5. Energies of Inhomogeneities, Dilute Reinforcements and Cracks

As in the previous chapter, we consider homogeneous inclusions and inhomogeneities in subvolumes \( {\Omega_r} \) of an infinitely extended homogeneous volume \( {\Omega_{{0}}} \) of a comparison medium or ‘matrix’ of stiffness \( {{L}_0} \); the total volume \( \Omega = {\Omega_{{0}}} + {\Omega_r} \). In Sects. 5.1.4 and in 5.2 and 5.3, we examine composite aggregates with dilute reinforcement, which may consist of many distinct inhomogeneities \( {{L}_r} \) in a matrix \( {{L}6557} \), as described in Sect.4.​4. Systems containing cracks are discussed in Sect. 5.4. Loads applied to both single and multiple inhomogeneity systems include displacement or traction fields acting at a remote boundary to generate uniform overall strain or stress, and piecewise uniform, physically based eigenstrains in both matrix and inhomogeneities. Those include thermal and moisture-induced strains, phase transformations, and inelastic strains. Low loading rates causing only small strains are assumed.

When applied separately to a homogeneous material, each of these loads generates a certain amount of potential energy \( {\mathcal V} \) defined in Sect. 3.​7. When applied together to a heterogeneous material, they generate the total potential energy, which is equal to the sum of the energies caused by the applied loads and of the potential energy generated by interactions between individual load components and/or inhomogeneities. Different interaction energies are derived for selected combinations of applied loads and material configurations. The results are useful in several applications, e.g., in estimating the energy released by interfacial decohesion of inhomogeneities from the surrounding matrix, or energy changes associated with phase transformations.