Showing papers on "Representative elementary volume published in 2002"

Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme

Abstract: A gradient-enhanced computational homogenization procedure, that allows for the modelling of microstructural size effects, is proposed within a general non-linear framework. In this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element (RVE). This enables us to incorporate the microstructural size and to account for non-uniform macroscopic deformation fields within the microstructural cell. Every microstructural constituent is modelled as a classical continuum and the RVE problem is formulated in terms of standard equilibrium and boundary conditions. From the solution of the microstructural boundary value problem, the macroscopic stress tensor and the higher-order stress tensor are derived based on an extension of the Hill-Mandel condition. This automatically delivers the microstructurally based constitutive response of the higher-order macro continuum and deals with the microstructural size in a natural way. Several examples illustrate the approach, particularly the microstructural size effects.

A numerical approximation to the elastic properties of sphere-reinforced composites

Abstract: Three-dimensional cubic unit cells containing 30 non-overlapping identical spheres randomly distributed were generated using a new, modified random sequential adsortion algorithm suitable for particle volume fractions of up to 50% The elastic constants of the ensemble of spheres embedded in a continuous and isotropic elastic matrix were computed through the finite element analysis of the three-dimensional periodic unit cells, whose size was chosen as a compromise between the minimum size required to obtain accurate results in the statistical sense and the maximum one imposed by the computational cost Three types of materials were studied: rigid spheres and spherical voids in an elastic matrix and a typical composite made up of glass spheres in an epoxy resin The moduli obtained for different unit cells showed very little scatter, and the average values obtained from the analysis of four unit cells could be considered very close to the “exact” solution to the problem, in agreement with the results of Drugan and Willis (J Mech Phys Solids 44 (1996) 497) referring to the size of the representative volume element for elastic composites They were used to assess the accuracy of three classical analytical models: the Mori–Tanaka mean-field analysis, the generalized self-consistent method, and Torquato's third-order approximation

Microstructural Randomness Versus Representative Volume Element in Thermomechanics

Abstract: Continuum thermomechanics hinges on the concept of a representative volume element (RVE), which is well defined in two situations only: (i) unit cell in a periodic microstructure, and (ii) statistically representative volume containing a very large (mathematically infinite) set of microscale elements (eg, grains) Response of finite domains of material, however, displays statistical scatter and is dependent on the scale and boundary conditions In order to accomplish stochastic homogenization of material response, scale-dependent hierarchies of bounds are extended to dissipative/irreversible phenomena within the framework of thermomechanics with internal variables In particular, the free-energy function and the dissipation function become stochastic functionals whose scatter tends to decrease to zero as the material volume is increased These functionals are linked to their duals via Legendre transforms either in the spaces of ensemble average velocities or ensemble-average dissipative forces In the limit of infinite volumes (RVE limit (ii) above) all the functionals become deterministic, and classical Legendre transforms of deterministic thermomechanics hold As an application, stochastic continuum damage mechanics of elastic-brittle solids is developed ©2002 ASME

Topological Characterization of Porous Media

Abstract: It is an attractive approach to predict flow and in based on direct investigations of their structure. The most crucial property is the of the structure because it is difficult to measure. This is true both at the pore scale, which may be represented as a binary structure, and at a larger scale defined by continuous macroscopic state variables as phase density or. At the pore scale a function is introduced which is defined by the as a function of the pore diameter. This function is used to generate of the porous structure that allow to predict bulk hydraulic properties of the material. At the continuum scale the structure is represented on a grey scale representing the porosity of the material with a given resolution. Here, topology is quantified by a connectivity function defined by the Euler characteristic as a function of a porosity threshold. Results are presented for the structure of natural soils measured by. The significance of topology at the continuum scale is demonstrated through numerical simulations. It is found that the effective permeabilities of two heterogeneous having the same auto-covariance but different topology differ considerably.

Multi-scale constitutive modeling of the small deformations of semi-crystalline polymers

Abstract: A multi-scale constitutive model for the small deformations of semi-crystalline polymers such as high density Polyethylene is presented. Each macroscopic material point is supposed to be the center of a representative volume element which is an aggregate of randomly oriented composite inclusions. Each inclusion consists of a stack of parallel crystalline lamellae with their adjacent amorphous layers. Micro-mechanically based constitutive equations are developed for each phase. A viscoelastic model is used for the crystalline lamellae. A new nonlinear viscoelastic model for the amorphous phase behavior is proposed. The model takes into account the fact that the presence of crystallites confines the amorphous phase in extremely thin layers where the concentration of chain entanglements is very high. This gives rise to a stress contribution due to elastic distortion of the chains. It is shown that the introduction of chains' elastic distortion can explain the viscoelastic behavior of crystalline polymers. The stress contribution from elastic stretching of the tic molecules linking the neighboring lamellae is also taken into account. Next, a constitutive model for a single inclusion considered as a laminated composite is proposed. The macroscopic stress-strain behavior for the whole RVE is found via a Sachs homogenization scheme (uniform stress throughout the material is assumed). Computational algorithms are developed based on fully implicit time-discretization schemes. (C) 2002 Elsevier Science Ltd. All rights reserved.

A Quantitative study of minimum sizes of representative volume elements of cubic polycrystals—numerical experiments

Abstract: The concept of representative volume element (RVE) plays a key role in correlating the properties of microscopically heterogeneous materials with those of their macroscopically homogenized ones. However, up to now little quantitative knowledge is available about RVE scales or sizes of various engineering materials, which have been becoming a necessity due to the rapid development of, for instance, microelectromechanical systems. A new and convenient definition of the minimum RVE size is introduced. Then more than 500 kinds of cubic polycrystalline material in the planar stress state are numerically tested. The major finding from these numerical experiments is that the RVE size for the effective shear modulus (as well as the Young's modulus) depends roughly linearly upon the anisotropy degree of the single crystal, while the effective area modulus does not. For the latter observation a theoretical proof is also given. With a maximum relative error 5%, all the materials tested (with one exception) have a minimal RVE size of 20 or less times as large as the grain size.

Effect of sampling volume on the measurement of soil physical properties: simulation with x-ray tomography data

Abstract: The dependence of macroscopic soil parameters on sampling volume is currently the object of renewed research focus. In this paper, x-ray computed tomography data related to cores obtained in two different locations in a field soil are used to simulate this dependence. Several integration methods are adopted, to mimic different measuring devices. Calculation results, relative to the volumetric water content, volumetric air content, gravimetric water content and dry bulk density, demonstrate that the size (up to 60×60×30 mm3), shape and positioning of sampling volumes influence significantly the measured values of soil parameters. In some cases, the instrumental dependence disappears within a range of sampling volumes, in agreement with a hypothesis underlying the so-called representative elementary volume concept. However, some parameters, like the soil bulk density, do not level off with increasing sampling volumes. These observations open new avenues for research on measurement processes in soils and other heterogeneous media.

3D finite element analysis of particle-reinforced aluminum

Abstract: Deformation in particle-reinforced aluminum has been simulated using three distinct types of finite element model: a three-dimensional repeating unit cell, a three-dimensional multi-particle model, and two-dimensional multi-particle models. The repeating unit cell model represents a fictitious periodic cubic array of particles. The 3D multi-particle (3D-MP) model represents randomly placed and oriented particles. The 2D generalized plane strain multi-particle models were obtained from planar sections through the 3D-MP model. These models were used to study the tensile macroscopic stress-strain response and the associated stress and strain distributions in an elastoplastic matrix. The results indicate that the 2D model having a particle area fraction equal to the particle representative volume fraction of the 3D models predicted the same macroscopic stress-strain response as the 3D models. However, there are fluctuations in the particle area fraction in a representative volume element. As expected, predictions from 2D models having different particle area fractions do not agree with predictions from 3D models. More importantly, it was found that the microscopic stress and strain distributions from the 2D models do not agree with those from the 3D-MP model. Specifically, the plastic strain distribution predicted by the 2D model is banded along lines inclined at 45 deg from the loading axis while the 3D model prediction is not. Additionally, the triaxial stress and maximum principal stress distributions predicted by 2D and 3D models do not agree. Thus, it appears necessary to use a multi-particle 3D model to accurately predict material responses that depend on local effects, such as strain-to-failure, fracture toughness, and fatigue life.

Equivalent-Continuum Modeling With Application to Carbon Nanotubes

Abstract: A method has been proposed for developing structure-property relationships of nano-structured materials. This method serves as a link between computational chemistry and solid mechanics by substituting discrete molecular structures with equivalent-continuum models. It has been shown that this substitution may be accomplished by equating the vibrational potential energy of a nano-structured material with the strain energy of representative truss and continuum models. As important examples with direct application to the development and characterization of single-walled carbon nanotubes and the design of nanotube-based devices, the modeling technique has been applied to determine the effective-continuum geometry and bending rigidity of a graphene sheet. A representative volume element of the chemical structure of graphene has been substituted with equivalent-truss and equivalent continuum models. As a result, an effective thickness of the continuum model has been determined. This effective thickness has been shown to be significantly larger than the interatomic spacing of graphite. The effective thickness has been shown to be significantly larger than the inter-planar spacing of graphite. The effective bending rigidity of the equivalent-continuum model of a graphene sheet was determined by equating the vibrational potential energy of the molecular model of a graphene sheet subjected to cylindrical bending with the strain energy of an equivalent continuum plate subjected to cylindrical bending.

Modelling of failure mode transition in ballistic penetration with a continuum model describing microcracking and flow of pulverized media

Abstract: An Erratum has been published for this article in International Journal for Numerical Methods in Engineering 2002; 55(4):499–501. A new continuum model to describe damage, fragmentation and large deformation of pulverized brittle materials is presented. The multiple-plane-microcracking (MPM) model, developed by Espinosa, has been modified to track microcracking on 13 orientations under high pressure, high strain rate and high deformation. This model provides the elastic and inelastic response of the material before massive crack coalescence. When pulverization occurs, the constitutive response is modelled by means of a visco-plastic model for granular material, which is a generalization to three dimensions of the double-sliding theory augmented by a consolidation mechanism. The initialization of the granular model is governed by a yield surface at the onset of massive crack coalescence. This is accomplished by examining a representative volume element, modelled using the MPM model, in compression-shear. The main advantage of this approach is to keep a continuum model at all stages of the deformation process and thus avoid the difficulties of crack representation in a discrete finite element code. This model has been implemented in LS-DYNA and used to examine interface defeat of long rod penetrators by a confined ceramic plate. The numerical simulations are compared to experiments in order to identify failure modes. The model parameters were obtained independently by simulating plate and rod impact experiments. The proposed model captures most of the physical observations as well as failure mode transition, from interface defeat to full penetration, with increasing impact velocity. Copyright © 2002 John Wiley & Sons, Ltd.

Continuum Modelling of Contaminant Transport in Fractured Porous Media

Abstract: This work is aimed towards deriving macroscopic models that describe pollutant migration through fractured porous media. A homogenisation method is used, that is, macroscopic models are deduced from the physical description over a representative elementary volume (REV), which consists of an open fracture surrounded by a porous matrix block. No specific geometry is at issue. The fractured porous medium is saturated by an incompressible fluid. At the REV's scale, the transport is assumed to be advective-diffusive in the porous matrix and due to convection and molecular diffusion in the fracture's domain. It is also assumed that there is no diffusion in the solid. We demonstrate that the macroscopic behaviour is described by a single-continuum model. Fluid flow is described by Darcy's law. Four macroscopic single-continuum models are obtained for the contaminant transport: a diffusive model, an advective-diffusive model and two advective-dispersive models. One of the two advective-dispersive models accounts for the advection process in the porous matrix. The domains of validity of these models are defined by means of the orders of magnitude of the local Peclet numbers in the porous matrix block and in the fracture's domain.

Towards Stochastic Continuum Thermodynamics

Abstract: We explore the relation of a line of studies over the last dozen years on mesoscale material response below the Representative Volume Element with the line of studies on mesoscopic continuum physics. The objective is the development of stochastic continuum thermodynamics able to grasp random microstructural features absent in deterministic continuum theories. We first compare two lines of studies ‐ those dealing with the response on a scale smaller than the Representative Volume Element (RVE) vis-a `-vis those on mesoscopic continuum physics ‐ and observe the dilemma of choice of the random field approximation. As a common basis for connecting both approaches, we propose a mesoscale Statistical Volume Element (SVE) and consider bounding its response via two admissible loadings. Employing the paradigm of thermal conductivity, we also present the relevant Legendre transformations. The latter are then generalized to the more general situation of fields governed by a quartet of Legendre transformations.

A computational mesoscale evaluation of material characteristics of porous shape memory alloys

Abstract: Computational techniques are used to estimate porous shape memory alloy material behavior in this study. Mesoscale analysis techniques based on a two-dimensional finite element method are used to evaluate a representative volume element (RVE) to determine effective material response. The porous shape memory alloy is defined as comprised of two constituents, dense NiTi shape memory alloy (SMA), and pores that are defined as free of all material and distributed randomly. An existing rate-independent type constitutive model is used to define the dense SMA material behavior. Open porosity is considered with the axis of generation perpendicular to the plane of the two-dimensional RVE. Pores are allowed to vary in size and shape with no limit on maximum pore size. Computational evaluations are completed for porous SMA for a range of pore volume fractions. Effective material behavior is calculated and results are compared with computationally determined material behavior based on the micromechanical unit cell finite element method. Calculated effective material elastic and transformation characteristics compare well between the two methods for low to moderate (≤0.3) pore volume fractions. Significant variations are seen for higher pore volume fractions (>0.3) due to spatial orientation and percolation effects that are not considered in unit cell analyses.

Thermodiffusion in Porous Media and Its Consequences

Abstract: This contribution presents a synthesis of experimental, theoretical and numerical studies performed on thermodiffusion and thermodiffusion-convection transport in porous media.

Energy dissipation and permeability in porous media

Abstract: We investigate flow through porous media by solving the Navier-Stokes equations in 3D porous structures using the lattice Boltzmann method. We analyse the distribution of local specific dissipation of mechanical energy and we use this quantity to investigate the microscopic origin of absolute permeability. The averaging of this quantity on a flow cross-section provides a methodology to locate energy losses and to spot the appropriate scale of the permeability Representative Elementary Volume (REV). The effectiveness of the approach is shown by a numerical study of the flow field in simplified porous media for which experimental results are available.

Stress fluctuations and macroscopic stick-slip in granular materials.

Abstract: This paper deals with the quasi-static regime of deformation of granular matter. It investigates the size of the Representative Elementary Volume (REV), which is the minimum packing size above which the macroscopic mechanical behaviour of granular materials can be defined from averaging. The first part uses typical results from recent literature and finds that the minimum REV contains in general 10 grains; this result holds true either for most experiments or for Discrete Element Method (DEM) simulation. This appears to be quite small. However, the second part gives a counterexample, which has been found when investigating uniaxial compression of glass spheres which exhibit stick-slip; we show in this case that the minimum REV becomes 107 grains. This makes the system not computable by DEM. Moreover, similarity between the Richter law of seism and the exponential statistics of stick-slip is stressed.

Predicting oak wood properties using X-ray inspection: representation, homogenisation and localisation. Part I: Digital X-ray imaging and representation by finite elements

Abstract: This paper is the first part of a complete modelling where wood is a 2D composite material. It proposes a comprehensive approach to predict the elastic and shrinkage properties of oak wood in the transverse plane from the local properties of the anatomical tissues and the actual morphology of those tissues. Part I is devoted to the methodology of representation. According to oak anatomy, the representative elementary volume used in this work consists of one annual ring limited in the tangential direction by large zones of ray cells. A high-resolution digital X-ray imaging device was built to represent the spatial distribution of tissues directly from cross-sections of wood. A complete image processing, in- cluding image segmentation and partitioned boundaries of tissues (ray cells, big vessels, etc.), was developed; which makes it possible to build a finite element mesh from these images. Thanks to the control of mesh refinement, the number of triangular elements is minimized while a good description of the anatomical structure is obtained. Using these F.E. meshes, Part II will present the homogenization principle and a few exam- ples of calculations of the properties. Macroscopic properties (mechanical and shrinkage) and some localization problems are computed and il- lustrated.

Low order inelastic micromorphic polycrystals

Abstract: The paper deals with some fundamental issues essential for constitutive modelling of plastic behaviour of metals. Geometric and kinematic aspects of intragranular as well as intergranular plastic deformation of polycrystals are discussed. Homogeneous grain strains are composed into the resulting behaviour of representative volume element (RVE). A homogenization of total, plastic and elastic strains has been done. Constitutive equations by a self consistent method have been discussed. A simplest case of higher gradient theory is discussed. Elastic strain is covered by the efiective field homogenization method inside a RVE. It is underlined that plastic stretching and plastic spin are not independent. .

Micromechanics Models for Mechanical Properties

Abstract: This chapter discusses micromechanics analysis of fiber reinforced composite materials, particularly those materials reinforced long fibers A microscopically inhomogeneous composite material can be idealized as a macroscopically homogeneous continuum when the behavior of engineering structures made of the material can be satisfactorily retained Such idealization can be realized over a representative sample of the composite material Selection of the dimensions of a representative volume is imperative The representative volume must be sufficiently large compared to the scale of the microstructure so that it contains a sufficient number of individual constituents and microstructural features It must also be small compared to the whole structural body so that it is entirely typical of the whole composite structure on average The general procedure to predict the mechanical properties of a textile composite using finite element method (FEM) consists of dividing the textile composite structure into a number of unit cells and analyzing the mechanical properties of a unit cell using FEM, and reconstructing the entire reinforcement geometry by assembling the unit cells for predicting mechanical properties of textile composites Thus, the ability of a FEM model to predict mechanical properties depends upon the accuracy of modeling the fiber geometry in a unit cell

Shakedown- and limit analysis of periodic composites

Abstract: A methodology for the assessment of periodic composites under variable repeated loads by means of the shakedown theory is presented and applied to metal-matrix-composites. The approach is based on the local shakedown analysis in a representative volume element of the composite and the use of averaging techniques to determine the domains of admissible stresses on the macro-level.