On the Rate-Independent Limit of Systems with Dry Friction and Small Viscosity
On the Rate-Independent Limit of Systems with Dry Friction and Small Viscosity
复制标题
干摩擦小粘度体系的速率无关极限
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
A. Mielke
中科院分区：
文献类型：
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作者：
M. Efendiev;A. Mielke
具有非凸能量的速率无关系统产生具有跳跃的解。为了求解跳跃路径，我们考虑了两种不同的正则化方法，即(I)小粘度和(Ii)时间离散化背景下的局部最小化。在通过弧长参数化重标解之后，我们得到了一个新的极限问题，它同样是率无关的。我们建立了粘性正则解和时间离散解的收敛结果。通常，极限函数不再通过弧长进行参数化；然而，另一种重新参数化会导致解的出现。利用Young-measure引理，我们证明了如果干摩擦势和粘性势满足结构相容条件，则后一种重新参数化是不必要的。
Rate-independent systems with nonconvex energies generate solutions with jumps. To resolve the full jump path we consider two difierent regularizations, namely (i) small viscosity and (ii) local minimization in the time discretized setting. After rescaling the solutions via arc-length parametrization we obtain a new limit problem, which is again rate independent. We establish convergence results for the viscously regularized solutions as well as for the time-discretized solutions. In general the limit function is no longer parametrized by arc length; however, another reparametrization leads to a solution. Using a Young-measure argument, we show that the latter reparamterization is not necessary if the dry-friction potential and the viscous potential satisfy a structural compatibility condition.