Finite Deformation Kinematics

Let us assume that we have access to data (at quadrature points) that includes:

• $\sigma(t)$: the current Cauchy stress
• $q^*$: state variables relevant to material state

As well as a constitutive model that provides a function

Review: Small Deformation Stress Update Procedure

For the case of small deformation, the update procedure is pretty straightforward, and the steps go something like this:

Given:

• the current stress state, $\mathbf{\sigma}(t)$
• a trial displacement increment, $\hat{u}$

1. Compute the increment in the small strain tensor increment
   $\hat{\varepsilon} = \frac{1}{2}(\nabla \hat{u} + (\nabla \hat{u})^\top)$

2. Pass $\hat{\varepsilon}$ to the constitutive to update stress
   $\sigma_{new} = \sigma(\sigma_{old}, \hat{\varepsilon})$

The function $\sigma(\cdot, \cdot)$ may be as simple as linear elasticity, or it might be more complicated to account for plasticity, hardening/softening, or damage effects. Regardless, if the displacements are small,