Unified Micromechanics Theory of Composites

TL;DR

The paper introduces AGIE-CAM, a universal, modular micromechanics framework that unifies local and nonlocal, linear and nonlinear behavior across random, periodic, and deterministic composite materials. By employing a body force with compact support (BFCS) loading and a novel RVE concept, it derives Additive General Integral Equations (AGIE) to solve for effective and surrogate parameters without relying on Green functions or predefined constitutive laws, enabling robust ML/NN integration. It extends the approach to peridynamics, strain-gradient, and coupled multi-physical problems, and provides translation-averaged datasets $\mathcal D^{\rm r}$ and $\mathcal D^{\rm p}$ (and $\mathcal D^{\rm d}$ for deterministic structures) to train surrogate nonlocal operators via four $\mathcal G_\gamma$ mappings. The framework eliminates sample-size, boundary-layer, and edge effects, delivering scale-independent surrogates and enabling efficient, accurate ML-driven micromechanics across an array of microtopologies, including nonlocal and nonlinear phase properties. This represents a paradigm shift in micromechanics, offering a modular, data-driven path from AGIE to RVE to surrogate operators, with broad implications for predictive modeling of heterogeneous media.

Abstract

We consider the matrix composite materials (CM) of either random (statistically homogeneous or inhomogeneous), periodic, or deterministic (neither random nor periodic) structures. CMs exhibit linear or nonlinear behavior, coupled or uncoupled multi-physical phenomena, locally elastic, weakly nonlocal (strain gradient and stress gradient), or strongly nonlocal (strain-type and displacement-type, peridynamics) phase properties. A modified Computational Analytical Micromechanics (CAM) approach introduces an exact Additive General Integral Equation (AGIE) for CMs of any structure and phase properties mentioned above. The unified iteration solution of static AGIEs is adapted to the body force with compact support serving as a fundamentally new universal training parameter. The approach also establishes a critical threshold for filtering out unsuitable sub-datasets of effective parameters through a novel Representative Volume Element (RVE) concept, which extends Hill's classical framework. This RVE concept eliminates sample size, boundary layer, and edge effects, making it applicable to CMs of any structure and phase properties, regardless of local or nonlocal, linear or nonlinear. Incorporating this new RVE concept into machine learning and neural network techniques enables the construction of any unpredefined surrogate nonlocal operators. The methodology is structured as a modular, block-based framework, allowing independent development and refinement of software components. This flexible, robust AGIE-CAM framework integrates data-driven, multi-scale, and multi-physics modeling, accelerating research in CM of any microtopology and phase properties considered. The AGIE-CAM framework represents a groundbreaking paradigm shift in the micromechanics of composites, redefining the very philosophy that underpins our understanding of their behavior at the microscopic level.