$\newcommand{\Dev}{\operatorname{Dev}}$ $\newcommand{\dev}{\operatorname{dev}}$ $\newcommand{\grad}{\operatorname{grad}}$ $\newcommand{\Grad}{\operatorname{Grad}}$ $\newcommand{\divg}{\operatorname{div}}$ $\newcommand{\Ei}{\operatorname{Ei}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\Divg}{\operatorname{Div}}$ $\newcommand{\cay}{\operatorname{cay}}$ $\newcommand{\rot}{\operatorname{rot}}$

The linear elastic material model as described in Section 5.1↑ is only valid for small strains and small rotations. A first modification to this model to the range of nonlinear deformations is given by the St. Venant-Kirchhoff model [23], which in FEBio is referred to as isotropic elasticity. This model is objective for large strains and rotations. For the isotropic case it can be derived from the following hyperelastic strain-energy function: $$\begin{equation} W=\frac{1}{2}\lambda\left(\tr\mathbf{E}\right)^{2}+\mu\mathbf{E}:\mathbf{E}\,.\label{eq411} \end{equation}$$ The second Piola-Kirchhoff stress can be derived from this: $$\begin{equation} \mathbf{S}=\frac{\partial W}{\partial\mathbf{E}}=\lambda\left(\tr\mathbf{E}\right)\mathbf{I}+2\mu\mathbf{E}\,.\label{eq412} \end{equation}$$ Note that these equations are similar to the corresponding equations in the linear elastic case, except that the small strain tensor is replaced by the Green-Lagrange elasticity tensor $\mathbf{E}$ . The material elasticity tensor is then given by, $$\begin{equation} \boldsymbol{\mathbb{C}}=\frac{\partial\mathbf{S}}{\partial\mathbf{E}}=\lambda\mathbf{I}\otimes\mathbf{I}+2\mu\mathbf{I}\,\overline{\underline{\otimes}}\,\mathbf{I}\,.\label{eq413} \end{equation}$$ It is important to note that although this model is objective, it should only be used for small strains. For large strains, the response can be somewhat strange if not completely unrealistic. For example, it can be shown that under uni-axial tension the stress becomes infinite and the volume tends to zero for finite strains. Therefore, for large strains it is highly recommended to avoid this material and instead use one of the other non-linear material models described below. The Cauchy stress is $$\begin{equation} \boldsymbol{\sigma}=\frac{1}{J}\left(\lambda\tr\mathbf{E}-\mu\right)\mathbf{b}+\frac{\mu}{J}\mathbf{b}^{2}\,,\label{eq414} \end{equation}$$ where $\tr\mathbf{E}=\left(\tr\mathbf{b}-3\right)/2$ , whereas the spatial elasticity tensor is $$\begin{equation} \boldsymbol{\mathcal{C}}=\frac{\lambda}{J}\mathbf{b}\otimes\mathbf{b}+\frac{2}{J}\mu\mathbf{b}\,\overline{\underline{\otimes}}\,\mathbf{b}\,.\label{eq415} \end{equation}$$