A material network consisting of discrete material nodes and their interactions can represent complex microstructure responses. Under this interaction viewpoint, the material network can be viewed as a trainable system involving fitting parameters including not only the weights of the material nodes but also the parameters characterizing their interactions. As opposed to the other existing works, this interaction-based material network does not rely on the micromechanics of multiple-phase laminates but on constraining all requirements of a truly microscopic boundary value problem including the stress and strain averaging principles and the Hill–Mandel energetically consistent condition. Consequently, the proposed framework can be applied to microstructures with the presence of voids, which is not achievable with the laminate theory. To make a material network become a surrogate of a full-field microscopic model, this work proposes two different training procedures to calibrate its fitting parameters. On the one hand, a nonlinear training procedure is proposed considering sequential data collected from finite element simulations on the full-field model subjected to proportional loading paths. On the other hand, a linear elastic training procedure considers only the elastic response of the heterogeneous material. The accuracy and efficiency of the proposed framework for microstructures with the presence of voids are demonstrated by comparing the predictions of the trained material networks with the ones of the direct numerical simulations in both contexts of virtual testing and multiscale simulations. It is also shown that the linear elastic training procedure requires a lower computational cost but could lead to less accurate predictions in comparison with the nonlinear training procedure.

由离散材料节点及其相互作用组成的材料网络可以表示复杂的微观结构响应。在这种交互观点下，材料网络可以被视为一个可训练的系统，它涉及拟合参数，不仅包括材料节点的权重，还包括表征它们相互作用的参数。与其他现有工作相反，这种基于相互作用的材料网络不依赖于多相层压板的微观力学，而是严格约束真正微观边界值问题的所有要求，包括应力和应变平均原理以及 Hill-Mandel一致的条件。因此，提议的框架可以应用于存在空隙的微观结构，这是层压理论无法实现的。为了使材料网络成为全场微观模型的替代品，这项工作提出了两种不同的训练程序来校准其拟合参数。一方面，考虑从受比例加载路径影响的全场模型的有限元模拟中收集的连续数据，提出了一种非线性训练程序。另一方面，线性弹性训练程序只考虑弹性响应的异质材料。通过在虚拟测试和多尺度模拟的背景下将训练过的材料网络的预测与直接数值模拟的预测进行比较，证明了所提出的存在空隙的微结构框架的准确性和效率。还表明，线弹性训练过程需要较低的计算成本，但与非线性训练过程相比，可能导致预测精度较低。