Modeling

In their most general form, material (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.

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

(67)\[ Q \bm{S} \left( \bm{E} \right) Q^T = \bm{S} \left( Q \bm{E} Q^T \right), \]

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 (67) are known as Cauchy elastic materials.

The strain energies may be expressed in terms of invariants of the symmetric Cauchy-Green tensors \(\bm{C}\) or \(\bm{b}\)

(68)\[ \begin{aligned} \mathbb{I}_1 (\bm{C}) &= \trace \bm{C}, \\ \mathbb{I}_2 (\bm{C}) &= \frac 1 2 \left( \mathbb{I}_1^2 - \bm{C} \tcolon \bm{C} \right) \\ \mathbb{I}_3 (\bm{C}) &= \operatorname{det} \bm{C}, \end{aligned} \]

Since \(\bm{C}\) and \(\bm{b}\) have the same eigenvalues, which are the square of the principal stretches \(\lambda_i^2\), we conclude that \(\mathbb{I}_i (\bm{C}) = \mathbb{I}_i (\bm{b}), \, i=1,2,3\). For the strain energy function \(\psi \left( \bm{C} \right) = \psi \left( \mathbb{I}_1, \mathbb{I}_2, \mathbb{I}_3 \right)\) we can determine the constitutive equations for isotopic hyperelastic materials by taking the gradient of strain energy as

(69)\[ \bm{S} = \frac{\partial \psi}{\partial \bm{E}}= 2 \frac{\partial \psi}{\partial \bm{C}} = 2 \sum_{i=1}^3 \frac{\partial \psi}{\partial \mathbb{I}_i} \frac{\partial \mathbb{I}_i}{\partial \bm{C}} \]

where \(\bm{S}\) is the general form of the stress relation in initial configuration and the derivative of invariants are

(70)\[ \begin{aligned} \frac{\partial \mathbb{I}_1}{\partial \bm{C}} &= \bm{I}, \\ \frac{\partial \mathbb{I}_2}{\partial \bm{C}} &= \mathbb{I}_1 \bm{I} - \bm{C}\\ \frac{\partial \mathbb{I}_3}{\partial \bm{C}} &= \mathbb{I}_3 \bm{C}^{-1}, \end{aligned} \]

The symmetric Kirchhoff stress tensor in current configuration \(\bm{\tau}\) can be obtained by representing strain energy in terms of \(\bm{b}\)’s invariants and taking the gradient as

(71)\[ \bm{\tau}= \frac{\partial \psi}{\partial \bm{e}} \bm{b} = 2 \frac{\partial \psi}{\partial \bm{b}} \bm{b} \]

or we can push-forward \(\bm{S}\) using (25) to define \(\bm{\tau}\).

The strain energy function may be written in terms of principal stretches \(\lambda_i\) as \(\psi \left( \bm{C} \right) = \psi \left( \lambda_1, \lambda_2, \lambda_3 \right)\) where we can define the constitutive relations by

(72)\[ \bm{S} = 2 \frac{\partial \psi}{\partial \bm{C}} = 2 \sum_{i=1}^3 \frac{\partial \psi}{\partial \lambda_i^2} \frac{\partial \lambda_i^2}{\partial \bm{C}} = \sum_{i=1}^3 S_i \hat{\bm{N}_i} \hat{\bm{N}_i}^T \]

and push-forward \(\bm{S}\)

(73)\[ \bm{\tau} = \bm{F} \bm{S} \bm{F}^T = \sum_{i=1}^3 \tau_i \hat{\bm{n}_i} \hat{\bm{n}_i}^T \]

where \(S_i\) and \(\tau_i\) are three principal stresses given by

(74)\[ \begin{aligned} S_i &= \frac{1}{\lambda_i} \frac{\partial \psi}{\partial \lambda_i}, & \tau_i &= {\lambda_i} \frac{\partial \psi}{\partial \lambda_i}. \end{aligned} \]

where we have used

(75)\[ \begin{aligned} \frac{\partial \lambda_i^2}{\partial \bm{C}} &= \hat{\bm{N}_i} \hat{\bm{N}_i}^T, & \frac{\partial \psi}{\partial \lambda_i^2} &= \frac{1}{2\lambda_i} \frac{\partial \psi}{\partial \lambda_i}. \end{aligned} \]

Note that the invariants can be written in terms of principal stretch as

(76)\[ \begin{aligned} \mathbb{I}_1 (\bm{C}) &= \lambda_1^2 + \lambda_2^2 + \lambda_3^2, \\ \mathbb{I}_2 (\bm{C}) &= \lambda_1^2 \lambda_2^2 + \lambda_2^2 \lambda_3^2 + \lambda_1^2 \lambda_3^2, \\ \mathbb{I}_3 (\bm{C}) &= \lambda_1^2 \lambda_2^2 \lambda_3^2, \end{aligned} \]

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

In the following sections, we introduce the most common constitutive models in the literature, as well as their numerical implementation in Ratel (weak formulation, linearization, etc.)

Boundary Conditions