Hyperelastic solid constitutive models are defined by a Helmholtz free energy density function of the deformation gradient.
```math
\mathbf{P}:\dot{\mathbf{F}} - \dot{a}(\mathbf{F}) \geq 0
```
Satisfying the second law of thermodynamics (here, equivalent to extremized or zero dissipation) yields a relation for the stress.
```math
\mathbf{P} = \frac{\partial a}{\partial\mathbf{F}}
```
Consequently, the tangent stiffness associated with the first Piola-Kirchhoff stress is symmetric for these constitutive models.
```math
\mathcal{C}_{iJkL} = \mathcal{C}_{kLiJ}
```