Micromechanics of composites/Introduction

The main aim of micromechanics is to find the properties of a continuum material point based on the microstructure. We assume that the microstructure can be described as a continuum even though it is heterogeneous and may contain voids.

The properties at a material point are computed using an averaging procedure. Many such averaging procedures exist, some of which assume the existence of a representative volume element (RVE). A RVE is usually chosen such that it represents adequately the local microstructure of the macroscopic scale continuum (in a statistical sense). [1]

We follow the following principle from Hill (1972):

Experimental determinations of mechanical behaviour rest ultimately on measured loads or mean displacements over pairs of opposite faces of a representative cube. Macro-variables intended for constitutive laws should thus be capable of definition in terms of surface data alone, either directly or indirectly. It is not necessary, by any means, that macro-variables so defined should be unweighted volume averages of their macroscopic counterparts. However, variables that do have this special property are naturally the easiest to handle analytically in the transition between levels. Accordingly, we approach the construction of macro-variables by first identifying some relevant averages that depend uniquely on surface data.

In this section, the main focus is on defining average quantities that may be used to describe the macroscopic constitutive behavior of composites. The average quantities are then shown to depend solely on boundary data. Both infinitesimal and finite deformations are considered.

Governing Equations

The equations that govern the motion of a solid include the balance laws for mass, momentum, and energy. Kinematic equations and constitutive relations are needed to complete the system of equations. Physical restrictions on the form of the constitutive relations are imposed by an entropy inequality that expresses the second law of thermodynamics in mathematical form.

The balance laws express the idea that the rate of change of a quantity (mass, momentum, energy) in a volume must arise from three causes:

1. the physical quantity itself flows through the surface that bounds the volume,
2. there is a source of the physical quantity on the surface of the volume, or/and,
3. there is a source of the physical quantity inside the volume.

Let $\Omega$ be an RVE and let $\partial{\Omega}$ be the surface of the RVE. Let the position vector of a point in space be given by $\mathbf{x}$. We can write

$\mathbf{x} = x_1~\mathbf{e}_1 + x_2~\mathbf{e}_2 + x_3~\mathbf{e}_3 \qquad \implies \qquad \mathbf{x} = x_i~\mathbf{e}_i$

where $(\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3)$ are the unit vectors that define a Cartesian coordinate system.

Let $f(\mathbf{x},t)$ be the physical quantity that is flowing through the RVE. Let $g(\mathbf{x},t)$ be sources on the surface of the RVE and let $h(\mathbf{x},t)$ be sources inside the RVE. Let $\mathbf{n}(\mathbf{x},t)$ be the outward unit normal to the surface $\partial{\Omega}$. Let $\mathbf{v}(\mathbf{x},t)$ be the velocity of the physical particles that carry the physical quantity that is flowing. Also, let the speed at which the bounding surface $\partial{\Omega}$ is moving be $u_n$ (in the direction $\mathbf{n}$).

Then, balance laws can be expressed in the general form

$\cfrac{d}{dt}\left[\int_{\Omega} f(\mathbf{x},t)~\text{dV}\right] = \int_{\partial{\Omega}} f(\mathbf{x},t)[u_n(\mathbf{x},t) - \mathbf{v}(\mathbf{x},t)\cdot\mathbf{n}(\mathbf{x},t)]~\text{dA} + \int_{\partial{\Omega}} g(\mathbf{x},t)~\text{dA} + \int_{\Omega} h(\mathbf{x},t)~\text{dV} ~.$

Note that the functions $f(\mathbf{x},t)$, $g(\mathbf{x},t)$, and $h(\mathbf{x},t)$ can be scalar valued, vector valued, or tensor valued - depending on the physical quantity that the balance equation deals with.

The balance laws of mass, momentum, and energy can be written as:

\begin{align} \dot{\rho} + \rho~\boldsymbol{\nabla} \bullet \mathbf{v} & = 0 & & \qquad\text{Balance of Mass} \\ \rho~\dot{\mathbf{v}} - \boldsymbol{\nabla} \bullet \boldsymbol{\sigma} - \rho~\mathbf{b} & = 0 & & \qquad\text{Balance of Linear Momentum} \\ \boldsymbol{\sigma} & = \boldsymbol{\sigma}^T & & \qquad\text{Balance of Angular Momentum} \\ \rho~\dot{e} - \boldsymbol{\sigma}:(\boldsymbol{\nabla}\mathbf{v}) + \boldsymbol{\nabla} \bullet \mathbf{q} - \rho~s & = 0 & & \qquad\text{Balance of Energy.} \end{align}

In the above equations $\rho(\mathbf{x},t)$ is the mass density (current), $\dot{\rho}$ is the material time derivative of $\rho$, $\mathbf{v}(\mathbf{x},t)$ is the particle velocity, $\dot{\mathbf{v}}$ is the material time derivative of $\mathbf{v}$, $\boldsymbol{\sigma}(\mathbf{x},t)$ is the Cauchy stress tensor, $\mathbf{b}(\mathbf{x},t)$ is the body force density, $e(\mathbf{x},t)$ is the internal energy per unit mass, $\dot{e}$ is the material time derivative of $e$, $\mathbf{q}(\mathbf{x},t)$ is the heat flux vector, and $s(\mathbf{x},t)$ is an energy source per unit mass.

The gradient and divergence operators are defined such that

$\boldsymbol{\nabla} \mathbf{v} = \sum_{i,j = 1}^3 \frac{\partial v_i}{\partial x_j}\mathbf{e}_i\otimes\mathbf{e}_j = v_{i,j}\mathbf{e}_i\otimes\mathbf{e}_j ~;~~ \boldsymbol{\nabla} \bullet \mathbf{v} =\sum_{i=1}^3 \frac{\partial v_i}{\partial x_i} = v_{i,i} ~;~~ \boldsymbol{\nabla} \bullet \boldsymbol{\sigma} = \sum_{i,j=1}^3 \frac{\partial \sigma_{ij}}{\partial x_j}~\mathbf{e}_i = \sigma_{ij,j}~\mathbf{e}_i ~.$

The contraction operation is given as

$\boldsymbol{A}:\boldsymbol{B} = \sum_{i,j=1}^3 A_{ij}~B_{ij} = A_{ij}~B_{ij} ~.$

References

1. See the introductory chapters from the following for a detailed description of a RVE: S. Nemat-Nasser and M. Hori, 1993, Micromechanics: Overall Properties of Heterogeneous Materials, North-Holland ; G. W. Milton, 2002, The Theory of Composites, Cambridge University Press ; S. Torquato, 2002, Random Heterogeneous Materials, Springer.