This work presents a novel formulation and numerical strategy for the simulation of geometrically nonlinear structures. First, a non-canonical Hamiltonian (Poisson) formulation is introduced by including the dynamics of the stress tensor. This framework is developed for von-K\'arm\'an nonlinearities in beams and plates, as well as finite strain elasticity with Saint-Venant material behavior. In the case of plates, both negligible and non-negligible membrane inertia are considered. For the former case the two-dimensional elasticity complex is leveraged to express the dynamics in terms of the Airy stress function. The finite element discretization employs a mixed approach, combining a conforming approximation for displacement and velocity fields with a discontinuous stress tensor representation. A staggered, linear implicit time integration scheme is proposed, establishing connections with existing explicit-implicit energy-preserving methods. The stress degrees of freedom are statically condensed, reducing the computational complexity to solving a system with a positive definite matrix. The methodology is validated through numerical experiments on the Duffing oscillator, a von-K\'arm\'an beam, and a column undergoing finite strain elasticity. Comparisons with fully implicit energy-preserving method and the explicit Newmark scheme demonstrate that the proposed approach achieves superior accuracy while maintaining energy stability. Additionally, it enables larger time steps compared to explicit schemes and exhibits computational efficiency comparable to the leapfrog method.

翻译：本文提出了一种用于模拟几何非线性结构的新颖公式与数值策略。首先，通过引入应力张量的动力学描述，建立了一种非正则哈密顿（泊松）公式体系。该框架针对梁和板的冯·卡门非线性问题以及具有圣维南材料行为的有限应变弹性力学展开。在板的分析中，同时考虑了可忽略与不可忽略的膜惯性效应。对于前者，利用二维弹性复形理论通过艾里应力函数表达动力学方程。有限元离散采用混合方法，将位移场和速度场的协调近似与不连续的应力张量表示相结合。提出了一种交错式线性隐式时间积分格式，建立了与现有显隐式能量守恒方法的理论联系。通过静态凝聚消除应力自由度，将计算复杂度简化为求解具有正定矩阵的线性系统。通过对杜芬振子、冯·卡门梁和经历有限应变弹性变形的柱体进行数值实验验证了该方法。与全隐式能量守恒方法和显式纽马克格式的对比表明，所提方法在保持能量稳定性的同时获得了更优的精度。此外，相较于显式格式允许采用更大时间步长，其计算效率与蛙跳方法相当。