There are many materials that the linear elasticity theory cannot predict their mechanical behavior against applied loads. This research proposes a comprehensive theoretical method to obtain the mechanical response of hyperelastic models (with nonlinear elastic deformations) such as polymers and rubbers. They are vital in the design phase of complicated engineering structures like engine mounts and structural bearings in aerospace and automotive industries. The presented theory is implemented in detail and has no limitations in analyzing geometrically and physically nonlinear materials. As a test case, the governing equations of a sheet made of nonlinear elastic material are derived within the framework of this new approach. The derived governing equations are completely nonlinear in all major directions. Consequently, results can be one of the most accurate mathematical simulation results for nonlinear elastic material structures. Then, the obtained equations are solved using a meshless solution method named the semi-analytical polynomial method. The stress and deformation results of the structure under uniform and non-uniform transverse external loadings are obtained for different types of boundary conditions, loading, and material properties. Consequently, this research can be widely used as a suitable essential reference for researchers studying the mechanical behavior of nonlinear elastic structures.

有许多材料的线弹性理论无法预测它们对施加载荷的机械行为。本研究提出了一种综合理论方法来获得超弹性模型（具有非线性弹性变形）如聚合物和橡胶的机械响应。它们在航空航天和汽车行业的发动机支架和结构轴承等复杂工程结构的设计阶段至关重要。所提出的理论得到了详细的实施，并且在分析几何和物理非线性材料方面没有限制。作为一个测试案例，由非线性弹性材料制成的板的控制方程是在这种新方法的框架内推导出来的。导出的控制方程在所有主要方向上都是完全非线性的。最后，结果可以是非线性弹性材料结构最准确的数学模拟结果之一。然后，使用称为半解析多项式方法的无网格求解方法对获得的方程进行求解。针对不同类型的边界条件、载荷和材料特性，获得了均匀和非均匀横向外载荷下结构的应力和变形结果。因此，该研究可广泛用作研究非线性弹性结构力学行为的研究人员的重要参考。针对不同类型的边界条件、载荷和材料特性，获得了均匀和非均匀横向外载荷下结构的应力和变形结果。因此，该研究可广泛用作研究非线性弹性结构力学行为的研究人员的重要参考。针对不同类型的边界条件、载荷和材料特性，获得了均匀和非均匀横向外载荷下结构的应力和变形结果。因此，该研究可广泛用作研究非线性弹性结构力学行为的研究人员的重要参考。