Problem 1: Part 6: Continuum elastic-plastic tangent modulus
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The continuum elastic-plastic tangent modulus is defined by the
following relation
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Derive an expression for the elastic plastic tangent modulus using the results you have
derived in the previous parts.
The stress rate is given by
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From the previous part
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Plug in expression for stress rate to get
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We have to express the above in the form
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Since the denominator is a scalar, we don't have to worry about it
for this calculation. In that case we can write
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In index notation, we can write
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Let us manipulate the numerator of the second term above so that we get what we need. Thus
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In the above
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and
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Therefore,
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This gives us
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or,
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Hence
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The continuum elastic-plastic tangent modulus is therefore
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