A higher-order nonlocal elasticity continuum model for deterministic and stochastic particle-based materials

Abstract

This paper proposes, for particle-based materials, a higher-order nonlocal elasticity continuum model that includes the Piola peridynamics and the Eringen nonlocal elasticity. When referring to particle-based materials, we denote systems that can be modeled as assemblies of material points (or particles). Note that this paper is not devoted to granular materials, then factors such as the topology of contacts, granulometry, grain sizes, shapes, and geometric structure are not considered. Additionally, when referring to Piola peridynamics, we specifically denote the particular peridynamic model developed by Piola, which differs from the commonly adopted approach to peridynamics. The proposed higher-order nonlocal elasticity continuum model offers several advantages. First, it can describe interactions between material points over longer ranges than those considered by Eringen nonlocal elasticity. Second, it exhibits similar characteristics to gradient-type theories and Piola peridynamics, enabling the consideration of more complex external and contact actions, including Nth order forces and stresses. Furthermore, the proposed deterministic model is developed to lay the foundation for a stochastic formulation applicable to uncertain particle-based materials. We want to emphasize that the aim of this paper is not to unify Eringen nonlocal elasticity with the various existing peridynamic models.

Appendix

In this appendix, we give some of the formulas presented in this paper for $N=2$, i.e., for 2nd gradient continua. In the following, summation is implied for repeated indices. Let us assume that $\textbf{r}$ at the point $\overline{\textbf{x}}$ can be approximated using its Taylor expansion in the neighborhood of $\textbf{x}$ truncated at the second order,

$$\begin{aligned} r_{i}(\overline{\textbf{x}})=r_{i}(\textbf{x})+\frac{\partial r_{i}(\textbf{x})}{\partial x_{j}}(\overline{x}_{j}-x_{j})+\frac{1}{2}\frac{\partial ^{2} r_{i}(\textbf{x})}{\partial x_{j}\partial x_{k}}(\overline{x}_{j}-x_{j})(\overline{x}_{k}-x_{k}). \end{aligned}$$

(64)

Let us define the tensor $\textbf{f}^{(1)}(\textbf{x})$ represented by the matrix $[\textbf{f}^{(1)}(\textbf{x})]$ such that

$$\begin{aligned}{}[\textbf{f}^{(1)}(\textbf{x})]_{ij}=\frac{\partial r_{i}(\textbf{x})}{\partial x_{j}}. \end{aligned}$$

(65)

The tensor $\textbf{f}(\overline{\textbf{x}},\textbf{x})$ represented by the matrix $[\textbf{f}(\overline{\textbf{x}},\textbf{x})]$ (see Eq. (5)) becomes

$$\begin{aligned}{}[\textbf{f}(\overline{\textbf{x}},\textbf{x})]_{ij}=[\textbf{f}^{(1)}(\textbf{x})]_{ij} +\frac{1}{2}\frac{\partial [\textbf{f}^{(1)}(\textbf{x})]_{ij}}{\partial x_{k}}(\overline{x}_{k}-x_{k}). \end{aligned}$$

(66)

Taking into account Eqs. (65) and (66), Eq. (64) can be rewritten as:

$$\begin{aligned} r_{i}(\overline{\textbf{x}})=r_{i}(\textbf{x})+[\textbf{f}(\overline{\textbf{x}}, \textbf{x})]_{ij}(\overline{x}_{j}-x_{j}), \end{aligned}$$

(67)

where $\mathrm {det(\textbf{f}(\overline{\textbf{x}},\textbf{x}))>0}$ under the hypothesis of orientation-preserving deformations. By taking into account Eq. (66), tensor $\varvec{\mathbbm {c}}(\overline{\textbf{x}},\textbf{x})$ represented by the matrix $[\varvec{\mathbbm {c}}(\overline{\textbf{x}},\textbf{x})]$ (see Eq. (12)) can be written in terms of components as

$$\begin{aligned}{}[\varvec{\mathbbm {c}}(\overline{\textbf{x}},\textbf{x})]_{pq}=[\textbf{c}^{(1)}(\textbf{x})]_{pq} +\frac{1}{2}c_{pqj}^{(12)}(\textbf{x})(\overline{x}_{j}-x_{j})+\frac{1}{4}c_{pqjk}^{(2)} (\textbf{x})(\overline{x}_{j}-x_{j})(\overline{x}_{k}-x_{k}), \end{aligned}$$

(68)

where $\textbf{c}^{(1)}(\textbf{x})$ is the right Cauchy–Green tensor (see Eq. (32)), $\textbf{c}^{(12)}(\textbf{x})=\nabla \textbf{c}^{(1)}(\textbf{x})$ is the third-order tensor whose components are

$$\begin{aligned} c_{pqj}^{(12)}(\textbf{x})=[\textbf{f}^{(1)}(\textbf{x})]_{ip}\frac{\partial [\textbf{f}^{(1)}(\textbf{x})]_{iq}}{\partial x_{j}}+\frac{\partial [\textbf{f}^{(1)}(\textbf{x})]_{ip}}{\partial x_{j}}[\textbf{f}^{(1)}(\textbf{x})]_{iq}, \end{aligned}$$

(69)

and $\textbf{c}^{(2)}(\textbf{x})$ is the fourth-order tensor whose components are

$$\begin{aligned} c_{pqjk}^{(2)}(\textbf{x})=\frac{\partial [\textbf{f}^{(1)}(\textbf{x})]_{ip}}{\partial x_{j}}\frac{\partial [\textbf{f}^{(1)}(\textbf{x})]_{iq}}{\partial x_{k}}. \end{aligned}$$

(70)

Taking into account Eq. (68), tensor $\varvec{\mathbbm {e}}(\overline{\textbf{x}},\textbf{x})$ represented by the matrix $[\varvec{\mathbbm {e}}(\overline{\textbf{x}},\textbf{x})]$ (see (14) can be rewritten in terms of components as

$$\begin{aligned}{}[\varvec{\mathbbm {e}}(\overline{\textbf{x}},\textbf{x})]_{pq}=[\textbf{e}^{(1)}(\textbf{x})]_{pq} +\frac{1}{4}c_{pqj}^{(12)}(\textbf{x})(\overline{x}_{j}-x_{j})+\frac{1}{8}c_{pqjk}^{(2)}(\textbf{x}) (\overline{x}_{j}-x_{j})(\overline{x}_{k}-x_{k}). \end{aligned}$$

(71)

Deriving Eq. (64) with respect to $\overline{\textbf{x}}$, it is not difficult to show that $[\varvec{\mathbbm {e}}(\overline{\textbf{x}},\textbf{x})]=[\varvec{\mathbbm {e}}(\textbf{x},\overline{\textbf{x}})]$ and, consequently, $[\varvec{\mathbbm {e}}(\overline{\textbf{x}},\textbf{x})]=[\varvec{\textrm{e}}(\overline{\textbf{x}},\textbf{x})]$. Thus, replacing Eq. (71) into Eq. (53), we obtain a novel deformation energy density suitable for particle-based materials under the hypothesis of second gradient continua. Further details and computational aspects will be addressed in the second part of the paper. It will be shown that tensor $[\varvec{\mathbbm {e}}(\overline{\textbf{x}},\textbf{x})]$ allows us to formulate even more general models, exploring the possibility to encompass gradient-type and integral nonlocal theories.