Analytical investigation of structurally stable configurations in shape memory alloy-actuated plates

Strains produced by active materials embedded in plates have been extensively used to manipulate the shape of surface-like engineering structures. Shape memory alloys (SMAs) are active materials that provide a significant amount of strain under large stresses, a characteristic of great utility in such morphing structures. In this work, an analytical approach to approximate the deformation of plates with SMA constituents is developed via the Rayleigh-Ritz method. An additive set of kinematically admissible displacement fields with unknown coefficients is used to describe the plate displacement field. The total potential energy is then calculated using the displacement field, loading conditions, and constitutive relations for the plate layer(s) composed of SMA wire meshes, dense SMA films, and/or elastic material. The unknown coefficients are then found via minimization of the total potential energy. This approach provides closed-form expressions for the approximate deformation of the plates including multistable configurations. The response of circular SMA-based plates is studied herein. The results show that temperature fields with a linear variation in the radial direction induce multistable configurations in which the plate Gaussian curvature is determined by the direction of the temperature gradient. An alternative inversion of the proposed approach is used to directly compute the temperature field required to morph a plate towards a prescribed goal shape. The obtained closed-form expressions show good agreement with detailed non-linear finite element analysis simulations. The proposed approach provides analysts with a low computational cost and relatively simple implementation to assess the potentially stable configurations of SMA-based plates under given loading conditions. Knowledge of such stable configurations is very valuable in the design of SMA-based morphing structures.