We present a mathematical model describing the equilibrium of a viscoelastic body with long-term memory in frictional contact with a sliding foundation. The process is quasistatic, and material damage resulting from excessive stress or strain is captured by a damage function. We assume the material is inhomogeneous, leading to multiple contact boundary conditions. The contact interface is partitioned into two segments: One part takes into account the wear of the contact surface, utilizing Archard's law. Here, contact is modeled with a normal compliance condition with unilateral constraints, coupled with a sliding version of Coulomb's law of dry friction. In the other part, contact is modeled with a nonmonotone condition involving normal compliance and a subdifferential frictional boundary condition. Variational formulation of the model is governed by a coupled system consisting of a variational–hemivariational inequality for the displacement field, a parabolic variational inequality for the damage field and an integral equation for the wear function. We study a fully discrete scheme for numerical approximation with an error estimation of the solution to this problem. Optimal error estimates for the linear finite element method are derived, followed by numerical simulations illustrating the behavior of the model.

中文翻译：

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具有损伤、长期记忆和磨损的准静态摩擦双边接触问题的数值分析和模拟


我们提出了一个数学模型，描述了具有长期记忆的粘弹性体在与滑动基础摩擦接触时的平衡。该过程是准静态的，由过度应力或应变引起的材料损坏由损坏函数捕获。我们假设材料是不均匀的，导致多个接触边界条件。接触界面分为两部分：一部分利用阿查德定律考虑接触表面的磨损。在这里，接触是用单边约束的正常柔度条件建模的，并耦合了库仑干摩擦定律的滑动版本。在另一部分，接触使用非单调条件进行建模，涉及法向柔度和次差分摩擦边界条件。该模型的变分公式由一个耦合系统控制，该系统由位移场的变分-半变分不等式、损伤场的抛物线变分不等式和磨损函数的积分方程组成。我们研究了一种用于数值近似的全离散方案，并对这个问题的解进行了误差估计。推导出线性有限元方法的最佳误差估计，然后进行数值仿真以说明模型的行为。