Optimal orientation of anisotropic material with given Kelvin moduli in FMO problems for plates and shells

In this work we investigate the compliance minimization problem of a plate subjected to the in-plane and transverse loadings acting simultaneously. In this way we attempt to generalize the classical Free Material Optimization considerations to the coupled membrane-bending loading case. In our approach we utilize the spectral representation of a fourth-rank constitutive (stiness) tensor in finding the optimal orientation of its second-rank eigentensors. The case under study is based on three assumptions: the plate is homogeneous with respect to its thickness, the design variables are not restricted by any isoperimetric condition and the Kelvin moduli values are kept fixed on the middle plane of a structure. Optimization task is thus reduced to the equilibrium problem of an eective hyperelastic plate with strictly convex eective nonlinear potential expressed in terms of strains. Corresponding constitutive equations are analytically and explicitly derived which allows determining all components of the optimized stiness tensor.