Given:
The strain energy density for a material undergoing small strain
![{\displaystyle {\text{(1)}}\qquad U({\boldsymbol {\varepsilon }})=\int _{0}^{\boldsymbol {\varepsilon }}{\boldsymbol {\sigma }}:d{\boldsymbol {\varepsilon }}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cab8a04ad0e2ccfba1942003f9402508aa2ba79)
Show:
For linear elastic deformations and small strains,
![{\displaystyle {\text{(2)}}\qquad U({\boldsymbol {\varepsilon }})={\frac {1}{2}}{\boldsymbol {\sigma }}:{\boldsymbol {\varepsilon }}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55a09922f3f0a78fe2743c174a0e0f35cb5c8112)
If the strain energy density is given by equation (1), then
(for linear elastic materials) the stress and strain can be related
using
![{\displaystyle {\text{(3)}}\qquad \sigma _{ij}={\frac {\partial U({\boldsymbol {\varepsilon }})}{\partial \varepsilon _{ij}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/642a172a929e685c43d9e4f808f5986407f93c01)
We will show that equation (2) is equivalent to equation (3).
We start off with equation (2) and work backward.
![{\displaystyle {\text{(4)}}\qquad U({\boldsymbol {\varepsilon }})={\frac {1}{2}}\sigma _{ij}\varepsilon _{ij}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21363e7bfd6e29cc2c9a3ecc9b5933f17e7201ca)
For linear elastic materials,
![{\displaystyle {\text{(5)}}\qquad \sigma _{ij}=C_{ijkl}\varepsilon _{kl}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b1f344331a3784310b17ae3c6b368cb9ed3ec0c)
Substituting equation (5) into equation (4), we get,
![{\displaystyle {\text{(6)}}\qquad U({\boldsymbol {\varepsilon }})={\frac {1}{2}}C_{ijkl}\varepsilon _{kl}\varepsilon _{ij}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e74a592ba6abee78ec720f5a9f37e0538c0f3fa1)
Recall that, for a second order tensor
,
![{\displaystyle {\frac {\partial A_{ij}}{\partial A_{kl}}}=\delta _{ik}\delta _{jl}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2451f79e597ffdc60e9c3f02b959f9cfe7bf3326)
and that for a fourth order rensor
(substitution rule),
![{\displaystyle C_{ijkl}\delta _{ir}=C_{rjkl}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2f376d361c75d47b1283323a3c8d4bdd7dbf374)
Differentiating equation (6) with respect to
,
we have,
![{\displaystyle {\begin{aligned}{\frac {\partial U({\boldsymbol {\varepsilon }})}{\partial \varepsilon _{rs}}}&={\frac {1}{2}}C_{ijkl}\varepsilon _{kl}\delta _{ir}\delta _{js}+{\frac {1}{2}}C_{ijkl}\varepsilon _{ij}\delta _{kr}\delta _{ls}\\&={\frac {1}{2}}C_{rskl}\varepsilon _{kl}+{\frac {1}{2}}C_{ijrs}\varepsilon _{ij}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8ef4609341cae81f9b290818879a4578ddae93d)
Using the symmetry of the stiffness tensor, we have,
![{\displaystyle {\begin{aligned}{\frac {\partial U({\boldsymbol {\varepsilon }})}{\partial \varepsilon _{rs}}}&={\frac {1}{2}}C_{rskl}\varepsilon _{kl}+{\frac {1}{2}}C_{rsij}\varepsilon _{ij}\\&={\frac {1}{2}}\sigma _{rs}+{\frac {1}{2}}\sigma _{rs}\\&=\sigma _{rs}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99224789d544eb1ec687bc3cdcb1280c5e228dbb)
Therefore,
![{\displaystyle \sigma _{ij}={\frac {\partial U({\boldsymbol {\varepsilon }})}{\partial \varepsilon _{ij}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59c86a22fbb109be23620d85bd301b6593b27392)
which is the same as equation (3). Hence shown.