Micromechanics of Composite Materials

Derivations and presentations of results in this book will appear in the tensor components, or in the related matrix notation. In the tensor component or subscript notation, vectors or first-order tensors are denoted by lower case italics with a single letter subscript, such as \( {n_i} \) or \( {\nu_j} \), while second, third and fourth-order tensors are written as \( {\varepsilon_{\textit{ij}}},\,\,{ \in_{\textit{ijk}}},\,\,{L_{\textit{ijkl}}} \), with the number of subscripts indicating the order or rank R of the tensor. The subscripts have a certain assigned range of values, which is i, j,… = 1, 2, 3, or \( \rho = 3 \) for tensorial quantities in the Cartesian coordinates \( {x_i} \). The number of tensor components is \( N = {R^{\rho }} \). It is then convenient to write the components of a first, second or fourth order tensors as \( {(3} \times {1),}\,\,{(3} \times {3)}\,\,{\text{\; or (9}} \times {9)} \) arrays, which need not conform to the rules of matrix algebra. The third order tensor can be displayed in three \( {(3} \times {3)} \) arrays.

This chapter provides a brief introduction to micromechanics. Following an overview of several descriptors of microstructural geometry is an outline of the procedures that predict overall response of a heterogeneous aggregate in terms of phase volume averages of local strain or stress fields. Applied loads include uniform overall strain or stress and a piecewise uniform distribution of eigenstrains or transformation strains in the phases. Derivations of theorems, formulae and connections that will frequently be used in subsequent chapters are presented in Sects. 3.7, 3.8 and 3.9. A summary of the overall and local response estimates appears in the concluding Sect. 3.10. Many symbols used in this and following chapters are summarized in Table 2.​5.

Overall mechanical properties and local strain and stress field averages, caused in individual phases of heterogeneous solids by remotely applied uniform strain or stress, are often derived from estimates of local fields in ellipsoidal homogeneous inclusions and inhomogeneities, bonded to a large volume of a surrounding matrix or ‘comparison medium’. The attraction of this approach lies in the relative simplicity of evaluation of the local fields, and in the adaptability of ellipsoidal shapes, such as prolate or oblate ellipsoids, spheroids, cylinders, spheres, penny-shaped discs or slits, to represent either short or long fibers, particles, voids and cracks of different shapes. Transition from local fields in a single inhomogeneity to those in interacting inhomogeneities comprising composite aggregates and polycrystals is accomplished, in part, by assigning certain properties to the comparison medium, as shown in Chaps.​ 6 and 7.

As in the previous chapter, we consider homogeneous inclusions and inhomogeneities in subvolumes \( {\Omega_r} \) of an infinitely extended homogeneous volume \( {\Omega_{{0}}} \) of a comparison medium or ‘matrix’ of stiffness \( {{L}_0} \); the total volume \( \Omega = {\Omega_{{0}}} + {\Omega_r} \). In Sects. 5.1.4 and in 5.2 and 5.3, we examine composite aggregates with dilute reinforcement, which may consist of many distinct inhomogeneities \( {{L}_r} \) in a matrix \( {{L}6557} \), as described in Sect.4.​4. Systems containing cracks are discussed in Sect. 5.4. Loads applied to both single and multiple inhomogeneity systems include displacement or traction fields acting at a remote boundary to generate uniform overall strain or stress, and piecewise uniform, physically based eigenstrains in both matrix and inhomogeneities. Those include thermal and moisture-induced strains, phase transformations, and inelastic strains. Low loading rates causing only small strains are assumed.

When applied separately to a homogeneous material, each of these loads generates a certain amount of potential energy \( {\mathcal V} \) defined in Sect. 3.​7. When applied together to a heterogeneous material, they generate the total potential energy, which is equal to the sum of the energies caused by the applied loads and of the potential energy generated by interactions between individual load components and/or inhomogeneities. Different interaction energies are derived for selected combinations of applied loads and material configurations. The results are useful in several applications, e.g., in estimating the energy released by interfacial decohesion of inhomogeneities from the surrounding matrix, or energy changes associated with phase transformations.

This chapter is concerned with composites and polycrystals, consisting of two or more distinct phases that have known stiffnesses L _r defined in the fixed overall coordinate system of a representative volume V. Phase volume fractions C _r \( \Sigma_{{r = 1}}^n\,{c_r} = 1, \) are no longer small, hence evaluation of both overall properties and local fields must reflect interactions between individual phase volumes. Spatial distribution of the phases in V is statistically homogeneous, as described in Sect. 3.​2.​2, and perfect bonding is assumed at all interfaces. Of interest are derivations of upper and lower bounds on the overall stiffness \( {L} = {{L}^{\text{T}}} \) and compliance \( {M} = {{L}^{{ - 1}}} \) of the aggregate, and of estimates of phase volume averages of strain and stress fields, caused in the heterogeneous system by application of uniform overall strain \( {{\varepsilon }^0} \) or stress \( {{\sigma }^0} \). Those are sought in terms of known volume fractions, elastic moduli, shape and alignment of the constituent phases, Sects. 6.1 and 6.2.

Together with the methods described in the previous chapter, overall moduli and local field averages in the phases can be estimated by one of several approximate methods, which use different models of the microstructure. Among those described here are variants of the average field approximation, or AFA, which rely on strain or stress field averages in solitary ellipsoidal inhomogeneities, embedded in large volumes of different comparison media L ₀. Among the most widely used procedures are the self-consistent and Mori-Tanaka methods, and the differential scheme, described in Sects. 7.1, 7.2 and 7.3. Those are followed by several double inclusion or double inhomogeneity models in Sect. 7.4, and by illustrative comparison with finite element evaluations for functionally graded materials in Sect. 7.5.

Together with the stresses caused by mechanical loads, composite materials must withstand stresses caused by distribution of transformation strains or eigenstrains in individual phases or subvolumes of each phase. As pointed out in Sect. 3.6.1, the former term applies here to all physically based deformations not caused by mechanical loads, including actual phase transformations. Frequent sources of transformation strains are changes in temperature and/or moisture content, piezoelectric and magneto-electro-elastic and pyroelectric effects, (Benveniste 1992, 1993; Benveniste and Milton 2003), as well as diffusive and displacive transformations involved in kinetics of structural change in crystals and polycrystals (Ashby and Jones 1986), or martensitic phase transformations in steels, and shape memory alloys (Entchev and Lagudas 2002; Levitas and Javanbakh 2011). Inelastic deformations associated with plasticity, viscoelasticity and viscoplasticity will join this list in Chap.​ 12, but those will be analyzed in an entirely different setting.

Several types of bonds may exist at the juncture between adjacent phases in contact. On the microscale of many composite materials, most desired is a perfect bond along a sharp spatial boundary S of vanishing thickness. It guarantees that both traction and displacement vectors remain continuous on S. Contact between phase surfaces may also involve presence of one or more interphases, thin bonded layers of additional homogeneous phases introduced, for example, as coatings on particles or fibers, or as products of an interfacial chemical reaction. During composites manufacture and/or loading, an interface is expected to transmit certain tractions between adjacent constituents. When the resolved tensile and/or shear stress reaches a high magnitude, the interface may become imperfect by allowing partial or complete decohesion, a displacement jump, possibly accompanied by a distribution of ‘adhesive’ tractions. In an opposite situation, a high compressive stress may cause radial cracking in one of the phases in contact, or in the surrounding matrix. While magnitudes of interface tractions determine material propensity to distributed damage, the work required by either decohesion or radial cracking must be provided by release of potential energy, which is proportional to phase volume Chap. 5. Therefore, small inhomogeneities are less likely sources of damage than large ones.

Laminated plates and shells are made by laying up and co-curing unidirectionally reinforced fibrous composite plies or laminae, which have different in-plane orientation and are ordered in a certain stacking sequence. Ply thicknesses are material system specific and their final magnitudes may depend on the fabrication procedure. Most polymer matrix composites are made using pre-impregnated or prepreg tapes or sheets, reinforced by tows consisting of many small diameter (<20 μm) fibers, which typically form ∼0.127 mm (0.005 in.) thick plies. Metal matrix laminates are often reinforced by monolayers of large diameter (150 μm) filaments, which yield ply thicknesses of ~0.200 mm (0.008 in.). Therefore, many plies are required to build up section thicknesses required in larger structures.

This chapter provides a short introduction to constitutive relations for materials that exhibit incremental elastic-plastic deformation in response to an applied loading path which extends beyond their initial yield surface. In a certain sense, it is analogous to Chap.​ 2 on Anisotropic Elastic Solids, with which it shares the results pertaining to isotropic elasticity. Moreover, the instantaneous tangential stiffness or compliance matrices may have as many as 21 nonzero coefficients, as in triclinic elastic materials. In preparation Chap.​ 12, attention is focused on those parts of incremental plasticity theory that are useful in modeling of metal matrix composites.

Initial applications of elastic–plastic and other inelastic constitutive relations in predicting overall response of heterogeneous materials had focused on polycrystalline metals, modeled as a multiphase system of randomly orientated single crystal grains which were assigned certain yield conditions and slip mechanisms. Early work includes the slip theory of Batdorf and Budiansky (1949), the rigid-plastic single crystal system of Bishop and Hill (1951), the elastic–plastic K.B.W. model of Kröner (1961) and the self-consistent approximation by Hershey (1954) and by Budiansky and Wu (1962). Further developed by Hill (1965c, 1967) and implemented by Hutchinson (1970), the SCM approximation extended the elasticity form of the method to polycrystals and two-phase composites. That and numerous other extensions of elastic micromechanical methods to inelastic systems provide an interface with the latter. However, they often assume uniform elastic and inelastic deformation in each grain, or in the entire matrix of a particulate or fibrous composite, according to a specified constitutive relation. Since local deformation is not uniform, the overall response predicted by such theories is not supported by experiments, as shown in Sect. 12.2.2. Nonuniform local deformation was examined on composite cylinders under axisymmetric and thermal loads, and in shakedown state, by Dvorak and Rao (1976a, b), Tarn, et al. (1975). General loading effects were investigated with models which constrained only longitudinal deformation by elastic fibers (Dvorak and Bahei-El-Din 1979, 1980, 1982). More recent work, supported by numerical methods, has focused on realistic aspects of deformation mechanisms of polycrystals and composites, as reviewed by Dawson, Hutchinson, Torquato and others in a report on research trends in solid mechanics (Dvorak 1999).