Showing papers on "Representative elementary volume published in 2000"

Pore scale study of flow in porous media: Scale dependency, REV, and statistical REV

Abstract: Flow in porous media is studied at the pore-scale with lattice Boltzmann simulations on pore geometries reconstructed from computed microtomographic images. Pore scale results are analyzed to give quantities such as permeability, porosity and specific surface area at various scales and at various locations. With this, some fundamental issues such as scale dependency and medium variability can be assessed quantitatively. More specifically, the existence and size of the well known concept, representative elementary volume (REV), can be quantified. It is found that the size of an REV varies spatially and depends on the quantity being represented. For heterogeneous media, a better measure may be the so called “statistical REV”, which has weaker requirements than does the deterministic REV.

Microstructure and transport properties of porous building materials. II: Three-dimensional X-ray tomographic studies

Abstract: Three-dimensional X-ray microtomography is used to obtain three-dimensional images of the microstructure of two types of brick. The images are processed to remove the noise (random and circular pattern) and then thresholded to match the porosity determined experimentally. The 3-D binary images are then analyzed to estimate their vapor diffusivity and air permeability to compare to experimental data published in part one of this report. Care must be taken in obtaining the tomographic images at a resolution that both enables isolation and quantification of the pores of interest and provides a representative elementary volume for the transport property calculations. In general, the agreement between computed and measured properties is reasonable, suggesting that X-ray microtomography can provide valuable information on the characteristics and properties of the pore networks developed in these porous building materials. A preliminary evaluation indicates that the Katz-Thompson relationship between permeability, diffusivity, and pore size is valid for these materials.

Study of poroelasticity material coefficients as response of microstructure

Abstract: Using an asymptotic homogenization method, the effect of the porous medium microstructure on the values of poroelastic coefficients is studied in this paper First, the Biot’s poroelasticity theory and general relations linking the macroscopic poroelastic coefficients with the averaged micromechanical solutions are recalled Considering a variational formulation of appropriate boundary values problems stated for the representative volume element, microstructural parameters affecting the values of poroelastic coefficients are identified In order to clarify specific roles of some relevant microstructure parameters, numerical investigations for some simplified pores geometries are presented The numerical results obtained clearly show a strong dependence of the poroelastic coefficients on the internal geometry of pores as well as on the global porosity of the medium In the last part, based on the micromechanics analysis, the definition of initial plastic yield condition for saturated porous media is introduced and a new interpretation of the effective stress concept in inelastic domain is proposed Copyright © 2000 John Wiley & Sons, Ltd

Micromechanics-based variational estimates for a higher-order nonlocal constitutive equation and optimal choice of effective moduli for elastic composites

Abstract: A generalization of the Hashin–Shtrikman variational formulation to random composites, due to J.R. Willis, is employed to derive micromechanics-based variational estimates for a higher-order nonlocal constitutive equation relating the ensemble averages of stress and strain, for a class of random linear elastic composite materials. We analyze two-phase composites with any isotropic and statistically uniform distribution of phases (which themselves may have arbitrary shape and anisotropy), within a formulation accounting for one- and two-point probabilities, and derive an explicit nonlocal constitutive equation that includes terms up through the fourth gradient of average strain. The analysis is carried out first for an arbitrary comparison medium. Then, a new approach is outlined and applied which employs the nonlocal correction to determine the optimal choice of comparison medium, and hence the optimal effective modulus tensor (as well as the optimal tensor coefficients of the nonlocal terms) for the amount of statistical information employed. The new higher order analysis provides a highly accurate nonlocal constitutive equation, valid down to quite small volume size scales and to rather strong variations of average strain with position. Among several applications illustrated, it permits accurate analytical assessment of the remarkably small predictions derived by Drugan and Willis (1996. Journal of the Mechanics and Physics of Solids 44, 497–524) of the minimum representative volume element (RVE) size needed for accuracy of the standard constant-effective-modulus macroscopic constitutive equation for elastic matrix-inclusion composites that have spherical inclusions/voids. It also affords an analytical assessment of the improved (i.e., reduced) minimum RVE size scale, compared to a standard constant-effective-modulus constitutive equation, to which the leading-order nonlocal constitutive equation derived by Drugan and Willis applies. This improvement is shown to be dramatic in some example cases.

Evaluation of laboratory dolomite core sample size using representative elementary volume concepts

Abstract: The adequacy for laboratory testing of four dolomite cores from the Culebra Dolomite of the Rustler Formation at the Waste Isolation Pilot Plant near Carlsbad, New Mexico, were evaluated using representative elementary volume (REV) theory. Gamma ray computerized tomography created three-dimensional grids of bulk density and macropore index over volumes from 1.4 × 10−7 to 1.6 L. Three different methods for both volume averaging and REV analysis were applied and compared. Both density and macropore index converged to single values with increasing volume, which meets the most common qualitative definition of a REV. Statistical test results for the relatively homogeneous samples indicate that volumes larger than 1 to 7 mL have constant properties. Contrarily, a highly varied sample required 250 and 373 mL to achieve invariant density and macropore characteristics, respectively. Prismatic volume averaging was found to be better than slice averaging, while a qualitative test for the REV provided similar results as a rigorous statistical method. All cores were larger than the REV but were significantly different from one another. This implies that multiple cores are necessary to determine the entire range of transport properties within the rock.

3D analysis of stress transfer in the micromechanics of fiber reinforced composites by using an eigen-function expansion method

Abstract: This paper presents an exact solution for an inhomogeneous, transversely isotropic, elastic circular cylinder subjected to axisymmetric force and displacement boundary conditions. The solution is obtained on the basis of an eigen-function expansion method and can satisfy all the boundary conditions prescribed on the curved and end surfaces of the cylinder. It can be used directly in the micromechanical analysis of fiber reinforced composites to investigate the typical Representative Volume Element (RVE). The element consists of a combined circular cylinder composed of a solid inner circular cylinder of transversely isotropic fiber and a concentric outer circular cylinder of isotropic matrix material. Using this solution, all the stress and displacement components of both the inner fiber and the outer matrix, and hence the stress transfer in the interface between the fiber and matrix, are expressed analytically. The numerical results presented show that stress concentration occurs near the ends of the cylinder where external forces are applied.

Micromechanical modeling of the martensitic transformation induced plasticity in steels

Abstract: A micromechanical model is developed to predict the overall behavior of a representative volume element (RVE) of a material undergoing non-thermoelastic martensitic phase transformation. The theoretical approach is based on the evaluation of the energy dissipation using the concept of moving boundaries. Assuming an ellipsoidal growth of martensitic microdomains and taking into account some physical aspects typical of martensitic phase transformation in ductile materials, the obtained dissipation is reduced to a more simple form leading one to choose the volume fractions of each possible martensitic variants as the internal variables describing the microstructure evolution. The nucleation and growth conditions of a martensitic microdomain are derived using, simultaneously, the classical inelastic inclusion problem together with interface operators. The obtained results are combined with kinetics and kinematics studies to derive the constitutive equation of an austenitic single crystal from which the overall behavior of a polycrystalline RVE is deduced using the self-consistent scale transition method. Comparison with experimental data shows good agreement.

Fluctuations, Correlation and Representative Elementary Volume (REV) in Granular Materials

Abstract: In general, the mechanics of granular matter is described using continuum mechanics approach; this requires to introduce the concepts of stress and strain, which are averaged quantities, so that this needs also to introduce the notion of representative elementary volume (REV) above which averaged quantities have some physical meaning. As local quantities fluctuate spatially in granular matter; a local measure of stress and strain shall exhibit fluctuations too, whose typical amplitude depends on the sampling size L. This paper discusses this problem and the causes for large scale correlation. Mean stress σ applied to a plane surface of size L² is calculated and its fluctuation amplitude δσ is found when local forces are not correlated; it is found that δσ/ σ ∝ 1/L . It is shown also that large scale fluctuations of stress can always be interpreted as an inhomogeneous stress field and that static equilibrium modifies the mean stress applied to a rod (in 2d), even if it does not perturb the contact force distribution. This last result is compared to experiment; which indicates that the number N of contacts per rod (in 2d) is 2

The Fiber Composite With Nonlinear Interface—Part I: Axial Tension

Abstract: This paper treats the effective axial tension response of a composite consisting of fibers that debond from the matrix according to nonlinear Needleman-type cohesive zones. A second, related paper (Part II) treats effective antiplane shear response. The composite cylinders representation of a representative volume element (RVE) is employed throughout. For axial tension loading a simple rotationally symmetric boundary value problem for a single composite cylinder is solved. Bounds on the total potential energy and the total complementary energy are shown to coincide and an exact solution for axial extension and Poisson contraction of an RVE of the composite is obtained. Nonlinear interfacial debonding, however, is shown to have a negligible effect on extensional response and only a small, though potentially destabilizing, effect on Poisson contraction response.

Local reaction and diffusion in porous media transport models.

Abstract: The problem of when advection-dispersion models apply for reactive transport in porous media is addressed. Assuming local mass balances, including arbitrary homogeneous and interfacial chemical reactions, are known, volume averaging is applied to obtain a set of equations for the average concentrations. It is shown that timescale constraints must be satisfied in addition to the well-known length-scale constraint needed for volume averaging. The timescale for simulation must be longer than a diffusion timescale in the representative elementary volume, t/TD ≫ 1. In addition, interfacial reaction timescales must be larger than meaningful diffusion timescales, Tr/TD ≫ 1. When these constraints are satisfied, the usual dispersion coefficient exists and is time-invariant and independent of reactions. Reaction rate expressions and all mass transfer fluxes can be expressed in terms of the average concentrations of the macroscopic model. Even when surface reactions are fast, it is shown that the fluid volume can be subdivided into small enough regions such that the appropriate time constraint Tr/TD ≫ 1 is satisfied. An average model can be obtained that includes mass transfer resistance expressed in terms of a mass transfer coefficient. The mass transfer coefficient is defined and is shown to depend only on the geometry of the porous medium and the flow field. This work provides a theoretical basis for the commonly used advection-dispersion models for reactive transport at the Darcy scale and provides both length-scale and timescale constraints for when they apply.

Multi-inclusion method for finite deformations : exact results and applications

Abstract: For linearly elastic heterogeneous solids, averaging theorems are developed by Nemat-Nasser and Hori (S. Nemat-Nasser, M. Hori, Micromechanics: Overall Properties of Heterogeneous Solids, Elsevier, Amsterdam, 1993; and S. Nemat-Nasser, M. Hori, J. Eng. Mater. Technol. 117 (1995) 412), using a multi-inclusion model. This model is based on the observation that the average strain and stress in any annulus of a nested sequence of ellipsoids embedded in an infinite homogeneous solid, can be computed exactly when each annulus undergoes a uniform transformation with an associated transformation strain different from the others; the central ellipsoid may have a non-uniform transformation strain. This result generalizes the pioneering observation of Tanaka and Mori (K. Tanaka, T. Mori, J. Elast. 2 (1972) 199; and T. Mori, K. Tanaka, Acta Met. 21 (1973) 571) who showed that the average strain and stress in the region between two similar and coaxial ellipsoids in an infinite uniform elastic solid, are zero for any transformation strain within the inner ellipsoid; see also Benveniste, and, and Mori and Wakashima (T. Benveniste, Mech. Mater. 6 (1987) 147; and T. Mori, K. Wakashima, Successive iteration method in the evaluation of average fields in elastically inhomogeneous materials, Micromechanics and Inhomogeneity — The T. Mura 65th Anniversary Volume, Springer, New York, 1990, pp. 269–282). Similar results apply to the finite deformation problems, provided that the nominal stress rate and the rate of change of the deformation gradient, (measured relative to any arbitrary state) are used as the dynamical and kinematical variables; see Nemat-Nasser (S. Nemat-Nasser, Mech. Mater. 31, (1999) 493) for a comprehensive account of a rigorous treatment of the transition from micro- to macro-variables of a representative volume element of a finitely deformed aggregate. An exact method for homogenization of an ellipsoidal heterogeneity in an unbounded finitely deformed homogeneous solid, is developed, using the generalized Eshelby tensor. It is shown that many results for single-, double-, and multi-inclusion problems in linear elasticity (see S. Nemat-Nasser, M. Hori, Micromechanics: Overall Properties of Heterogeneous Solids, Elsevier, Amsterdam, 1993; and S. Nemat-Nasser, M. Hori, J. Eng. Mater. Technol. 117 (1995) 412), also apply to the finite-deformation rate problems, provided suitable kinematical and dynamical variables are used. The problem of the double inclusion is considered and exact expressions are given for the average field quantities, taken over the region between the two, as well as within each ellipsoidal domains, one containing the other, when arbitrary eigenvelocity gradients are prescribed within an arbitrary region contained in the inner ellipsoid. This generalizes to the fully nonlinear, finitely-deformed, elastoplastic case, the Tanaka–Mori (K. Tanaka, T. Mori, J. Elast. 2 (1972) 199) result, and the double inclusion result of Nemat-Nasser and Hori (S. Nemat-Nasser, M. Hori, Micromechanics: Overall Properties of Heterogeneous Solids, Elsevier, Amsterdam, 1993; and S. Nemat-Nasser, M. Hori, J. Eng. Mater. Technol. 117 (1995) 412), which have been developed for linearly elastic solids. The application of the exact results to the problem of estimating the overall mechanical response of a finitely deformed heterogeneous representative volume element (RVE) is outlined and the overall effective pseudo-modulus tensor of the RVE is calculated for rate-independent elastoplastic materials.

Parameter identification for a macroscopic granular soil model applied to dense Berlin Sand

Abstract: The present paper outlines the macroscopic stress-strain behaviour of cohesionless soils by the example of dense Berlin Sand. Granular materials as sand show very complex stress-strain relations depending on the stress state and the load history. Examination of these relations is performed at a representative elementary volume (REV) consisting of a sufficient amount of particles. In triaxial tests quasi-static loads are applied to axisymmetric samples, representing the REV, with simultaneous measurement of the response of the granular structure. Within the constitutive equations of the elasto-plastic model, the elastic response is described by a materially non-linear elasticity law. Plastic deformations are considered in the context of a single-surface yield function with isotropic hardening properties. Non-associated flow is realized with an additional plastic potential function. Isotropic behaviour is assumed and the study is restricted to small strains. The parameter identification for the model under study is shown for dense Berlin Sand on the basis of triaxial tests.

Unified Micromechanical Model for Estimating Elastic, Elasto-Plastic and Strength Behaviors of Knitted Fabric Reinforced Composites

Abstract: This work presents a unified micromechanical model for estimating the three-dimensional elastic, elasto-plastic, and strength properties of a plain weft knitted fabric reinforced composite. A bridging matrix is used to correlate the stresses generated in fiber and matrix phases in a representative volume element (RVE) of the composite. With this bridging matrix, the stress in each constituent phase is explicitly expressed as a function of the overall applied stress and all the mechanical properties of the composite are easily derived from the knowledge of the physical and geometric properties of the constituents. The total stress-strain behavior of the composite is obtained based on the stress-strain behaviors of the constituent materials. The composite strength is then defined as the overall applied stress corresponding to which the maximum normal stress in either the fiber or the matrix attains its ultimate value. Good correlation between theoretical and experimental results has been found.

Modelling of solid‐phase sintering of hardmetal using a mesomechanics approach

Abstract: The mesomechanics approach presented in this paper aims at enhancing the understanding of, as well as providing a predicting capability for, the densification process in cemented carbides due to solid-phase sintering. The major mesostructural constituents are tungsten carbide (WC) particles and large pores, which are embedded in a contiguous cobolt (Co) matrix. A preprocessor code, which is based on Voronoi polygonization, was developed to generate the morphology with prescribed area fraction and size distribution of the constituents. In a continuum model, the ‘driving force’ that brings about the densification is the sintering stress, which is given a rational thermodynamic definition in the paper. This stress represents the boundary loading of a representative volume element (RVE) at free sintering, i.e. in the absence of macroscopic stresses. In such a volume element (or unit cell) the constituents WC and Co are assumed as viscoplastic non-porous solids. A generalized Bingham model (of Norton-type with hardening) seems to be sufficient to represent the creep properties, which are assumed to be of dislocation as well as of diffusion type. The temperature dependence of certain material parameters is discussed. Thermal expansion is accounted for. The developed algorithm was implemented in the commercial FE-code ABAQUS. Finally, the simulation results are compared with experimental results from the sintering of free as well as uniaxially loaded specimens. Copyright © 2000 John Wiley & Sons, Ltd.

Finite Element Investigation of the Effect of Particle Distribution on the Uniaxial Stress–Strain Behavior of Particulate‐Reinforced Metal‐Matrix Composites

Abstract: A multi-particle 2D finite element model of a 20% particulate reinforced metal-matrix composite was developed on a statistical basis taking into account the correlations between the position, size and orientation of the ceramic particles in the matrix. The stress–strain curves in tension and compression given by the clustered multi-particle model are compared with the curves obtained from one-particle unit cell simulations. It is shown that clustering of particles increases the plastic strain accumulated in the matrix leading to a higher strain hardening and thus to a higher flow stress. The size of the representative volume element (RVE) should be at least equal to the correlation length of the geometrically relevant correlation functions, which was 2.4 times larger than the average interparticle distance for the experimentally studied case. Reasonable agreement is obtained between computed residual strains and data available in the literature. © 2001 Elsevier Science B.V. All rights reserved.

Effective properties of piezoelectric polycrystals

Abstract: Effective elastic, dielectric, and piezoelectric properties of piezoceramics are calculated with the finite element method using a representative volume element. A polycrystal is considered which consists of single-domain crystals defined by a 2D random Voronoi tessellation. The random orientations of the domains in the unpoled state are described by an isotropic orientation distribution. To model poled stats, simple switching criteria are applied, so that e.g. the polarization direction of the individual grains fall within a cone of a specific angle about the poling direction. Homogeneous boundary conditions are imposed and the volume averages of the local mechanical and electrical fields are calculated. The effective material properties can then be calculated as a function of the single crystal moduli and the poling degree. The main advantages of this procedure are the more realistic random geometry and the fact that interactions among the grains are taken into account. Interest is focused on basic questions such as the influence of the boundary conditions, the necessary size of the representative volume element, and the variation of the effective moduli due to the stochastic model, in particular due to the orientation distribution. The numerical results are compared to those of analytical models.

The Fiber Composite With Nonlinear Interface—Part II: Antiplane Shear

Abstract: This paper treats the effective antiplane shear response of a composite consisting of fibers that interact with the matrix through nonlinear Needleman-type cohesive zones. The first paper (Part I) examines effective axial tension response. The composite cylinders representation of a representative volume element (RVE) is employed throughout. For antiplane shear loading the elastic field solution for a single composite cylinder is found in the form of a series expansion whose coefficients are governed by an infinite set of nonlinear equations. Bounds on the total potential energy and the total complementary energy of an RVE do not coincide although they are shown to differ by a term of order O(c 4 ) where c is the fiber volume concentration. Interaction effects due to finite volume concentration, coupled with nonlinear interface characterization, are shown to precipitate instability in composite response.

Simulation of the thermomechanical behavior of shape memory alloys under multiaxial nonproportional loading

Abstract: The paper presents a mathematical description of shape memory alloys within the framework of continuum mechanics, where the spatially multidimensional case is considered. A new simple model is introduced, which correctly describes not only all well-studied shape memory effects but also the more complex behavior. The key idea is a new set of internal state variables, which are averaged values for a representative volume element of the polycrystalline material. A tensor-valued variable describes the state of orientation of martensite. The relative volume fraction of stress induced martensite is defined by taking a certain tensor norm of this internal variable. A free energy is chosen and thermodynamical forces are derived. These forces are sufficient to define the onset of the phase transitions, so that we do not need to introduce transition surfaces explicitly within the evolution equations. Finite element discretizations for the approximation of the field variables and finite difference approximations for the time integration of local variables are explained.© (2000) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Failure Investigation of Fiber-Reinforced Composite Materials by Shakedown Analysis

Abstract: A methodology is proposed to investigate failure of composite materials under thermo-mechanical variable loads. By using the homogenization technique of periodic media, the plastically admissible ranges of macroscopic stresses can be found from the shakedown analysis of representative volume elements of the considered composite.

Integrated Micro/Macro Approach for Laminate Composite Analysis: Applications to Turbine Blades

Abstract: An integrated micro/macro mechanical procedure for structural analysis of unidirectional metal matrix composites is proposed. The micromechanical analysis is performed with the composite cylinder assemblage model or the representative volume element approach. The macroscopic stress-strain relation is based on a modification of the vanishing fiber diameter theory and the concept of a smeared finite element. The fibers are considered to be linear elastic, and the matrix viscoplastic behavior is described by the Bodner and Partom model (Bodner, S. R., Review of Unified Elastic-Viscoplastic Theory, Unified Constitutive Equations for Plastic Deformation and Creep of Engineering Alloys, edited by A. Miller, Elsevier Science, New York, 1987, pp. 77-106). The overall composite behavior is assumed to be elastic-viscoplastic. Two types of material systems are used: SCS-6/Ti-15-3 and SCS-6/Tiβ21-S. Both sets of material systems exhibit isotropic and/or kinematic hardening under specific conditions of temperature and loading. The proposed methodology is validated with experimental and analytical results available in the literature. After the validation process, this methodology is applied to a three-dimensional finite element model of a [0/90] 2s SCS-6/Tiβ21-S turbine blade tip. Aerodynamic and centrifugal loading are considered to be acting at the same time. The turbine blade tip inelastic strain field is presented.

Elastic Modulus of Particulate Composites Using a Multiphase Model

Abstract: In this paper a theoretical model for the evaluation of the elastic modulus of the particulate composites is presented. The model takes into account the influence of neighbouring spherical inclusions on the stiffness of the composite material consisting of the matrix and filler. A microstructural composite model reproducing the basic cell of the composite at a microscopic scale was transformed to a 4-phase representative volume element in order to apply the classical theory of elasticity to this simplified model of a composite-unit cell.Theoretical results based on this model were compared with experimental results carried out with iron-particle filled epoxy resin composites and also with other theoretical results derived from other works.

On Porous Soil Materials Saturated with a Compressible Pore-fluid Mixture

Abstract: Geomaterials can be understood as consisting of a porous solid skeleton, with the pores saturated by a fluid. The fluid may be water, air or, a water-air mixture. The mechanical behaviour of the whole soil body, consisting of the soil skeleton and the pore-fluid, is governed by the properties of its constituents. Since the exact structure of the pore-space is not known, a homogenization over a representative elementary volume is required in order to obtain a macroscopic approach. The description of the motion of a multiphase-mixture of immiscible constituents can be performed by the Theory of Porous Media (TPM). Based on the classical mixture theory, the TPM deals with superimposed continua. Additionally, the information of the structure is included via the concept of volume fractions [1–3]. Here, the soil skeleton material is assumed to be incompressible, whereas the pore-fluid mixture is compressible according to its composition of water and air by variable volume fractions. The pressure-density-function of the pore-fluid is derived within the framework of the TPM. Furthermore, the viscous pore-fluid allows for a linear momentum exchange (interaction force) between the constituents, which, under special assumptions, leads to the well-known Darcy law.