neck” in the gage length of the specimen such as that seen below.

Until the neck forms, the deformation is essentially uniform throughout the specimen, but after necking all subsequent deformation takes place in the neck. The neck becomes smaller and smaller, local true stress increases all the time, until the specimen fails. This will be the failure mode for most ductile metals. As the neck shrinks, the nonuniform geometry there alters the uniaxial stress state to a complex one involving shear components as well as normal stresses.

=== Drawing ===
[[File:Drawing.PNG|thumb|Necking and drawing in a stress-strain curve]]
This figure shows the engineering stress-strain curve for a semi-crystalline thermoplastic. The response of this material is similar to that of typical metals, in that it shows a proportional limit followed by a maximum in the curve at which necking takes place. (This is commonly described in plastics as the yield stress, although plastic flow has begun at earlier strains.) The polymer, however, differs dramatically in that the neck does not continue shrinking until the specimen fails. Rather, the material in the neck stretches only to a “natural draw ratio” which is a function of temperature and specimen processing, beyond which the material in the neck stops stretching and new material at the neck shoulders necks down. The neck then propagates until it spans the full gage length of the specimen, a process called drawing. This process can be observed without the need for a testing machine, by stretching a “six-pack holder.” Not all polymers are able to sustain this drawing process. Drawing occurs when the necking process produces a strengthened microstructure whose breaking load is greater than that needed to induce necking in the untransformed material just outside the neck.

== Failure ==
[[File:BrittleDuctile.PNG|thumb|Smooth brittle fracture (left) and "cup and cone" shaped ductile fracture (right)]]
Ductile specimens often fail with a “cup and cone” geometry as seen below, in which the outer regions fail in shear and the interior in tension. When the specimen fractures, the engineering strain at break <math> (\epsilon_f) </math> will include the deformation in the necked region and the un-necked region together. Since the true strain in the neck is larger than that in the un-necked material, the value of <math> (\epsilon_f) </math> will depend on the fraction of the gage length that has necked. Therefore, <math> (\epsilon_f) </math> is a function of the specimen geometry as well as the material, and thus is only a crude measure of material ductility. Brittle specimen failure is characterized by very rapid cracking under stress while little or no evidence of ductility or plastic degradation before the fracture occurs. When a piece of equipment suffers a brittle fracture, there is no thinning or necking down. Rather, this damage mechanism often causes cracking without warning, sometimes fracturing equipment into many pieces. Brittle fracture is often caused by low temperatures. If a metal is at or below its brittle-to-ductile transition temperature, then it will be susceptible to brittle fracture. Critical sized flaws, metallurgical degradation, large grain sizes, and high thickness significantly increase the likelihood of a sample experiencing brittle fracture.

== True Stress-Strain Curve ==
[[File:TrueSSC.PNG|thumb|A true stress-strain curve compared its engineering counterpart <ref name="fracture"> United States Naval Academy. (n.d.). FRACTURE OF MATERIALS. </ref> ]]
During elastic deformation, the change in cross-sectional area is less than 1% for most metals and ceramics; thus <math> \sigma_e \approx \sigma_t </math>. However, during plastic deformation, the differences between the true and the engineering values become progressively larger as the specimen dimensions experience substantial change from their original values. Due to this, the engineering stress-strain curve must be interpreted with caution beyond the elastic limits. Using the true stress <math> \sigma_t = \frac{F}{A} </math> rather than the engineering stress <math> \sigma_e = \frac{F}{A_0} </math> can give a more direct measure of the material’s response in the plastic flow range, although this requires using the following additional equations:

<math> \epsilon_t =ln(1+\epsilon_e) </math>
<math> \sigma_t = \sigma_e(1+\epsilon_e) </math>

Ductile metals often have true stress-strain relations that can be described by a simple power-law relation of the form:

<math> \sigma_t = A\epsilon_tn \rightarrow log(\sigma_t) = log(A) + nlog(\epsilon_t)</math>

The strain hardening parameter (n), is the slope of the log-log plot, and is useful as a measure of the resistance to necking. Ductile metals at room temperature usually exhibit values of n from 0.02 to 0.5.
[[File:Toughness.PNG|thumb|Resilience and toughness shown graphically]]

== Strain Energy, Resilience, & Toughness ==
[[File:EnergyDissipated.PNG|thumb|How to graphically determine energy dissipated in a plastically deformed (but not fractured) specimen ]]
The area under the engineering stress-strain curve up to a given value of strain is the total mechanical energy per unit volume consumed by the material in straining it to that value. This is easily shown as follows:

<math> U^* = \frac {1}{V} \int FdL= \int_0^L \frac{FdL}{A_0L_0}= \int_0^{\epsilon}\sigma d\epsilon </math>

In the absence of molecular slip and other mechanisms for energy dissipation, this mechanical energy is stored reversibly within the material as strain energy. The area up to the yield point is termed the modulus of resilience, and the total area up to fracture is termed the modulus of toughness.

The term “resilience” alludes to the concept that up to the point of yielding, the material is unaffected by the applied stress and upon unloading will return to its original shape. But when the strain exceeds the yield point, the material is deformed irreversibly, so that some residual strain will persist even after unloading. The modulus of resilience is then the quantity of energy the material can absorb without suffering damage. Similarly, the modulus of toughness is the energy needed to completely fracture the material. Materials showing good impact resistance, as well as ductile materials, are generally those with high moduli of toughness. The resilience & toughness integrals can be calculated from stress-strain data via methods such as [https://www.youtube.com/watch?v=R8g3B2iySA4&ab_channel=StephanieGlen trapezoidal Riemann sums in Excel].

During loading, the area under the stress-strain curve is the strain energy per unit volume absorbed by the material. Conversely, the area under the unloading curve is the energy released by the material. In the elastic range, these areas are equal and no net energy is absorbed. But if the material is loaded into the plastic range as shown above, the energy absorbed exceeds the energy released and the difference is dissipated as heat.

== Compression ==
Materials that exhibit high tensile strength tend to (but do not always!) exhibit low compressive strength. Likewise, materials high in compressive strength tend to exhibit low tensile strength. Therefore, compression testing is often used on brittle materials such as concrete, metals, plastics, ceramics, composites, and corrugated materials like cardboard. These materials are often used in a load-bearing capacity where their integrity under compressive forces is critical.

Unlike tensile tests, which are usually conducted to determine the tensile properties of a specific material, compression tests are often performed on finished products. Common items such as tennis balls, golf balls, water bottles, protective cases, plastic pipes, and furniture are all examples of products that need to be evaluated for their compressive strength. For example, an engineer may want to conserve plastic by creating water bottles with thinner walls, but the bottles must still be strong enough to be packed in pallets and stacked on top of each other for transport. Compression testing can help the engineer fine tune the balance between product strength and material conservation.
[[File:Instroncompression.jpg|thumb|285x285px|Compression Test on Instron Instrument<ref>Borden, O. 11 June 2025</ref>]]
As long as the loads are sufficiently small (stresses less than the proportional limit), in many materials the relations outlined for tensile testing apply equally well in compression. The expression for deformation <math> (\delta) </math> from a given load

<math> \delta = \frac{FL}{AE} </math>,

applies just as in tension, with negative values for δ and F indicating compression. Further, the modulus E is the same in tension and compression to a good approximation, and the stress-strain curve simply extends as a straight line into the third quadrant as shown below.

There are some practical difficulties in performing stress-strain tests in compression. If excessively large loads are mistakenly applied in a tensile test, perhaps by wrong settings on the testing machine, the specimen simply breaks and the test must be repeated with a new specimen. But in compression, a mistake can easily damage the load cell or other sensitive components, since even after specimen failure the loads are not necessarily relieved. This is why when designing a method for the Instron tester, there must always be a second stop condition at 28kN load. The ultimate compressive strength of a material is the value of compressive stress reached when the material fails completely. When brittle materials reach their ultimate compressive strength they are crushed, and the load drops drastically. Materials with higher ductility, (most plastics) do not rupture, but instead continue deforming until the load is no longer being applied to the specimen, but rather between the two compression platens. In these cases, compressive strength can be reported as specific deformations such as 1%, 5%, or 10% of the test specimen's original height. <ref name="Instron"> Instron. (n.d.). What is Compression Testing? https://www.instron.us/en-us/our-company/library/test-types/compression-test. </ref>

== Bend Testing ==
[[File:BendVMDiagram.PNG|thumb|Diagram of a three point bend test, with graphs of bending moment ''M'', shear ''Q'' and deflection ''w'' <ref name="Penn"> Penn Center for Musculoskeletal Disorders. Three Point and Four Point Bending | Penn Center for Musculoskeletal Disorders | Perelman School of Medicine at the University of Pennsylvania. (n.d.). https://www.med.upenn.edu/pcmd/3pt-4pt-bending.html. </ref>]]
In bending tests, standardized, mostly cylindrical specimens are placed centrally on a test fixture. <ref name="GOM"> Bending test: Suitable for both brittle and ductile materials. (n.d.). https://www.gom.com/en/topics/bend-test. </ref> The rounded support rollers (bearings) are arranged parallel to each other at a certain distance (support width). The diameter of the cylindrical specimen is proportional to the support width of the bearings. The test punch, which moves down slowly and at a constant speed, loads the sample with increasing force until it breaks or reaches the previously determined deformation. Flexural modulus (elastic modulus in bending), stress-strain behavior, and failure limits can all be obtained using a bend test. <ref name="Bruker"> 3-Point Bend Testing. Bruker. (n.d.). https://www.bruker.com/en/products-and-solutions/test-and-measurement/tribometers-and-mechanical-testers/3-point-bend-testing.html. </ref> The 3-point bending test is the most frequently performed bending test. However, it has the disadvantage that in addition to the compressive and tensile forces exerted, transverse forces are also effective in the material.

During the test, the values of the bending force and deflection are recorded, which are then used to determine the material characteristics. The test sequence is plotted in a stress-strain curve and can also be recorded with a video camera. Bending tests are performed to obtain information about the bending behavior of the tested material from the single-axis bending stress. In case of brittle materials, the bending strength is determined this way. For ductile materials, the limit yield point, greatest possible bending angle, and Young’s modulus are determined.

Calculation of the flexural stress <math>\sigma_f</math>

:<math>\sigma_f = \frac{3 F L}{2 b d^2}</math> for a rectangular cross section

:<math>\sigma_f = \frac{F L}{\pi R^3}</math> for a circular cross section<ref name=Biomaterials>"Chapter 4 Mechanical Properties of Biomaterials". Biomaterials – The intersection of Biology and Material Science. New Jersey, United States: Pearson Prentice Hall. 2008. p. 152.</ref>

Calculation of the flexural strain <math>\epsilon_f</math>

:<math>\epsilon_f = \frac{6Dd}{L^2}</math>

Calculation of flexural modulus <math>E_f</math><ref name="Zweben"> Zweben, C., W. S. Smith, and M. W. Wardle (1979), "Test methods for fiber tensile strength, composite flexural modulus, and properties of fabric-reinforced laminates", Composite Materials: Testing and Design (Fifth Conference), ASTM International</ref>
[[File:Mark10 3point.jpg|thumb|3 Point Bend Test set up on Mark 10 Machine<ref>Borden, O. 11 June 2025</ref>]]
:<math>E_f = \frac{L^3 m}{4 b d^3}</math>

in these formulas the following parameters are used:
* <math>\sigma_f</math> = Stress in outer fibers at midpoint, (MPa)
* <math>\epsilon_f</math> = Strain in the outer surface, (mm/mm)
* <math>E_f</math> = flexural Modulus of elasticity,(MPa)
* <math>F</math> = load at a given point on the load deflection curve, (N)
* <math>L</math> = Support span, (mm)
* <math>b</math> = Width of test beam, (mm)
* <math>d</math> = Depth or thickness of tested beam, (mm)
* <math>D</math> = maximum deflection of the center of the beam, (mm)
* <math>m</math> = The gradient (i.e., slope) of the initial straight-line portion of the load deflection curve, (N/mm)

* <math>R</math> = The radius of the beam, (mm)

In a bending test, the bending stress is greatest in the center of the specimen (highest deflection). At this point, the greatest bending moment is always present. From the central pressure point, the bending moment decreases linearly in both directions toward the bearings. The material is subjected to pressure on its inner side and tension on its outer side. In the outer fibers of the specimen, the bending stress (tensile and compressive stress) is greatest and decreases inwards toward the neutral fiber.

If a partially plastically deformed specimen is relieved during the bending test by raising the test punch, only the internal stresses (residual stresses) present in the material and the resulting torque are still effective. This will partially reshape the sample.

=== Bending Behavior of Ductile Materials ===
If the bending stress in the sample made of a ductile material is lower than the limit stress of the plastic deformation, the bending stress is exclusively elastic. As the bending stress increases, the yield strength is exceeded first in the peripheral areas of the specimen. These areas then deform plastically (the so-called material flow). The limit yield point is the limit bending stress up to which easily deformable materials can be loaded by bending without permanent deformation in the marginal area.

The moment this kind of deformation occurs can be determined directly from the deflection-force diagram, in a similar manner to how the yield strength is found in a tensile test. With deflection increasing steadily, more and more internal areas of the specimen are involved in the plastic deformation. This is a result of the stress increase. For steels, for example, the limit yield point is between 10 to 20 percent higher than the yield strength due to the linear stress increase. If the yield strength is exceeded in the edge fibers during the bending test, the inner and exclusively elastically stressed fibers impede the flow movement. Bending tests with ductile materials are different from those performed with brittle materials: Tough materials can be subjected to extreme plastic deformation, but they cannot be broken, no matter how strong the applied force is. Therefore, a bending test with a ductile sample is finished when the yield point is exceeded. The bending strength of ductile materials is determined by the point in time at which the plastic deformation occurs.

=== Bending Behavior of Brittle Materials ===
Specimens made of brittle materials show a different bending behavior during materials testing. They break without clearly visible material flow behavior. Therefore, the determination of the limit yield point is more complicated for brittle materials. In order to be able to determine the bending strength nevertheless, the maximum bending stress at which the sample breaks is determined. However, the bending strength is a fictitious value that is not identical to the bending stress actually occurring in the material. Another characteristic of bending tests with brittle materials is the fracture deflection. This technical term describes the greatest possible deflection of a specimen shortly before fracture.

The fracture deflection depends on the support width: Larger distances between the bearings allow for greater deflections. In order to check the strength of brittle materials, the bending test is often more suitable than the tensile test, because the materials are subjected to bending stress only. If this sample were to be subjected to a tensile test, it would break prematurely and measurement problems would occur. For certain brittle materials, the tensile test is therefore replaced by the bending test. These materials include thermosetting sheets and molding materials, thermoplastic injection molding compounds, and fiber-reinforced plastics.

== Related Theory ==
* [[Cyclic Loading]]
* [[Stress Theory]]

== References ==
<references />

[[Index.php?title=Category:Equipment|meas]]

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ObordenMEASURE:

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== Summary ==
Image of Instron being set up for use in a compression test

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Image of a Tensile test preformed on the Instron with an ASTM dog bone 3D print.

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ObordenMEASURE: ObordenMEASURE uploaded a new version of File:Mark10 3point.jpg

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Image of 3-point bend test set up on Mark 10

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ObordenMEASURE: Condensed image of Instron Tensile Test

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Mechanical Testing

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ObordenMEASURE: /* Compression */ Added images of compression and bend test set ups.

{|class="infobox"
!
!style="font-size:135%;text-align:center;" | Mark-10 ESM303 [https://sites.gatech.edu/mse-mill/sop/mark-10/ [SOP<nowiki>]</nowiki>]
!style="font-size:135%;text-align:center;" | Instron 5967 [https://sites.gatech.edu/mse-mill/sop/instron/ [SOP<nowiki>]</nowiki>]
|-
|                          ||[[File:Mark10-052621.png|center|250px]]||[[File:Instron-052621.png|center|250px]]
|-
|'''Load Cells''':||500 mN, 25 N, 500 N, 1500 N||30 kN
|-
|'''Load Capacity''':||1.5 kN @ up to 610 mm/min||30 kN
|-
|'''Speed range''':||13 - 330 mm/min||0.001 mm/min - 1000 mm/min
|-
|'''Travel range''':||45.7 cm||1.212 m
|-
|'''Max width''':||n/a||0.418 m
|}

The Mark-10 ESM303 Mechanical Tester is a mechanical testing frame capable of testing materials under tensile, cyclic, compressive, and 3-point bending modes. This machine has several load cells: a 1500 N, a 500 N, a 25 N, and a 500 mN load cell.

The Instron Mechanical Tester is a mechanical testing frame capable of testing materials under tensile, compressive, and 3-point-bending modes. This machine is equipped with a 30kN load cell.

== Loading ==
[[File:Loading.PNG|thumb|199x199px|left|1. Compression 2. Tension 3. Shear 4. Torsion 5. Bending]]

A material in service is subjected to different forces. Depending on the magnitude of these forces, the metal may or may not change its shape. The act of applying the force is known as loading. There are several different ways in which these forces may be applied on a part, including compression, tension, and bending. <ref name="Velling"> Velling, A. (2021, May 5). Stress-Strain Curve: How to Read the Graph? Fractory. https://fractory.com/stress-strain-curve/. </ref> <ref name="Meyers">Meyers, M., & Chawla, K. (2016). Hardness. In Mechanical Behavior of Materials (2nd ed., pp. 214–225). chapter, Cambridge University Press.</ref>

== Stress and Strain ==

=== True Stress/Strain ===
Stress is defined as the ratio of applied force to the cross-sectional area of the material it is applied to. The formula for calculating (true) material stress is:

<math> \sigma_t =\frac{F}{A}, </math> Where ''F'' is force, ''A'' is area, and <math> \sigma </math> is stress.

Stress can be understood as an internal force induced in the material in response to an externally applied force (Newton’s 3rd law). It will try to resist any change in dimension caused by the external force. Strain is defined as the ratio of the change in dimension to the initial dimension of the metal. It does not have a unit (sometimes designated as length/length). The formula for calculating (true) strain is:

<math> \epsilon_t = \int_{l_0} ^l \frac{dl}{l} = ln(\frac{l}{l_0}), </math> Where <math> l_0 </math> is starting or initial length, ''l'' is current length, and <math> \epsilon </math> is strain.

=== Engineering Stress/Strain ===
[[File:SScurve.PNG|thumb|Engineering stress-strain curve of a typical ductile material <ref name="Roylance"> Roylance, D. (2001, August 23). STRESS-STRAIN CURVES.</ref>]]In many applications, a simpler form of strain, commonly called engineering or nominal strain, is used. This type of strain is defined as

<math> \epsilon_e = \frac{(l-l_0)}{l_0} </math>

When the strains are reasonably small, the engineering and true strains are approximately the same. The elastic deformation in metals and ceramics rarely exceeds 0.005, and for this value, the difference between <math> \epsilon_t </math> and <math> \epsilon_e </math> can be neglected. In a likewise fashion, engineering (nominal) stress is defined as:

<math> \sigma_e = \frac{F}{A_0} </math>, Where <math> A_0 </math> is the original area of cross-section.

Normal strain may be positive or negative depending on the external force’s directions and therefore effect on the original length. Tensile stresses and strains are considered to be positive while their compressive counterparts are considered to be negatives. While there are multiple types of stresses and strains, this article focuses on their normal longitudinal variants.

== Tensile Testing & Engineering Stress-Strain Curves ==
[[File:Tensile Test Instron.jpeg|thumb|207x207px|Tensile Test Set Up on Instron tester<ref>Borden, O. 11 June 2025</ref>]]
The most common method to obtain a stress-strain curve is to subject a sample to a tensile test. When setup for tensile testing, the mechanical testers have two claws which hold the two extremes of the sample and pull it at a uniform rate. The force applied and the strain produced is recorded until a fracture occurs. The two parameters are then plotted on an X-Y graph to get a stress-strain curve. Note that the testers produces a force-displacement plot (with the same general shape) that must be manually converted to a stress-strain curve using the aforementioned equations in conjunction with the sample dimensions (or entering the sample dimensions into the software on the Instron). There are various sections on the stress and strain curve (described below) that describe different behavior of a material depending on the amount of stress induced. Stress and strain curves for brittle and ductile materials are different.

A stress-strain curve always has the same 3 sections: elastic deformation, plastic deformation, and fracture. These three section occur differently in different materials and have subcategories as described below.

[[File:BrittleSSCurve.PNG|thumb|300x300px|Typical stress-strain curve of a brittle material <ref name="MechaniCalc> "Mechanical Properties of Materials. MechaniCalc. (n.d.). https://mechanicalc.com/reference/mechanical-properties-of-materials.</ref>]]

== Elastic Deformation ==
In the early (low strain) portion of the curve, most materials obey Hooke’s Law <math> (F_s=kx) </math> to a reasonable approximation, with the constant of proportionality being the modulus of elasticity or Young’s modulus (E), described by the following equation:

<math> \sigma_e = E\epsilon_e \rightarrow E= \frac{\sigma_e}{\epsilon_e} </math>

Also known as the modulus of resilience, it is analogous to the stiffness of a spring (''k''). That’s also why Hooke’s Law includes a spring constant. Materials exhibit elastic flow (ability to return to their original shape upon unloading) up to a certain strain.

== Plastic Deformation ==
As strain is increased, many materials eventually deviate from the linear proportionality of Hooke’s Law, the point of departure being termed the proportional limit. This is usually associated with stress-induced “plastic” flow in the specimen. Here the material is undergoing a rearrangement of its internal molecular or microscopic structure, in which atoms are being moved to new equilibrium positions. This plasticity requires a mechanism for molecular mobility, which in crystalline materials can arise from dislocation motion. Materials lacking this mobility, for instance by having internal microstructures that block dislocation motion, are usually brittle rather than ductile (fail at lower rather than higher strains). The stress-strain curve for brittle materials are typically linear over their full range of strain, eventually terminating in fracture with no to minimal plastic flow (permanent deformation).

=== Yield Stress ===
[[File:0.2Offset.PNG|thumb|Graphical depiction of finding 0.2% offset yield strength (stress)]]
The stress needed to increase the strain beyond the proportional limit in a ductile material continues to rise beyond the proportional limit; the material requires an ever-increasing stress to continue straining, a mechanism termed strain hardening. Microstructural rearrangements associated with plastic flow are usually not reversed. The residual strain induced by a given stress can be determined by drawing an unloading line from the highest point reached on the engineering stress strain curve at that stress back to the strain axis, drawn with a slope equal to that of the initial elastic loading line. This is done because the material unloads elastically, there being no force driving the molecular structure back to its original position. A closely related term is the yield stress <math> (\sigma_y) </math>; this is the stress needed to induce plastic deformation in the specimen. Since it is often difficult to pinpoint the exact stress at which plastic deformation begins, the yield stress is often taken to be the stress needed to induce a specified amount of permanent strain, typically 0.2%. The construction used to find this “offset yield stress” is shown below, where a line of slope E is drawn from the strain axis at <math> \epsilon = 0.2\% </math>; this is the unloading line that would result in the specified permanent strain. The stress at the point of intersection with the curve is the offset yield stress. Engineering calculations typically use a material’s yield strength for determining its ability to resist a load because if the load is greater than the yield strength, the result will be unwanted plastic deformation. See [https://www.youtube.com/watch?v=p0TQ8SV0g2U&t=586s this video] for a detailed explanation of how to find the 0.2% offset yield stress & strain in Excel.

=== Ultimate Strength ===
It appears that the rate of strain hardening decreases up to the ultimate strength (UTS), the greatest level of stress on the curve. Beyond that point, the material appears to strain soften, so that each increment of additional strain requires a smaller stress. The apparent change from strain hardening to strain softening is an artifact of the plotting procedure (using engineering rather than true values), as is the maximum observed in the curve at the UTS. Even though the UTS is perhaps the materials property most commonly reported in tensile tests, it is not a direct measure of the material due to the influence of geometry, and should be used with caution. The yield stress is usually preferred to the UTS in designing with ductile metals, although the UTS is a valid design criterion for brittle materials that do not exhibit flow-induced reductions in cross-sectional area.

=== Necking ===
[[File:Necking.PNG|thumb|Necking in a tensile specimen]]
The true stress is not quite uniform throughout the specimen, and there will always be some location - perhaps a nick or some other defect at the surface - where the local stress is maximum. Once the maximum in the engineering curve has been reached, the localized flow at this site cannot be compensated by further strain hardening, so the area there is reduced further. This increases the local stress even more, which accelerates the flow further. This localized and increasing flow soon leads to a “neck” in the gage length of the specimen such as that seen below.

Until the neck forms, the deformation is essentially uniform throughout the specimen, but after necking all subsequent deformation takes place in the neck. The neck becomes smaller and smaller, local true stress increases all the time, until the specimen fails. This will be the failure mode for most ductile metals. As the neck shrinks, the nonuniform geometry there alters the uniaxial stress state to a complex one involving shear components as well as normal stresses.

=== Drawing ===
[[File:Drawing.PNG|thumb|Necking and drawing in a stress-strain curve]]
This figure shows the engineering stress-strain curve for a semi-crystalline thermoplastic. The response of this material is similar to that of typical metals, in that it shows a proportional limit followed by a maximum in the curve at which necking takes place. (This is commonly described in plastics as the yield stress, although plastic flow has begun at earlier strains.) The polymer, however, differs dramatically in that the neck does not continue shrinking until the specimen fails. Rather, the material in the neck stretches only to a “natural draw ratio” which is a function of temperature and specimen processing, beyond which the material in the neck stops stretching and new material at the neck shoulders necks down. The neck then propagates until it spans the full gage length of the specimen, a process called drawing. This process can be observed without the need for a testing machine, by stretching a “six-pack holder.” Not all polymers are able to sustain this drawing process. Drawing occurs when the necking process produces a strengthened microstructure whose breaking load is greater than that needed to induce necking in the untransformed material just outside the neck.

== Failure ==
[[File:BrittleDuctile.PNG|thumb|Smooth brittle fracture (left) and "cup and cone" shaped ductile fracture (right)]]
Ductile specimens often fail with a “cup and cone” geometry as seen below, in which the outer regions fail in shear and the interior in tension. When the specimen fractures, the engineering strain at break <math> (\epsilon_f) </math> will include the deformation in the necked region and the un-necked region together. Since the true strain in the neck is larger than that in the un-necked material, the value of <math> (\epsilon_f) </math> will depend on the fraction of the gage length that has necked. Therefore, <math> (\epsilon_f) </math> is a function of the specimen geometry as well as the material, and thus is only a crude measure of material ductility. Brittle specimen failure is characterized by very rapid cracking under stress while little or no evidence of ductility or plastic degradation before the fracture occurs. When a piece of equipment suffers a brittle fracture, there is no thinning or necking down. Rather, this damage mechanism often causes cracking without warning, sometimes fracturing equipment into many pieces. Brittle fracture is often caused by low temperatures. If a metal is at or below its brittle-to-ductile transition temperature, then it will be susceptible to brittle fracture. Critical sized flaws, metallurgical degradation, large grain sizes, and high thickness significantly increase the likelihood of a sample experiencing brittle fracture.

== True Stress-Strain Curve ==
[[File:TrueSSC.PNG|thumb|A true stress-strain curve compared its engineering counterpart <ref name="fracture"> United States Naval Academy. (n.d.). FRACTURE OF MATERIALS. </ref> ]]
During elastic deformation, the change in cross-sectional area is less than 1% for most metals and ceramics; thus <math> \sigma_e \approx \sigma_t </math>. However, during plastic deformation, the differences between the true and the engineering values become progressively larger as the specimen dimensions experience substantial change from their original values. Due to this, the engineering stress-strain curve must be interpreted with caution beyond the elastic limits. Using the true stress <math> \sigma_t = \frac{F}{A} </math> rather than the engineering stress <math> \sigma_e = \frac{F}{A_0} </math> can give a more direct measure of the material’s response in the plastic flow range, although this requires using the following additional equations:

<math> \epsilon_t =ln(1+\epsilon_e) </math>
<math> \sigma_t = \sigma_e(1+\epsilon_e) </math>

Ductile metals often have true stress-strain relations that can be described by a simple power-law relation of the form:

<math> \sigma_t = A\epsilon_tn \rightarrow log(\sigma_t) = log(A) + nlog(\epsilon_t)</math>

The strain hardening parameter (n), is the slope of the log-log plot, and is useful as a measure of the resistance to necking. Ductile metals at room temperature usually exhibit values of n from 0.02 to 0.5.
[[File:Toughness.PNG|thumb|Resilience and toughness shown graphically]]

== Strain Energy, Resilience, & Toughness ==
[[File:EnergyDissipated.PNG|thumb|How to graphically determine energy dissipated in a plastically deformed (but not fractured) specimen ]]
The area under the engineering stress-strain curve up to a given value of strain is the total mechanical energy per unit volume consumed by the material in straining it to that value. This is easily shown as follows:

<math> U^* = \frac {1}{V} \int FdL= \int_0^L \frac{FdL}{A_0L_0}= \int_0^{\epsilon}\sigma d\epsilon </math>

In the absence of molecular slip and other mechanisms for energy dissipation, this mechanical energy is stored reversibly within the material as strain energy. The area up to the yield point is termed the modulus of resilience, and the total area up to fracture is termed the modulus of toughness.

The term “resilience” alludes to the concept that up to the point of yielding, the material is unaffected by the applied stress and upon unloading will return to its original shape. But when the strain exceeds the yield point, the material is deformed irreversibly, so that some residual strain will persist even after unloading. The modulus of resilience is then the quantity of energy the material can absorb without suffering damage. Similarly, the modulus of toughness is the energy needed to completely fracture the material. Materials showing good impact resistance, as well as ductile materials, are generally those with high moduli of toughness. The resilience & toughness integrals can be calculated from stress-strain data via methods such as [https://www.youtube.com/watch?v=R8g3B2iySA4&ab_channel=StephanieGlen trapezoidal Riemann sums in Excel].

During loading, the area under the stress-strain curve is the strain energy per unit volume absorbed by the material. Conversely, the area under the unloading curve is the energy released by the material. In the elastic range, these areas are equal and no net energy is absorbed. But if the material is loaded into the plastic range as shown above, the energy absorbed exceeds the energy released and the difference is dissipated as heat.

== Compression ==
Materials that exhibit high tensile strength tend to (but do not always!) exhibit low compressive strength. Likewise, materials high in compressive strength tend to exhibit low tensile strength. Therefore, compression testing is often used on brittle materials such as concrete, metals, plastics, ceramics, composites, and corrugated materials like cardboard. These materials are often used in a load-bearing capacity where their integrity under compressive forces is critical.

Unlike tensile tests, which are usually conducted to determine the tensile properties of a specific material, compression tests are often performed on finished products. Common items such as tennis balls, golf balls, water bottles, protective cases, plastic pipes, and furniture are all examples of products that need to be evaluated for their compressive strength. For example, an engineer may want to conserve plastic by creating water bottles with thinner walls, but the bottles must still be strong enough to be packed in pallets and stacked on top of each other for transport. Compression testing can help the engineer fine tune the balance between product strength and material conservation.
[[File:Instroncompression.jpg|thumb|285x285px|Compression Test on Instron Instrument<ref>Borden, O. 11 June 2025</ref>]]
As long as the loads are sufficiently small (stresses less than the proportional limit), in many materials the relations outlined for tensile testing apply equally well in compression. The expression for deformation <math> (\delta) </math> from a given load

<math> \delta = \frac{FL}{AE} </math>,

applies just as in tension, with negative values for δ and F indicating compression. Further, the modulus E is the same in tension and compression to a good approximation, and the stress-strain curve simply extends as a straight line into the third quadrant as shown below.

There are some practical difficulties in performing stress-strain tests in compression. If excessively large loads are mistakenly applied in a tensile test, perhaps by wrong settings on the testing machine, the specimen simply breaks and the test must be repeated with a new specimen. But in compression, a mistake can easily damage the load cell or other sensitive components, since even after specimen failure the loads are not necessarily relieved. This is why when designing a method for the Instron tester, there must always be a second stop condition at 28kN load. The ultimate compressive strength of a material is the value of compressive stress reached when the material fails completely. When brittle materials reach their ultimate compressive strength they are crushed, and the load drops drastically. Materials with higher ductility, (most plastics) do not rupture, but instead continue deforming until the load is no longer being applied to the specimen, but rather between the two compression platens. In these cases, compressive strength can be reported as specific deformations such as 1%, 5%, or 10% of the test specimen's original height. <ref name="Instron"> Instron. (n.d.). What is Compression Testing? https://www.instron.us/en-us/our-company/library/test-types/compression-test. </ref>

== Bend Testing ==
[[File:BendVMDiagram.PNG|thumb|Diagram of a three point bend test, with graphs of bending moment ''M'', shear ''Q'' and deflection ''w'' <ref name="Penn"> Penn Center for Musculoskeletal Disorders. Three Point and Four Point Bending | Penn Center for Musculoskeletal Disorders | Perelman School of Medicine at the University of Pennsylvania. (n.d.). https://www.med.upenn.edu/pcmd/3pt-4pt-bending.html. </ref>]]
In bending tests, standardized, mostly cylindrical specimens are placed centrally on a test fixture. <ref name="GOM"> Bending test: Suitable for both brittle and ductile materials. (n.d.). https://www.gom.com/en/topics/bend-test. </ref> The rounded support rollers (bearings) are arranged parallel to each other at a certain distance (support width). The diameter of the cylindrical specimen is proportional to the support width of the bearings. The test punch, which moves down slowly and at a constant speed, loads the sample with increasing force until it breaks or reaches the previously determined deformation. Flexural modulus (elastic modulus in bending), stress-strain behavior, and failure limits can all be obtained using a bend test. <ref name="Bruker"> 3-Point Bend Testing. Bruker. (n.d.). https://www.bruker.com/en/products-and-solutions/test-and-measurement/tribometers-and-mechanical-testers/3-point-bend-testing.html. </ref> The 3-point bending test is the most frequently performed bending test. However, it has the disadvantage that in addition to the compressive and tensile forces exerted, transverse forces are also effective in the material.

During the test, the values of the bending force and deflection are recorded, which are then used to determine the material characteristics. The test sequence is plotted in a stress-strain curve and can also be recorded with a video camera. Bending tests are performed to obtain information about the bending behavior of the tested material from the single-axis bending stress. In case of brittle materials, the bending strength is determined this way. For ductile materials, the limit yield point, greatest possible bending angle, and Young’s modulus are determined.

Calculation of the flexural stress <math>\sigma_f</math>

:<math>\sigma_f = \frac{3 F L}{2 b d^2}</math> for a rectangular cross section

:<math>\sigma_f = \frac{F L}{\pi R^3}</math> for a circular cross section<ref name=Biomaterials>"Chapter 4 Mechanical Properties of Biomaterials". Biomaterials – The intersection of Biology and Material Science. New Jersey, United States: Pearson Prentice Hall. 2008. p. 152.</ref>

Calculation of the flexural strain <math>\epsilon_f</math>

:<math>\epsilon_f = \frac{6Dd}{L^2}</math>

Calculation of flexural modulus <math>E_f</math><ref name="Zweben"> Zweben, C., W. S. Smith, and M. W. Wardle (1979), "Test methods for fiber tensile strength, composite flexural modulus, and properties of fabric-reinforced laminates", Composite Materials: Testing and Design (Fifth Conference), ASTM International</ref>
[[File:Mark10 3point.jpg|thumb|3 Point Bend Test set up on Mark 10 Machine<ref>Borden, O. 11 June 2025</ref>]]
:<math>E_f = \frac{L^3 m}{4 b d^3}</math>

in these formulas the following parameters are used:
* <math>\sigma_f</math> = Stress in outer fibers at midpoint, (MPa)
* <math>\epsilon_f</math> = Strain in the outer surface, (mm/mm)
* <math>E_f</math> = flexural Modulus of elasticity,(MPa)
* <math>F</math> = load at a given point on the load deflection curve, (N)
* <math>L</math> = Support span, (mm)
* <math>b</math> = Width of test beam, (mm)
* <math>d</math> = Depth or thickness of tested beam, (mm)
* <math>D</math> = maximum deflection of the center of the beam, (mm)
* <math>m</math> = The gradient (i.e., slope) of the initial straight-line portion of the load deflection curve, (N/mm)

* <math>R</math> = The radius of the beam, (mm)

In a bending test, the bending stress is greatest in the center of the specimen (highest deflection). At this point, the greatest bending moment is always present. From the central pressure point, the bending moment decreases linearly in both directions toward the bearings. The material is subjected to pressure on its inner side and tension on its outer side. In the outer fibers of the specimen, the bending stress (tensile and compressive stress) is greatest and decreases inwards toward the neutral fiber.

If a partially plastically deformed specimen is relieved during the bending test by raising the test punch, only the internal stresses (residual stresses) present in the material and the resulting torque are still effective. This will partially reshape the sample.

=== Bending Behavior of Ductile Materials ===
If the bending stress in the sample made of a ductile material is lower than the limit stress of the plastic deformation, the bending stress is exclusively elastic. As the bending stress increases, the yield strength is exceeded first in the peripheral areas of the specimen. These areas then deform plastically (the so-called material flow). The limit yield point is the limit bending stress up to which easily deformable materials can be loaded by bending without permanent deformation in the marginal area.

The moment this kind of deformation occurs can be determined directly from the deflection-force diagram, in a similar manner to how the yield strength is found in a tensile test. With deflection increasing steadily, more and more internal areas of the specimen are involved in the plastic deformation. This is a result of the stress increase. For steels, for example, the limit yield point is between 10 to 20 percent higher than the yield strength due to the linear stress increase. If the yield strength is exceeded in the edge fibers during the bending test, the inner and exclusively elastically stressed fibers impede the flow movement. Bending tests with ductile materials are different from those performed with brittle materials: Tough materials can be subjected to extreme plastic deformation, but they cannot be broken, no matter how strong the applied force is. Therefore, a bending test with a ductile sample is finished when the yield point is exceeded. The bending strength of ductile materials is determined by the point in time at which the plastic deformation occurs.

=== Bending Behavior of Brittle Materials ===
Specimens made of brittle materials show a different bending behavior during materials testing. They break without clearly visible material flow behavior. Therefore, the determination of the limit yield point is more complicated for brittle materials. In order to be able to determine the bending strength nevertheless, the maximum bending stress at which the sample breaks is determined. However, the bending strength is a fictitious value that is not identical to the bending stress actually occurring in the material. Another characteristic of bending tests with brittle materials is the fracture deflection. This technical term describes the greatest possible deflection of a specimen shortly before fracture.

The fracture deflection depends on the support width: Larger distances between the bearings allow for greater deflections. In order to check the strength of brittle materials, the bending test is often more suitable than the tensile test, because the materials are subjected to bending stress only. If this sample were to be subjected to a tensile test, it would break prematurely and measurement problems would occur. For certain brittle materials, the tensile test is therefore replaced by the bending test. These materials include thermosetting sheets and molding materials, thermoplastic injection molding compounds, and fiber-reinforced plastics.

== Related Theory ==
* [[Cyclic Loading]]
* [[Stress Theory]]

== References ==
<references />

[[Index.php?title=Category:Equipment|meas]]

File:Mark10 3point.jpg

2025-06-17T13:47:17Z

ObordenMEASURE: Image of 3-point bend test set up on Mark 10

== Summary ==
Image of 3-point bend test set up on Mark 10

File:Instroncompression.jpg

2025-06-17T13:45:41Z

ObordenMEASURE: Image of Instron being set up for use in a compression test

== Summary ==
Image of Instron being set up for use in a compression test

Mechanical Testing

2025-06-17T13:44:48Z

ObordenMEASURE: /* Tensile Testing & Engineering Stress-Strain Curves */ Added image of Tensile Test

{|class="infobox"
!
!style="font-size:135%;text-align:center;" | Mark-10 ESM303 [https://sites.gatech.edu/mse-mill/sop/mark-10/ [SOP<nowiki>]</nowiki>]
!style="font-size:135%;text-align:center;" | Instron 5967 [https://sites.gatech.edu/mse-mill/sop/instron/ [SOP<nowiki>]</nowiki>]
|-
|                          ||[[File:Mark10-052621.png|center|250px]]||[[File:Instron-052621.png|center|250px]]
|-
|'''Load Cells''':||500 mN, 25 N, 500 N, 1500 N||30 kN
|-
|'''Load Capacity''':||1.5 kN @ up to 610 mm/min||30 kN
|-
|'''Speed range''':||13 - 330 mm/min||0.001 mm/min - 1000 mm/min
|-
|'''Travel range''':||45.7 cm||1.212 m
|-
|'''Max width''':||n/a||0.418 m
|}

The Mark-10 ESM303 Mechanical Tester is a mechanical testing frame capable of testing materials under tensile, cyclic, compressive, and 3-point bending modes. This machine has several load cells: a 1500 N, a 500 N, a 25 N, and a 500 mN load cell.

The Instron Mechanical Tester is a mechanical testing frame capable of testing materials under tensile, compressive, and 3-point-bending modes. This machine is equipped with a 30kN load cell.

== Loading ==
[[File:Loading.PNG|thumb|199x199px|left|1. Compression 2. Tension 3. Shear 4. Torsion 5. Bending]]

A material in service is subjected to different forces. Depending on the magnitude of these forces, the metal may or may not change its shape. The act of applying the force is known as loading. There are several different ways in which these forces may be applied on a part, including compression, tension, and bending. <ref name="Velling"> Velling, A. (2021, May 5). Stress-Strain Curve: How to Read the Graph? Fractory. https://fractory.com/stress-strain-curve/. </ref> <ref name="Meyers">Meyers, M., & Chawla, K. (2016). Hardness. In Mechanical Behavior of Materials (2nd ed., pp. 214–225). chapter, Cambridge University Press.</ref>

== Stress and Strain ==

=== True Stress/Strain ===
Stress is defined as the ratio of applied force to the cross-sectional area of the material it is applied to. The formula for calculating (true) material stress is:

<math> \sigma_t =\frac{F}{A}, </math> Where ''F'' is force, ''A'' is area, and <math> \sigma </math> is stress.

Stress can be understood as an internal force induced in the material in response to an externally applied force (Newton’s 3rd law). It will try to resist any change in dimension caused by the external force. Strain is defined as the ratio of the change in dimension to the initial dimension of the metal. It does not have a unit (sometimes designated as length/length). The formula for calculating (true) strain is:

<math> \epsilon_t = \int_{l_0} ^l \frac{dl}{l} = ln(\frac{l}{l_0}), </math> Where <math> l_0 </math> is starting or initial length, ''l'' is current length, and <math> \epsilon </math> is strain.

=== Engineering Stress/Strain ===
[[File:SScurve.PNG|thumb|Engineering stress-strain curve of a typical ductile material <ref name="Roylance"> Roylance, D. (2001, August 23). STRESS-STRAIN CURVES.</ref>]]In many applications, a simpler form of strain, commonly called engineering or nominal strain, is used. This type of strain is defined as

<math> \epsilon_e = \frac{(l-l_0)}{l_0} </math>

When the strains are reasonably small, the engineering and true strains are approximately the same. The elastic deformation in metals and ceramics rarely exceeds 0.005, and for this value, the difference between <math> \epsilon_t </math> and <math> \epsilon_e </math> can be neglected. In a likewise fashion, engineering (nominal) stress is defined as:

<math> \sigma_e = \frac{F}{A_0} </math>, Where <math> A_0 </math> is the original area of cross-section.

Normal strain may be positive or negative depending on the external force’s directions and therefore effect on the original length. Tensile stresses and strains are considered to be positive while their compressive counterparts are considered to be negatives. While there are multiple types of stresses and strains, this article focuses on their normal longitudinal variants.

== Tensile Testing & Engineering Stress-Strain Curves ==
[[File:Tensile Test Instron.jpeg|thumb|207x207px|Tensile Test Set Up on Instron tester<ref>Borden, O. June 2025</ref>]]
The most common method to obtain a stress-strain curve is to subject a sample to a tensile test. When setup for tensile testing, the mechanical testers have two claws which hold the two extremes of the sample and pull it at a uniform rate. The force applied and the strain produced is recorded until a fracture occurs. The two parameters are then plotted on an X-Y graph to get a stress-strain curve. Note that the testers produces a force-displacement plot (with the same general shape) that must be manually converted to a stress-strain curve using the aforementioned equations in conjunction with the sample dimensions (or entering the sample dimensions into the software on the Instron). There are various sections on the stress and strain curve (described below) that describe different behavior of a material depending on the amount of stress induced. Stress and strain curves for brittle and ductile materials are different.

A stress-strain curve always has the same 3 sections: elastic deformation, plastic deformation, and fracture. These three section occur differently in different materials and have subcategories as described below.

[[File:BrittleSSCurve.PNG|thumb|300x300px|Typical stress-strain curve of a brittle material <ref name="MechaniCalc> "Mechanical Properties of Materials. MechaniCalc. (n.d.). https://mechanicalc.com/reference/mechanical-properties-of-materials.</ref>]]

== Elastic Deformation ==
In the early (low strain) portion of the curve, most materials obey Hooke’s Law <math> (F_s=kx) </math> to a reasonable approximation, with the constant of proportionality being the modulus of elasticity or Young’s modulus (E), described by the following equation:

<math> \sigma_e = E\epsilon_e \rightarrow E= \frac{\sigma_e}{\epsilon_e} </math>

Also known as the modulus of resilience, it is analogous to the stiffness of a spring (''k''). That’s also why Hooke’s Law includes a spring constant. Materials exhibit elastic flow (ability to return to their original shape upon unloading) up to a certain strain.

== Plastic Deformation ==
As strain is increased, many materials eventually deviate from the linear proportionality of Hooke’s Law, the point of departure being termed the proportional limit. This is usually associated with stress-induced “plastic” flow in the specimen. Here the material is undergoing a rearrangement of its internal molecular or microscopic structure, in which atoms are being moved to new equilibrium positions. This plasticity requires a mechanism for molecular mobility, which in crystalline materials can arise from dislocation motion. Materials lacking this mobility, for instance by having internal microstructures that block dislocation motion, are usually brittle rather than ductile (fail at lower rather than higher strains). The stress-strain curve for brittle materials are typically linear over their full range of strain, eventually terminating in fracture with no to minimal plastic flow (permanent deformation).

=== Yield Stress ===
[[File:0.2Offset.PNG|thumb|Graphical depiction of finding 0.2% offset yield strength (stress)]]
The stress needed to increase the strain beyond the proportional limit in a ductile material continues to rise beyond the proportional limit; the material requires an ever-increasing stress to continue straining, a mechanism termed strain hardening. Microstructural rearrangements associated with plastic flow are usually not reversed. The residual strain induced by a given stress can be determined by drawing an unloading line from the highest point reached on the engineering stress strain curve at that stress back to the strain axis, drawn with a slope equal to that of the initial elastic loading line. This is done because the material unloads elastically, there being no force driving the molecular structure back to its original position. A closely related term is the yield stress <math> (\sigma_y) </math>; this is the stress needed to induce plastic deformation in the specimen. Since it is often difficult to pinpoint the exact stress at which plastic deformation begins, the yield stress is often taken to be the stress needed to induce a specified amount of permanent strain, typically 0.2%. The construction used to find this “offset yield stress” is shown below, where a line of slope E is drawn from the strain axis at <math> \epsilon = 0.2\% </math>; this is the unloading line that would result in the specified permanent strain. The stress at the point of intersection with the curve is the offset yield stress. Engineering calculations typically use a material’s yield strength for determining its ability to resist a load because if the load is greater than the yield strength, the result will be unwanted plastic deformation. See [https://www.youtube.com/watch?v=p0TQ8SV0g2U&t=586s this video] for a detailed explanation of how to find the 0.2% offset yield stress & strain in Excel.

=== Ultimate Strength ===
It appears that the rate of strain hardening decreases up to the ultimate strength (UTS), the greatest level of stress on the curve. Beyond that point, the material appears to strain soften, so that each increment of additional strain requires a smaller stress. The apparent change from strain hardening to strain softening is an artifact of the plotting procedure (using engineering rather than true values), as is the maximum observed in the curve at the UTS. Even though the UTS is perhaps the materials property most commonly reported in tensile tests, it is not a direct measure of the material due to the influence of geometry, and should be used with caution. The yield stress is usually preferred to the UTS in designing with ductile metals, although the UTS is a valid design criterion for brittle materials that do not exhibit flow-induced reductions in cross-sectional area.

=== Necking ===
[[File:Necking.PNG|thumb|Necking in a tensile specimen]]
The true stress is not quite uniform throughout the specimen, and there will always be some location - perhaps a nick or some other defect at the surface - where the local stress is maximum. Once the maximum in the engineering curve has been reached, the localized flow at this site cannot be compensated by further strain hardening, so the area there is reduced further. This increases the local stress even more, which accelerates the flow further. This localized and increasing flow soon leads to a “neck” in the gage length of the specimen such as that seen below.

Until the neck forms, the deformation is essentially uniform throughout the specimen, but after necking all subsequent deformation takes place in the neck. The neck becomes smaller and smaller, local true stress increases all the time, until the specimen fails. This will be the failure mode for most ductile metals. As the neck shrinks, the nonuniform geometry there alters the uniaxial stress state to a complex one involving shear components as well as normal stresses.

=== Drawing ===
[[File:Drawing.PNG|thumb|Necking and drawing in a stress-strain curve]]
This figure shows the engineering stress-strain curve for a semi-crystalline thermoplastic. The response of this material is similar to that of typical metals, in that it shows a proportional limit followed by a maximum in the curve at which necking takes place. (This is commonly described in plastics as the yield stress, although plastic flow has begun at earlier strains.) The polymer, however, differs dramatically in that the neck does not continue shrinking until the specimen fails. Rather, the material in the neck stretches only to a “natural draw ratio” which is a function of temperature and specimen processing, beyond which the material in the neck stops stretching and new material at the neck shoulders necks down. The neck then propagates until it spans the full gage length of the specimen, a process called drawing. This process can be observed without the need for a testing machine, by stretching a “six-pack holder.” Not all polymers are able to sustain this drawing process. Drawing occurs when the necking process produces a strengthened microstructure whose breaking load is greater than that needed to induce necking in the untransformed material just outside the neck.

== Failure ==
[[File:BrittleDuctile.PNG|thumb|Smooth brittle fracture (left) and "cup and cone" shaped ductile fracture (right)]]
Ductile specimens often fail with a “cup and cone” geometry as seen below, in which the outer regions fail in shear and the interior in tension. When the specimen fractures, the engineering strain at break <math> (\epsilon_f) </math> will include the deformation in the necked region and the un-necked region together. Since the true strain in the neck is larger than that in the un-necked material, the value of <math> (\epsilon_f) </math> will depend on the fraction of the gage length that has necked. Therefore, <math> (\epsilon_f) </math> is a function of the specimen geometry as well as the material, and thus is only a crude measure of material ductility. Brittle specimen failure is characterized by very rapid cracking under stress while little or no evidence of ductility or plastic degradation before the fracture occurs. When a piece of equipment suffers a brittle fracture, there is no thinning or necking down. Rather, this damage mechanism often causes cracking without warning, sometimes fracturing equipment into many pieces. Brittle fracture is often caused by low temperatures. If a metal is at or below its brittle-to-ductile transition temperature, then it will be susceptible to brittle fracture. Critical sized flaws, metallurgical degradation, large grain sizes, and high thickness significantly increase the likelihood of a sample experiencing brittle fracture.

== True Stress-Strain Curve ==
[[File:TrueSSC.PNG|thumb|A true stress-strain curve compared its engineering counterpart <ref name="fracture"> United States Naval Academy. (n.d.). FRACTURE OF MATERIALS. </ref> ]]
During elastic deformation, the change in cross-sectional area is less than 1% for most metals and ceramics; thus <math> \sigma_e \approx \sigma_t </math>. However, during plastic deformation, the differences between the true and the engineering values become progressively larger as the specimen dimensions experience substantial change from their original values. Due to this, the engineering stress-strain curve must be interpreted with caution beyond the elastic limits. Using the true stress <math> \sigma_t = \frac{F}{A} </math> rather than the engineering stress <math> \sigma_e = \frac{F}{A_0} </math> can give a more direct measure of the material’s response in the plastic flow range, although this requires using the following additional equations:

<math> \epsilon_t =ln(1+\epsilon_e) </math>
<math> \sigma_t = \sigma_e(1+\epsilon_e) </math>

Ductile metals often have true stress-strain relations that can be described by a simple power-law relation of the form:

<math> \sigma_t = A\epsilon_tn \rightarrow log(\sigma_t) = log(A) + nlog(\epsilon_t)</math>

The strain hardening parameter (n), is the slope of the log-log plot, and is useful as a measure of the resistance to necking. Ductile metals at room temperature usually exhibit values of n from 0.02 to 0.5.
[[File:Toughness.PNG|thumb|Resilience and toughness shown graphically]]

== Strain Energy, Resilience, & Toughness ==
[[File:EnergyDissipated.PNG|thumb|How to graphically determine energy dissipated in a plastically deformed (but not fractured) specimen ]]
The area under the engineering stress-strain curve up to a given value of strain is the total mechanical energy per unit volume consumed by the material in straining it to that value. This is easily shown as follows:

<math> U^* = \frac {1}{V} \int FdL= \int_0^L \frac{FdL}{A_0L_0}= \int_0^{\epsilon}\sigma d\epsilon </math>

In the absence of molecular slip and other mechanisms for energy dissipation, this mechanical energy is stored reversibly within the material as strain energy. The area up to the yield point is termed the modulus of resilience, and the total area up to fracture is termed the modulus of toughness.

The term “resilience” alludes to the concept that up to the point of yielding, the material is unaffected by the applied stress and upon unloading will return to its original shape. But when the strain exceeds the yield point, the material is deformed irreversibly, so that some residual strain will persist even after unloading. The modulus of resilience is then the quantity of energy the material can absorb without suffering damage. Similarly, the modulus of toughness is the energy needed to completely fracture the material. Materials showing good impact resistance, as well as ductile materials, are generally those with high moduli of toughness. The resilience & toughness integrals can be calculated from stress-strain data via methods such as [https://www.youtube.com/watch?v=R8g3B2iySA4&ab_channel=StephanieGlen trapezoidal Riemann sums in Excel].

During loading, the area under the stress-strain curve is the strain energy per unit volume absorbed by the material. Conversely, the area under the unloading curve is the energy released by the material. In the elastic range, these areas are equal and no net energy is absorbed. But if the material is loaded into the plastic range as shown above, the energy absorbed exceeds the energy released and the difference is dissipated as heat.

== Compression ==
Materials that exhibit high tensile strength tend to (but do not always!) exhibit low compressive strength. Likewise, materials high in compressive strength tend to exhibit low tensile strength. Therefore, compression testing is often used on brittle materials such as concrete, metals, plastics, ceramics, composites, and corrugated materials like cardboard. These materials are often used in a load-bearing capacity where their integrity under compressive forces is critical.

Unlike tensile tests, which are usually conducted to determine the tensile properties of a specific material, compression tests are often performed on finished products. Common items such as tennis balls, golf balls, water bottles, protective cases, plastic pipes, and furniture are all examples of products that need to be evaluated for their compressive strength. For example, an engineer may want to conserve plastic by creating water bottles with thinner walls, but the bottles must still be strong enough to be packed in pallets and stacked on top of each other for transport. Compression testing can help the engineer fine tune the balance between product strength and material conservation.

As long as the loads are sufficiently small (stresses less than the proportional limit), in many materials the relations outlined for tensile testing apply equally well in compression. The expression for deformation <math> (\delta) </math> from a given load

<math> \delta = \frac{FL}{AE} </math>,

applies just as in tension, with negative values for δ and F indicating compression. Further, the modulus E is the same in tension and compression to a good approximation, and the stress-strain curve simply extends as a straight line into the third quadrant as shown below.

There are some practical difficulties in performing stress-strain tests in compression. If excessively large loads are mistakenly applied in a tensile test, perhaps by wrong settings on the testing machine, the specimen simply breaks and the test must be repeated with a new specimen. But in compression, a mistake can easily damage the load cell or other sensitive components, since even after specimen failure the loads are not necessarily relieved. This is why when designing a method for the Instron tester, there must always be a second stop condition at 28kN load. The ultimate compressive strength of a material is the value of compressive stress reached when the material fails completely. When brittle materials reach their ultimate compressive strength they are crushed, and the load drops drastically. Materials with higher ductility, (most plastics) do not rupture, but instead continue deforming until the load is no longer being applied to the specimen, but rather between the two compression platens. In these cases, compressive strength can be reported as specific deformations such as 1%, 5%, or 10% of the test specimen's original height. <ref name="Instron"> Instron. (n.d.). What is Compression Testing? https://www.instron.us/en-us/our-company/library/test-types/compression-test. </ref>

== Bend Testing ==
[[File:BendVMDiagram.PNG|thumb|Diagram of a three point bend test, with graphs of bending moment ''M'', shear ''Q'' and deflection ''w'' <ref name="Penn"> Penn Center for Musculoskeletal Disorders. Three Point and Four Point Bending | Penn Center for Musculoskeletal Disorders | Perelman School of Medicine at the University of Pennsylvania. (n.d.). https://www.med.upenn.edu/pcmd/3pt-4pt-bending.html. </ref>]]
In bending tests, standardized, mostly cylindrical specimens are placed centrally on a test fixture. <ref name="GOM"> Bending test: Suitable for both brittle and ductile materials. (n.d.). https://www.gom.com/en/topics/bend-test. </ref> The rounded support rollers (bearings) are arranged parallel to each other at a certain distance (support width). The diameter of the cylindrical specimen is proportional to the support width of the bearings. The test punch, which moves down slowly and at a constant speed, loads the sample with increasing force until it breaks or reaches the previously determined deformation. Flexural modulus (elastic modulus in bending), stress-strain behavior, and failure limits can all be obtained using a bend test. <ref name="Bruker"> 3-Point Bend Testing. Bruker. (n.d.). https://www.bruker.com/en/products-and-solutions/test-and-measurement/tribometers-and-mechanical-testers/3-point-bend-testing.html. </ref> The 3-point bending test is the most frequently performed bending test. However, it has the disadvantage that in addition to the compressive and tensile forces exerted, transverse forces are also effective in the material.

During the test, the values of the bending force and deflection are recorded, which are then used to determine the material characteristics. The test sequence is plotted in a stress-strain curve and can also be recorded with a video camera. Bending tests are performed to obtain information about the bending behavior of the tested material from the single-axis bending stress. In case of brittle materials, the bending strength is determined this way. For ductile materials, the limit yield point, greatest possible bending angle, and Young’s modulus are determined.

Calculation of the flexural stress <math>\sigma_f</math>

:<math>\sigma_f = \frac{3 F L}{2 b d^2}</math> for a rectangular cross section

:<math>\sigma_f = \frac{F L}{\pi R^3}</math> for a circular cross section<ref name=Biomaterials>"Chapter 4 Mechanical Properties of Biomaterials". Biomaterials – The intersection of Biology and Material Science. New Jersey, United States: Pearson Prentice Hall. 2008. p. 152.</ref>

Calculation of the flexural strain <math>\epsilon_f</math>

:<math>\epsilon_f = \frac{6Dd}{L^2}</math>

Calculation of flexural modulus <math>E_f</math><ref name="Zweben"> Zweben, C., W. S. Smith, and M. W. Wardle (1979), "Test methods for fiber tensile strength, composite flexural modulus, and properties of fabric-reinforced laminates", Composite Materials: Testing and Design (Fifth Conference), ASTM International</ref>

:<math>E_f = \frac{L^3 m}{4 b d^3}</math>

in these formulas the following parameters are used:
* <math>\sigma_f</math> = Stress in outer fibers at midpoint, (MPa)
* <math>\epsilon_f</math> = Strain in the outer surface, (mm/mm)
* <math>E_f</math> = flexural Modulus of elasticity,(MPa)
* <math>F</math> = load at a given point on the load deflection curve, (N)
* <math>L</math> = Support span, (mm)
* <math>b</math> = Width of test beam, (mm)
* <math>d</math> = Depth or thickness of tested beam, (mm)
* <math>D</math> = maximum deflection of the center of the beam, (mm)
* <math>m</math> = The gradient (i.e., slope) of the initial straight-line portion of the load deflection curve, (N/mm)

* <math>R</math> = The radius of the beam, (mm)

In a bending test, the bending stress is greatest in the center of the specimen (highest deflection). At this point, the greatest bending moment is always present. From the central pressure point, the bending moment decreases linearly in both directions toward the bearings. The material is subjected to pressure on its inner side and tension on its outer side. In the outer fibers of the specimen, the bending stress (tensile and compressive stress) is greatest and decreases inwards toward the neutral fiber.

If a partially plastically deformed specimen is relieved during the bending test by raising the test punch, only the internal stresses (residual stresses) present in the material and the resulting torque are still effective. This will partially reshape the sample.

=== Bending Behavior of Ductile Materials ===
If the bending stress in the sample made of a ductile material is lower than the limit stress of the plastic deformation, the bending stress is exclusively elastic. As the bending stress increases, the yield strength is exceeded first in the peripheral areas of the specimen. These areas then deform plastically (the so-called material flow). The limit yield point is the limit bending stress up to which easily deformable materials can be loaded by bending without permanent deformation in the marginal area.

The moment this kind of deformation occurs can be determined directly from the deflection-force diagram, in a similar manner to how the yield strength is found in a tensile test. With deflection increasing steadily, more and more internal areas of the specimen are involved in the plastic deformation. This is a result of the stress increase. For steels, for example, the limit yield point is between 10 to 20 percent higher than the yield strength due to the linear stress increase. If the yield strength is exceeded in the edge fibers during the bending test, the inner and exclusively elastically stressed fibers impede the flow movement. Bending tests with ductile materials are different from those performed with brittle materials: Tough materials can be subjected to extreme plastic deformation, but they cannot be broken, no matter how strong the applied force is. Therefore, a bending test with a ductile sample is finished when the yield point is exceeded. The bending strength of ductile materials is determined by the point in time at which the plastic deformation occurs.

=== Bending Behavior of Brittle Materials ===
Specimens made of brittle materials show a different bending behavior during materials testing. They break without clearly visible material flow behavior. Therefore, the determination of the limit yield point is more complicated for brittle materials. In order to be able to determine the bending strength nevertheless, the maximum bending stress at which the sample breaks is determined. However, the bending strength is a fictitious value that is not identical to the bending stress actually occurring in the material. Another characteristic of bending tests with brittle materials is the fracture deflection. This technical term describes the greatest possible deflection of a specimen shortly before fracture.

The fracture deflection depends on the support width: Larger distances between the bearings allow for greater deflections. In order to check the strength of brittle materials, the bending test is often more suitable than the tensile test, because the materials are subjected to bending stress only. If this sample were to be subjected to a tensile test, it would break prematurely and measurement problems would occur. For certain brittle materials, the tensile test is therefore replaced by the bending test. These materials include thermosetting sheets and molding materials, thermoplastic injection molding compounds, and fiber-reinforced plastics.

== Related Theory ==
* [[Cyclic Loading]]
* [[Stress Theory]]

== References ==
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File:Tensile Test Instron.jpeg

2025-06-17T13:40:07Z

ObordenMEASURE: Image of a Tensile test preformed on the Instron with an ASTM dog bone 3D print.

== Summary ==
Image of a Tensile test preformed on the Instron with an ASTM dog bone 3D print.