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, i.e. 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.

为了进一步提高结构效率，复合材料在工程结构中的使用正在增加。同时，对更薄组件的推动导致结构行为受弹性非线性影响，例如大挠度和不稳定性。为实现高效可靠的设计，数值模型必须准确且以最小的计算成本预测非线性位移以及相应的应力和应变响应。在这项工作中，我们提出了一种新的基于张量的可变运动学连续壳 (VKCS) 公式，该公式在总拉格朗日意义上是几何非线性的。主要贡献是开发和验证非线性连续壳模型，该模型在几何和运动学描述方面完全通用。控制方程以张量形式导出和呈现，这使得具有复杂曲率的模型能够直接进行空间映射。“变量运动学”功能意味着可以以分层和正交各向异性的方式细化元素场变量，IE 可以使用具有任意扩展阶数的任何多项式函数独立地离散平面内和厚度方向的位移。有了这个特性，模型配置可以针对特定的非线性问题进行定制，同时还可以通过使用高阶基函数实现快速的求解收敛速度。为了验证，VKCS 模型已针对文献中存在的非线性问题进行了基准测试，这些问题具有复杂平衡路径的大位移。此外，我们提出了两个新基准来研究折断的浅屋顶中的 3D 柯西应力，以及风力涡轮机叶片截面的后屈曲行为。VKCS 公式被证明是一种多功能工具，允许用户在多种模型配置之间轻松切换，并且因此可以适应不同设计阶段所需的不同分析保真度。此外，我们的基准测试表明，与传统的低阶连续单元相比，可变运动学模型需要更少的自由度和运行时间来跟踪复杂的 3D 应力。