Showing papers on "Representative elementary volume published in 1992"

A comparison of homogenization and standard mechanics analyses for periodic porous composites

Abstract: Composite material elastic behavior has been studied using many approaches, all of which are based on the concept of a Representative Volume Element (RVE). Most methods accurately estimate effective elastic properties when the ratio of the RVE size to the global structural dimensions, denoted here as ν, goes to zero. However, many composites are locally periodic with finite ν. The purpose of this paper was to compare homogenization and standard mechanics RVE based analyses for periodic porous composites with finite ν. Both methods were implemented using a displacement based finite element formulation. For one-dimensional analyses of composite bars the two methods were equivalent. Howver, for two- and three-dimensional analyses the methods were quite different due to the fact that the local RVE stress and strain state was not determined uniquely by the applied boundary conditions. For two-dimensional analyses of porous periodic composites the effective material properties predicted by standard mechanics approaches using multiple cell RVEs converged to the homogenization predictions using one cell. In addition, homogenization estimates of local strain energy density were within 30% of direct analyses while standard mechanics approaches generally differed from direct analyses by more than 70%. These results suggest that homogenization theory is preferable over standard mechanics of materials approaches for periodic composites even when the material is only locally periodic and ν is finite.

On the homogenization and the simulation of random materials

Abstract: It has become possible to compute the homogenized behaviour by simulation of a representative volume element submitted to homogeneous stress and strain boundary conditions. It is established with the aid of a recent ergodic theorem that the overall properties of a sufficiently large domain in an ergodic random medium are deterministic.

Compressive strength of unidirectional composites

Abstract: A combined analytical and semiempirical approach is used to obtain a simple equation for predicting the compressive strength of unidirectional composites. The formulation is based on the concept of microbuckling of a representative volume element in the composite, with the effect of shear deformation included. The validity of the equation proposed here is supported by good correlation with experimental data for E-glass, carbon, and boron fiber composites. 129 refs.

Residual Stresses in a Composite with Continuously Varying Young's Modulus in the Fiber/Matrix Interphase

Abstract: The incorporation of a realistic interphasial region into the micromechani cal analyses of composite systems is critical to the understanding of composite behavior. The interphase is usually modeled as a homogeneous region, despite the fact that it may have spatial property variations. A representative volume element with three cylinders (concentric cylinder assemblage) is considered here for the determination of local thermal stresses in a composite. Three different expressions are used to simulate the Young's modulus variations in the interphase, while the Poisson's ratio and coefficient of thermal expansion of the interphase region are chosen to be constant. The governing field equa tions in terms of displacements are solved in closed form. It is found that, though the solu tion is "dilute," the Young's modulus variations have a distinct effect on the local thermal stresses.

Analytical Models of Stress Transfer in Unidirectional Composites and Cross-Ply Laminates, and Their Application to the Prediction of Matrix/Transverse Cracking

Abstract: Many unidirectional composites are made using carbon fibres which have anisotropic thermo-mechanical properties. There is a need, therefore, to take account of this anisotropy when making predictions of the properties of damaged composites. For the more general case when the fibres and matrix are both transverse isotropic solids, a relatively simple shear-lag approach to understanding stress transfer between fibres and matrix is presented. A similar approach is used to develop a shear-lag model of stress transfer between neighbouring plies in a cross-ply laminate containing transverse cracks. As to be expected stress transfer is governed by second order ordinary differential equations which are easily solved. It is shown how a more realistic model of stress transfer for unidirectional composites must be modified when the fibres and matrix of the composite are transverse isotropic solids. Reference is made to more realistic models of stress transfer in cross-ply laminates containing transverse cracks in the 90 ply. The more realistic models lead to fourth order differential equations. Such models are thus more flexible than shear lag models in that a greater variety of boundary conditions can be satisfied.

A Three-Dimensional Statistical Micromechanical Theory for Brittle Solids with Interacting Microcracks:

Abstract: A three-dimensional statistical micromechanical theory is presented to in- vestigate effective elastic moduli of brittle solids with many randomly located, penny- shaped microcracks. The macroscopic constitutive relations are statistically and microme- chanically derived by taking the ensemble average over all possible realizations which feature the same statistical distribution of microcracks. Approximate analytical solutions of a two-microcrack interaction model are presented to account for pairwise microcrack interaction among many randomly located, aligned microcracks. Therefore, the ensemble- averaged stress perturbations due to microcrack interaction can be constructed in closed- form. The overall effective compliances of microcrack-weakened brittle solids are derived by further taking the volume average of the ensemble-averaged stress-strain relations over the entire mesostructural domain of a representative volume element. Some numerical ex- amples are given to illustrate the behavior of the proposed method. Comparison with some existing methods is also appended. Finally, a higher-order ensemble-average formulation of microcrack interaction is briefly discussed. The proposed framework is fundamentally different from existing "effective medium methods" which do not depend on microcrack locations and configurations. It is emphasized that no Monte Carlo simulations are neces- sary in the proposed framework.

Axisymmetric Micromechanical Stress Fields in Composites

Abstract: Using Reissner’s variational theorem in conjunction with an equilibrium stress field in which the r-dependence is assumed, we formulate an approximate model to define the thermoelastic response of a concentric cylindrical body under axisymmetric boundary conditions The interfaces between continguous cylinders may be either continuous or subjected to mixed traction and displacement boundary conditions The external surfaces may be subjected to mixed boundary conditions that are consistent with the model assumptions but otherwise arbitrary The model is designed to analyze experiments such as pullout tests and also to represent the concentric cylinder model of a composite representative volume element and it contains the capability to enhance the accuracy of a given numerical solution An illustrative thermal stress problem is solved and used to compare with an existing elasticity solution and to examine some of the details regarding sensitivity to model parameters

Analytical Modeling of Micromechanical Stress Variations in Continuos Fiber-Reinforced Composites

Abstract: The micromechanical stresses that arise within each phase (i.e., fiber, matrix, interphase) of a fiber-reinforced composite material under three-dimensional mechanical and hygrothermal loading are investigated analytically. An elasticity solution is developed for the case in which the fibers are arranged in a regular, periodic array. The model accounts for fiber-fiber interactions, multiple interphases, cylindrically-orthotropic and temperature- and moisture-dependent material properties. In addition to modeling micromechanical stress states, the model is used to calculate composite properties using knowledge of the stress variation in the individual phases present in the material system. The elasticity solutions developed herein are more accurate, more computationally efficient, and less numerically sensitive (in the modeling of thin interphases) than an equivalent finite element model. Results presented in this paper include the effect of interphase stiffness on the stress distribution in a composite lamina subjcted to tranverse loading. The study suggests that there is an “optimum” interphase stiffness which minimizes the stress concentration in the matrix region. Comparisons performed with a concentric cylinder model suggest that an appropriate interphase region can reduce the extreme stress variations caused by fiber-fiber interaction.

Phenomenological Aspects of Damage

Abstract: The damage of materials is the progressive physical process by which they break. The mechanics of damage is the study, through mechanical variables, of the mechanisms involved in this deterioration when the materials are subjected to loading. At the microscale level this is the accumulation of microstresses in the neighborhood of defects or interfaces and the breaking of bonds, which both damage the material. At the mesoscale level of the representative volume element this is the growth and the coalescence of microcracks or microvoids which together initiate one crack. At the macroscale level this is the growth of that crack. The two first stages may be studied by means of damage variables of the mechanics of continuous media defined at the mesoscale level. The third stage is usually studied using fracture mechanics with variables defined at the macroscale level.

Some Aspects of Continuum Damage Mechanics Applied to Polymer and Ceramic Matrix Composites

Abstract: Certain recently observed characteristics in the response of cracked cross ply laminates of new toughened polymer matrix composites and of ceramic matrix composites are described. A continuum damage mechanics analysis of these aspects of damage-response relationships is presented as an extension of this author’s previous work.

Hydrodynamics and dispersion near bounding surfaces of porous media

Abstract: This chapter provides an overview of hydrodynamics and dispersion near bounding surfaces of a porous media. There are many problems in which fluid flow and heat transfer occur through the bounding surfaces of the porous media. These velocity and heat flux vectors may be parallel or perpendicular to the bounding surfaces or may be at an arbitrary angle. The chapter reviews the recent treatments of fluid and heat flow near the interface of a porous and a plain medium. The plain medium is occupied by a solid or a fluid. The chapter also discusses the Brinkman continuum treatment of the interface, the Beavers-Joseph slip treatment, and a direct simulation of the flow for a two-dimensional porous medium. The chapter also discusses the recent results for dispersion in the porous media, and reviews the anisotropy in the dispersion tensor and its nonuniformity near the solid-bounding surfaces. A general energy equation for porous media that includes the dispersion effects can be obtained by averaging the local energy equation over a representative elementary volume.. The chapter discusses a semi-empirical treatment based on the velocity slip, a theoretical treatment for interfacial hydrodynamics, another semi-empirical treatment based on an effective viscosity, and a general, two-dimensional direct simulation. Their experimental results showed that the slip velocity could be correlated with the square root of the permeability and with the velocity gradient in the fluid layer evaluated at the interface. The idea behind the construction of such a correlation was the notion of continuity in the shear stress across the interface. The nonuniformities in the phase distribution at and near the bounding surface, and its effects on the fluid flow and heat transfer were most significant if the primary heat transfer was through these surfaces.

On plastic zones and fracture strengths in some metal matrix composites

Abstract: Based on a Dugdale-type approach for cancelling the singularity at the crack tip, an attempt was made to predict the overall stress distributions, as well as the size of the plastic zone found in notched metal matrix composite plate. The work was developed by considering notched, unidirectionally reinforced fibrous metal matrix composite plate under uniform normal tension load. Predictions of the proposed method were compared with the experimental results, and a fairly good agreement was observed. Simple closed form expressions for the local stresses and fracture strength are provided.

Micromechanics as a Basis for Damage Mechanics

Abstract: A review of the micromechanics of composites is given herein. This includes the concepts of geometric scales, the representative volume element, and volume averaging of state variables in the representative volume element. These concepts are first reviewed for composites composed of linear elastic constituents and then extended to include the case of composites with time dependent microcracks. A brief review is then given of recent micromechanics solutions which include the effects of damage. Results are discussed for both laminated composites and composites with one or more inelastic phases.

Predicting properties of composite materials

Abstract: Composite materials offer significant advantages over conventional materials in certain automotive applications. For example, metal-matrix composite of SiC particles in Al can be used to reduce the weight of connecting rods and pistons, while maintaining the required strength, resulting in improved fuel economy. Polymer composite with glass fibers may offer advantageous substitutes for various automobile parts and is an ongoing research activity in the automotive industry. The potential widespread use of a variety of composites has focused attention on predicting their mechanical properties from a knowledge of the constituents’ properties. What one would actually like to predict is the average of the constituents’ moduli; these are called effective moduli of the medium.

Green’s Function Method for Calculation of Stress Fields in Composite Materials

Abstract: A Green’s function method is described for calculating the displacement and the stress fields in an elastic composite solid. The Green’s function is obtained by solving the equations of elastic equilibrium (Christoffel’s equation) for a delta function force with prescribed boundary conditions. The Green’s function then gives the solution of the Christoffel’s equation for any integrable force distribution. The solution gives the displacement field from which the stress field is calculated- The discontinuities or defects in the solid are represented by appropriate boundary conditions. The method is illustrated by applying it to a composite solid containing a plane interface and a (45/-45) fiber-reinforced composite having a free surface normal to the interface.

A Continuum Damage Model for the Description of High Strain Rate Deformations

Abstract: The behaviour of metals under dynamic loading is described by introducing a continuum damage theory, which is also valid for nonisothermal processes. Microstructural changes are modelled by internal variables, e.g. damage due to shear bands is described with a second order tensor. Dislocation induced viscoplastic deformations are modelled using a flow rule of the overstress-type.