On eigenstrains without displacements

The present paper is concerned with an elastic body in a static intermediate configuration, the body being loaded by an additional distribution of transient eigenstrain. For the sake of convenience, we use the notion of actuation stress in the present paper in order to describe the action of eigenstrain. It is assumed that the incremental displacements produced by the additional actuation stress can be prescribed within the framework of the theory of infinitesimally small dynamic deformations superimposed upon a static strain. The intermediate configuration thus may represent a possibly large deformation from an undistorted state of the body. It is the scope of the present contribution to derive distributions of actuation stress, which do not produce any incremental displacements from the intermediate configuration. Solutions of this problem are not unique in general. In the present paper, we particularly seek for a displacement-free distribution of actuation stress that acts within a sub-region of the body only. We show that any statically admissible stress, which is produced in the sub-region by a possibly transient eigenstrain, and which satisfies appropriate dynamic boundary conditions, may be used as actuation stress in order that the incremental displacements vanish throughout the whole body. The body then remains in the static intermediate configuration for all times. Appropriate boundary-value problems are suggested for computing the statically admissible stress in the sub-domain. Other than elastic constitutive equations might be used for selecting a suitable boundary-value problem for the actuation stress. It is furthermore shown that the incremental first Piola-Kirchhoff stress of the elastic body is equal to the actuation stress inside the sub-region under consideration, and that it vanishes outside. As an example, we present successful Finite Element computations for the vibrations of a rectangular domain in a state of plane strain, where the rectangle is fixed at two adjacent edges, the sub-region also being taken as rectangular. [PUBLICATION ABSTRACT]