THREE-DIMENSIONAL NUMERICAL ANALYSIS OF SPONTANEOUS LARGE-DEFORMATION OF PLATE-LIKE STRUCTURES

Plate-like structures in nature, such as flowers and leaves, tend to have a curvaceous shape as a result of a large deformation process. Similar phenomena can also be observed in plate-like structures in various engineering fields. Here, plate-like structure signifies a special three-dimensional structure that is the stack of identical planar structures and the thickness size is hence far less than the planar ones. Stimulated by the incompatible deformation due to factors such as growth and external stimuli, a plate-like structure possesses internal stresses. In this paper, a spontaneous deformation behavior produced by the existing internal stresses is studied. To this end, the strain energy of the plate-like structure is firstly decomposed into two contents, that is, the stretch-like deformation energy and the remaining deformation energy, respectively. To evaluate these two different energies, we suggest a numerical approach based on a three-dimensional (3D) large-deformation finite element analysis (FEA). A condition for buckling instability of such a plate-like structure is then proposed to be the crossover point at which the remaining deformation energy goes beyond the stretch-like deformation energy from none. With this condition, the concept of threshold thickness is further introduced in order to characterize the crossover point, which is verified by comparing the values from the FEA with those from the classical plate theory for a simply-supported square plate. Finally, several spontaneous large-deformation problems modeled by typical power-law thermal expansions are studied through a 3D large-deformation FEA, and the effects of incompatible factors such as the magnitude and the power index on internal stress fields and the threshold thickness are also examined. Our present work shows that the large deformation of plate-like structures is a spontaneous process such that the remaining deformation energy increases from zero to a value larger than the stretch-like deformation energy. In particular, the 3D large-deformation FEA is an effective method to solve the buckling instability of plate-like structures stimulated by a complex internal stress field.