Higher gradient expansion for linear isotropic peridynamic materials

Abstract

Peridynamics is a non-local continuum mechanics that replaces the differential operator embodied by the stress term $\mathrm{div}\mathit{S}$ in Cauchy’s equation of motion by a non-local force functional $\mathfrak{L}$ to take into account long-range forces. The resulting equation of motion reads

$$\rho \stackrel{··}{\mathit{u}}=\mathfrak{L}\mathit{u}+\mathit{b}(\mathit{u}=\mathrm{displacement},\mathit{b}=\mathrm{body\; force},\rho =\mathrm{density}).$$

If the characteristic length $\delta$ of the interparticle interaction approaches 0, the operator $\mathfrak{L}$ admits an expansion in $\delta$ that, for a linear isotropic material, reads

$$\mathfrak{L}\mathit{u}=(\lambda +\mu )\nabla \mathrm{div}\mathit{u}+\mu \Delta \mathit{u}+{\delta }^2{\Theta }_2·{\nabla }^4\mathit{u}+{\delta }^4{\Theta }_3·{\nabla }^6\mathit{u}+\dots ,$$

where $\lambda$ and $\mu$ are the Lamé moduli of the classical elasticity, and the remaining higher-order corrections contain products of the type ${\mathit{T}}_s\mathit{u}:={\Theta }_s·{\nabla }^{2s}\mathit{u}$ of even-order gradients ${\nabla }^{2s}\mathit{u}$ (i.e., the collections of all partial derivatives of $\mathit{u}$ of order 2s) and constant coefficients ${\Theta }_s$ collectively forming a tensor of order 2s. Symmetry arguments show that the terms ${\mathit{T}}_s\mathit{u}$ have the form

$${\delta }^{2s-2}({\lambda }_s+{\mu }_s){\Delta }^{s-1}\nabla \mathrm{div}\mathit{u}+{\delta }^{2s-2}{\mu }_s{\Delta }^s\mathit{u},$$

where ${\lambda }_s$ and ${\mu }_s$ are scalar constants. This article explicitly determines ${\lambda }_s$ and ${\mu }_s$ in terms of the properties of the material (i.e., of the operator $\mathfrak{L}$ ) in all dimensions n (typically, $n=1,2$ or $3$ ).