Investigated in this paper are comprehensive static, dynamic and internal-resonance analyses on a wide range of hyperelastic shell structures, including cylindrical, spherical, doubly-curved, and hyperbolic hyperelastic shallow shells. Donnell's nonlinear shell theory and the Mooney-Rivlin strain energy density model are used to formulate the hyperelastic shell structure. Coupled equations of motion are obtained using Hamilton's principle with highly nonlinear terms due to the curvature in the structure, together with the material nonlinearity and large deformations. The coupled equations of motion are converted to a large set of equations using a two-dimensional Galerkin technique and are solved by employing the Newton-Raphson approach and dynamic equilibrium technique. The strength of the current methodology and model is first verified by comparing the static response of the structure with those obtained by using a finite element software. After verifying the model, a detailed analysis of the bending behaviour of the structure under a time-independent pressure is presented. Moreover, the free and forced vibration responses of the shell structure are presented for different cases, showing that the curvature terms play a significant role in changing the mechanical response of the hyperelastic shell. Furthermore, it is shown that for specific sets of curvatures, internal resonances are present which leads to a complicated, rich nonlinear responses. The nonlinear forced vibration response of the hyperelastic shell is also presented for different shell types and resonances.

本文研究了对各种超弹性壳结构（包括圆柱形、球形、双曲形和双曲线超弹性浅壳）的综合静态、动态和内部共振分析。Donnell 的非线性壳理论和 Mooney-Rivlin 应变能密度模型用于制定超弹性壳结构。由于结构中的曲率，以及材料的非线性和大变形，使用具有高度非线性项的哈密顿原理获得耦合运动方程。使用二维 Galerkin 技术将耦合的运动方程转换为大量方程，并使用 Newton-Raphson 方法和动态平衡技术求解。当前方法和模型的强度首先通过将结构的静态响应与使用有限元软件获得的静态响应进行比较来验证。在验证模型后，对结构在与时间无关的压力下的弯曲行为进行了详细分析。此外，针对不同情况给出了壳结构的自由振动响应和受迫振动响应，表明曲率项在改变超弹性壳的机械响应方面起着重要作用。此外，研究表明，对于特定的曲率集，存在内部共振，这会导致复杂、丰富的非线性响应。还针对不同的壳类型和共振给出了超弹性壳的非线性受迫振动响应。