On dynamic stress equivalence in force and eigenstrain induced linear elastic and viscoelastic wave propagation: A uni-axial study considering compressive shock waves

Considering the example of uni-axial compressive shock waves in a linear viscoelastic half space, this study demonstrates the possibility of equivalence between the stresses due to body forces and due to eigenstrain loadings in an advanced dynamic setting. The different loading cases that are compared for equivalence may be accompanied by equal or different constitutive behavior; in the latter case, initial elastic moduli are taken as equal, but the time rates of the relaxation functions may be different. A straightforward procedure for deriving the eigenstrain loadings that are necessary for stress equivalence is developed. The Green function problem of a concentrated body force of the Dirac delta function type travelling into the half-space is used as force loading, and a closed expression of the necessary eigenstrain loadings in integral form is presented, the eigenstrains acting in the absence of force loadings. Due to the linearity of the problem at hand, applying negative eigenstrains together with the force loading in case of equal constitutive behavior produces complete stress compensation, an effect that is of particular interest in smart structures and materials. Appropriateness of the method is demonstrated in two examples concerning a viscoelastic behavior of the Poynting-Thomson type in the force loading problem, the auxiliary constitutive behavior in the eigenstrain loading problem being either also of the Poyning-Thomson type or purely elastic.