Novel HW mixed zig-zag theory accounting for transverse normal deformability and lower-order counterparts assessed by old and new elastostatic benchmarks

Mixed zig-zag plate theories are derived from a recently developed 3-D five d.o.f. zig-zag “adaptive” theory ZZA (a priori fulfillment of interfacial stress constraints) under steadily growing limiting assumptions on displacement, strain and stress fields. The intended aim is trying to save computational costs simultaneously preserving accuracy. Lower-order theories assume a uniform or a polynomial transverse displacement, out-of-plane stresses are derived from local equilibrium equations or retaken from higher-order theories. The focus of this study is twofold: (i) to assess whether and for which cases a higher-order through-thickness zig-zag transverse displacement representation is essential, or vice versa a simpler kinematics can be assumed; (ii) to compare accuracy of theories based on Murakami's and Di Sciuva's zig-zag functions with the same expansion order across the thickness. A number of challenging benchmarks are retaken from literature and new benchmarks with a strong variation of material properties (damaged layers), distributed or localized step loading and some different boundary conditions are considered to assess accuracy of theories and of FEA 3-D (constituting the reference solution in lack of exact results). For these benchmarks, closed-form solutions are obtained assuming the same expansion order across the thickness and the same in-plane trial functions for all theories. The numerical illustrations show that lower order and Murakami's based theories fail for cases having the strongest layerwise and transverse anisotropy effects, or a marked through-thickness asymmetry. The Hu-Washizu highest-order theory HWZZ is shown to be always the only as accurate as ZZA, despite it reduces the computational effort.