Gwinner Joachim
Department of Aerospace Engineering, Universität der Bundeswehr München, Neubiberg 85577, Germany.
Philos Trans A Math Phys Eng Sci. 2022 Nov 14;380(2236):20210356. doi: 10.1098/rsta.2021.0356. Epub 2022 Sep 26.
This note is devoted to a novel frictional unilateral contact problem in finite-strain elasticity. Here, we adopt the Tresca friction model from linear elasticity. Our analysis relies on the polyconvexity approach to nonlinear elasticity due to J. Ball. We include the delicate case where the elastic body neither is fixed nor has a deformation prescribed along some part of its boundary, but rests on a rigid foundation with a free boundary and is only submitted to forces and loads acting in the interior (like gravity) and at the boundary, respectively. This leads to a loss of coercivity and necessitates an extra condition that prevents the body from escaping by the geometry of the obstacle. This new condition extends a similar condition of Ciarlet and Nečas from the frictionless case to the case of Tresca friction. In addition, as a first step towards a numerical treatment of such nonlinear problems, we present a smoothing procedure that tackles the non-smooth term from Tresca friction and provide a convergence result for the novel smoothing method. This article is part of the theme issue 'Non-smooth variational problems and applications'.
本笔记致力于研究有限应变弹性中的一个新型摩擦单边接触问题。在此，我们采用线性弹性中的特雷斯卡摩擦模型。我们的分析依赖于J. 鲍尔提出的非线性弹性的多凸性方法。我们考虑了一种微妙的情况，即弹性体既未被固定，其边界的某些部分也没有规定的变形，而是放置在具有自由边界的刚性基础上，并且仅分别受到作用于内部（如重力）和边界的力与载荷。这导致了强制性的丧失，因此需要一个额外的条件来防止物体因障碍物的几何形状而逃逸。这个新条件将Ciarlet和Nečas在无摩擦情况下的类似条件扩展到了特雷斯卡摩擦的情况。此外，作为对这类非线性问题进行数值处理的第一步，我们提出了一种平滑过程，该过程处理来自特雷斯卡摩擦的非光滑项，并为这种新型平滑方法提供了一个收敛结果。本文是“非光滑变分问题及其应用”主题特刊的一部分。
