A body-force analogy for dynamics of elastic bodies with eigenstrains

In the linear static theory of thermoelasticity, the body force analogy dates back to Duhamel. In its classical form, it reads: Consider the static deformation of an isotropic linear thermoelastic body under the action of a given temperature. Then the thermal stresses can be obtained by addition of an imaginary pressure to the isothermal stresses, which follow by solving the isothermal governing equations with certain imaginary body forces and surface tractions. Moreover, the thermal displacements due to the given temperature are identical to the isothermal displacements due to the imaginary body forces and surface tractions. In the present paper, a dynamic extension of this body force analogy is presented in the framework of the three-dimensional theory of linear anisotropic elastodynamics with eigenstrains. Our formulation thus includes not only effects such as thermal expansion strains or piezoelectric expansion strains, but also inelastic parts of strains in the framework of a geometrically linear theory. We treat two problems, namely a problem with a given distribution of eigenstrains and with assigned body and surface forces, and a second problem without eigenstrains, but with an auxiliary system of body and surface forces, which we determine such that the required body force analogy holds. It turns out that all what is needed for an extension of the classical static analogy to dynamics is the requirement of identical initial conditions and displacement boundary conditions in the two problems under consideration. We finally present the proper form of the jump conditions for balance of momentum that must be taken into account in the auxiliary problem in order that the body force analogy holds in the presence of a singular surface also.