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首页> 外文期刊>SIAM Journal on Mathematical Analysis >Two-scale homogenization for evolutionary variational inequalities via the energetic formulation
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Two-scale homogenization for evolutionary variational inequalities via the energetic formulation

机译：通过能量公式对演化变分不等式进行两尺度均质化

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摘要

This paper is devoted to two-scale homogenization for a class of rate-independent systems described by the energetic formulation or equivalently by an evolutionary variational inequality. In particular, we treat the classical model of linearized elastoplasticity with hardening. The associated nonlinear partial differential inclusion has periodically oscillating coefficients, and the aim is to find a limit problem for the case in which the period tends to 0. Our approach is based on the notion of energetic solutions, which is phrased in terms of a stability condition and an energy balance of an energy-storage functional and a dissipation functional. Using the recently developed method of weak and strong two-scale convergence via periodic unfolding, we show that these two functionals have a suitable two-scale limit, but now involving the macroscopic variable in the physical domain as well as the microscopic variable in the periodicity cell. Moreover, relying on an abstract theory of Gamma-convergence for the energetic formulation using so-called joint recovery sequences, it is possible to show that the solutions of the problem with periodicity converge to the energetic solution associated with the limit functionals.
机译：本文致力于通过能量公式或等效地由演化变分不等式描述的一类速率无关系统的两尺度均质化。特别是，我们使用硬化处理线性弹性弹塑性的经典模型。相关的非线性偏微分包含具有周期性的振荡系数，目的是在周期趋于0的情况下找到一个极限问题。我们的方法基于能量解的概念，即用稳定性来表达储能功能和耗散功能的状态和能量平衡。使用最近开发的通过周期性展开的弱和强两尺度收敛方法，我们显示这两个函数具有合适的两尺度极限，但现在涉及物理域中的宏观变量以及周期性中的微观变量细胞。此外，对于使用所谓的联合恢复序列的高能公式，依赖于伽玛收敛的抽象理论，有可能表明具有周期性的问题的解收敛到与极限功能相关的高能解。

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