Showing papers on "Representative elementary volume published in 2010"

A constitutive model for plastically anisotropic solids with non-spherical voids

Abstract: Plastic constitutive relations are derived for a class of anisotropic porous materials consisting of coaxial spheroidal voids, arbitrarily oriented relative to the embedding orthotropic matrix. The derivations are based on nonlinear homogenization, limit analysis and micromechanics. A variational principle is formulated for the yield criterion of the effective medium and specialized to a spheroidal representative volume element containing a confocal spheroidal void and subjected to uniform boundary deformation. To obtain closed form equations for the effective yield locus, approximations are introduced in the limit-analysis based on a restricted set of admissible microscopic velocity fields. Evolution laws are also derived for the microstructure, defined in terms of void volume fraction, aspect ratio and orientation, using material incompressibility and Eshelby-like concentration tensors. The new yield criterion is an extension of the well known isotropic Gurson model. It also extends previous analyses of uncoupled effects of void shape and material anisotropy on the effective plastic behavior of solids containing voids. Preliminary comparisons with finite element calculations of voided cells show that the model captures non-trivial effects of anisotropy heretofore not picked up by void growth models.

Characterizing and Modeling Mechanical Properties of Nanocomposites-Review and Evaluation

Abstract: This paper presents a critical review of the current work of experiment, theory of micro-nanomechanics, and numerical analysis on characterizing mechanical properties of nanocomposites. First, the classifications of nanomaterials are presented. Then nanoindentation testing and the corresponding finite element modeling are discussed, followed by analytical modeling stiffness of nanocomposites. The analytical models discussed include Voigt and Reuss bounds, Hashin and Shtrikman bounds, Halpin–Tsai model, Cox model, and various Mori and Tanaka models. These micromechanics models predict stiffness of nanocomposites with both aligned and randomly oriented fibers. The emphasis is on numerical modeling includes molecular dynamics modeling and finite element modeling. Three different approaches are discussed in finite element modeling, i.e. multiscale representative volume element (RVE) modeling, unit cell modeling, and object-oriented modeling. Finally, the mechanism of nanocomposite mechanical property enhancement and the ways to improve stiffness and fracture toughness for nanocomposites are discussed.

Regularized variational theories of fracture: A unified approach

Abstract: The fracture pattern in stressed bodies is defined through the minimization of a two-field pseudo-spatial-dependent functional, with a structure similar to that proposed by Bourdin–Francfort–Marigo (2000) as a regularized approximation of a parent free-discontinuity problem, but now considered as an autonomous model per se. Here, this formulation is altered by combining it with structured deformation theory, to model that when the material microstructure is loosened and damaged, peculiar inelastic (structured) deformations may occur in the representative volume element at the price of surface energy consumption. This approach unifies various theories of failure because, by simply varying the form of the class for admissible structured deformations, different-in-type responses can be captured, incorporating the idea of cleavage, deviatoric, combined cleavage-deviatoric and masonry-like fractures. Remarkably, this latter formulation rigorously avoid material overlapping in the cracked zones. The model is numerically implemented using a standard finite-element discretization and adopts an alternate minimization algorithm, adding an inequality constraint to impose crack irreversibility (fixed crack model). Numerical experiments for some paradigmatic examples are presented and compared for various possible versions of the model.

Computational homogenization for heterogeneous thin sheets

Abstract: In this paper, a computational homogenization technique for thin-structured sheets is proposed, based on the computational homogenization concepts for first- and second-order continua. The actual three-dimensional (3D) heterogeneous sheet is represented by a homogenized shell continuum for which the constitutive response is obtained from the nested analysis of a microstructural representative volume element (RVE), incorporating the full thickness of the sheet and an in-plane representative cell of the macroscopic structure. At an in-plane integration point of the macroscopic shell, the generalized strains, i.e. the membrane deformation and the curvature, are used to formulate the boundary conditions for the microscale RVE problem. At the RVE scale, all microstructural constituents are modeled as an ordinary 3D continuum, described by the standard equilibrium and the constitutive equations. Upon proper averaging of the RVE response, the macroscopic generalized stress and the moment resultants are obtained. In this way, an in-plane homogenization is directly combined with a through thickness stress integration. From a macroscopic point of view, a (numerical) generalized stress-strain constitutive response at every macroscopic in-plane integration point is obtained. Additionally, the simultaneously resolved microscale RVE local deformation and stress fields provide valuable information for assessing the reliability of a particular microstructural design.

Topological derivative for multi‐scale linear elasticity models applied to the synthesis of microstructures

Abstract: This paper proposes an algorithm for the synthesis/optimization of microstructures based on an exact formula for the topological derivative of the macroscopic elasticity tensor and a level set domain representation. The macroscopic elasticity tensor is estimated by a standard multi-scale constitutive theory where the strain and stress tensors are volume averages of their microscopic counterparts over a representative volume element. The algorithm is of simple computational implementation. In particular, it does not require artificial algorithmic parameters or strategies. This is in sharp contrast with existing microstructural optimization procedures and follows as a natural consequence of the use of the topological derivative concept. This concept provides the correct mathematical framework to treat topology changes such as those characterizing microstuctural optimization problems. The effectiveness of the proposed methodology is illustrated in a set of finite element-based numerical examples.Copyright © 2010 John Wiley & Sons, Ltd.

Stochastic laminate analogy for simulating the variability in modulus of discontinuous composite materials

Abstract: Recent composite technology research and development efforts have focused on discontinuous carbon fiber/epoxy molding systems derived from chopped aerospace-grade unidirectional tape prepreg. Although the average elastic modulus of this material has been shown to be as high as that of the continuous tape quasi-isotropic benchmark, experimental measurement by means of strain gage or extensometer has shown variation as high as 20%. Digital Image Correlation can be used successfully to obtain a full-field strain measurement, and it shows that a highly non-uniform strain distribution exists on the surface of the specimen, with distinct peaks and valleys. This pattern of alternating regions of high and low strain gradients, and which exhibit a characteristic shape and size, can be described in terms of Random Representative Volume Element (RRVE). The RRVE proposed here exhibits random elastic properties, which are assigned based on stochastic distributions. This approach leads to the analysis method proposed here, which is designed to compensate for the fact that traditional methods cannot capture the experimentally observed variation in modulus within a specimen and among different specimens. The method utilizes a randomization process to generate statistical distributions of fractions and orientations of chips within the RRVE, and then applies Classical Laminated Plate Theory to an equivalent quasi-isotropic tape laminate to calculate its average elastic properties. Validation of this method is shown as it applies to a finite element model that discretizes the structure in multiple RRVEs, whose properties are generated independently of the neighboring ones, and then are solved simultaneously. The approach generates accurate predictions of the strain distribution on the surface of the specimen.

Cosserat model for periodic masonry deduced by nonlinear homogenization

Abstract: The paper deals with the problem of the determination of the in-plane behavior of periodic masonry material. The macromechanical equivalent Cosserat medium, which naturally accounts for the absolute size of the constituents, is derived by a rational homogenization procedure based on the Transformation Field Analysis. The micromechanical analysis is developed considering a Cauchy model for masonry components. In particular, a linear elastic constitutive relationship is considered for the blocks, while a nonlinear constitutive law is adopted for the mortar joints, accounting for the damage and friction phenomena occurring during the loading history. Some numerical applications are performed on a Representative Volume Element characterized by a selected commonly used texture, without performing at this stage structural analyses. A comparison between the results obtained adopting the proposed procedure and a nonlinear micromechanical Finite Element Analysis is presented. Moreover, the substantial differences in the nonlinear behavior of the homogenized Cosserat material model with respect to the classical Cauchy one, are illustrated.

Hole-expansion formability of dual-phase steels using representative volume element approach with boundary-smoothing technique

Abstract: A qualitative analysis was carried out on the formability of dual-phase (DP) steels by introducing a realistic microstructure-based finite element approach. The present microstructure-based model was constructed using a mesh generation process with a boundary-smoothing algorithm after proper image processing. The developed model was applied to hole-expansion formability tests for DP steel sheets having different volume fractions and morphological features. On the basis of the microstructural inhomogeneity observed in the scanning electron micrographs of the DP steel sheets, it was inferred that the localized plastic deformation in the ferritic phase might be closely related to the macroscopic formability of DP steel. The experimentally observed difference between the hole-expansion formability of two different microstructures was reasonably explained by using the present finite element model.

A semi-continuum finite element approach to evaluate the Young’s modulus of single-walled carbon nanotube reinforced composites

Abstract: This paper describes a micromechanical finite element approach for the estimation of the effective Young’s modulus of single-walled carbon nanotube reinforced composites. These composite materials consist of aligned carbon nanotubes that are uniformly distributed within the matrix. Based on micromechanical theory, the Young’s modulus of the nanocomposite is estimated by considering a representative cylindrical volume element. Within the representative volume element, the reinforcement is modeled according to its atomistic microstructure while the matrix is modeled as a continuum medium. Spring-based finite elements are employed to simulate the discrete geometric structure and behavior of each single-walled carbon nanotube. The load transfer conditions between the carbon nanotubes and the matrix are modeled using joint elements of changeable stiffness that connect the two materials, simulating the interfacial region. The proposed model has been tested numerically and yields reasonable results for variable stiffness values of the joint elements. The effect of the interface on the performance of the composite is investigated for various volume fractions. The numerical results are compared with experimental and analytical predictions.

Crystal plasticity modeling of cyclic deformation for a polycrystalline nickel-based superalloy at high temperature

Abstract: Cyclic deformation at elevated temperature has been modeled for a polycrystalline nickel-based superalloy using the crystal-plasticity constitutive formulations. Finite element analyses were carried out for a representative volume element (RVE), consisting of randomly oriented grains and subjected to periodic boundary constraints. Model parameters were determined by fitting the strain-controlled cyclic test data at 650 °C for three different loading rates. Simulated results are in good agreement with the experimental data for both stress–strain loops and cyclic hardening behavior. The model was utilized to predict the stress relaxation behavior during the hold periods at the maximum and minimum strain levels, and the prediction compares well with the experimental results. Localized stress and strain concentrations were observed due to the heterogeneous nature of grain microstructure and the mismatch of the mechanical properties of individual grains.

Application of the multiscale FEM to the modeling of cancellous bone.

Abstract: This paper considers the application of multiscale finite element method (FEM) to the modeling of cancellous bone as an alternative for Biot’s model, the main intention of which is to decrease the extent of the necessary laboratory tests. At the beginning, the paper gives a brief explanation of the multiscale concept and thereafter focuses on the modeling of the representative volume element and on the calculation of the effective material parameters, including an analysis of their change with respect to increasing porosity. The latter part of the paper concentrates on the macroscopic calculations, which is illustrated by the simulation of ultrasonic testing and a study of the attenuation dependency on material parameters and excitation frequency. The results endorse conclusions drawn from the experiments: increasing excitation frequency and material density cause increasing attenuation.

Periodic homogenization and consistent estimates of transport parameters through sphere and polyhedron packings in the whole porosity range.

Abstract: This paper presents a study of transport parameters (diffusion, dynamic permeability, thermal permeability, trapping constant) of porous media by combining the homogenization of periodic media (HPM) and the self-consistent scheme (SCM) based on a bicomposite spherical pattern. The link between the HPM and SCM approaches is first established by using a systematic argument independent of the problem under consideration. It is shown that the periodicity condition can be replaced by zero flux and energy through the whole surface of the representative elementary volume. Consequently the SCM solution can be considered as a geometrical approximation of the local problem derived through HPM for materials such that the morphology of the period is "close" to the SCM pattern. These results are then applied to derive the estimates of the effective diffusion, the dynamic permeability, the thermal permeability and the trapping constant of porous media. These SCM estimates are compared with numerical HPM results obtained on periodic arrays of spheres and polyhedrons. It is shown that SCM estimates provide good analytical approximations of the effective parameters for periodic packings of spheres at porosities larger than 0.6, while the agreement is excellent for periodic packings of polyhedrons in the whole range of porosity.

Micromechanics and effective elastic moduli of particle-reinforced composites with near-field particle interactions

Abstract: A micromechanical framework is proposed to predict effective elastic moduli of particle-reinforced composites. First, the interacting eigenstrain is derived by making use of the exterior-point Eshelby tensor and the equivalence principle associated with the pairwise particle interactions. Then, the near-field particle interactions are accounted for in the effective elastic moduli of spherical-particle-reinforced composites. On the foundation of the proposed interacting solution, the consistent versus simplified micromechanical field equations are systematically presented and discussed. Specifically, the focus is upon the effective elastic moduli of two-phase composites containing randomly distributed isotropic spherical particles. To demonstrate the predictive capability of the proposed micromechanical framework, comparisons between the theoretical predictions and the available experimental data on effective elastic moduli are rendered. In contrast to higher-order formulations in the literature, the proposed micromechanical formulation can accommodate the anisotropy of reinforcing particles and can be readily extended to multi-phase composites.

Estimation of Influence Tensors for Eigenstressed Multiphase Elastic Media with Nonaligned Inclusion Phases of Arbitrary Ellipsoidal Shape

Abstract: The analysis of microheterogeneous materials exhibiting eigenstressed and/or eigenstrained phases requires an estimation of eigenstrain influence tensors. Within the framework of continuum micromechanics, we here derive these tensors from extended Eshelby-Laws matrix-inclusion problems, considering, as a new feature, an auxiliary matrix eigenstress. The auxiliary matrix eigenstress is a function of all phase eigenstresses and, hence, accounts for eigenstress interaction. If all material phases are associated with one and the same Hill tensor, the proposed method degenerates to the well-accepted transformation field analysis. Hence, the proposed concept can be interpreted as an extension of the transformation field analysis toward consideration of arbitrarily many Hill tensors, i.e., as an extension toward heterogeneous elastic media comprising inclusion phases with an arbitrary ellipsoidal shape and with arbitrary spatial orientation. This is of particular interest when studying heterogeneous media consisting of constituents with nonspherical phase shapes, e.g., cement-based materials including concrete, or bone. As for polycrystals studied by means of the self-consistent scheme, the auxiliary matrix eigenstress turns out to be equal to the eigenstresses homogenized over the representative volume element (RVE), which is analogous to the self-consistent assumption that the auxiliary stiffness is the average stiffness of the RVE. The proposed method opens the door for micromechanics-based modeling of a great variety of composite phase behaviors characterized by eigenstresses or eigenstrains, e.g., thermoelasticity, poroelasticity, drying-shrinkage, as well as general forms of inelastic behavior, damage, fatigue, and fracture.