Regularizations and relaxations of time-continiuous problems in plasticity

近十年来，有限应变弹塑性理论得到了迅速的发展。主要的推动力是发现，时间增量问题可以制定为最小化问题和最近的事态发展，在微结构领域的infimizing序列的泛函。虽然大多数的数学理论处理通过全局最小化静态问题的微观结构的形成，它是可取的，以获得模型的微观结构的演变缓慢变化的负载。将离散时间步长的层合板演化理论推广到连续时间尺度，本项目致力于塑性和一般微结构系统的时间演化模型的研究。使用空间正则化通过更高的梯度和时间正则化通过粘度，我们推导出数学模型，允许的存在理论的解决方案没有微观结构。时间正则化将导致粘性，时间连续的解决方案，收敛到所谓的BV解决方案在消失的粘度限制。虽然这种限制过程是在有限维系统和简单的半线性偏微分方程中理解的，但有限应变塑性的应用是本研究的主题。一系列增量最小化问题的松弛将使用加权能量耗散泛函进行攻击，并通过！-收敛最后，计划导出离散位错密度和钉扎位点的合适比例，使得动力学问题导致！-描述线张力极限中宏观位错模型的极限。

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.