Given:
The strain energy density for a material undergoing small strain
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Show:
For linear elastic deformations and small strains,
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If the strain energy density is given by equation (1), then
(for linear elastic materials) the stress and strain can be related
using
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We will show that equation (2) is equivalent to equation (3).
We start off with equation (2) and work backward.
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For linear elastic materials,
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Substituting equation (5) into equation (4), we get,
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Recall that, for a second order tensor ,
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and that for a fourth order rensor (substitution rule),
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Differentiating equation (6) with respect to ,
we have,
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Using the symmetry of the stiffness tensor, we have,
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Therefore,
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which is the same as equation (3). Hence shown.