Contact phenomena arise in a variety of industrial process and engineering applications. For this reason, contact mechanics has attracted substantial attention from research communities. Mathematical problems from contact mechanics have been studied extensively for over half a century. Effort was initially focused on variational inequality formulations, and in the past ten years considerable effort has been devoted to contact problems in the form of hemivariational inequalities. This article surveys recent development in studies of hemivariational inequalities arising in contact mechanics. We focus on contact problems with elastic and viscoelastic materials, in the framework of linearized strain theory, with a particular emphasis on their numerical analysis. We begin by introducing three representative mathematical models which describe the contact between a deformable body in contact with a foundation, in static, history-dependent and dynamic cases. In weak formulations, the models we consider lead to various forms of hemivariational inequalities in which the unknown is either the displacement or the velocity field. Based on these examples, we introduce and study three abstract hemivariational inequalities for which we present existence and uniqueness results, together with convergence analysis and error estimates for numerical solutions. The results on the abstract hemivariational inequalities are general and can be applied to the study of a variety of problems in contact mechanics; in particular, they are applied to the three representative mathematical models. We present numerical simulation results giving numerical evidence on the theoretically predicted optimal convergence order; we also provide mechanical interpretations of simulation results.

中文翻译：

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接触力学中半变分不等式的数值分析

接触现象出现在各种工业过程和工程应用中。因此，接触力学引起了研究界的广泛关注。接触力学的数学问题已经被广泛研究了半个多世纪。最初的努力集中在变分不等式的公式上，在过去的十年中，相当大的努力已经投入到以半变分不等式的形式出现的接触问题上。本文调查了接触力学中出现的半变分不等式研究的最新进展。我们在线性应变理论的框架内专注于弹性和粘弹性材料的接触问题，特别强调它们的数值分析。我们首先介绍三个具有代表性的数学模型，这些模型描述了在静态、历史相关和动态情况下可变形物体与基础接触的接触。在弱公式中，我们考虑的模型会导致各种形式的半变分不等式，其中未知数是位移场或速度场。基于这些例子，我们介绍和研究了三个抽象的半变分不等式，我们给出了它们的存在性和唯一性结果，以及数值解的收敛性分析和误差估计。抽象半变分不等式的结果具有普遍性，可应用于接触力学中各种问题的研究；特别是，它们被应用于三个具有代表性的数学模型。我们提出了数值模拟结果，为理论上预测的最佳收敛顺序提供了数值证据；我们还提供模拟结果的机械解释。