Abstract We study two quasistatic contact problems which describe the frictionless contact between a body and deformable foundation on an infinite time interval. The contact is modelled by the normal compliance condition with limited penetration and memory. The first problem deals with evolution of a body made of a viscoplastic material and in the second problem the material is viscoelastic with long memory. The constitutive functions of these materials have a non-polynomial growth. For each problem we derive a variational formulation that has the form of an almost history-dependent variational inequality for the displacement field. We demonstrate existence and uniqueness results of abstract almost history-dependent inclusion and variational inequality in the reflexive Orlicz–Sobolev space. Finally, we apply the abstract results to prove existence of the unique weak solution to the contact problems.

中文翻译：

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用于无摩擦接触问题的具有几乎历史相关算子的变分不等式

摘要 我们研究了两个准静态接触问题，它们描述了物体与可变形基础在无限时间间隔上的无摩擦接触。接触由具有有限穿透力和记忆力的正常柔顺条件建模。第一个问题涉及由粘塑性材料制成的物体的演化，第二个问题涉及具有长记忆的粘弹性材料。这些材料的本构函数具有非多项式增长。对于每个问题，我们推导出一个变分公式，它具有位移场的几乎与历史相关的变分不等式的形式。我们证明了自反 Orlicz-Sobolev 空间中抽象的几乎依赖于历史的包含和变分不等式的存在性和唯一性结果。最后，