# Constitutive Modeling#

In their most general form, constitutive models define stress, \(\bm{S}\) (\(\bm{\tau}\) in Eulerian coordinate), in terms of state variables. For the materials used in Ratel, the state variables are constituted by the vector displacement field \(\bm{u}\), and its gradient \(\bm{H} = \nabla_X \bm{u}\). We consider constitutive models for stress in terms of strain due to deformation, given by \(\bm{S} \left( \bm{E} \right)\) (\(\bm{\tau} \left( \bm{e} \right) \) in Eulerian coordinate), which is a tensor-valued function of a tensor-valued input.

Note

The general Seth-Hill formula of strain measures in the Lagrangian and Eulerian coordinates are defined as

where \(n\) is a real number and \(\textbf{U}, \textbf{v}\) are unique SPD right (or material) and left (or spatial) stretch tensors defined by the *unique polar decomposition* of the deformation gradient (5) \(\bm{F} = \textbf{R} \textbf{U} = \textbf{v} \textbf{R}\) with \(\textbf{R}^T \textbf{R} = \bm{I}_3\) as the rotation tensor.

For the special case \(n=2\), we define right and left Cauchy-Green tensors

and Green-Lagrange and Green-Euler strains

A physical model must not depend on the coordinate system chosen to express it; an arbitrary choice of function will generally not be invariant under orthogonal transformations and thus will not admissible. Given an orthogonal transformation \(Q\), we desire

which means that we can change our reference frame before or after computing \(\bm{S}\) and get the same result. Constitutive relations in which \(\bm{S}\) is uniquely determined by \(\bm{E}\) while satisfying the invariance property (11) are known as Cauchy elastic materials.

For our constitutive relationships, we define a strain energy density functional \(\psi \left( \bm{E} \right) \in \mathbb R\) (\(\psi \left( \bm{e} \right) \in \mathbb R \)) and obtain the strain energy from its gradient,

Note

The strain energy density functional cannot be an arbitrary function \(\psi \left( \bm{E} \right)\).
It can only depend on *invariants*, scalar-valued functions \(\gamma\) satisfying

for all orthogonal matrices \(Q\).