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 shear components of the shell resultants and tangent operators, whilst showing no size dependency on the fine scale model. To date, a proper extension of second-order homogenisation to a thick shell model, such as the 5-parameter formulation, remains non-trivial due to the difficulties in projecting the macroscopic transverse shear strains to the fine scale whilst satisfying the stress boundary conditions on the top and bottom faces. To overcome this, the paper proposes a novel volumetric constraint on the fluctuation moment field, that is used in conjunction with a set of constraints obtained through an orthogonality condition. The result is a consistent downscaling procedure that can properly downscale all the macroscopic shell strains. More notably, a pure transverse shear deformation can be achieved, thus producing a uniform parabolic shear stress distribution in homogeneous materials. The key equations for upscaling are obtained through the modified Hill–Mandel condition. In particular, the shell tangent operators are derived in closed form as a function of the Taylor upper bound and softening terms coming from the fluctuation matrices. The proposed framework is general and can incorporate full geometric and material nonlinearities across all the length scales of interest. Through a series of benchmarks, it is demonstrated that the constitutive tangents computed through the present framework correspond well with analytical solutions. Finally, excellent agreements in model responses, including through-thickness stress distributions are found between the multi-scale (${\text{FE}}^2$) and the full-scale models in the numerical benchmarks, featuring the nonlinear loading of thin and thick heterogeneous panels.

本文提出了一种用于剪切变形壳的运动学一致的二阶计算均匀化方案。所提出的框架可以准确地评估壳合成和切线算子的膜、弯曲和横向剪切分量，同时对精细比例模型没有尺寸依赖性。迄今为止，由于在满足应力边界条件的同时难以将宏观横向剪切应变投影到精细尺度，因此将二阶均匀化适当扩展到厚壳模型（例如 5 参数公式）仍然很重要在顶面和底面上。为了克服这个问题，本文提出了一种新的波动矩场体积约束，它与通过正交条件获得的一组约束结合使用。结果是一致的缩小程序，可以正确缩小所有宏观壳应变。更值得注意的是，可以实现纯横向剪切变形，从而在均质材料中产生均匀的抛物线剪切应力分布。放大的关键方程是通过修改后的 Hill-Mandel 条件获得的。特别是，壳切线算子以封闭形式导出，作为泰勒上限和来自波动矩阵的软化项的函数。所提出的框架是通用的，可以在所有感兴趣的长度尺度上包含完整的几何和材料非线性。通过一系列基准测试，证明通过本框架计算的本构切线与解析解很好地对应。最后，${\text{FE}}^2$）和数值基准中的全尺寸模型，具有薄和厚异质面板的非线性加载。