Generalized Mori-Tanaka Approach in Peridynamic Micromechanics of Multilayered Composites of Random Structure

Abstract

The basic feature of the peridynamics (PD) is a continuum description of material behavior as the integrated nonlocal force interactions between material points. Besides the conventional local theory, the PD equation of motion introduced by Silling (J Mech Phys Solids 48: 175-209, 2000) has no spatial derivatives of displacement. A linearized bond-based PD model is used for the analysis of random structure CMs subjected to the remote volumetric homogeneous boundary conditions. Effective properties are expressed through the local stress polarization tensor averaged over the external interaction interface of inclusions rather than in an entire space. Any spatial derivatives of displacement fields are not required. Inclusions are considered as identical aligned layers with a statistically homogeneous distribution in the space. For one infinite layer in the infinite homogeneous matrix, 3D PD equilibrium equation is reduced to the 1D integral equation with 1D micromodulus obtained by integrations of the original 3D micromodulus over the cross-sections (parallel to layers) of a horizon region. One estimates the average strain and stress fields in the extended layer by the use of averaging displacement and traction over the external interaction interface. Effective moduli for PD multilayered CM are estimated in the matrix form representations usually used in locally elastic multilayered CM and based on the consideration of the normal and tangential parts of the effective moduli matrix.

Appendix

We shortly reproduce (following Chatzigeorgiou [25]) several mathematical formulae used in locally elastic micromechanics of the multilayered CMs and based on the normal and tangential parts (with respect to the axis $x^1$) of the second-order tensors separately. For a strain tensor $\varvec{\varepsilon }$, and a stress tensor $\varvec{\sigma }$, written in slightly modified Voigt notation as

$$\begin{aligned} \varvec{\varepsilon }=\begin{bmatrix} \varepsilon _{11}\\ \varepsilon _{22}\\ \varepsilon _{33}\\ 2\varepsilon _{12}\\ 2\varepsilon _{13}\\ 2\varepsilon _{23} \end{bmatrix}, \ \ \ \varvec{\sigma }= \begin{bmatrix} \sigma _{11}\\ \sigma _{22}\\ \sigma _{33}\\ \sigma _{12}\\ \sigma _{13}\\ \sigma _{23} \end{bmatrix} \end{aligned}$$

(A.1)

these normal and tangential parts are expressed through the $3\times 1$ vectors

$$\begin{aligned} \varvec{\varepsilon }^n=\begin{bmatrix} \varepsilon _{11}\\ 2\varepsilon _{12}\\ 2\varepsilon _{13}\\ \end{bmatrix}, \ \ \ \varvec{\varepsilon }^t=\begin{bmatrix} \varepsilon _{22}\\ 2\varepsilon _{33}\\ 2\varepsilon _{23}\\ \end{bmatrix}, \end{aligned}$$

(A.2)

$$\begin{aligned} \varvec{\sigma }^n= \begin{bmatrix} \sigma ^{11}\\ \sigma _{12}\\ \sigma _{13}\\ \end{bmatrix},\ \ \ \varvec{\sigma }^t= \begin{bmatrix} \sigma _{22}\\ \sigma _{33}\\ \sigma _{23} \end{bmatrix}, \end{aligned}$$

(A.3)

respectively. A fourth-order tensor $\textbf{F}$ presenting minor symmetries ($F_{ijkl} = F_{jikl} = F_{ijlk}$) and written in the Voigt form

$$\begin{aligned} \ \ \ \textbf{F}= \begin{bmatrix} F_{11}&{}F_{12}&{}F_{13}&{}F_{14}&{}F_{15}&{}F_{16}\\ F_{21}&{}F_{22}&{}F_{23}&{}F_{24}&{}F_{25}&{}F_{26}\\ F_{31}&{}F_{32}&{}F_{33}&{}F_{34}&{}F_{35}&{}F_{36}\\ F_{41}&{}F_{42}&{}F_{43}&{}F_{44}&{}F_{45}&{}F_{46}\\ F_{51}&{}F_{52}&{}F_{53}&{}F_{54}&{}F_{55}&{}F_{56}\\ F_{61}&{}F_{62}&{}F_{63}&{}F_{64}&{}F_{65}&{}F_{66}\\ \end{bmatrix} \end{aligned}$$

(A.4)

can be represented by the $3\times 3$ matrices

$$\begin{aligned} \ \ \ \textbf{F}^{nn}= \begin{bmatrix} F_{11}&{}F_{14}&{}F_{15}\\ F_{41}&{}F_{44}&{}F_{45}\\ F_{51}&{}F_{54}&{}F_{55}\\ \end{bmatrix},\ \ \ \textbf{F}^{nt}= \begin{bmatrix} F_{12}&{}F_{13}&{}F_{16}\\ F_{42}&{}F_{43}&{}F_{46}\\ F_{52}&{}F_{53}&{}F_{56}\\ \end{bmatrix} \end{aligned}$$

(A.5)

$$\begin{aligned} \textbf{F}^{tn}= \begin{bmatrix} F_{21}&{}F_{24}&{}F_{25}\\ F_{31}&{}F_{34}&{}F_{35}\\ F_{61}&{}F_{64}&{}F_{65}\\ \end{bmatrix}, \ \ \ \textbf{F}^{tt}= \begin{bmatrix} F_{22}&{}F_{23}&{}F_{26}\\ F_{32}&{}F_{33}&{}F_{36}\\ F_{62}&{}F_{63}&{}F_{66}\\ \end{bmatrix} \end{aligned}$$

(A.6)

and expressed through the matrix-type relations

$$\begin{aligned} \textbf{F}= \varvec{\mathcal {I}}_n\cdot \textbf{F}^{nn}\cdot \varvec{\mathcal {I}}^{\top }_n +\varvec{\mathcal {I}}_n\cdot \textbf{F}^{nt}\cdot \varvec{\mathcal {I}}^{\top }_t +\varvec{\mathcal {I}}_t\cdot \textbf{F}^{tn}\cdot \varvec{\mathcal {I}}^{\top }_n +\varvec{\mathcal {I}}_t\cdot \textbf{F}^{tt}\cdot \varvec{\mathcal {I}}^{\top }_t. \end{aligned}$$

(A.7)

Here, the $6\times 3$ matrices

$$\begin{aligned} \varvec{\mathcal {I}}_n= \begin{bmatrix} 1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0\\ 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 1\\ 0 &{} 0 &{} 0\\ \end{bmatrix},\ \ \ \varvec{\mathcal {I}}_t= \begin{bmatrix} 0 &{} 0 &{} 0\\ 1 &{} 0 &{} 0\\ 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 1\\ \end{bmatrix} \end{aligned}$$

(A.8)

have the properties

$$\begin{aligned} \varvec{\mathcal {I}}_n^{\top }\cdot \varvec{\varepsilon }=\varvec{\varepsilon }^n, \ \ \ \varvec{\mathcal {I}}_t^{\top }\cdot \varvec{\varepsilon }=\varvec{\varepsilon }^t, \end{aligned}$$

(A.9)

$$\begin{aligned} \varvec{\mathcal {I}}_n^{\top }\cdot \varvec{\mathcal {I}}_t=\varvec{\mathcal {I}}_t^{\top }\cdot \textbf{I}_n=\textbf{0}, \ \ \varvec{\mathcal {I}}_n^{\top }\cdot \varvec{\mathcal {I}}_n\varvec{\mathcal {I}}_t^{\top }\cdot \textbf{I}_t={\textbf{I}}, \end{aligned}$$

(A.10)

$$\begin{aligned} \varvec{\mathcal {I}}_n\cdot \textbf{I}\cdot \varvec{\mathcal {I}}_n^{\top }+\varvec{\mathcal {I}}_t\cdot \textbf{I}\cdot \varvec{\mathcal {I}}_t^{\top }=\varvec{\mathcal {I}}, \end{aligned}$$

(A.11)

where

$$\begin{aligned} \varvec{\mathcal {I}}=\textrm{diag}(1,1,1,1,1,1),\ \ \ \textbf{I}^{33}=\textrm{diag}(1,1,1). \end{aligned}$$

(A.12)

Let us consider the tensor $\textbf{F}$ connecting two symmetric second-order tensors $\textbf{a}$ and $\textbf{b}$ (e.g., strain or stress type). It can be presented in either the indicial (Einstein’s) or matrix notations

$$\begin{aligned} a_{ij}=F_{ijkl}b_{kl}, \end{aligned}$$

(A.13)

$$\begin{aligned} \textbf{a}^n=\textbf{F}^{nn}\cdot \textbf{b}^n+\textbf{F}^{nt}\cdot \textbf{b}^t, \ \ \textbf{a}^t=\textbf{F}^{tn}\cdot \textbf{b}^n+\textbf{F}^{tt}\cdot \textbf{b}^t, \end{aligned}$$

(A.14)

respectively. In particular, the linear stress–strain relations are expressed in either the indicial (Einstein’s) or matrix notations

$$\begin{aligned} \sigma _{ij}=L_{ijkl}\varepsilon _{kl}, \end{aligned}$$

(A.15)

$$\begin{aligned} \varvec{\sigma }^n=\textbf{L}^{nn}\cdot \varvec{\varepsilon }^n+\textbf{L}^{nt}\cdot \varvec{\varepsilon }^t, \ \ \varvec{\sigma }^t=\textbf{L}^{tn}\cdot \varvec{\varepsilon }^n+\textbf{L}^{tt}\cdot \varvec{\varepsilon }^t, \end{aligned}$$

(A.16)

respectively, where the stiffness matrix $\textbf{L}$ posses major symmetries ($L_{ijkl}=L_{klij}$):

$$\begin{aligned} \textbf{L}^{nn}=(\textbf{L}^{nn})^{\top },\ \ \ \textbf{L}^{tn}=(\textbf{L}^{nt})^{\top }, \ \ \ \textbf{L}^{tt}=(\textbf{L}^{tt})^{\top }. \end{aligned}$$

(A.17)

Due to the form of the structure, all fields vary spatially only in the $x^1$ direction, transforming the equilibrium Eq. (2) into a 1D problem. For simplicity, the elasticity tensor is decomposed with the help of representations (A.5) and (A.6). Since all fields depend only on $x^1$, the equilibrium equation is reduced to

$$\begin{aligned} \frac{\textrm{d} }{\textrm{d}{x^1}}\Big ({\textbf{L}}^{nn}\cdot \frac{\textrm{d}{\textbf{u}} }{\textrm{d}{x^1}}\Big )=\textbf{0}, \end{aligned}$$

(A.18)

where

$$\begin{aligned} \textbf{u}= \begin{bmatrix} u_1\\ u_2\\ u_3 \end{bmatrix},\ \ \ \frac{\textrm{d}\textbf{u}}{\textrm{d}x^1}=\varvec{\varepsilon }^n, \end{aligned}$$

(A.19)

$$\begin{aligned} {\textbf{u}}=(u_1,u_2.u_3)^{\top }. \end{aligned}$$

(A.20)