Elasticity/Energy methods example 1

Example 1
Given:



\Pi^c[\boldsymbol{\sigma}(\mathbf{x}) + \Delta\boldsymbol{\sigma}(\mathbf{x})] - \Pi^c[\boldsymbol{\sigma}(\mathbf{x})] = \int_{\mathcal R} [U^c(\boldsymbol{\sigma}+\Delta\boldsymbol{\sigma}) - U^c(\boldsymbol{\sigma}) - \boldsymbol{\varepsilon}:\Delta\boldsymbol{\sigma}]~dV $$

Show:

\Pi^c[\boldsymbol{\sigma}(\mathbf{x}) + \Delta\boldsymbol{\sigma}(\mathbf{x})] - \Pi^c[\boldsymbol{\sigma}(\mathbf{x})] = \int_{\mathcal R} U^c(\Delta\boldsymbol{\sigma})~dV $$

Solution
For a linear elastic material, the complementary strain energy density is given by

U^c(\boldsymbol{\sigma}) = \frac{1}{2}\boldsymbol{\sigma}:\text{S}:\boldsymbol{\sigma} $$ where $$\text{S}$$ is the compliance tensor.

Therefore,

U^c(\boldsymbol{\sigma}+\Delta\boldsymbol{\sigma}) = \frac{1}{2}(\boldsymbol{\sigma}+\Delta\boldsymbol{\sigma}):\text{S}:(\boldsymbol{\sigma}+\Delta\boldsymbol{\sigma}) = \frac{1}{2}(\sigma_{ij}+\Delta\sigma_{ij})S_{ijkl}(\sigma_{kl}+\Delta\sigma_{kl}) $$ or (using the symmetry of the compliance tensor),
 * $$\begin{align}

U^c(\boldsymbol{\sigma}+\Delta\boldsymbol{\sigma}) & = \frac{1}{2}\left[\sigma_{ij}\sigma_{kl}+\sigma_{ij}\Delta\sigma_{kl}+ \sigma_{kl}\Delta\sigma_{ij}+\Delta\sigma_{ij}\Delta\sigma_{kl}\right] S_{ijkl} \\ &= \frac{1}{2}\left[\sigma_{ij}S_{ijkl}\sigma_{kl}+ \sigma_{ij}S_{ijkl}\Delta\sigma_{kl}+ \sigma_{kl}S_{ijkl}\Delta\sigma_{ij}+ \Delta\sigma_{ij}S_{ijkl}\Delta\sigma_{kl}\right] \\ &= \frac{1}{2}\left[\sigma_{ij}S_{ijkl}\sigma_{kl}+ \varepsilon_{kl}\Delta\sigma_{kl}+ \varepsilon_{ij}\Delta\sigma_{ij}+ \Delta\sigma_{ij}S_{ijkl}\Delta\sigma_{kl}\right] \\ &= \frac{1}{2}\left[\sigma_{ij}S_{ijkl}\sigma_{kl}+ 2\varepsilon_{kl}\Delta\sigma_{kl}+ \Delta\sigma_{ij}S_{ijkl}\Delta\sigma_{kl}\right] \\ &= \frac{1}{2}\boldsymbol{\sigma}:\text{S}:\boldsymbol{\sigma} + \boldsymbol{\varepsilon}:\Delta\boldsymbol{\sigma} + \frac{1}{2}\Delta\boldsymbol{\sigma}:\text{S}: \Delta\boldsymbol{\sigma} \\ &= U^c(\boldsymbol{\sigma}) + \boldsymbol{\varepsilon}:\Delta\boldsymbol{\sigma} + U^c(\Delta\boldsymbol{\sigma}) \end{align}$$ Therefore,

U^c(\boldsymbol{\sigma}+\Delta\boldsymbol{\sigma}) = U^c(\boldsymbol{\sigma}) + \boldsymbol{\varepsilon}:\Delta\boldsymbol{\sigma} + U^c(\Delta\boldsymbol{\sigma}) $$ Plugging into the given equation

\Pi^c[\boldsymbol{\sigma}(\mathbf{x}) + \Delta\boldsymbol{\sigma}(\mathbf{x})] - \Pi^c[\boldsymbol{\sigma}(\mathbf{x})] = \int_{\mathcal R} [U^c(\boldsymbol{\sigma}) + \boldsymbol{\varepsilon}:\Delta\boldsymbol{\sigma} + U^c(\Delta\boldsymbol{\sigma}) - U^c(\boldsymbol{\sigma}) - \boldsymbol{\varepsilon}:\Delta\boldsymbol{\sigma}]~dV $$ or,

{ \Pi^c[\boldsymbol{\sigma}(\mathbf{x}) + \Delta\boldsymbol{\sigma}(\mathbf{x})] - \Pi^c[\boldsymbol{\sigma}(\mathbf{x})] = \int_{\mathcal R} U^c(\Delta\boldsymbol{\sigma})~dV } $$ Hence shown.