A geometrically nonlinear variable-kinematics continuum shell element for the analyses of laminated composites

To facilitate further gains in structural efficiency, the use of composite materials in engineering structures is on the rise. Simultaneously, a drive for thinner components is leading to structural behaviour that is governed by elastic nonlinearities such as large deflections and instabilities. For efficient and reliable design, numerical models must predict the nonlinear displacements, as well as the corresponding stress and strain responses, both accurately and at minimal computational cost. In this work, we present a novel tensor-based variable kinematics continuum shell (VKCS) formulation that is geometrically nonlinear in a total Lagrangian sense. The key contribution is the development and validation of a nonlinear continuum shell model that is completely general in terms of its geometric and kinematic descriptions. The governing equations are derived and presented in tensorial form, which enables a straightforward spatial mapping for models with complex curvatures. The `variable-kinematics' capability means that the element field variables can be refined in a hierarchical and orthotropic manner, \ie the in-plane and through-thickness displacements can be independently discretised using any polynomial functions with arbitrary orders of expansion. With this feature, the model configurations can be tailored for specific nonlinear problems, whilst also achieving fast solution convergence rate through the use of higher-order basis functions. For validation, the VKCS model has been benchmarked against existing nonlinear problems in the literature that feature large displacements with complex equilibrium paths. In addition, we have proposed two new benchmarks to investigate the 3D Cauchy stress in a snapped shallow roof, and the postbuckling behaviour of a wind turbine blade section. The VKCS formulation is shown to be a versatile tool that allows the user to easily switch between a multitude of model configurations, and can thus accommodate the varying fidelity of analyses required across different design stages. Furthermore, our benchmarks have demonstrated that the variable-kinematics model requires fewer degrees of freedom and run time to track complex 3D stresses when compared to conventional low-order continuum elements.