We consider an initial and boundary value problem $ {{\mathcal{P}}} $ which describes the frictionless contact of a viscoplastic body with an obstacle made of a rigid body covered by a layer of elastic material. The process is quasistatic and the time of interest is $ \mathbb{R}_+ = [0,+\infty) $. We list the assumptions on the data and derive a variational formulation ${{\mathcal{P}}}_V $ of the problem, in a form of a system coupling an implicit differential equation with a time-dependent variational-hemivariational inequality, which has a unique solution. We introduce the concept of Tykhonov triple $ {{\mathcal{T}}} = (I,\Omega, {{\mathcal{C}}}) $ where $ I $ is set of parameters, $ \Omega $ represents a family of approximating sets and ${{\mathcal{C} }} $ is a set of sequences, then we define the well-posedness of Problem ${{\mathcal{P}}}_V $ with respect to $ {{\mathcal{T}}}$. Our main result is Theorem 3.4, which provides sufficient conditions guaranteeing the well-posedness of $ {{\mathcal{P} }}_V $ with respect to a specific Tykhonov triple. We use this theorem in order to provide the continuous dependence of the solution with respect to the data. Finally, we state and prove additional convergence results which show that the weak solution to problem $ {{\mathcal{P}}} $ can be approached by the weak solutions of different contact problems. Moreover, we provide the mechanical interpretation of these convergence results.

中文翻译：

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Tykhonov粘塑性接触问题的适定性^†

我们考虑一个初始值和边界值问题$ {{\ mathcal {P}}} $，它描述了粘塑性物体与障碍物的无摩擦接触，该障碍物由一层弹性材料覆盖的刚体制成。该过程是准静态的，感兴趣的时间是$ \ mathbb {R} _ + = [0，+ \ infty）$。我们列出数据上的假设，并以系统的形式导出问题的变分公式$ {{\ mathcal {P}}} _ V $，形式是隐式微分方程与时变变半偏不等式耦合，有一个独特的解决方案。我们介绍Tykhonov三元的概念$ {{\ mathcal {T}}} =（I，\ Omega，{{\ mathcal {C}}}}）$其中$ I $是参数集，$ \ Omega $代表一个一组近似集，而$ {{\ mathcal {C}}} $是一组序列，然后我们定义问题$ {{\ mathcal {P}}} _ V $相对于$ {{\ mathcal {T}}} $$的适定性。我们的主要结果是定理3.4，它提供了充分的条件来保证$ {{\ mathcal {P}}} _ V $相对于特定的Tykhonov三元组的适定性。我们使用该定理来提供解决方案相对于数据的连续依赖性。最后，我们陈述并证明了其他收敛性结果，这些结果表明问题$ {{\ mathcal {P}}} $的弱解可以通过不同接触问题的弱解来解决。此外，我们提供了这些收敛结果的机械解释。这就提供了足够的条件来保证$ {{{\ mathcal {P}}} _ V $相对于特定的Tykhonov三元组的适定性。我们使用该定理来提供解决方案相对于数据的连续依赖性。最后，我们陈述并证明了其他收敛性结果，这些结果表明问题$ {{\ mathcal {P}}} $的弱解可以通过不同接触问题的弱解来解决。此外，我们提供了这些收敛结果的机械解释。这就提供了足够的条件来保证$ {{{\ mathcal {P}}} _ V $相对于特定的Tykhonov三元组的适定性。我们使用该定理来提供解对数据的连续依赖性。最后，我们陈述并证明了其他收敛性结果，这些结果表明问题$ {{\ mathcal {P}}} $的弱解可以通过不同接触问题的弱解来解决。此外，我们提供了这些收敛结果的机械解释。