Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 588-638, March 2024.
Abstract. We study the large-scale behavior of a small-strain lattice model for a network composed of elastoplastic springs with random material properties. We formulate the model as an evolutionary rate independent system. In an earlier work we derived a homogenized continuum model, which has the form of linearized elastoplasticity, as an evolutionary [math]-limit as the lattice parameter tends to zero. In the present paper we introduce a periodic representative volume element (RVE) approximation for the homogenized system. As a main result we prove convergence of the RVE approximation as the size of the RVE tends to infinity. We also show that the hysteretic stress-strain relation of the effective system can be described with the help of a generalized Prandtl–Ishlinskii operator, and we prove convergence of a periodic RVE approximation for that operator. We combine the RVE approximation with a numerical scheme for rate-independent systems and obtain a computational scheme that we use to numerically investigate the homogenized system in the specific case when the original network is given by a two-dimensional lattice model. We simulate the response of the system to cyclic and uniaxial, monotonic loading, and numerically investigate the convergence rate of the periodic RVE approximation. In particular, our simulations show that the RVE error decays with the same rate as the RVE error in the static case of linear elasticity.

中文翻译：

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弹塑性弹簧网络中的代表性体积元近似

多尺度建模与仿真，第 22 卷，第 1 期，第 588-638 页，2024 年 3 月。
摘要。我们研究由具有随机材料属性的弹塑性弹簧组成的网络的小应变晶格模型的大规模行为。我们将该模型表述为一个与进化率无关的系统。在早期的工作中，我们推导出了一个均质连续体模型，该模型具有线性弹塑性的形式，作为晶格参数趋于零时的演化[数学]极限。在本文中，我们介绍了均质系统的周期代表体积元（RVE）近似。作为主要结果，我们证明了当 RVE 的大小趋于无穷大时，RVE 近似的收敛性。我们还表明，有效系统的滞后应力-应变关系可以借助广义 Prandtl-Ishlinskii 算子来描述，并且我们证明了该算子的周期性 RVE 近似的收敛性。我们将 RVE 近似与速率无关系统的数值方案相结合，并获得了一种计算方案，当原始网络由二维晶格模型给出时，我们用该方案对特定情况下的均质系统进行数值研究。我们模拟系统对循环和单轴、单调载荷的响应，并数值研究周期性 RVE 近似的收敛速度。特别是，我们的模拟表明，RVE 误差以与线弹性静态情况下的 RVE 误差相同的速率衰减。