Most existing semi-analytical models to analyse multistable structures are based on the principle of minimum potential energy, which inherently does not satisfy the in-plane equilibrium equations in a strong sense. Although such models predict the multistable shapes with reasonable accuracy, there is a significant discrepancy in predicting the snap-through behaviour when compared to the finite element (FE) or experimental results. In this work, a refined analytical model is derived to analyse a bistable cross-ply elliptical disc using Fppl von Krmn kinematics. The Rayleigh–Ritz approach with a decoupled energy formulation is used, where the stretching and bending energies are separated using the semi-inverse constitutive equation. The in-plane stress resultants are expressed in terms of curvatures using the compatibility and the in-plane equilibrium equations. A closed-form solution for the curvatures is derived to predict the cured bistable shapes of elliptical laminates. A layer of macro fibre composite actuator is added to trigger the snap-through between the stable shapes. Consequently, the critical voltage at which the snap-through occurs is predicted using the analytical model. The model is further extended to square plates, where the solutions of the differential equations emanating from the compatibility and in-plane equilibrium equations are obtained by converting the resulting differential equations into the form of standard FE elasticity problem. The in-plane stress resultants for the square plates are calculated by numerically solving the resulting elasticity problem using a standard FE discretization approach. The solutions are further compared with previous experimental studies and the results from a fully geometrically non-linear FE model.

大多数现有的分析多稳态结构的半解析模型都基于最小势能原理，本质上并不能很好地满足面内平衡方程。尽管此类模型以合理的精度预测多稳态形状，但与有限元 (FE) 或实验结果相比，在预测快速通过行为方面存在显着差异。在这项工作中，使用 Fppl von Krmn 运动学推导出了一个精细的分析模型来分析双稳态交叉层椭圆盘。使用具有解耦能量公式的 Rayleigh-Ritz 方法，其中使用半逆本构方程分离拉伸和弯曲能量。面内应力合力使用兼容性和面内平衡方程以曲率表示。导出了曲率的封闭形式解，以预测椭圆层压板的固化双稳态形状。添加了一层宏纤维复合致动器以触发稳定形状之间的快速通过。因此，使用分析模型预测发生快速通过的临界电压。该模型进一步扩展到方板，其中通过将所得微分方程转换为标准有限元弹性问题的形式，获得了源自兼容性方程和面内平衡方程的微分方程的解。方形板的面内应力合力是通过使用标准有限元离散化方法对所得弹性问题进行数值求解来计算的。将这些解决方案与之前的实验研究和完全几何非线性 FE 模型的结果进行了进一步比较。