ABSTRACT
This chapter focuses exclusively on the meshfree approach of Silling and Askari [33]. To date, the vast majority of peridynamic simulations have utilized this approach. The decision to focus on this particular computational strategy is not meant to imply that it is superior to alternative approaches. Applying the finite element technique to peridynamics, for example, may provide superior results in some cases. Nonetheless, the meshfree approach of Silling and Askari has proven to be a reliable and efficient strategy for addressing problems in solid mechanics with pervasive material failure and fracture. Advantages include ease of implementation, computational efficiency, and the presence of a natural mechanism for material separation. The method of Silling and Askari is a direct discretization of the strong form of the peridynamic balance of linear momentum [34],
ρ (x) u¨(x, t) = ∫ Hx
(T [x, t]〈q−x〉−T [q, t]〈x−q〉)dVq+b(x, t) , (5.1)
where ρ is the material density, u¨ is acceleration, t is time, x and q are material points, Hx is a spherical neighborhood of radius δ centered at x, T denotes a peridynamic force state, dVq denotes the infinitesimal volume associated with q, and b is a body force density. For problems in three-dimensional space, the approach of Silling and Askari requires evaluation of a three-dimensional integral. In contrast, solution of the weak form of the peridynamic balance of linear momentum, for example using a finite element discretization, requires evaluation of a six-dimensional integral. For this reason, solution strategies for peridynamic models based on the weak form entail significant geometric complexity and computational expense relative to strong-form strategies.
