Modeling of consistent second‐order plate theories for anisotropic materials

Abstract

AbstractBy Fourier‐series expansion in thickness direction of the plate with respect to a basis of scaled Legendre polynomials, several equivalent (and therefore exact) two‐dimensional formulations of the three‐dimensional boundary‐value problem of linear elasticity in weak formulation for a plate with constant thickness are derived. These formulations are sets of countably many PDEs, which are power series in the squared plate parameter. For the special case of a homogeneous monoclinic material, we obtain an approximative plate theory in finitely many PDEs and unknown variables by the truncation approach of the uniform‐approximation technique. The PDE system is reduced to a scalar PDE expressed in the mid‐plane displacement. The resulting second‐order theory, considered as a first‐order theory, is equivalent to the classical Kirchhoff theory for the special case of an isotropic material and equivalent to Huber's classical theory for an actual monoclinic material. However it remains shear‐rigid as a second‐order theory. Therefore, it is modified by an a‐priori assumption to a theory for monoclinic materials, that presumes the former equivalences, considered as a first‐order theory, but is in addition equivalent to Kienzler's theory as a second‐order theory for the special case of isotropy, which implies further equivalences to established shear‐deformable theories, especially the Reissner‐Mindlin theory and Zhilin's plate theory. The presented new second‐order plate theory for monoclinic materials is finally a system of two coupled PDEs of differentiation order six in two variables.