In this work, we develop a theoretical formulation for describing dissipative material behaviors in a stochastic setting, using the framework of Generalized Standard Materials (GSM). Our goal is to capture the variability inherent in the material model while ensuring thermodynamic consistency, by employing the mathematical framework of stochastic programming. We first show how average behaviors can be computed using the expected value of the free energy and dissipation pseudo-potentials. We then introduce the concept of a risk-averse effective measure, which provides both an optimistic and a pessimistic estimate of the uncertain material behavior. To this end, we utilize the Conditional Value-at-Risk, a widely used risk measure in mathematical finance. We also demonstrate how these concepts can be extended to variational problems at the structure scale, allowing us to compute the effective response of a structure composed of a stochastic material.

中文翻译：

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随机广义标准材料和规避风险的有效行为


在这项工作中，我们使用广义标准材料 （GSM） 的框架开发了一种理论公式，用于描述随机环境中的耗散材料行为。我们的目标是通过采用随机规划的数学框架来捕获材料模型中固有的可变性，同时确保热力学一致性。我们首先展示了如何使用自由能和耗散赝势的期望值来计算平均行为。然后，我们引入了规避风险的有效措施的概念，它对不确定的材料行为提供了乐观和悲观的估计。为此，我们利用了条件风险值，这是数学金融中广泛使用的风险衡量标准。我们还演示了如何将这些概念扩展到结构尺度的变分问题，从而使我们能够计算由随机材料组成的结构的有效响应。