Higher gradient expansion for linear isotropic peridynamic materials

Peridynamics is a non-local continuum mechanics that replaces the differential operator embodied by the stress term div S in Cauchy’s equation of motion by a non-local force functional L to take into account long-range forces. The resulting equation of motion reads ρ u ·· = L u + b ( u = displacement , b = body force , ρ = density ) . If the characteristic length δ of the interparticle interaction approaches 0, the operator L admits an expansion in δ that, for a linear isotropic material, reads L u = ( λ + μ ) ∇ div u + μ Δ u + δ 2 Θ 2 · ∇ 4 u + δ 4 Θ 3 · ∇ 6 u + … , where λ and μ are the Lamé moduli of the classical elasticity, and the remaining higher-order corrections contain products of the type T s u : = Θ s · ∇ 2 s u of even-order gradients ∇ 2 s u (i.e., the collections of all partial derivatives of u of order 2s) and constant coefficients Θ s collectively forming a tensor of order 2s. Symmetry arguments show that the terms T s u have the form δ 2 s − 2 ( λ s + μ s ) Δ s − 1 ∇ div u + δ 2 s − 2 μ s Δ s u , where λ s and μ s are scalar constants. This article explicitly determines λ s and μ s in terms of the properties of the material (i.e., of the operator L ) in all dimensions n (typically, n = 1 , 2 or 3 ).