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.


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


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 \(Q\).