Micromechanics techniques can be employed to model the individual constituents within the composite material. Typically a repeating unit cell (RUC) in the com- posite microstructure is identified, and analysis is performed on that RUC assuming periodic boundary conditions. In addition, representative volume element (RVE) methodologies exist which incorporate applying non-periodic boundary conditions to a subvolume that accurately represents the composite microstructure. Microme- chanics can be utilized to provide the homogenized composite stiffness or to model damage and failure within the constituents and provide the resulting homogenized composite response. If utilized for the latter, the global mechanisms can arise through the natural evolution and interaction of the mechanisms in the constituents in the micromechanics model. Numerous micromechanics frameworks exist that encompass analytical, semi-analytical, and numerical techniques. An expansive review of many micromechanics theories is given by Aboudi et al. (2012) and Kanoute et al. (2009).
The first micromechanics models were used to calculate the elastic stiffness of the composite from the properties of its constituents. The simplest approximations by Voigt (1887) and Reuss (1929), are commonly referred to as the rule of mixtures and calculate the elastic stiffness, or compliance, tensor, by through a weighted sum of the stiffness (compliance) tensors of the constituents using the volume fraction of the constituents. Work by Hill (1952) proved that the actual stiffness tensor was
(CSA) model for spherical inclusions, and the concentric cylinder assemblage (CCA) model for long cylindrical fibers, provided more accurate estimates on the stiffness of the composite by assuming the composite was composed of a distribution of spheres, or cylinders, containing a inner fiber core and outer matrix shell [Hashin (1962); Christensen and Waals (1972)]. The homogenized properties of single sphere, or cylinder, can be determined by solving a set boundary value problems (BVPs), and those properties can be averaged over the desired distribution of orientations. The generalized-self consistent scheme assumed the spherical, or cylindrical, fiber and ma- trix were embedded in an effective medium representing the homogenized composite [Christensen and Lo (1979)]. The properties of the effective medium could then be calculated. This method can be used to provide the transverse shear modulus which cannot be calculated using the CSA or CCA.
Mori and Tanaka (1973) originally developed a method for calculating the average fields in a fiber contained in an infinite volume of matrix by assuming the fields in the matrix are equivalent to the applied far fields and calculating concentration matrices for the fields in the fiber. The Eshelby inclusion method provides concentration matrices for calculating the average fields in both of the constituents by assuming the presence of an eigenstrain in an ellipsoidal inclusion [Eshelby (1957); Mura (1982); Timoshenko and Goodier (1970)]. The availability of average, constituent level fields provided by these methods make them amenable for damage and failure modeling by prescribing the appropriate constitutive laws in the constituents. However, these methods neglect the variation of the local fields within the constituent materials.
Approaches developed by Nemat-Nasser et al. (1982); Walker et al. (1989) dis- cretized an RUC of the composite into a number of subvolumes. Global constitutive laws were formulated in terms of the constitutive behavior of the subvolumes in the form of a set of integral equations. These integral equations were solved approxi- mately using a Green’s function approach or a Fourier series approach.
The transformation field analysis (TFA) assumes the fields in a discretized RUC are piecewise uniform [Dvorak (1992)]. Furthermore, the local stresses and strains contain contributions from the eigenstrains and eigenstresses, respectively. The eigen- fields may include thermal, inelastic ad damage effects. Elastic strain concentration tensors (calculated using other methods) are used to relate the global fields to the local fields for the purely elastic case, and account for the shape and volume fraction of the phases in the RUC. These concentration tensors simply need to be calculated once, and nonlinearity is achieved through the evolution of the local eigenstrains and eigenstresses. Thus, the number of unknowns that need to be solved throughout the problem are reduced considerably. TFA was later extended to non-uniform trans- formation field analysis (NTFA) by Michel and Suquet (2003) to incorporate fully non-uniform local fields.
The method of cells (MOC) developed by Aboudi (1991) discretized a rectangu- lar composite RUC into four subvolumes, called subcells. One of the subcells was occupied by the fiber material and the rest were occupied by the matrix. Linear displacement fields were assumed in each of the subcells. Displacement and trac- tion continuity conditions were enforced, in an average integral sense, at the subcell interfaces, along with periodic boundary conditions at the RUC boundaries to de- rive a set of equations that yield a strain concentration matrix. This could, in turn, be used to obtain the local subcell strains from the applied fields. Following deter- mination of the subcell strains, the subcell stress are readily calculated using the local constitutive laws, and volume averaging can be used to obtain the homogenized thermomechanical properties of the composite. MOC was later extended to the gen- eralized method of cells (GMC) by Paley and Aboudi (1992) which accommodated any number of subcells and constituents in two periodic directions. Aboudi (1995) adapted the formulation to accommodate triply-periodic materials. Finally, Aboudi et al. (2001) developed the high-fidelity generalized method of cells (HFGMC) which
utilized second order displacement field approximations in the subcells, rather than linear. Aboudi et al. (2003); Haj-Ali and Aboudi (2009) showed the local field accuracy produced by HFGMC corresponded very well to FEM for elastic and inelastic phases. Bednarcyk et al. (2004) utilized HFGMC to model fiber-matrix debonding in metal matrix composites (MMCs), and Bednarcyk et al. (2010) implemented a multi-axial damage model in HFGMC. Reformulations, of GMC and HFGMC, which reduced the total number of unknowns in the problem were introduced by Pindera and Bednarcyk (1999) and Bansal and Pindera (2004), respectively.
The semi-analytical methods (Green’s function/Fourier series approach, TFA, GMC, HFGMC) offer a distinct advantage over the analytical, mean-field approaches, in that, spatially varying local fields can be determined. This robustness allows for better representation of the damage and failure mechanisms at the constituent scale. Furthermore, their semi-analytical formulations retain a computational advantage over fully numerical methods
However, as computational power increases, detailed fully numerical microme- chanics simulations are more feasible. Recent works by Gonzal´ez and Llorca (2007a); Totry et al. (2008, 2010) modeled multiple random fibers using a two-dimensional (2-D) FEM model. The matrix was modeled using plasticity, and the fiber and ma- trix was allowed to debond using CZM elements at the fiber-matrix interfaces. The response of the RUC was investigated under combinations of transverse compression and transverse shear loading. Cid Alfaro et al. (2010) modeled fiber-matrix debonding and matrix cracking under global transverse tension in a single-fiber RUC by including CZM elements between every matrix-matrix or fiber-matrix interface. Mishnaevsky Jr. and Brønsted (2007) simulated fiber-matrix pullout in a three-dimensional (3-D) RVE by modeling the fiber-matrix interface as a damaging solid.