Regularizations and relaxations of time-continiuous problems in plasticity

可塑性时间连续问题的正则化和松弛

• 批准号: 35737369
• 负责人: Professor Dr. Alexander Mielke
• 金额: --
• 依托单位: Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)
• 依托单位国家: 德国
• 项目类别: Research Units
• 财政年份: 2007
• 资助国家: 德国
• 起止时间: 2006-12-31 至 2014-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/35737369?language=en
• 关键词: Regularizations; relaxations; time; continiuous; problems

During the last decade, the theory of finite-strain elastoplasticity has been developed quite rapidly. The major impulses were the discovery that time-incremental problems can be formulated as minimization problems and the recent developments in the field of microstructures in infimizing sequences of functionals. While most of the mathematical theory treats the formation of microstructure via global minimization in static problems, it is desirable to derive models for the evolution of microstructure under slowly varying loads. The theory for laminate evolution in discrete time steps will be extended to the continuous time-level.This project is devoted to the study of temporal evolution models for plasticity and for systems with microstructures in general. Using spatial regularization via higher gradients and temporal regularization via viscosity we derive mathematical models that allow for an existence theory for solutions without microstructure. The temporal regularization will lead to viscous, time-continuous solutions, which converge to so-called BV solutions in the vanishing-viscosity limit. While this limiting procedure is understood in finite- dimensional systems and simple semilinear partial differential equations, the application to finite-strain plasticity is topic of the present research.The relaxation of a sequence of incremental minimization problems will be attacked using weighted energy-dissipation functionals and a development by !-convergence. Finally, it is planned to derive suitable scalings for discrete dislocation densities and pinning sites, such that the dynamic problem leads to a !-limit describing macroscopic dislocation models in the line-tension limit.

近十年来，有限应变弹塑性理论得到了较快的发展。主要的动机是发现时间增量问题可以表示为最小化问题，以及泛函加密化序列中微结构领域的最新发展。虽然大多数数学理论都是通过静力问题中的全局极小化来处理微结构的形成，但需要推导出在缓慢变化的载荷下微结构的演化模型。层合板在离散时间步长下的演化理论将扩展到连续时间水平。本项目致力于塑性和微结构系统的时间演化模型的研究。利用高梯度的空间正则化和粘性的时间正则化，我们得到了不存在微观结构解的存在理论的数学模型。时间正则化将导致粘性的、时间连续的解，这些解在消失粘性极限下收敛到所谓的BV解。虽然这个极限过程在有限维系统和简单的半线性偏微分方程组中是被理解的，但是它在有限应变塑性中的应用是本研究的主题。最后，计划为离散位错密度和钉扎位导出合适的标度，使得动力学问题导致在线-张力极限中描述宏观位错模型。