Mean-field methods are a common procedure for characterizing random heterogeneous materials. However, they typically provide only mean stresses and strains, which do not always allow predictions of failure in the phases since exact localization of these stresses and strains requires exact microscopic knowledge of the microstructures involved, which is generally not available. In this work, the maximum entropy method pioneered by Kreher and Pompe (Internal Stresses in Heterogeneous Solids, Physical Research, vol. 9, 1989) is used for estimating one-point probability distributions of local stresses and strains for various classes of materials without requiring microstructural information beyond the volume fractions. This approach yields analytical formulae for mean values and variances of stresses or strains of general heterogeneous linear thermoelastic materials as well as various special cases of this material class. Of these, the formulae for discrete-phase materials and the formulae for polycrystals in terms of their orientation distribution functions are novel. To illustrate the theory, a parametric study based on Al-Al$_{2}$O$_{3}$ composites is performed. Polycrystalline copper is considered as an additional example. Through comparison with full-field simulations, the method is found to be particularly suited for polycrystals and materials with elastic contrasts of up to 5. We see that, for increasing contrast, the dependence of our estimates on the particular microstructures is increasing, as well.

中文翻译：

---

基于最大熵的热弹性随机异质材料应力和应变估计

平均场方法是表征随机异质材料的常用方法。然而，它们通常仅提供平均应力和应变，这并不总是允许预测相中的失效，因为这些应力和应变的精确定位需要所涉及的微观结构的精确微观知识，而这通常是不可用的。在这项工作中，Kreher 和 Pompe 首创的最大熵方法（异构固体的内部应力，物理研究，第 9 卷，1989 年）用于估计各类材料的局部应力和应变的单点概率分布，而不需要超出体积分数的微观结构信息。这种方法产生了一般异质线性热弹性材料的应力或应变的平均值和方差的分析公式，以及该材料类别的各种特殊情况。其中，离散相材料的公式和多晶的取向分布函数公式是新颖的。为了说明该理论，进行了基于 Al-Al$_{2}$O$_{3}$ 复合材料的参数研究。多晶铜被认为是一个额外的例子。通过与全场模拟的比较，发现该方法特别适用于弹性对比度高达 5 的多晶和材料。我们看到，为了增加对比度，我们的估计对特定微结构的依赖性也在增加.