This chapter introduces micromechanics-based progressive damage simulations, wherein local failures occur and accumulate as damage progresses through a composite material, resulting in a nonlinear stress-strain response. In contrast, Chapter 4 considered only damage initiation, without any damage propagation. A subvolume elimination method is employed, where subcell or constituent material stiffnesses are instantaneously reduced to a very low value once a specified failure criterion is reached within the subvolume. For laminates, this is done at the microscale within each ply. The pathological mesh dependence introduced by this method (when used within HFGMC or finite-element analysis), is illustrated and discussed. Although significant portions of the MATLAB code from previous chapters are preserved, due to the added complexities associated with progressive damage, new driver scripts that include the ability to apply loading incrementally and perform iterations within a given increment are required. Because of the nonlinearity present in progressive damage, which is controlled by the local fields and their redistribution as damage progresses, the impact of the chosen micromechanics theory, local failure criterion, and microstructural representation is significantly amplified compared to linear elastic analysis. Consequently, these influences are studied in detail for PMC and CMC composite materials and laminates through example problems. It is demonstrated that HFGMC, with its normal-shear coupling and accurate local fields, enables capture of the details of damage progression, which are particularly influential for disordered (random) microstructures. Of course, since detailed progressive damage simulations are computationally expensive, maximizing computational efficiency is highly desirable. If only damage initiation is of interest, then the margin of safety predictions from Chapter 4 can be employed without the need for more computationally expensive progressive damage analysis. Furthermore, in fiber-dominated situations in PMCs, the present chapter demonstrates that simpler, more efficient, micromechanics theories like MOC and MT are sufficient to predict the composite ultimate strength. This is because the effect of local fields is suppressed in fiber-dominated situations. Finally, for CMCs, it is shown that the progressive damage behavior is highly influenced by damage initiation location (i.e., interface vs. matrix), which is controlled by their relative strengths.

This chapter presents the generalized method of cells (GMC) micromechanics theory. The theory is a generalization of the method of cells (MOC) in the sense that it allows an arbitrary number of subcells in the repeating unit cell (RUC), rather than the MOC's four subcell RUC. Consequently, this generalization enables more than two constituent materials and different microstructures to be considered. After the theory and associated MATLAB code are presented, example problems are given focusing on PMCs, as well as CMCs, which include an explicit interfacial material. GMC's capabilities are illustrated by examining the predicted effective properties, local fields, and failure envelopes.

This chapter presents the high-fidelity generalized method of cells (HFGMC) micromechanics theory. By employing a second-order displacement field in the subcells, the method includes shear coupling that is absent in the method of cells, the generalized method of cells, and analytical models like the Mori-Tanaka method. The presence of shear coupling provides HFGMC predictions with higher-fidelity local stress and strain fields and the ability to account for microstructural effects correctly. However, this increased fidelity comes at the cost of computational efficiency and subcell grid independence. HFGMC therefore represents a step further away from fully analytical micromechanics towards the fully numerical finite element method. The HFGMC theory, including thermomechanical effects, is presented, followed by example problems that highlight the theory's capabilities (for example, consideration of different ordered and disordered fiber-packing arrangements and statistical effective property distributions). HFGMC's predictions are compared to those of the previously presented simpler theories as well as the finite element method.

This chapter addresses failure of unidirectional composites and laminates by determining the load at which allowables are first exceeded at some point in the composite. Stress and strain allowables (for either effective ply materials or constituent materials) quantify limits on the stress and strain components, below which the material can be expected to safely operate (with no loss of structural integrity). The maximum stress, maximum strain, Tsai-Hill, and Tsai-Wu failure criteria are employed to assess failure, at the ply level in laminates, at the constituent scale in unidirectional composites, and at the constituent scale within laminates. The equations enabling calculation of margins of safety (MoS) for each of these criteria are presented, both in the case of pure mechanical loading and in the case of thermomechanical loading, wherein the thermal loading is treated as a preload. The MATLAB implementation is then presented enabling calculation of the laminate or composite allowable loads by proportionally scaling the applied loading to the point at which failure would first occur somewhere within the laminate or composite. Code is also provided that generates multiaxial failure envelopes. Finally, example problems are presented that illustrate the ability to assess failure of laminates and unidirectional composites based on effective ply material allowables and constituent allowables.

This chapter presents the fundamental concepts related to micromechanics, most importantly the strain and stress concentration tensors relating the local fields to the applied composite-level loading. Then four classical micromechanics theories (the Voigt approximation, Reuss approximation, Mori-Tanaka (MT) method, and method of cells (MOC)) are presented in terms of their unique concentration tensors, from which all effective mechanical properties can be determined. While the Voigt, Reuss, and MT theories provide average fields for each constituent, the MOC enables prediction of the variations of the stress and strain fields within the matrix constituent. Furthermore, in contrast to MT and MOC, the simple Voigt and Reuss approaches, while illustrative from a theoretical standpoint, do not provide good approximations of the effective composite properties nor the local fields. All four theories are extended to accommodate thermomechanical loading, wherein the effective coefficients of thermal expansion (CTEs) are predicted, along with thermal strain concentration tensors, which enable the calculation of local fields in the presence of thermal loading. The MATLAB implementation of these classical micromechanics theories (for solving stand-alone micromechanics and multiscale (micromechanics-based) laminate analysis problems), along with associated flowcharts, are provided and discussed. Finally, a number of example problems comparing the micromechanics theory results are presented for both stand-alone micromechanics and micromechanics-based laminate simulations.

This chapter presents classical lamination theory, wherein the composite laminate plies are represented using effective anisotropic properties (rather than using micromechanics to calculate their effective properties). Towards this end, the chapter begins with a review of linear, thermoelastic, constitutive behavior for isotopic, transversely isotropic, and orthotropic materials. Then the mechanics of a ply, along with the ply stress-strain coordinate transformations and strain-displacement relations, are introduced. This culminates in the development of the thermomechanical laminate constitutive relations, that is, the construction of the ABD matrix along with the mechanical and thermal force and moment resultant expressions. The associated MATLAB implementation of these classical laminate expressions, as well as other common functions and solution flow charts, are then provided. Next, several example problems addressing different types of laminates (unidirectional, cross-ply, and angle-ply) and classes of thermoelastic composite materials (PMCs, MMCs, and CMCs) are presented. Finally, a brief overview of laminate notation and classification and common design rules are provided.