Theory Manual Version 3.4
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Subsection 2.7.1: Governing Equations Up Chapter 2: Continuum Mechanics Section 2.9: Equilibrium Swelling

2.8 Mixture of Solids

A solid material may consist of a heterogeneous mixture of various solid constituents that are constrained to move together. If each constituent is denoted by the superscript , a constrained mixture satisfies for all , where is the velocity of the solid mixture. For example, a fiber-reinforced material may consist of a mixture of fibers and a ground matrix. In general, the constitutive relation for such a constrained mixture of solids may be a complex function of the mass fraction of each constituent as well as the ultrastructure of the constituents and their mutual interactions. The mass fraction of each constituent may be represented by the apparent density , which is the ratio of the mass of to the volume of the mixture in the reference configuration, in an elemental region. In the framework of hyperelasticity, the general representation for the strain energy density for such a solid mixture may have the form where is the deformation gradient of constituent and is the number of solid constituents in the mixture. Though the solid constituents are constrained to move together, their deformation gradients are not necessarily the same, depending on how the various solid constituents of a constrained mixture were assembled [6].
With no loss of generality, it may be assumed that the strain energy density of the mixture is the summation of the strain energy densities of all the constituents, where is the strain energy density of constituent .
Now, as a special case, we may assume that the simplest form of the constitutive relation for a mixture of constrained solids is This special form assumes that there are no explicit dependencies among the various solid constituents of the mixture. Thus, depends only on the deformation gradient and mass content of .
Furthermore, if we assume that for all (implying no residual stresses in the solid mixture), then the general form for further reduces to Consequently, the stress tensor for the mixture becomes In other words, the stress in the solid mixture may be evaluated from the sum of the stresses in each mixture constituent using the same hyperelasticity relation as for a single, pure solid constituent. The fact that also depends on implies that the material properties appearing in the constitutive relation for are dependent on the mass content of solid in the mixture.
For nearly-incompressible solids, using a reasoning similar to that which led to (2.8-4), the uncoupled strain energy density for the solid mixture may be of the form where is the volumetric energy component, is the distortional energy component, and is the distortional part of the deformation gradient, as described in Section 2.4.3↑.

Subsection 2.7.1: Governing Equations Up Chapter 2: Continuum Mechanics Section 2.9: Equilibrium Swelling