G.J. Weng

Average stress in the matrix and effective moduli of randomly oriented composites

Abstract By considering the variation of average stress in the matrix, some elastic properties of randomly oriented composites are established as a function of aspect ratio. Both three- and two-dimensional random orientations, resulting, respectively, in a complete and transverse isotropy, are considered. As the shape of inclusions changes, the isotropic bulk and shear moduli are shown to vary within the Hashin-Shtrikman bounds. The aspect ratio dependence of the five in-plane and out-of-plane moduli with planar orientations are also explicitly given; these results suggest that the in-plane properties are most effectively reinforced by fibrous inclusions, whereas the out-of-plane ones are more responsive to the disc type. The accompanying variations of average stress in the matrix are seen to be closely related to the corresponding variations of the moduli. Comparisons with some limited experimental data also show a reasonable agreement.

Elastic moduli for a class of porous materials

SummaryThe effective elastic moduli for a class of porous materials with various distributions of spheroidal voids are given explicitly. The distributions considered include the unidirectionally aligned voids, three-dimensionally and two-dimensionally, randomly oriented voids, and voids with two types of biased orientations. While the 3-d random orientation results in a macroscopically isotropic solid, the porous media associated with the other arrangements are transversely isotropic. The five independent elastic constants for each arrangement, as well as the two for the isotropic case, are derived by means of Mori-Tanakas mean field theory in conjunction with Eshelbys solution. Specific results for long, cylindrical pores and for thin cracks with the above orientations are also obtained, the latter being expressed in terms of the crack-density parameter. Before we set out the analysis, it is further proven that, in the case of long, circular inclusions, the five effective moduli of a fiber composite derived from the Mori-Tanaka method coincide with Hills and Hashins lower bounds if the matrix is the softer phase, and coincide with their upper bounds if the matrix is the harder.

Plasticity of a two-phase composite with partially debonded inclusions

The effective elastoplastic behavior of a two-phase composite consisting of partially debonded elastic inclusions and a ductile matrix is investigated by a homogenization method. The method drew information from a recent study by the authors on the effective elastic moduli of the said composite and from an energy approach suggested by Qui and Weng, J. Appl. Mech., 59, 261 [1992] to address the homogenized plastic state of the heterogeneously deformed ductile matrix. Two types of partial debonding configuration are considered; the first is on the top and bottom of the aligned oblate inclusions and the other is on the lateral surface of the prolate ones, with special reference to spherical inclusions for both types of debonding. The transversely isotropic elastoplastic properties of the partially debonded composite are found to be highly dependent upon the debonding mode and the volume concentration and shape of inclusions. A damage mechanics based on Weibulls statistical function is also proposed to study the progressive partial debonding of the initially bonded composite under pure tension and under biaxial tension, respectively, for these two types of partial debonding. It is found that the interfacial strength, particle concentration, inclusion shape and debonding mode all play significant role in the overall response of the heterogeneous system during the progressive debonding process.

Plastic potential and yield function of porous materials with aligned and randomly oriented spheroidal voids

Abstract Based on an energy approach, the plastic potential and yield function of a porous material containing either aligned or randomly oriented spheroidal voids are developed at a given porosity and pore shape. The theory is applicable to both elastically compressible and incompressible matrix and, it is proved that, in the incompressible case, the theory with spherical and aligned spheroidal voids also coincides with Ponte Castanedas bounds of the Hashin-Shtrikman and Willis types, respectively. Comparison is also made between the present theory and those of Gurson and Tvergaard, with a result giving strong overall support of this new development. For the influence of pore shape, the yield function and therefore the stress-strain curve of the isotropic porous material are found to be stiffest when the voids are spherical, and those associated with other pore shapes all fall below these values, the weakest one being caused by the disc-shaped voids. The transversely isotropic nature of the yield function and stress-strain curves of a porous material containing aligned pores are also demonstrated as a function of porosity and pore shape, and it is further substantiated with a comparison with an exact, local analysis when the void shape becomes cylindrical.

A Micromechanics-Based Hysteresis Model for Ferroelectric Ceramics

Based on the mechanics of domain switch and irreversible thermodynamics, a micromechanics-based model that incorporates the effect of polarization strain and electric polarization in the switched domain is developed to predict the evolution of new domain and the associated hysteresis loops of a ferroelectric ceramic. The new domain concentration c r associated with the remanent polarization P r , and the new domain concentration c c at the coercive field E c , are also found in terms of the saturation polarization P s , the coercive field E c , and the dielectric permittivity of the parent domain. The theory is developed with a homogenization technique for a coupled, dual-phase electromechanical system with an evolving microstructure, whose driving force arises from the reduction of Gibbs’ free energy and whose resistance force comes from the energy dissipation due to domain wall movement. The developed theory is applied to a lanthanum doped lead zirconate titanate (PLZT), first for the calculation of its hysteresis electric displacement versus electric field (D vs. E) relation and the butterfly-shaped longitudinal strain versus electric field (∊ vs. E) relation, and then for its nonlinear compressive stress-strain relation and nonlinear depolarization. The results are shown to provide the essential features of the hysteresis behavior and found to be in quantitative accord with the experimental data.

Some reflections on the Mori-Tanaka and Ponte Castañeda-Willis methods with randomly oriented ellipsoidal inclusions

SummaryFor a two-phase isotropic composite consisting of an isotropic matrix and oriented isotropic ellipsoidal inclusions, Mori-Tanakas (MT) [6] method and the more recent Ponte Castañeda-Willis (PCW) [1] method are perhaps the only two methods that deliver explicit results for its effective moduli. An attractive feature of the MT method is that it always stays within the Hashin-Shtrikman [3] bounds, while the novel part of the PCW approach is that it has a well defined microstructure. In this paper, we made a comparative study on these two models, for both elasticity and their applications to plasticity. Over the entire range of inclusion shapes, the PCW estimates are found to be consistently stiffer than the MT estimates. An investigation of the possibility of a PCW microstructure for the MT model indicates that the MT moduli could be found from the PCW formulation, but this would require a spatial distribution that is identical to the oriented inclusion shape. Such a requirement implies that the underlying two-point joint probability density function is not symmetric, and thus it is not permissible. One is led to conclude that, unlike the aligned case, the MT model cannot be realized from the PCW microstructure with randomly oriented inclusions.

Anisotropic hardening in single crystals and the plasticity of polycrystals

Abstract Based on the dislocation structures developed during plastic deformation, an anisotropic hardening law is developed to describe the latent hardening behavior of slip systems under multislip. This theory incorporates the concept of isotropic hardening, kinematic hardening, and the two-parameter representation; it automatically includes the strength differential between the forward and reversed slips and between the acute and obtuse cross slips. The self-hardening modulus of a slip system is found to be “associated” with the latent hardening law involved, and, based on some experimental evidence, two specific sets of self-hardening modulus are suggested. An important feature of this associated modulus is that the slip system with a soft latent hardening (e.g., the reversed system with a Bauschinge effect) will have an enhanced self-hardening modulus. This newly developed hardening law, together with its associated latent hardening moduli, is then applied to examine the strain-hardening behavior of a polycrystal. Although crystals with a stronger latent hardening will, in general, also lead to a stranger strain-hardening for the polycrystal, the stress-strain behavior of the polycrystal using the kinematic hardening law of single crystals is found to be not necessarily softer than that using the isotropic hardening law. Within the range of experimentally measured latent hardening ratio of slip systems, the anisotropic theory is also used to calculate the motion of yield surface of a polycrystal. The general results, employing four selected types of anisotropic hardening, all show the essential features of experimental observations by Phillips and his co-workers. The application is highlighted with a reasonably successful quantitative modeling of initial and subsequent yield surfaces of an aluminum.

A unified approach from elasticity to viscoelasticity to viscoplasticity of particle-reinforced solids

Abstract This paper presents a novel approach to determine the overall elastic-viscoplastic behavior of a particle-reinforced metal-matrix composite. It is built upon the linkage from elasticity to viscoelasticity through the correspondence principle, and then from viscoelasticity to viscoplasticity by means of the concept of secant viscosity and an energy approach. Albeit approximate, this theory is analytically tractable, and capable of delivering the stress-strain relations of the composite sittem under various applied strain rates. The theory embodies the nonlinear, rate-dependent, and work-hardening nature of the constitutive equations of the ductile matrix, as well as the elasticity of both matrix and inclusions. The influence of particle concentration, elastic stiffness, and applied strain rate on the overall dilatational and deviatoric stress-strain behaviors are examined in detail. The theory is finally applied to predict the response of a silicon-carbide/aluminum system and the result is found to be in accord with experimental observations.

Dislocation theories of work hardening and yield surfaces of single crystals

SummaryThe nature of existing dislocation hardening mechanisms are analyzed. This analysis shows that the hardening mechanisms can be classified into two categories: isotropic and kinematic; consequently a general physical hardening law, which incorporates the concept of the degree of isotropy in work hardening, is proposed. The corresponding motions of yield surface in both stress and strain space are examined on the basis of this general hardening law and its two extreme cases: isotropic and kinematic hardening.ZusammenfassungEs werden die Eigenschaften vorhandener Versetzungsverfestigungstheorien untersucht. Diese Analyse zeigt, daß die Verfestigungsmechanismen in zwei Gruppen eingeteilt werden können; nämlich isotrope und kinematische. Als Folge davon wird ein allgemeines, physikalisches Verfestigungsgesetz vorgeschlagen, welches die Begriffe des Grades der Isotropie und der Arbeitsverfestigung vereinigt. Die dazugehörigen Bewegungen der Fließfläche werden sowohl im Spannungs-als auch im Verzerrungsraum auf der Basis dieses allgemeinen Verfestigungsgesetzes und seiner beiden extremen Fälle, der isotropen und der kinematischen Verfestigung, untersucht.

A self-consistent relation for the time-dependent creep of polycrystals

Abstract A self-consistent relation with a weakening constraint power in the matrix is derived for the primary and steady-state creep of polycrystals. This derivation makes use of a linear viscoelastic comparison material, under which the constraint power of the creeping matrix is found to decrease exponentially with the ratio of the elastic shear modulus to the secant creep modulus, or when expressed alternately, with the ratio of the creep strain to the elastic strain. Such a dramatic decrease leads one to believe that the overall creep strain of the polycrystal as calculated by the traditional elastic-constraint model could be far too low; a direct comparison between the two, however, quickly reveals that the accuracy of the elastic-constraint model is better than what is initially thought, and certainly far better than in the rate-independent plasticity. With this new relation, the creep heterogeneity among the constituent grains are then studied in details. It is demonstrated that while the creep strains of the more favorably oriented grains tend to increase, and those of the less favorably oriented ones decrease, their creep rates become virtually uniform during the long-term, steady-state creep. This suggests that grain compatibility, instead of stress equilibrium, is the more dominant factor governing the grain-boundary condition during the steady-state process.