Realizing the Willis equations with pre-stresses

•The presence of inhomogeneous pre-stresses results in the Willis equations.•The material length scale is naturally incorporated in the Willis equations.•The Willis equations are generally valid for both quasi-static and dynamic problems.

This paper proves that the linear elastic behavior of the material with inhomogeneous pre-stresses can be described by the Willis equations. In this case, the additional terms in the Willis equations, compared with the classical linear elastic equations for homogeneous media, are related to the gradient of pre-stresses. In this way, the material length scale is naturally incorporated in the framework of continuum mechanics. All these findings also coincide with the results of transformation elastodynamics, so that they can meet the requirement of the principle of material objectivity and the principle of general invariance.