Buckling analysis of a composite shell with multiple delaminations based on a higher order zig-zag theory

A new three-node triangular shell element based on a higher order zig-zag theory is developed for laminated composite shells with multiple delaminations. The baseline higher order zig-zag composite shell theory for multiple delaminations has been developed in a general curvilinear coordinate system and in general tensor notation. All the complicated curvatures of the surface including twisting curvatures can be described in an exact manner since the developed shell finite element is based on the geometrically exact surface representation. It is therefore possible to apply the present finite element formulation to the composite structures with arbitrary shaped multiple delaminations, general layup configurations and arbitrary boundary conditions. The slopes of deflections are included as independent degrees of freedom, which require the normal slope continuities between element interfaces. The nonconforming shape functions of Specht's three-node triangular plate bending element are employed to interpolate out-of-plane displacements. The developed element passes the bending and twisting patch tests for flat surface configurations. The element is then used to analyze the linear buckling problem of composite cylindrical shells with multiple delaminations. Throughout the numerical examples, it is demonstrated that the developed shell finite element is numerically efficient because it has minimal nodal degrees of freedom to describe the composite shells having multiple delaminations. The finite element realization of a higher order zig-zag shell theory with multiple delaminations can be served as a powerful numerical tool to analyze the complicated shell structures for the critical buckling loads as well as the corresponding mode shapes.