Introduction to Anisotropic Hyperelasticity

Before jumping into the main topic, let’s quickly review isotropic material models. A linear elastic material model can be written \(\sigma_{ij} = f([E,\nu]; \varepsilon_{ij})\), which can be expressed in words as: the stress is given by a function that depends on two material parameters (elastic constants), and the applied strain state. This equation can also be written: \( \sigma_{ij} = 2\mu \varepsilon_{ij} + \lambda \varepsilon_{kk} \delta_{ij}\).

For finite strains it is better to use a hyperelastic material model. The most common way to write the stress response in this case is: \(\boldsymbol{\sigma}=f(\Psi(p_i; I_1^*, I_2^*, J); \mathbf{F})\). Note that this equation shows that the stress depends on another function \(\Psi\) which depends on material model parameters (\(p_i\)) and invariants of the applied deformation, and then also on the deformation gradient. This is fundamentally different than for the case with linear elasticity. The Cauchy stress can in this case be calculated from (see Chapter 5 in my book):

\[\boldsymbol{\sigma} = \frac{2}{J}
\left[ \frac{\partial\Psi}{\partial I_1^*} + \frac{\partial\Psi}{\partial I_2^*} I_1^* \right] \mathbf{b}^*
– \frac{2}{J} \frac{\partial\Psi}{\partial I_2^*} \left(\mathbf{b}^*\right)^2
+ \left[ \frac{\partial\Psi}{\partial J}
– \frac{2 I_1^*}{3J} \frac{\partial\Psi}{\partial I_1^*}
– \frac{4 I_2^*}{3J} \frac{\partial\Psi}{\partial I_2^*}
\right] \mathbf{I}\]