Theoretical framework¶

CabanaPD uses the peridynamic formulation^{1 2}. Given a bounded body \(\mathcal{B}\subset \mathbb{R}^3\), the equation of motion for a material point \({\bf x} \in \mathcal{B}\) at time \(t\geqslant 0\) is:

where \(\mathbf{x}\) denotes the position in the reference configuration, \(\rho\) is the mass density, \({\bf u}\) is the displacement field, \({\bf f}\) is the bond force function, \({\bf b}\) is a prescribed body force density, and \(\mathcal{H}_{\bf x}\) is the nonlocal neighborhood of \({\bf x}\) defined by $$ \mathcal{H}_{\bf x}:=\lbrace {\bf x}^\prime \in \mathcal{B}: \lVert{\bf x}^\prime -{\bf x}\rVert\leqslant \delta \rbrace, $$ where \(\|\cdot\|\) denotes the Euclidean norm, \(\delta>0\) is the maximum distance over which nonlocal interactions occur (called the horizon), and \({\bf x}^\prime\in\mathcal{H}_{\bf{x}}\) denotes a neighboring material point.

References¶