Peridynamics

Peridynamics is a formulation of continuum mechanics that is oriented toward deformations with discontinuities, especially fractures. Peridynamics is a formulation of continuum mechanics that is oriented toward deformations with discontinuities, especially fractures. The peridynamic theory is based on integral equations, in contrast with the classical theory of continuum mechanics, which is based on partial differential equations. Since partial derivatives do not exist on crack surfaces and other singularities, the classical equations of continuum mechanics cannot be applied directly when such features are present in a deformation. The integral equations of the peridynamic theory can be applied directly, because they do not require partial derivatives. The ability to apply the same equations directly at all points in a mathematical model of a deforming structure helps the peridynamic approach avoid the need for the special techniques of fracture mechanics. For example, in peridynamics, there is no need for a separate crack growth law based on a stress intensity factor. The basic equation of peridynamics is the following equation of motion: where x {displaystyle x} is a point in a body R {displaystyle R} , t {displaystyle t} is time, u {displaystyle u} is the displacement vector field, and ρ {displaystyle ho } is the mass density in the undeformed body. x ′ {displaystyle x'} is a dummy variable of integration. The vector valued function f {displaystyle f} is the force density that x ′ {displaystyle x'} exerts on x {displaystyle x} . This force density depends on the relative displacement and relative position vectors between x ′ {displaystyle x'} and x {displaystyle x} . The dimensions of f {displaystyle f} are force per volume squared. The function f {displaystyle f} is called the 'pairwise force function' and contains all the constitutive (material-dependent) properties. It describes how the internal forces depend on the deformation. The interaction between any x {displaystyle x} and x ′ {displaystyle x'} is called a 'bond.' The physical mechanism in this interaction need not be specified.It is usually assumed that f {displaystyle f} vanishes whenever x ′ {displaystyle x'} is outside a neighborhood of x {displaystyle x} (in the undeformed configuration) called the horizon. The term 'peridynamic,' an adjective, was proposed in the year 2000 and comes from the prefix peri, which means all around, near, or surrounding; and the root dyna, which means force or power. The term 'peridynamics,' a noun, is a shortened form of the phrase peridynamic model of solid mechanics.