George J. Dvorak

Transformation Field Analysis of Inelastic Composite Materials

A new method is proposed for evaluation of local fields and overall properties of composite materials subjected to incremental thermomechanical loads and to transformation strains in the phases. The composite aggregate may consist of many perfectly bonded inelastic phases of arbitrary geometry and elastic material symmetry. In principle, any inviscid or time-dependent inelastic constitutive relation that complies with the additive decomposition of total strains can be admitted in the analysis. The governing system of equations is derived from the representation of local stress and strain fields by novel transformation influence functions and concentration factor tensors, as discussed in the preceding paper by G. J. Dvorak and Y. Benveniste. The concentration factors depend on local and overall thermoelastic moduli, and can be evaluated with a selected micromechanical model. Applications to elastic-plastic, viscoelastic, and viscoplastic systems are discussed. The new approach is contrasted with some presently accepted procedures based on the self-consistent and Mori—Tanaka approximations, which are shown to violate exact relations between local and overall inelastic strains.

Progressive Transverse Cracking In Composite Laminates

HE DEVELOPMENT OF a satisfactory theory for cross-ply laminates which have been damaged by transverse matrix cracking under monotonic loading has attracted a substantial number of investigators. The formulation of a shear lag model appears to have been first proposed in a series of papers by Bailey and his co-workers [1,2,3,4,5,6]. This work, in turn, relies on some studies of unidirectional composites by Aveston and Kelly [6]. Subsequent contributions to the theory have geen given by Wang [7], Highsmith and Reifsnider [8], Flaggs and Kural [9], Nuismer and Tan [10], Manders, Chou, Jones and Rock [11], Fukunaga, Chou, Peters and Schulte [12], Flaggs [13], Ohira [14] and Ogin, Smith and Beaumont [15,16]. Doubtless a diligent search of the literature would disclose other related work.

On transformation strains and uniform fields in multiphase elastic media

The effect of local eigenstrain and eigenstress fields, or transformation fields, on the local strains and stresses is explored in multiphase elastic solids of arbitrary geometry and material symmetry. The residual local fields caused by such transformation fields are sought in terms of certain transformation influence functions and transformation concentration factor tensors. General properties of these functions and concentration factors, and their relation to the analogous mechanical influence functions and concentration factors, are established, in part, with the help of uniform strain fields in multiphase media. Specific estimates of the transformation concentration factor tensors are evaluated by the self-consistent and Mori-Tanaka methods. It is found here that although the two methods use different constraint tensors in solutions of the respective dilute problems, their estimates of the mechanical, thermal, and transformation concentration factor tensors, and of the overall stiffness of multiphase media have a similar structure. Proofs that guarantee that these methods comply with the general properties of the transformation influence functions, and provide diagonally symmetric estimates of the overall elastic stiffness, are given for two-phase and multiphase systems consisting of, or reinforced by, inclusions of similar shape and alignment. One of the possible applications of the results, in analysis of overall instantaneous properties and local fields in inelastic composite materials, is described in the following paper.

MICROMECHANICAL MODELS FOR GRADED COMPOSITE MATERIALS

Abstract Elastic response of selected plane-array models of graded composite microstructures is examined under both uniform and linearly varying boundary tractions and displacements, by means of detailed finite element studies of large domains containing up to several thousand inclusions. Models consisting of piecewise homogeneous layers with equivalent elastic properties estimated by Mori-Tanaka and selfconsistent methods are also analysed under similar boundary conditions. Comparisons of the overall and local fields predicted by the discrete and homogenized models are made using a C/SiC composite system with very different Youngs moduli of the phases, and relatively steep composition gradients. The conclusions reached from these comparisons suggest that in those parts of the graded microstructure which have a well-defined continuous matrix and discontinuous second phase, the overall properties and local fields are predicted by Mori-Tanaka estimates. On the other hand, the response of graded materials with a skeletal microstructure in a wide transition zone between clearly defined matrix phases is better approximated by the self-consistent estimates. Certain exceptions are noted for loading by overall transverse shear stress. The results suggest that the averaging methods originally developed for statistically homogeneous aggregates may be selectively applied, with a reasonable degree of confidence, to aggregates with composition gradients, subjected to both uniform and nonuniform overall loads.

Stiffness changes in unidirectional composites caused by crack systems

Abstract This paper is concerned with the development of constitutive equations for fibrous composites which contain a family of longitudinal slit cracks. The general theory in which the crack length is of the same order as the fiber diameter gives rise to a three-phase model. When the fiber diameter is much smaller than the crack length, we obtain the associated two-phase model. Detailed numerical results are given for selected systems.

Stress fields in composites with coated inclusions

Abstract A micromechanics model is presented for the prediction of stress fields in coated fiber composites. The method is based on the “average stress in the matrix” concept of Mori and Tanaka and is formulated for the case of thermoelastic loading. A general description of the model is first given for three-phase materials and then specialized to the case of coated fiber composites. Results are presented for typical coated fiber composite systems under a variety of mechanical loading situations and uniform temperature change.

Bounds on overall instantaneous properties of elastic-plastic composites

Abstract M inimum principles of plasticity are used to derive upper and lower bounds on ordered eigenvalues and on diagonal terms of instantaneous stiffness and compliance matrices of elastic-plastic composite aggregates subjected to uniform overall stress or strain increments. The method is implemented with the help of a periodic model of binary fibrous composites ; overall properties are derived from incremental solutions of an inclusion problem in a small unit cell. Actual results are found with displacement and equilibrium formulations of the finite element method. In this implementation the instantaneous properties of the inelastic phases of the composite are known only in terms of the approximate values calculated at each step, hence the bounds are valid for an aggregate in which the actual properties have been replaced by the approximate ones. Examples are presented for the fibrous B-A1 system. The technique can be used to evaluate instantaneous element properties in a finite element program for analysis of metal matrix composite structures.

Analysis of progressive matrix cracking in composite laminates. II: First ply failure

The mechanics of transverse cracking in an elastic fibrous composite ply is explored for the case of low crack density. Cracks are assumed to initiate from a nucleus created by localized fiber debonding and matrix cracking. Conditions for onset of unstable cracking from such nuclei are evaluated with regard to interaction of cracks with adjacent plies of different elastic properties. It is found that cracks may propagate in two directions on planes which are parallel to the fiber axis and perpendicular to the midplane of the ply. In general, crack propagation in the direction of the fiber axis controls the strength of thin plies, while cracking in the direction perpendicular to the fiber axis determines the strength of thick plies. The theory relates ply thickness, crack geometry, and ply tough ness to ply strength. It predicts a significant increase in strength with decreasing ply thick ness in constrained thin plies. The strength of thick plies is found to be constant, but it may be reduced by preexisting damage. Strength of plies of intermediate thickness, and of unconstrained thick plies is evaluated as well. Results are illustrated by comparison with experimental data.

Analysis of progressive matrix cracking in composite laminates. I: Thermoelastic properties of a ply with cracks

Overall stiffnesses and compliances, thermal expansion coefficients, and stress and strain averages are evaluated for a fibrous composite lamina which contains a given density of open transverse cracks and is subjected to uniform mechanical loads and thermal changes. The evaluation procedure is based on the self-consistent method and is similar, in principle, to that used in finding elastic constants of unidirectional com posites.

On diagonal and elastic symmetry of the approximate effective stiffness tensor of heterogeneous media

Abstract T he existence of diagonal symmetry in estimates of overall stiffness tensors of heterogeneous media is examined for several micromechanical models. The dilute approximation gives symmetric estimates for all matrix-based multiphase media. The Mori-Tanaka and the self-consistent methods do so for all two-phase systems, but only for those multiphase systems where the dispersed inclusions have a similar shape and alignment. However, the differential schemes associated with the self-consistent method can predict diagonally symmetric overall stiffness and compliance for multiphase systems of arbitrary phase geometry. A related question is raised about the equivalence of two possible approaches to evaluation of the overall thermal stress and strain tensors. A direct estimate follows from each of the above models, whereas L evin s results [ Mechanics of Solids 2 , 58 (1967)] permit an indirect evaluation in terms of the estimated overall mechanical properties or concentration factors and phase thermoelastic moduli. These two results are shown to coincide for those systems and models which return diagonally symmetric estimates of the overall stiffness. Finally, model predictions of the overall elastic symmetry of composite media are discussed with regard to the spatial distribution of the phases.