A kinematically consistent second-order computational homogenisation framework for thick shell models

This paper presents a kinematically consistent second-order computational homogenisation scheme for shear deformable shells. The proposed framework can accurately evaluate the membrane, bending, and transverse components of shell resultants tangent operators, whilst showing no size dependency on fine scale model. To date, proper extension to thick model, such as 5-parameter formulation, remains non-trivial due difficulties in projecting macroscopic strains satisfying stress boundary conditions top bottom faces. overcome this, proposes novel volumetric constraint fluctuation moment field, that is used conjunction with set constraints obtained through an orthogonality condition. result downscaling procedure properly downscale all strains. More notably, pure deformation be achieved, thus producing uniform parabolic distribution homogeneous materials. key equations upscaling are modified Hill–Mandel In particular, operators derived closed form function Taylor upper bound softening terms coming from matrices. general incorporate full geometric material nonlinearities across length scales interest. Through series benchmarks, it demonstrated constitutive tangents computed present correspond well analytical solutions. Finally, excellent agreements model responses, including through-thickness distributions found between multi-scale (FE2) full-scale models numerical featuring nonlinear loading thin heterogeneous panels.