Problem 1: Part 7: Flow rule
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The theory of plasticity also states that the material
satisfies the von Mises yield condition
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where is the deviatoric part of the stress . Derive an expression for in terms of the normal to the yield surface
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The von Mises yield function is
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We can alternatively write the yield function as
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in which case the following equations take a slightly different form.
Therefore,
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The deviatoric part of is
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Therefore,
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Now,
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and
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Therefore,
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The derivative with respect to is
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Let us use index notation to find the derivatives. In index notation,
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Hence, the components of the second-order tensor are
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Therefore,
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Similarly,
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Hence, the components of the second-order tensor are
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Therefore,
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Plugging the above results into the expression for the derivative
of we get
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Hence, we get
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The required expression is
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