论文标题
在热弹性中产生的准变量接触问题的分析
Analysis of a quasi-variational contact problem arising in thermoelasticity
论文作者
论文摘要
我们制定并研究了涉及膜和模具作为隐式障碍问题的热成型过程的两个数学模型。特别是，膜耦合取决于模具的热位移，而模具的热位移依赖于膜通过接触区域。所考虑的两个模型是固定（或椭圆形）模型和一个进化（或准危）的模型。对于第一个模型，我们通过求解与椭圆方程相连的椭圆形变体不等式来证明弱解的存在。通过探索在非分类数据下设置的触点变化的良好特性，我们为存在常规溶液的存在提供了足够的条件，在某些收缩条件下，也是独特的结果。我们将这些结果应用于一系列的半危险问题，这些问题是进化或准危问题的常规解决方案的近似值。在这里，在某些条件下，我们能够证明存在进化问题和特殊情况，以及时间依赖性解决方案的独特性。
We formulate and study two mathematical models of a thermoforming process involving a membrane and a mould as implicit obstacle problems. In particular, the membrane-mould coupling is determined by the thermal displacement of the mould that depends in turn on the membrane through the contact region. The two models considered are a stationary (or elliptic) model and an evolutionary (or quasistatic) one. For the first model, we prove the existence of weak solutions by solving an elliptic quasi-variational inequality coupled to elliptic equations. By exploring the fine properties of the variation of the contact set under non-degenerate data, we give sufficient conditions for the existence of regular solutions, and under certain contraction conditions, also a uniqueness result. We apply these results to a series of semi-discretised problems that arise as approximations of regular solutions for the evolutionary or quasistatic problem. Here, under certain conditions, we are able to prove existence for the evolutionary problem and for a special case, also the uniqueness of time-dependent solutions.
