Showing papers on "Representative elementary volume published in 1993"

A Micromechanics Approach for Constructing Locally Averaged Damage Dependent Constitutive Equations in Inelastic Composites

Abstract: Homogenization techniques are utilized herein to construct locally aver aged mechanical constitutive equations for composites. Techniques are first reviewed for elastic composites, followed by a review of elastic composites with cracks and inelastic composites without cracks. These reviews are followed by a development of homogenized equations for inelastic composites with cracks. All equations are constructed for the case wherein a representative volume element is subjected to spatially uniform boundary strains. Several examples are presented for a representative SiC/Ti continuous fiber com posite subjected to tensile deformations transverse to the fiber direction. It is demonstrated that as the damage and matrix material model are increased in complexity, the computa tional requirements needed to obtain homogenized properties increase to the point where homogenization techniques may not be computationally tenable when applied recursively in macroscale structural analyses.

Self-similar heterogeneity in granular porous media at the representative elementary volume scale

Abstract: Geometrical measurements of sandstone pore microstructure (eg, Katz and Thompson, 1985) indicated that both the surface and volume of pores are self-similar and exhibit the same fractal dimension This result yields the microscale hydraulic conductivity to be given as a function of the inner and outer cutoffs and fractal dimension Accordingly, the fractal dimension can be viewed as the imprint of the process responsible for shaping the structure of voids in a porous medium We show that the same approach can be extended to different porous media, eg, to granular aggregates, and it can also be used to describe the soil structure at the scale of the representative elementary volume For this purpose the fractal rock fragmentation model proposed by Turcotte (1986) has been used to derive the particle size distribution for a fractal mixture Sedimentation of this mixture is then simulated using a ballistic aggregation algorithm, and both the solid phase and the void space geometry of the resulting aggregate are analyzed The results show that the same fractal dimension embedded in the fragmentation process is preserved by the sedimentation mechanism and it also characterizes the volume of the pore space and the spatial distribution of the particles

Accuracy of the Generalized Self-Consistent Method in Modelling the Elastic Behaviour of Periodic Composites

Abstract: Local stress and strain fields in the unit cell of an infinite, two-dimensional, periodic fibrous lattice have been determined by an integral equation approach. The effect of the fibres is assimilated to an infinite two-dimensional array of fictitious body forces in the matrix constituent phase of the unit cell. By subtracting a volume averaged strain polarization term from the integral equation we effectively embed a finite number of unit cells in a homogenized medium in which the overall stress and strain correspond to the volume averaged stress and strain of the constrained unit cell. This paper demonstrates that the zeroth term in the governing integral equation expansion, which embeds one unit cell in the homogenized medium, corresponds to the generalized self-consistent approximation. By comparing the zeroth term approximation with higher order approximations to the integral equation summation, both the accuracy of the generalized self-consistent composite model and the rate of convergence of the integral summation can be assessed. Two example composites are studied. For a tungsten/copper elastic fibrous composite the generalized self-consistent model is shown to provide accurate, effective, elastic moduli and local field representations. The local elastic transverse stress field within the representative volume element of the generalized self-consistent method is shown to be in error by much larger amounts for a composite with periodically distributed voids, but homogenization leads to a cancelling of errors, and the effective transverse Young's modulus of the voided composite is shown to be in error by only 23% at a void volume fraction of 75%.

Predicting High Temperature Ultimate Strength of Continuous Fiber Metal Matrix Composites

Abstract: A model to predict the high temperature ultimate strength of a continuous fiber metal matrix composite (CFMMC) has been developed. The model extends the work of Rosen by including high temperature processes such as matrix creep, fiber-matrix de bond, and the effects of randomly spaced fiber breaks which typically exist in the MMC prior to loading. A finite element model (FEM), developed in the form of a representative volume element (RVE), is used to calculate the time-dependent stress field surrounding a fiber break. Variables included in the calculation are process-related parameters such as the fiber diameter, the fiber-matrix interface strength, and interface roughness. Statistical analysis is used to infer the strength of a large composite sample from the stress analysis of a single break provided by the FEM.

Bundle pullout—a failure mechanism limiting the tensile strength of continuous fiber reinforced brittle matrix composites— and its implications for strength dependence on volume and type of loading

Abstract: T he mechanics of tensile failure of brittle matrix composites, uniaxially reinforced with continuous strong fibers, are investigated. A representative volume element description is used to establish a statistical theory for the strength of such composites. The load carrying capacity of such composite components is limited by a phenomenon called “bundle pullout”; a concentration of fiber failures in a small volume element enables whole clusters of fibers and matrix to be pulled out of the surrounding material due to its relatively low shear strength. The probability of initiating failure by this mechanism is governed by the stochastic distribution of flaws and fiber breaks. A certain number of failures have to occur in one (any) small volume element, the size of which is dependent on the global strain applied. This mechanism explains the dependence of the strength of components on both size and shape as well as on the type of loading. The theory presented is used to calculate the strength of a certain material in tension and in both 3-point and 4-point bending of two different cross-sections. The results agree very well in both level and range with reported experiments.

Characterization and mechanism of fibre matrix materials

Abstract: Composites which are used to form engineering materials and which consist of strong stiff fibres in a polymer resin require scientific understanding from which design procedures may be developed. The mechanical and physical properties of the composite are clearly controlled by their constituent properties and by the micro structural configurations. It is therefore necessary to be able to predict properties when parameter variations take place.

Application of Homogenization Theory and Limit Analysis to the Evaluation of the Macroscopic Strength of Fiber Reinforced Composite Materials

Abstract: The macroscopic strength domain of a composite material reinforced by long, parallel fibers is, in general, unknown but for its theoretical definition. In this note it is shown how a homogenization technique applied to yield design theory allows the derivation of two domains (in the space of macroscopic stresses) which are a lower and an upper bound to the composite strength domain. The dependence of these domains on the fiber content and on the shape of the fiber array is pointed out. Analytical equations for the approximate uniaxial macroscopic strength of composites with Drucker-Prager or Von Mises type matrix are derived. For more complex stress conditions, the relevant strength domains are numerically evaluated as well. The discrepancy between the two bounds is in many cases relatively small. In particular, the two bounds yield the same value for the uniaxial strength of the composite along the fiber direction, which by consequence is exactly determined.

Constitutive models for composite materials with interfaces

Abstract: A micromechanical model for composite materials is proposed in which the components are assumed to obey continuum laws, and where the interfaces between the components are modelled with nonlinear springs. A homogenization procedure is used to obtain a constitutive relation of a fictitious homogeneous material body, that best represents the real heterogeneous composite. The latter relation shows a strain-softening behaviour which gives rise to mesh-dependent finite element solutions. A nonlocal formulation of this macroscopic relation can solve these problems.

An anisotropic damage model for laminated composites

Abstract: A continuum damage model for laminated fiber-reinforced composite materials subjected to quasistatic loadings is presented. The damage state is characterized by a second order tensor. The model requires the establishment of the stress-strain relation for damaged materials and a damage evolution law in conjunction with a criterion for damage growth. The governing equations are solved with the finite element method. Quasistatic loading of laminate structures is investigated. A comparison with available experimental data shows good agreement.