Stress Theory - Revision history

https://millwiki.mse.gatech.edu/index.php?action=history&feed=atom&title=Stress_Theory Stress Theory - Revision history 2026-10-17T09:04:04Z Revision history for this page on the wiki MediaWiki 1.42.7 https://millwiki.mse.gatech.edu/index.php?title=Stress_Theory&diff=38660&oldid=prev Amarin32 at 20:56, 14 June 2021 2021-06-14T20:56:43Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:56, 14 June 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l4">Line 4:</td> <td colspan="2" class="diff-lineno">Line 4:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== True vs. Engineering Stress ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== True vs. Engineering Stress ==</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Beyond the yield point, molecular flow causes a substantial reduction in the specimen cross-sectional area A, so the true stress <math> \sigma_t= \frac{F}{A} </math> actually borne by the material is larger than the engineering stress computed from the original cross-sectional area (<math> \sigma_e= \frac{F}{A_0} </math>). The load must equal the true stress times the actual area (<math> F =\sigma_t A </math>), and as long as strain hardening can increase σt enough to compensate for the reduced area A, the load and therefore the engineering stress will continue to rise as the strain increases. <ref name="Roylance"> Roylance, D. (2001, August 23). STRESS-STRAIN CURVES.</ref<del style="font-weight: bold; text-decoration: none;">> <math> \sigma_e = \frac{F}{A_0} </math</del>> Eventually, however, the decrease in area due to flow becomes larger than the increase in true stress due to strain hardening, and the load begins to fall. This is a geometrical effect, and if the true stress rather than the engineering stress were plotted no maximum would be observed in the curve. The strain hardening rate is the slope of the stress-strain curve, also called the tangent modulus. At the UTS the differential of the load F is zero, giving an analytical relation between the true stress and the area at necking:</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Beyond the yield point, molecular flow causes a substantial reduction in the specimen cross-sectional area A, so the true stress <math> \sigma_t= \frac{F}{A} </math> actually borne by the material is larger than the engineering stress computed from the original cross-sectional area (<math> \sigma_e= \frac{F}{A_0} </math>). The load must equal the true stress times the actual area (<math> F =\sigma_t A </math>), and as long as strain hardening can increase σt enough to compensate for the reduced area A, the load and therefore the engineering stress will continue to rise as the strain increases. <ref name="Roylance"> Roylance, D. (2001, August 23). STRESS-STRAIN CURVES.</ref> Eventually, however, the decrease in area due to flow becomes larger than the increase in true stress due to strain hardening, and the load begins to fall. This is a geometrical effect, and if the true stress rather than the engineering stress were plotted no maximum would be observed in the curve. The strain hardening rate is the slope of the stress-strain curve, also called the tangent modulus. At the UTS the differential of the load F is zero, giving an analytical relation between the true stress and the area at necking:</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math> F =\sigma_t A \rightarrow dF = 0 =\sigma_t dA + A d\sigma_t \rightarrow -\frac{dA}{A} = \frac{d\sigma_t}{\sigma_t} </math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math> F =\sigma_t A \rightarrow dF = 0 =\sigma_t dA + A d\sigma_t \rightarrow -\frac{dA}{A} = \frac{d\sigma_t}{\sigma_t} </math></div></td></tr>  </table>

Amarin32 https://millwiki.mse.gatech.edu/index.php?title=Stress_Theory&diff=38658&oldid=prev Amarin32 at 20:55, 14 June 2021 2021-06-14T20:55:21Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:55, 14 June 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l4">Line 4:</td> <td colspan="2" class="diff-lineno">Line 4:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== True vs. Engineering Stress ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== True vs. Engineering Stress ==</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Beyond the yield point, molecular flow causes a substantial reduction in the specimen cross-sectional area A, so the true stress <math> \sigma_t= \frac{F}{A} </math> actually borne by the material is larger than the engineering stress computed from the original cross-sectional area (<math> \sigma_e= \frac{F}{A_0} </math>). The load must equal the true stress times the actual area (<math> F =\sigma_t A </math>), and as long as strain hardening can increase σt enough to compensate for the reduced area A, the load and therefore the engineering stress will continue to rise as the strain increases. <ref name="Roylance"> Roylance, D. (2001, August 23). STRESS-STRAIN CURVES.</ref><del style="font-weight: bold; text-decoration: none;">]] </del><math> \sigma_e = \frac{F}{A_0} </math> Eventually, however, the decrease in area due to flow becomes larger than the increase in true stress due to strain hardening, and the load begins to fall. This is a geometrical effect, and if the true stress rather than the engineering stress were plotted no maximum would be observed in the curve. The strain hardening rate is the slope of the stress-strain curve, also called the tangent modulus. At the UTS the differential of the load F is zero, giving an analytical relation between the true stress and the area at necking:</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Beyond the yield point, molecular flow causes a substantial reduction in the specimen cross-sectional area A, so the true stress <math> \sigma_t= \frac{F}{A} </math> actually borne by the material is larger than the engineering stress computed from the original cross-sectional area (<math> \sigma_e= \frac{F}{A_0} </math>). The load must equal the true stress times the actual area (<math> F =\sigma_t A </math>), and as long as strain hardening can increase σt enough to compensate for the reduced area A, the load and therefore the engineering stress will continue to rise as the strain increases. <ref name="Roylance"> Roylance, D. (2001, August 23). STRESS-STRAIN CURVES.</ref> <math> \sigma_e = \frac{F}{A_0} </math> Eventually, however, the decrease in area due to flow becomes larger than the increase in true stress due to strain hardening, and the load begins to fall. This is a geometrical effect, and if the true stress rather than the engineering stress were plotted no maximum would be observed in the curve. The strain hardening rate is the slope of the stress-strain curve, also called the tangent modulus. At the UTS the differential of the load F is zero, giving an analytical relation between the true stress and the area at necking:</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math> F =\sigma_t A \rightarrow dF = 0 =\sigma_t dA + A d\sigma_t \rightarrow -\frac{dA}{A} = \frac{d\sigma_t}{\sigma_t} </math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math> F =\sigma_t A \rightarrow dF = 0 =\sigma_t dA + A d\sigma_t \rightarrow -\frac{dA}{A} = \frac{d\sigma_t}{\sigma_t} </math></div></td></tr>  </table>

Amarin32 https://millwiki.mse.gatech.edu/index.php?title=Stress_Theory&diff=38657&oldid=prev Amarin32 at 20:55, 14 June 2021 2021-06-14T20:55:01Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:55, 14 June 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l4">Line 4:</td> <td colspan="2" class="diff-lineno">Line 4:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== True vs. Engineering Stress ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== True vs. Engineering Stress ==</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Beyond the yield point, molecular flow causes a substantial reduction in the specimen cross-sectional area A, so the true stress <math> \sigma_t= \frac{F}{A} </math> actually borne by the material is larger than the engineering stress computed from the original cross-sectional area (<math> \sigma_e= \frac{F}{A_0} </math>). The load must equal the true stress times the actual area (<math> F =\sigma_t A </math>), and as long as strain hardening can increase σt enough to compensate for the reduced area A, the load and therefore the engineering stress will continue to rise as the strain increases. Eventually, however, the decrease in area due to flow becomes larger than the increase in true stress due to strain hardening, and the load begins to fall. This is a geometrical effect, and if the true stress rather than the engineering stress were plotted no maximum would be observed in the curve. The strain hardening rate is the slope of the stress-strain curve, also called the tangent modulus. At the UTS the differential of the load F is zero, giving an analytical relation between the true stress and the area at necking:</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Beyond the yield point, molecular flow causes a substantial reduction in the specimen cross-sectional area A, so the true stress <math> \sigma_t= \frac{F}{A} </math> actually borne by the material is larger than the engineering stress computed from the original cross-sectional area (<math> \sigma_e= \frac{F}{A_0} </math>). The load must equal the true stress times the actual area (<math> F =\sigma_t A </math>), and as long as strain hardening can increase σt enough to compensate for the reduced area A, the load and therefore the engineering stress will continue to rise as the strain increases. <ins style="font-weight: bold; text-decoration: none;"><ref name="Roylance"> Roylance, D. (2001, August 23). STRESS-STRAIN CURVES.</ref>]] <math> \sigma_e = \frac{F}{A_0} </math> </ins>Eventually, however, the decrease in area due to flow becomes larger than the increase in true stress due to strain hardening, and the load begins to fall. This is a geometrical effect, and if the true stress rather than the engineering stress were plotted no maximum would be observed in the curve. The strain hardening rate is the slope of the stress-strain curve, also called the tangent modulus. At the UTS the differential of the load F is zero, giving an analytical relation between the true stress and the area at necking:</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math> F =\sigma_t A \rightarrow dF = 0 =\sigma_t dA + A d\sigma_t \rightarrow -\frac{dA}{A} = \frac{d\sigma_t}{\sigma_t} </math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math> F =\sigma_t A \rightarrow dF = 0 =\sigma_t dA + A d\sigma_t \rightarrow -\frac{dA}{A} = \frac{d\sigma_t}{\sigma_t} </math></div></td></tr>  </table>

Amarin32 https://millwiki.mse.gatech.edu/index.php?title=Stress_Theory&diff=38656&oldid=prev Amarin32 at 20:54, 14 June 2021 2021-06-14T20:54:04Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:54, 14 June 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== When is <math> \sigma= \frac{F}{A} </math> Valid? ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== When is <math> \sigma= \frac{F}{A} </math> Valid? ==</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The equation <math> \sigma= \frac{F}{A} </math> is valid only if the stress is uniformly distributed over the cross-section of the bar. This condition is realized if the axial force F acts through the centroid of the cross-sectional area. Uniform stress exists throughout the length of the bar except near the ends. The stress distribution at the end of a bar depends upon how the load F is transmitted to the bar. If the load is distributed uniformly over the end, the stress pattern at the end is the same as everywhere else. However, it is more likely that the load is transmitted through a pin or a bolt, producing high localized stresses called stress concentrations. As a practical rule, the formula <math> \sigma= \frac{F}{A} </math> may be used with good accuracy at any point within a prismatic bar that is at least as far away from the stress concentration as the largest lateral dimension of the bar. Of course, even when the stress is not uniformly distributed, the equation <math> \sigma= \frac{F}{A} </math> may still be useful because it gives the average normal stress on the cross section.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The equation <math> \sigma= \frac{F}{A} </math> is valid only if the stress is uniformly distributed over the cross-section of the bar. This condition is realized if the axial force F acts through the centroid of the cross-sectional area. Uniform stress exists throughout the length of the bar except near the ends. The stress distribution at the end of a bar depends upon how the load F is transmitted to the bar. If the load is distributed uniformly over the end, the stress pattern at the end is the same as everywhere else. However, it is more likely that the load is transmitted through a pin or a bolt, producing high localized stresses called stress concentrations. As a practical rule, the formula <math> \sigma= \frac{F}{A} </math> may be used with good accuracy at any point within a prismatic bar that is at least as far away from the stress concentration as the largest lateral dimension of the bar. Of course, even when the stress is not uniformly distributed, the equation <math> \sigma= \frac{F}{A} </math> may still be useful because it gives the average normal stress on the cross section. <ins style="font-weight: bold; text-decoration: none;"><ref name="Goodno"> Goodno, B. J., & Gere, J. M. (2018). Mechanics of materials (9th ed.). Cengage Learning. </ref></ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== True vs. Engineering Stress ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== True vs. Engineering Stress ==</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l9">Line 9:</td> <td colspan="2" class="diff-lineno">Line 9:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This expression states that the load and therefore the engineering stress will reach a maximum as a function of strain when the fractional decrease in area becomes equal to the fractional increase in true stress.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This expression states that the load and therefore the engineering stress will reach a maximum as a function of strain when the fractional decrease in area becomes equal to the fractional increase in true stress.</div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== References == </ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><references /> </ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>__NOTOC__</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>__NOTOC__</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Theory]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Theory]]</div></td></tr>  </table>

Amarin32 https://millwiki.mse.gatech.edu/index.php?title=Stress_Theory&diff=38631&oldid=prev Amarin32 at 17:27, 10 June 2021 2021-06-10T17:27:16Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:27, 10 June 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l9">Line 9:</td> <td colspan="2" class="diff-lineno">Line 9:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This expression states that the load and therefore the engineering stress will reach a maximum as a function of strain when the fractional decrease in area becomes equal to the fractional increase in true stress.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This expression states that the load and therefore the engineering stress will reach a maximum as a function of strain when the fractional decrease in area becomes equal to the fractional increase in true stress.</div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">__NOTOC__</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Theory]]</ins></div></td></tr>  </table>

Amarin32 https://millwiki.mse.gatech.edu/index.php?title=Stress_Theory&diff=38630&oldid=prev Amarin32 at 17:25, 10 June 2021 2021-06-10T17:25:49Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:25, 10 June 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>== When is <math> \sigma= \frac{F}{A} Valid? ==</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>== When is <math> \sigma= \frac{F}{A} <ins style="font-weight: bold; text-decoration: none;"></math> </ins>Valid? ==</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The equation <math> \sigma= \frac{F}{A} </math> is valid only if the stress is uniformly distributed over the cross-section of the bar. This condition is realized if the axial force F acts through the centroid of the cross-sectional area. Uniform stress exists throughout the length of the bar except near the ends. The stress distribution at the end of a bar depends upon how the load F is transmitted to the bar. If the load is distributed uniformly over the end, the stress pattern at the end is the same as everywhere else. However, it is more likely that the load is transmitted through a pin or a bolt, producing high localized stresses called stress concentrations. As a practical rule, the formula <math> \sigma= \frac{F}{A} </math> may be used with good accuracy at any point within a prismatic bar that is at least as far away from the stress concentration as the largest lateral dimension of the bar. Of course, even when the stress is not uniformly distributed, the equation <math> \sigma= \frac{F}{A} </math> may still be useful because it gives the average normal stress on the cross section.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The equation <math> \sigma= \frac{F}{A} </math> is valid only if the stress is uniformly distributed over the cross-section of the bar. This condition is realized if the axial force F acts through the centroid of the cross-sectional area. Uniform stress exists throughout the length of the bar except near the ends. The stress distribution at the end of a bar depends upon how the load F is transmitted to the bar. If the load is distributed uniformly over the end, the stress pattern at the end is the same as everywhere else. However, it is more likely that the load is transmitted through a pin or a bolt, producing high localized stresses called stress concentrations. As a practical rule, the formula <math> \sigma= \frac{F}{A} </math> may be used with good accuracy at any point within a prismatic bar that is at least as far away from the stress concentration as the largest lateral dimension of the bar. Of course, even when the stress is not uniformly distributed, the equation <math> \sigma= \frac{F}{A} </math> may still be useful because it gives the average normal stress on the cross section.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>  </table>

Amarin32 https://millwiki.mse.gatech.edu/index.php?title=Stress_Theory&diff=38629&oldid=prev Amarin32 at 17:25, 10 June 2021 2021-06-10T17:25:17Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:25, 10 June 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>== When is <del style="font-weight: bold; text-decoration: none;">σ</del>=F<del style="font-weight: bold; text-decoration: none;">/</del>A Valid? ==</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>== When is <ins style="font-weight: bold; text-decoration: none;"><math> \sigma</ins>= <ins style="font-weight: bold; text-decoration: none;">\frac{</ins>F<ins style="font-weight: bold; text-decoration: none;">}{</ins>A<ins style="font-weight: bold; text-decoration: none;">} </ins>Valid? ==</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The equation <math> \sigma= \frac{F}{A} </math> is valid only if the stress is uniformly distributed over the cross-section of the bar. This condition is realized if the axial force F acts through the centroid of the cross-sectional area. Uniform stress exists throughout the length of the bar except near the ends. The stress distribution at the end of a bar depends upon how the load F is transmitted to the bar. If the load is distributed uniformly over the end, the stress pattern at the end is the same as everywhere else. However, it is more likely that the load is transmitted through a pin or a bolt, producing high localized stresses called stress concentrations. As a practical rule, the formula <math> \sigma= \frac{F}{A} </math> may be used with good accuracy at any point within a prismatic bar that is at least as far away from the stress concentration as the largest lateral dimension of the bar. Of course, even when the stress is not uniformly distributed, the equation <math> \sigma= \frac{F}{A} </math> may still be useful because it gives the average normal stress on the cross section.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The equation <math> \sigma= \frac{F}{A} </math> is valid only if the stress is uniformly distributed over the cross-section of the bar. This condition is realized if the axial force F acts through the centroid of the cross-sectional area. Uniform stress exists throughout the length of the bar except near the ends. The stress distribution at the end of a bar depends upon how the load F is transmitted to the bar. If the load is distributed uniformly over the end, the stress pattern at the end is the same as everywhere else. However, it is more likely that the load is transmitted through a pin or a bolt, producing high localized stresses called stress concentrations. As a practical rule, the formula <math> \sigma= \frac{F}{A} </math> may be used with good accuracy at any point within a prismatic bar that is at least as far away from the stress concentration as the largest lateral dimension of the bar. Of course, even when the stress is not uniformly distributed, the equation <math> \sigma= \frac{F}{A} </math> may still be useful because it gives the average normal stress on the cross section.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>  </table>

Amarin32 https://millwiki.mse.gatech.edu/index.php?title=Stress_Theory&diff=38628&oldid=prev Amarin32: /* True vs. Engineering Stress */ 2021-06-10T17:15:16Z

<p><span dir="auto"><span class="autocomment">True vs. Engineering Stress</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:15, 10 June 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l8">Line 8:</td> <td colspan="2" class="diff-lineno">Line 8:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math> F =\sigma_t A \rightarrow dF = 0 =\sigma_t dA + A d\sigma_t \rightarrow -\frac{dA}{A} = \frac{d\sigma_t}{\sigma_t} </math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math> F =\sigma_t A \rightarrow dF = 0 =\sigma_t dA + A d\sigma_t \rightarrow -\frac{dA}{A} = \frac{d\sigma_t}{\sigma_t} </math></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The last </del>expression states that the load and therefore the engineering stress will reach a maximum as a function of strain when the fractional decrease in area becomes equal to the fractional increase in true stress.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">This </ins>expression states that the load and therefore the engineering stress will reach a maximum as a function of strain when the fractional decrease in area becomes equal to the fractional increase in true stress.</div></td></tr>  </table>

Amarin32 https://millwiki.mse.gatech.edu/index.php?title=Stress_Theory&diff=38627&oldid=prev Amarin32 at 17:14, 10 June 2021 2021-06-10T17:14:28Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:14, 10 June 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l6">Line 6:</td> <td colspan="2" class="diff-lineno">Line 6:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Beyond the yield point, molecular flow causes a substantial reduction in the specimen cross-sectional area A, so the true stress <math> \sigma_t= \frac{F}{A} </math> actually borne by the material is larger than the engineering stress computed from the original cross-sectional area (<math> \sigma_e= \frac{F}{A_0} </math>). The load must equal the true stress times the actual area (<math> F =\sigma_t A </math>), and as long as strain hardening can increase σt enough to compensate for the reduced area A, the load and therefore the engineering stress will continue to rise as the strain increases. Eventually, however, the decrease in area due to flow becomes larger than the increase in true stress due to strain hardening, and the load begins to fall. This is a geometrical effect, and if the true stress rather than the engineering stress were plotted no maximum would be observed in the curve. The strain hardening rate is the slope of the stress-strain curve, also called the tangent modulus. At the UTS the differential of the load F is zero, giving an analytical relation between the true stress and the area at necking:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Beyond the yield point, molecular flow causes a substantial reduction in the specimen cross-sectional area A, so the true stress <math> \sigma_t= \frac{F}{A} </math> actually borne by the material is larger than the engineering stress computed from the original cross-sectional area (<math> \sigma_e= \frac{F}{A_0} </math>). The load must equal the true stress times the actual area (<math> F =\sigma_t A </math>), and as long as strain hardening can increase σt enough to compensate for the reduced area A, the load and therefore the engineering stress will continue to rise as the strain increases. Eventually, however, the decrease in area due to flow becomes larger than the increase in true stress due to strain hardening, and the load begins to fall. This is a geometrical effect, and if the true stress rather than the engineering stress were plotted no maximum would be observed in the curve. The strain hardening rate is the slope of the stress-strain curve, also called the tangent modulus. At the UTS the differential of the load F is zero, giving an analytical relation between the true stress and the area at necking:</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>F = <del style="font-weight: bold; text-decoration: none;">σtA → </del>dF =0= <del style="font-weight: bold; text-decoration: none;">σtdA </del>+ <del style="font-weight: bold; text-decoration: none;">Adσt → −dA/</del>A = <del style="font-weight: bold; text-decoration: none;">dσt</del>/<del style="font-weight: bold; text-decoration: none;">σt</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><math> </ins>F =<ins style="font-weight: bold; text-decoration: none;">\sigma_t A \rightarrow </ins>dF = 0 =<ins style="font-weight: bold; text-decoration: none;">\sigma_t dA </ins>+ A <ins style="font-weight: bold; text-decoration: none;">d\sigma_t \rightarrow -\frac{dA}{A} </ins>= <ins style="font-weight: bold; text-decoration: none;">\frac{d\sigma_t}{\sigma_t} <</ins>/<ins style="font-weight: bold; text-decoration: none;">math></ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The last expression states that the load and therefore the engineering stress will reach a maximum as a function of strain when the fractional decrease in area becomes equal to the fractional increase in true stress.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The last expression states that the load and therefore the engineering stress will reach a maximum as a function of strain when the fractional decrease in area becomes equal to the fractional increase in true stress.</div></td></tr>  </table>

Amarin32 https://millwiki.mse.gatech.edu/index.php?title=Stress_Theory&diff=38626&oldid=prev Amarin32 at 17:11, 10 June 2021 2021-06-10T17:11:25Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:11, 10 June 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l4">Line 4:</td> <td colspan="2" class="diff-lineno">Line 4:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== True vs. Engineering Stress ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== True vs. Engineering Stress ==</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Beyond the yield point, molecular flow causes a substantial reduction in the specimen cross-sectional area A, so the true stress <del style="font-weight: bold; text-decoration: none;">σt </del>= F/<del style="font-weight: bold; text-decoration: none;">A </del>actually borne by the material is larger than the engineering stress computed from the original cross-sectional area (<math> \sigma_e= \frac{F}{A_0} </math>). The load must equal the true stress times the actual area (<math> F =\sigma_t A </math>), and as long as strain hardening can increase σt enough to compensate for the reduced area A, the load and therefore the engineering stress will continue to rise as the strain increases. Eventually, however, the decrease in area due to flow becomes larger than the increase in true stress due to strain hardening, and the load begins to fall. This is a geometrical effect, and if the true stress rather than the engineering stress were plotted no maximum would be observed in the curve. The strain hardening rate is the slope of the stress-strain curve, also called the tangent modulus. At the UTS the differential of the load F is zero, giving an analytical relation between the true stress and the area at necking:</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Beyond the yield point, molecular flow causes a substantial reduction in the specimen cross-sectional area A, so the true stress <ins style="font-weight: bold; text-decoration: none;"><math> \sigma_t</ins>= <ins style="font-weight: bold; text-decoration: none;">\frac{</ins>F<ins style="font-weight: bold; text-decoration: none;">}{A} <</ins>/<ins style="font-weight: bold; text-decoration: none;">math> </ins>actually borne by the material is larger than the engineering stress computed from the original cross-sectional area (<math> \sigma_e= \frac{F}{A_0} </math>). The load must equal the true stress times the actual area (<math> F =\sigma_t A </math>), and as long as strain hardening can increase σt enough to compensate for the reduced area A, the load and therefore the engineering stress will continue to rise as the strain increases. Eventually, however, the decrease in area due to flow becomes larger than the increase in true stress due to strain hardening, and the load begins to fall. This is a geometrical effect, and if the true stress rather than the engineering stress were plotted no maximum would be observed in the curve. The strain hardening rate is the slope of the stress-strain curve, also called the tangent modulus. At the UTS the differential of the load F is zero, giving an analytical relation between the true stress and the area at necking:</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>F = σtA → dF =0= σtdA + Adσt → −dA/A = dσt/σt</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>F = σtA → dF =0= σtdA + Adσt → −dA/A = dσt/σt</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The last expression states that the load and therefore the engineering stress will reach a maximum as a function of strain when the fractional decrease in area becomes equal to the fractional increase in true stress.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The last expression states that the load and therefore the engineering stress will reach a maximum as a function of strain when the fractional decrease in area becomes equal to the fractional increase in true stress.</div></td></tr>  </table>

Amarin32