Eigenstrain controlled deformation- and stress-states

Abstract

Abstract (Quasi-)Static shape control of large space structures and of smart structures are the main topics of current interest. Imposed strains (eigenstrains) are of thermal nature or mainly make use of the piezoelectric effect in composites containing conventional ferroelectric polycrystals, natural crystals or special polymers. In the present paper, the direct solution of the quasistatic shape control problem in the context of the multiple field approach (i.e., considering the given structure in the background) is discussed in detail and the proper extension to dynamic problems (be connected with vibration suppression) is sketched. The imposed eigenstrain load is not assumed regionally concentrated, but distributed throughout the structure, however, with its intensity unlimited. The inverse problem of shape control can be exactly solved under these conditions. The results derived so far for static and dynamic piezoelectric actuation of isotropic or anisotropic linear elastic beam-, plate- and shell-type structures in the context of static stress-free eigenstrains (so called impotent eigenstrains) provide deep insight into the characteristic features of deformation control. It is demonstrated, that such an eigenstrain analysis can be applied directly and successfully to ‘intelligent’, ‘smart’, or ‘adaptive’ structures, which utilize piezoelectricity for the sake of structural actuation, sensing and control in an integrated circuit. However, the inverse problem associated with shape control is ill-posed. So called nilpotent eigenstrains (also known in the field of thermoelasticity) produce stresses but no deformations and thus, when properly defined, can be used to redistribute the load stresses without influencing shape control. The stress control procedure, however, is subjected to severe constraints given by the local conditions of equilibrium. For discretized or discrete structures (e.g., trusses) the general solution is worked out in detail and the two orthogonal subspaces of the impotent and nilpotent eigenstrains in Hilbert space are mentioned. Further, vibration suppression is discussed briefly under the condition of separation in space and time of the forcing function. In those cases, knowledge of the quasistatic load deformation suffices to define the distributed actuators producing impotent eigenstrain.