A micromechanical scheme with nonlinear concentration functions by physics-guided neural network

The mechanical behavior of heterogeneous materials has been reported to be significantly influenced by the nonlinear properties of both the matrix and interface. However, the micromechanical homogenization methods for predicting the effective properties are challenged in nonlinear problems due to the difficulties in solving the analytical form of the concentration tensors. In this study, we develop a nonlinear micromechanical scheme for heterogeneous materials with complex interfacial behaviors, where the key component, namely nonlinear concentration functions, is determined by the devised physics -guided neural network. In particular, the nonlinear Mori-Tanaka method (NMT) implemented within this new micromechanical scheme yields accurate solutions to axisymmetric nonlinear homogenization problems considering the effects of finite deformation, loading conditions, volume fraction, etc. Furthermore, the NMT is equivalent to the linear Mori-Tanaka method in the condition of the small deformation. Notably, this neural -network -based micromechanical scheme shows good generalization for different types of nonlinear interfaces, while the corresponding approach for generating training data via the finite element method (FEM) is cost-effective. This theoretical framework introduces a novel approach to nonlinear physical modeling, namely, not by the direct regression from the dataset but by deeply embedding neural networks in physical laws.