# 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

\begin{aligned} &\frac{1}{n} \left( \textbf{U}^n - \bm{I}_3 \right), \quad &\frac{1}{n} \left( \textbf{v}^n - \bm{I}_3 \right), \quad \text{if} \quad n \neq 0,\\ &\log \textbf{U}, \quad &\log \textbf{v}, \quad \text{if} \quad n = 0, \end{aligned}

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

$\bm{C} = \bm{F}^T \bm{F} = \textbf{U}^2, \quad \bm{b} = \bm{F} \bm{F}^T = \textbf{v}^2$

and Green-Lagrange and Green-Euler strains

(9)#$\bm{E} = \frac{1}{2} \left( \bm{C} - \bm{I}_3 \right) = \frac{1}{2} \left( \bm{H} + \bm{H}^T + \bm{H}^T \bm{H} \right).$
(10)#$\bm{e} = \frac{1}{2} \left( \bm{b} - \bm{I}_3 \right) = \frac{1}{2} \left( \bm{H} + \bm{H}^T + \bm{H} \bm{H}^T \right).$

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

(11)#$Q \bm{S} \left( \bm{E} \right) Q^T = \bm{S} \left( Q \bm{E} Q^T \right),$

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,

(12)#$\bm{S} \left( \bm{E} \right) = \frac{\partial \psi}{\partial \bm{E}}, \quad \bm{\tau} \left( \bm{e} \right) = \frac{\partial \psi}{\partial \bm{e}} \bm{b}$

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

$\gamma \left( \bm{E} \right) = \gamma \left( Q \bm{E} Q^T \right)$

for all orthogonal matrices $$Q$$.

Material Models