On Necessary and Sufficient Conditions for Eigenstrain-Type Control of Stresses in the Dynamics of Force-Loaded Elastic Bodies

In the present contribution, the possibility of controlling dynamic stresses in force-loaded bodies by means of actuating eigenstrain fields is addressed. The action of eigenstrains, such as thermal or piezoelectric actuating strains, is subsequently gathered under the notion of actuating stresses. Our study is performed in the framework of the theory of small incremental dynamic deformations superimposed upon a state of possibly large static pre-deformation of a hyperelastic body. Particularly, we present a solution for the general problem of producing certain incremental stress trajectories by means of specifically tailored actuation stresses that are superimposed onto the force-loaded body. This we shortly call the stress tracking problem. The problem of suppressing incremental stresses is contained as a special case. Subsequently, particular emphasis is given to the systematic derivation of necessary and sufficient conditions that must be satisfied in order to solve the stress tracking problem. Necessary conditions are presented that must be satisfied by the intermediate configuration and by the desired incremental stress field that shall be tracked, and sufficient conditions are derived that must be satisfied by the incremental actuating stresses. As an illustrative example, our three-dimensional formulation is eventually applied to the one-dimensional dynamic case of a straight homogeneous rod with a support excitation at one end and a single point-mass at the other end.