Nonlocal thermo-elastic constitutive relation of fibre-reinforced composites

Abstract

The effective properties of composite materials have been predicted by various micromechanical schemes. For composite materials of constituents which are described by the classical governing equations of the local form, the conventional micromechanical schemes usually give effective properties of the local form. However, it is recognized that under general loading conditions, spatiotemporal nonlocal constitutive equations may better depict the macroscopic behavior of these materials. In this paper, we derive the thermo-elastic dynamic effective governing equations of a fibre-reinforced composite in a coupled spatiotemporal integral form. These coupled equations reduce to the spatial nonlocal peridynamic formulation when the microstructural inertial effects are neglected. For static deformation and steady-state heat conduction, we show that the integral formulation is superior at capturing the variations of the average displacement and temperature in regions of high gradients than the conventional micromechanical schemes. The approach can be applied to analogous multi-field coupled problems of composites.

Appendix: Relation of the nonlocal description to Saint−Venant’s principle

Fig. A1

Characteristic lengths of nonlocal formulation and Saint–Venant’s principle

We note that Choi and Horgan [29] have studied Saint–Venant’s end effect for a semi-infinite composite strip in which a material is embedded between two layers of a different material, and the strip is subjected to self-equilibrating loads at the end. This is the structure in the paper of Silling [12], namely, the two-dimensional counterpart of the composite cylinder in the present work. The solutions of the two-dimensional strip are also given by expression (62b), but with $\chi _{u}$ expressed as

$$\begin{aligned} \chi _u^{2\mathrm{D}}=\frac{1}{r_0}\sqrt{\frac{3{\overline{E}}\mu _{\mathrm{m}}}{E_{\mathrm{f}}E_{\mathrm{m}}V_{\mathrm{f}}(1-V_{\mathrm{f}})^2}}, \end{aligned}$$

(A.1)

in which $r_0$ is the half width of the composite strip. By the solution of Ref. [29] for a semi-infinite composite strip, the Airy stress function, from which the displacement, strain, and stress can be derived, can be expressed as

$$\begin{aligned} \varphi =F{\mathrm{e}}^{-{\hat{\chi }}|z-L|} , \end{aligned}$$

(A.2)

where we have used $|z-L|$ to denote the distance away from the loaded end. F is a function of the coordinate in the transverse direction. The solution of ${\hat{\chi }}$ is generally complicated. However, for the case when $E_{\mathrm{f}}/E_{\mathrm{m}}\gg 1$, ${\hat{\chi }}$ can be solved from a transcendental equation as [29]

$$\begin{aligned} {\hat{\chi }}\approx \frac{1}{r_0}\sqrt{\frac{4\mu _{\mathrm{m}}(1-\nu _{\mathrm{f}}^2)(V_{\mathrm{f}}^2-3V_{\mathrm{f}}+3)}{E_{\mathrm{f}}V_{\mathrm{f}}^3(1-V_{\mathrm{f}})}}. \end{aligned}$$

(A.3)

With $E_{\mathrm{f}}/E_{\mathrm{m}}=10$, the Poisson ratio of the fibre and matrix $\nu _{\mathrm{f}}=\nu _{\mathrm{m}}=0.3$, ${\hat{\chi }}$ and $\chi _u^{2\mathrm{D}}$ depend on the volume fraction of the fibre $V_{\mathrm{f}}$ only. Therefore, we compare the values of $\chi _u^{2\mathrm{D}}$ and ${\hat{\chi }}$ in Fig. A1, for different values of $V_{\mathrm{f}}$. In Fig. A1, $L_{\mathrm{c}}$ denotes $1/\chi _u^{2\mathrm{D}}$ and $1/{\hat{\chi }}$. The orders of magnitude of $\chi _u^{2\mathrm{D}}$ and ${\hat{\chi }}$ are comparable, but the values are generally different, though they are close around $V_{\mathrm{f}}\approx 0.5$.