Peridynamic Theory

Abstract

At any instant of time, every point in the material denotes the location of a material particle, and these infinitely many material points (particles) constitute the continuum. In an undeformed state of the body, each material point is identified by its coordinates, ${{\mathbf{x}}_{(k) }}$ with $(k=1,2,\ldots,\infty )$, and is associated with an incremental volume, ${V_{(k) }}$, and a mass density of $\rho ({{\mathbf{x}}_{(k) }}).$ Each material point can be subjected to prescribed body loads, displacement, or velocity, resulting in motion and deformation. With respect to a Cartesian coordinate system, the material point ${{\mathbf{x}}_{(k) }}$ experiences displacement, ${{\mathbf{u}}_{(k) }}$, and its location is described by the position vector ${{\mathbf{y}}_{(k) }}$ in the deformed state. The displacement and body load vectors at material point ${{\mathbf{x}}_{(k) }}$ are represented by $\mathbf{u}_{(k) }({{\mathbf{x}}_{(k) }},t)$ and $\mathbf{b}_{(k) }({{\mathbf{x}}_{(k) }},t)$, respectively. The motion of a material point conforms to the Lagrangian description.