Neo-Hookean solid

Example

The incompressible neo-Hookean solid is the hyperelastic material with the strain energy density function below. $$W = \frac{\mu}{2}(I_1 - 3), \qquad I_1 = \tr(F^T F), \qquad \det F = 1.$$ Here $F$ is the deformation gradient, $I_1$ is the first invariant of the right Cauchy–Green tensor and $\mu > 0$ is a constant. Since $\partial I_1 / \partial F = 2F$, the first Piola–Kirchhoff stress is $P = \mu F - p F^{-T}$. The Cauchy stress is $\sigma = \mu F F^T - p I$, with $p$ a pressure fixed by equilibrium and boundary conditions. Check in simple tension with stretch $\lambda$: $F = \operatorname{diag}(\lambda, \lambda^{-1/2}, \lambda^{-1/2})$, and the lateral faces are free, so $\sigma_{22} = \mu \lambda^{-1} - p = 0$. Then the axial stress is $\sigma_{11} = \mu(\lambda^2 - \lambda^{-1})$. For $\lambda = 1 + \epsilon$ with small $\epsilon$, this is about $3\mu\epsilon$, which matches linear elasticity with shear modulus $\mu$ and Young's modulus $3\mu$ of an incompressible material. Compressible versions add a term depending on $J = \det F$; several different forms are in use.