A Simple FEM Formulation Applied to Nonlinear Problems of Impact with Thermomechanical Coupling

Abstract The thermal effects of problems involving deformable structures are essential to describe the behavior of materials in feasible terms. Verifying the transformation of mechanical energy into heat it is possible to predict the modifications of mechanical properties of materials due to its temperature changes. The current paper presents the numerical development of a finite element method suitable for nonlinear structures coupled with thermomechanical behavior; including impact problems. A simple and effective alternative formulation is presented, called FEM positional, to deal with the dynamic nonlinear systems. The developed numerical is based on the minimum potential energy written in terms of nodal positions instead of displacements. The effects of geometrical, material and thermal nonlinearities are considered. The thermodynamically consistent formulation is based on the laws of thermodynamics and the Helmholtz free-energy, used to describe the thermoelastic and the thermoplastic behaviors. The coupled thermomechanical model can result in secondary effects that cause redistributions of internal efforts, depending on the history of deformation and material properties. The numerical results of the proposed formulation are compared with examples found in the literature.

摘要：涉及可变形结构的热效应问题，是从工程可行视角精准描述材料力学行为的关键基础。通过验证机械能向热能的转化过程，可预测材料因温度变化所产生的力学性能演变。本文提出了一种适用于耦合热机械行为的非线性结构的有限元法（Finite Element Method, FEM）数值建模方法，研究范畴涵盖冲击问题。同时，本文提出了一种简洁高效的替代建模方案，命名为位置型有限元法（FEM positional），用于求解动态非线性系统。所开发的数值模型基于以节点位置而非位移为变量的最小势能原理。该模型充分考虑了几何非线性、材料非线性与热非线性三类非线性效应。符合热力学一致性的建模框架以热力学定律与亥姆霍兹自由能（Helmholtz free-energy）为理论基础，用于描述热弹性与热塑性行为。耦合热机械模型可能引发二次效应，该效应会依据变形历史与材料属性的差异，导致系统内部力发生重分布。本文将所提建模方法的数值结果与现有文献中的经典算例进行了对比验证。