Theory

Governing Equations

At its core, CALIBR8 is concerned with solving the balance of linear momentum in the absence of inertial terms for finite-deformation mechanics in a total-Lagrangian setting. This can be represented as

$$\begin{array}{r}\begin{array}{c}\{\begin{array}{llll}-\mathrm{\nabla }\cdot \mathit{P}& =0,& & \text{in}\mathcal{B},\\ \mathit{u}& =\mathit{g},& & \text{on}{\mathrm{\Gamma }}_g,\\ \mathit{P}\cdot \mathit{N}& =\mathit{h},& & \text{on}{\mathrm{\Gamma }}_h.\end{array}\end{array}\end{array}$$

where $\mathit{P}$ denotes the first Piola-Kirchhoff stress tensor $\mathcal{B}$ denotes the domain of interest in the reference configuration, ${\mathrm{\Gamma }}_g$ denotes the portion of the domain boundary on which Dirichlet boundary conditions $\mathit{u}$ are prescribed, and ${\mathrm{\Gamma }}_h$ denotes the portion of the domain boundary on which tractions $\mathit{P}\cdot \mathit{N}$ are prescribed,