We consider the dynamics of an elastic continuum under large deformation but small strain. Such systems can be described by the equations of geometrically nonlinear elastodynamics in combination with the St. Venant-Kirchhoff material law. The velocity-stress formulation of the problem turns out to have a formal port-Hamiltonian structure. In contrast to the linear case, the operators of the problem are modulated by the displacement field which can be handled as a passive variable and integrated along with the velocities. A weak formulation of the problem is derived and essential boundary conditions are incorporated via Lagrange multipliers. This variational formulation explicitly encodes the transfer between kinetic and potential energy in the interior as well as across the boundary, thus leading to a global power balance and ensuring passivity of the system. The particular geometric structure of the weak formulation can be preserved under Galerkin approximation via appropriate mixed finite elements. In addition, a fully discrete power balance can be obtained by appropriate time discretization. The main properties of the system and its discretization are shown theoretically and demonstrated by numerical tests.

翻译：我们考虑弹性体在大变形但小应变条件下的动力学问题。这类系统可通过几何非线性弹性动力学方程结合圣维南-基尔霍夫材料定律进行描述。问题的速度-应力公式展现出形式上的端口-哈密顿结构。与线性情况不同，该问题的算子受到位移场的调制，其中位移场可作为被动变量处理，并与速度一同积分。推导出问题的弱形式，并通过拉格朗日乘子引入本质边界条件。该变分形式明确编码了内部及边界处的动能与势能传递，从而建立全局功率平衡并确保系统的无源性。通过适当的混合有限元进行伽辽金近似，可保留弱形式的特定几何结构。此外，通过合适的时间离散化可获得完全离散的功率平衡。本文从理论上证明了系统及其离散化的主要特性，并通过数值实验进行了验证。