Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

DoITPoMS Teaching & Learning Packages Analysis of Deformation Processes • Notched bar in tension
Analysis of Deformation Processes AimsBefore you startIntroductionLévy-Mises Equations• Plane stress• Plane strainSlip Line Field TheoryWork Formula MethodLimit Analysis• Notched bar in tension• Notched bar in plane bending• Hodographs I• Hodographs IIFinite Element MethodSummaryQuestionsGoing further
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• Notched bar in tension

The plane strain condition is satisfied when breath, b » h, the depth of the bar.


Lower-Bound:

Find a stress system, e.g. σ = 0 in the length of the bar where there is the notch, σ = 2k elsewhere.


Therefore, for a breadth b, P = 2khb = load = stress × area


Upper-Bound:

Postulate a suitable simple deformation pattern.


Assume yielding by slip on 45º shear planes with shear yield stress k. Let displacement along shear plane AB = δx.

Then internal work done = \( {k.\left| {AB} \right|b\delta x} = k\sqrt 2 bh\delta x\), where the force is \( k\left| {AB} \right|b \) acting on the shear plane AB.

Distance moved by the external load \(P = \delta x\cos {45^ \circ }\) = \(\frac{{\delta x}}{{\sqrt 2 }}\)

\( \Rightarrow P \) \(\frac{{\delta x}}{{\sqrt 2 }}\) = \(k\sqrt 2 bh\delta x\)

\( \Rightarrow P = 2kbh\)


So, here we obtain the same result for the upper bound and lower bound \( \Rightarrow P = 2kbh\) is the true failure load, the load required to cause plastic flow.

 

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