Showing papers on "Representative elementary volume published in 1994"

Micromechanics and effective moduli of elastic composites containing randomly dispersed ellipsoidal inhomogeneities

Abstract: A micromechanical framework is proposed to investigate effective mechanical properties of elastic multiphase composites containing many randomly dispersed ellipsoidal inhomogeneities. Within the context of the representative volume element (RVE), four governing micromechanical ensemble-volume averaged field equations are presented to relate ensemble-volume averaged stresses, strains, volume fractions, eigenstrains, particle shapes and orientations, and elastic properties of constituent phases of a linear elastic particulate composite. A renormalization procedure is employed to render absolutely convergent integrals. Therefore, the micromechanical equations and effective elastic properties of a statistically homogeneous composite are independent of the shape of the RVE. Various micromechanical models can be developed based on the proposed ensemble-volume averaged constitutive equations. As a special class of models, inter-particle interactions are completely ignored. It is shown that the classical Hashin-Shtrikman bounds, Walpole's bounds, and Willi's bounds for isotropic or anisotropic elastic multiphase composites are related to the “noninteracting” solutions. Further, it is demonstrated that the Mori-Tanaka methodcoincides with the Hashin-Shtrikman bounds and the “noninteracting” micromechanical model in some cases. Specialization to unidirectionally aligned penny-shaped microcracks is also presented. An accurate, higher order (in particle concentration), probabilistic pairwise particle interaction formulation coupled with the proposed ensemble-volume averaged equations will be presented in a companion paper.

A Micromechanics Model for Thermoelastic Properties of Plain Weave Fabric Composites

Abstract: A micromechanics model is presented to predict thermoelastic properties of composites reinforced with plain weave fabrics. A representative volume element is chosen for analysis and the fiber architecture is described by a few simple functions. Equations are developed to calculate various phase fractions from geometric parameters that can be measured on a cross section. Effective elastic moduli and effective thermal expansion coefficients are determined under the assumption of uniform strain inside the representative volume element. The resulting model is similar to the classical laminated theory, and hence is easier to use than other models available in the literature. An experimental correlation is provided for a number of Nicalon SiC/CVI SiC and Graphite/CVI SiC composite laminates.

Micromechanics as a Basis of Continuum Random Fields

Abstract: Department of Materials Science and Mechanics, Michigan State University, East Lansing MI 48824-1226 Many problems in solid and geomechanics require the concept of a meso-continuum, which allows a resolution of stress and other dependent fields over scales not infinitely larger than the typical microscale. Passage from the microstructure to such a meso-continuum is based on a scale depen­dent window playing the role of a Representative Volume Element (RVE). It turns out that the material properties at the mesoscale cannot be uniquely approximated by a random field of stiffness with continuous realizations, but, rather, two random continuum fields, corresponding to essential and natural boundary conditions on RVE, need to be introduced to bound the material response from above and from below. In this paper Monte Carlo simulations are used to obtain the first- and second-order one- and two-point characteristics of these two random fields for random chessboards and matrix-inclusion composites. Special focus is on the correlation functions describing the auto-covariances and crosscovariances of effective random meso-scale conductivity tensor Q; and its dual Sjj. Following issues are investigated: i) scale-dependence of noise-to-signal ratios of various components of Cy and Sjj, ii) spatial structure of the correlation function, iii) uniform strain versus exact calculations in determination of the correlation function, iv) correlation structure of compos­ites with inclusions without and with overlap. 1. INTRODUCTION One of the classical goals of micromechanics is the derivation of macroscopic properties on scales practically infinitely larger than the lengthscales at the microlevel. There are, how­ever, situations where description of material response at intermediate levels - so-called meso-level (or meso-scale) -is necessary. These are, for example, i) necessity to resolve local fields (of stress, say), ii) materials with spatially-depen­dent statistics, iii) solution of boundary value problems at macro-scales. Case i) is exemplified by almost any one of many problems in soil mechanics where a meso-scale has to be introduced to define a continuum over a granular mass in the first place (Ostoja-Starzewski, 1992). Case ii) is a situation where a pas­sage to an infinite scale would be beyond interest, such as for instance in a heterogenous interface problem (Ostoja-Starze­wski and Jasiuk, 1992; Jasiuk and Ostoja-Starzewski, 1994). A paradigm of case iii) is provided by so-called Stochastic Finite Elements (SFE) - a field of research developing over the past fifteen years, in which a micromechanical basis in set­ting up of a stochastic stiffness matrix is sorely needed (Ostoja-Starzewski, 1993a and 1993b). Most of the past research in SFE concerned linear elastic structural responses and relied on a straightforward generali­zation of Hooke's law (see e.g. Contreras, 1980; Liu, et al, 1986; Benaroya and Rehak, 1988; Shinozuka, 1988; Ghanem andSpanos, 1991), that is

Fiber fracture during the consolidation of metal matrix composites

Abstract: A detailed study of conditions leading to fiber fracture during the consolidation of Ti14wt%Al21wt%Nb/SiC (SCS-6) composite monotapes has been conducted. For this continuous fiber reinforced composite system, the incidence of fracture increases with consolidation rate at higher process temperatures. Increasing consolidation temperature at a fixed pressure reduces the number of breaks per unit length of fiber. Examination of partially densified compacts has revealed the existence of significant fiber bending and ultimately fracture due to monotape surface roughness (asperities) which places the fibers in three point bending. A representative volume element has been defined for the consolidating lay-up and its response analyzed to predict the fiber deflection (and hence probability of failure) when the surface asperities deform either by plasticity or by steady state creep. The relationships between fiber fracture and process conditions predicted using the volume element are similar to those observed experimentally. The cell analysis suggests that fiber fracture is decreased by increases in fiber stiffness, strength, and diameter and by decreases in matrix yield and creep strength and monotape surface roughness.

Thermo-inelastic analysis of functionally graded materials: inapplicability of the classical micromechanics approach

Abstract: The standard micromechanical approach used to analyse the response of functionally graded materials is to decouple the local and global effects by assuming the existence of a representative volume element (RVE) at every point within the composite. In this paper, a new higher order micromechanical theory for the inelastic response of functionally graded materials (HOTFGM), which couples the local and global effects, is used to assess the limits of applicability of the standard uncoupled approach. Comparison of local stresses in the fiber and matrix phases of composites with a finite number of uniformly and nonuniformly spaced fibers in the through-thickness direction subjected to uniform and nonuniform temperature fields, obtained with HOTFGM and the standard RVE-based approach, illustrates the inaccuracy of the standard analysis in the presence of temperature-dependent and inelastic behavior of the matrix phase.

Modeling the Effect of Oxidation on Damage in SiC/Ti-15-3 Metal Matrix Composites

Abstract: In this paper, a micromechanical analysis is performed on a single ply continuous fiber SiC/Ti-15V-3Al-3Sn-3Cr (Ti-15-3) metal matrix composite to study the complex interactions between the composite microstructural components and the surrounding environment at high temperatures. Finite elements are incorporated to model oxygen diffusing into the free surface of a representative volume element (RVE) during cool down from the processing temperature. The resulting residual stress distribution is investigated assuming thermoelastic material models for the matrix, oxide layer, and fiber. Results indicate that the oxidized surface layer is prone to cracking upon subsequent mechanical loading, and this effect is strongly temperature dependent.

The Role of Effective Moduli in the Elastic Analysis of Composite Laminates

Abstract: The purpose of this introductory chapter is to review some of the salient features of micromechanics of composite media. In contrast to the extensive literature reviews given by Hashin (1964) and Chamis and Sendeckyj (1968) which treat the various approaches used in the determination of effective properties of heterogeneous bodies, our emphasis will be placed on the interpretation and application of effective elastic moduli. Physical and mathematical definitions of effective moduli are compared and their use in the analysis of engineering composite laminates will be discussed. Finally, we shall present an approach to treat nonuniform (linear variation) macroscopic membrane stresses within composite laminates.

2-d damage model for unidirectional composites under transverse tension and /or shear

Abstract: SUMMARY Under transverse loading, unidirectional brittle matrix composites such as those having ceramic and glass-ceramic matrices, are susceptible to matrix/coating cracking, interface debonding (probably with friction), and possible fibre breaks. In order to predict the influence of these forms of damage, including their possible interactions, we have developed a model which attempts to accurately represent the stress field in their presence. The present formulation is based on the use of Reissner's variational theorem in conjunction with an equilibrium stress field in which the r-dependence is assumed. The model is based on the geometry of a sector composed of a cylindrical core surrounded by a number of concentric cylindrical wedges which, in turn, may simulate a composite representative volume element. Each one of the wedges or the core can be mathematically subdivided in the radial direction to enhance the accuracy of a given numerical solution. The proposed model is then utilized to evaluate the ef...

Flow In Porous Media

Abstract: The flow in porous media is hard to describe. Based on a representative elementary volume, we use conservation laws of porous media to construct one-phase models (which models the flow through a porous medium where only one liquid or gas is present) and two-phase models. Both lead to the same partial differential equation for the saturation of a phase, which has an equilibrium and a non-equilibrium form. We analytically solve the equilibrium form using similarity solutions, this gives us useful results. For the non-equilibrium form we use a numerical approach to find a similarity solution. With the results we can say how the water distributes in some porous media.

Physical approach to averaging theorems on phase interfaces in a dispersed multiphase medium

Abstract: Within the framework of a procedure for scale-changing by averaging a representative elementary volume, theorems are developed to relate the averages of derivatives to the derivatives of averages over a surface, using elementary differential calculus. These theorems form the basis of a general macroscopic balance equation for a given quantity over interfaces of a dispersed multiphase medium. The equations of phase interfaces complement equations related to bulk phases describing transport in dispersed multiphase media.

The Anisotropy Effect on the Stiffness and Toughness of Encapsulated Fiber Composites

Abstract: The variation of the effective moduli in encapsulated fiber composites, which consist of anisotropic phases, is presented in this paper. The strain energy rate released by an edge crack in a rectangular composite thin sheet was evaluated and provided information for the toughness of the material prior to fracture. A typical composite material was macroscopically assumed as a transversely isotropic medium, whereas the representative volume element used for its description is formed by a cylindrical fiber of either a transversely isotropic, or a purely isotropic material, a transversely isotropic cylindrical annulus as coating of the fiber, and an annulus of the matrix covering the encapsulated fiber. The matrix consisted either of isotropic or transversely isotropic material, which is further surrounded by the equivalent composite, which averaged the actual properties of the bulk of the composite containing the dispersed encapsulated fibers. Solutions for the longitudinal and transverse elastic moduli E Lc and E Tc as well as the shear moduli G Lc and G Tc of the composite are defined in a closed form. Their variation in terms of the relative extent of the matrix is also evaluated. Finally, interesting information concerning the variation of the elastic stiffness of the composite due to anisotropy of the matrix was found. Furthermore, the strain energy density released by an edge crack in a rectangular composite sheet was determined for the various models with anisotropic fibers or matrices studied in this paper

Modelling Interfacial Debonding in Titanium Matrix Composites

Abstract: The results of an experimental program in which multiaxial loads were applied to [0 4 ] and [±45] 8 silicon carbide/titanium (SiC/Ti) tubes are reviewed showing that stress coupling, matrix viscoplasticity (including room temperature creep) and fiber/matrix interfacial damage all contribute to nonlinear response and permanent strains in titanium matrix composites (TMC). A micromechanical model that explicitly considers the aforementioned phenomena is presented herein. The model assumes a periodic microstructure and uses finite elements to analyze a representative volume element. The composite is assumed to be in a state of generalized plain strain making it possible to discretize only a generic transverse plane while still being able to apply three-dimensional loading though appropriate boundary conditions. The response of laminated composites is predicted using the lamina response predicted by the micromechanical model in nonlinear lamination theory. Predictions are presented to show the influence of the model parameters on the effective composite response of unidirectional [0 4 ] and angle-ply [± 45] s TMC laminates.

Modeling the flow of molten silicon in porous carbon preforms and the subsequent formation of silicon carbide

Abstract: In this investigation, the authors have modeled the formation of silicon carbide during the infiltration of porous carbon preforms, and predicted the amount of SiC formed only due to reaction between Si and C, coupled with diffusion. For a two-dimensional representative volume element (RVE) of a carbon preform with 30% volume fraction of carbon, they have numerically predicted the concentration profiles of Si and SiC, based on coupled reaction and diffusion. Consideration of only reaction and diffusion as a mechanism of formation of SiC in the model is not adequate for an efficient conversion of Si to SiC, leading thereby to the presence of residual Si. Finally, a two-dimensional approach to predict the transient permeability in the preform in relation to the transient change in porosity in an RVE, was discussed.

Micromechanics of BMC

Abstract: In the mechanics of composites of the graphite-epoxy class, at least for the usual structural applications, there has been little need for studies of micromechanics. Although the fiber stiffness properties can vary significantly in this class, characterization of the unidirectional composite has become routine and inexpensive. The quality of bonding between fibers and matrix was an issue at first, the chemical problems associated with strong bonding were quickly solved, and in general, there have been few applications demanding weak bonding. Finally, epoxy resins have demonstrated little significant variations in mechanical properties. All of these facts contribute to negate the need for sophisticated micromechanical analysis — strong, stiff fibers, a matrix having high ductibliity (strain-to-failure), and strong bonding are targets that lead to good mechanical performance. All of these points are clear from physical reasoning. With the advent of ceramic-and glass-ceramic composites, however, things are not so straightforward. In this case, there is a plethora of matrix materials, at least until sorting based on material compatibility has been accomplished and wide variations exist in the significant thermomechanical properties, such as modulus of elasticity, thermal expansion coefficient, ultimate strength, and fracture toughness: Perfectly bonded interfaces are not deemed desirable and there is a wider range of ambient temperatures, leading to the need for considerations of thermal stress effect. Furthermore, failure modes involving only individual constituents and/or interfaces are predominant. Finally, constituent materials are expensive, scarce, and difficult to test with precision. All these facts suggest that micromechanical analyses are of paramount importance.