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

\[\begin{split}\begin{aligned} \begin{cases} - \nabla \cdot \boldsymbol{P} &= 0, &&\text{in} \quad \mathcal{B}, \\ \boldsymbol{u} &= \boldsymbol{g}, &&\text{on} \quad \Gamma_g, \\ \boldsymbol{P} \cdot \boldsymbol{N} &= \boldsymbol{h}, &&\text{on} \quad \Gamma_h. \end{cases} \end{aligned}\end{split}\]

where \(\boldsymbol{P}\) denotes the first Piola-Kirchhoff stress tensor \(\mathcal{B}\) denotes the domain of interest in the reference configuration, \(\Gamma_g\) denotes the portion of the domain boundary on which Dirichlet boundary conditions \(\boldsymbol{u}\) are prescribed, and \(\Gamma_h\) denotes the portion of the domain boundary on which tractions \(\boldsymbol{P} \cdot \boldsymbol{N}\) are prescribed,