The elastic–viscoelastic correspondence principle for non-homogeneous materials with time translation non-invariant properties

This paper revisits the elastic–viscoelastic correspondence principle for non-homogeneous materials. Several recent publications discussed this principle for functionally graded materials (FGMs) with time translation invariant viscoelastic properties. It was demonstrated that the correspondence principle is valid only for the FGMs with separable relaxation moduli (moduli in separable form in space and time). This paper reconsiders this issue. It shows that the correspondence principle is valid even for non-homogeneous materials with separable relaxation moduli even if the time-dependences of the relaxation moduli in shear and dilatation are not necessarily time translation invariant. The property of similarity of Volterra operators is used to obtain the corresponding elastic solution. The correspondence is established between the elastic solution and the operator-transformed viscoelastic solution. The transformation operators are combinations of the Laplace transform operator and additional integral operators.