Polymer Engineering question related to Von Mises yield criterion

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kennyyip
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It is a proof question. Does anyone can tell me how to do this question?question 22.2
it is the question for extra credit for my final exam.
thank you so much.
 
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  • Photo of a printed textbook page titled Yield and Crazing, showing equation e = b/(1-b), problem 22.2 with a pressure-dependent von Mises yield criterion A[(σ1+σ2+σ3)] + B[(σ1-σ2)^2 + (σ2-σ3)^2 + (σ3-σ1)^2] = 1, and text about hydrostatic stress and polyethylene twinning below.
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Discussion

You can find A and B with two simultaneously equations.
For the first equation assume uniaxial tension yielding - i.e. take σ1 as fty and σ2=σ3=0.
For the second equation assume uniaxial compression yielding. This is a polymer so it makes sense that fty and fcy would be different values - and they have to be different values otherwise you'd find that A=0.

Once you have A and B, then consider the hydrostatic case - i.e. take σ1=σ2=σ3=σ. Solve for σ.
 

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