Showing papers on "Representative elementary volume published in 2006"

Fundamentals of Micromechanics of Solids

Abstract: Preface. 1 Introduction. 1.1 Background and Motivation. 1.2 Objectives. 1.3 Organization of Book. 1.4 Notation Conventions. References. 2 Basic Equations of Continuum Mechanics. 2.1 Displacement and Deformation. 2.2 Stresses and Equilibrium. 2.3 Energy, Work, and Thermodynamic Potentials. 2.4 Constitutive Laws. 2.5 Boundary Value Problems for Small-Strain Linear Elasticity. 2.6 Integral Representations of Elasticity Solutions. Problems. Appendix 2.A. Appendix 2.B. Appendix 2.C. References. Suggested Readings. 3 Eigenstrains. 3.1 Definition of Eigenstrains. 3.2 Some Examples of Eigenstrains. 3.3 General Solutions of Eigenstrain Problems. 3.4 Examples. Problems. Appendix 3.A. Appendix 3.B. References. Suggested Readings. 4 Inclusions and Inhomogeneities. 4.1 Definitions of Inclusions and Inhomogeneities. 4.2 Interface Conditions. 4.3 Ellipsoidal Inclusion with Uniform Eigenstrains (Eshelby Solution). 4.4 Ellipsoidal Inhomogeneities. 4.5 Inhomogeneous Inhomogeneities. Problems. Appendix 4.A. Appendix 4.B. Suggested Readings. 5 Definitions of Effective Moduli of Heterogeneous Materials. 5.1 Heterogeneity and Length Scales. 5.2 Representative Volume Element. 5.3 Random Media. 5.4 Macroscopic Averages. 5.5 Hill's Lemma. 5.6 Definitions of Effective Modulus of Heterogeneous Media. 5.7 Concentration Tensors and Effective Properties. Problems. Suggested Readings. 6 Bounds for Effective Moduli. 6.1 Classical Variational Theorems in Linear Elasticity. 6.2 Voigt Upper Bound and Reuss Lower Bound. 6.3 Extensions of Classical Variational Principles. 6.4 Hashin-Shtrikman Bounds. Problems. Appendix 6.A. References. Suggested Readings. 7 Determination of Effective Moduli. 7.1 Basic Ideas of Micromechanics for Effective Properties. 7.2 Eshelby Method. 7.3 Mori-Tanaka Method. 7.4 Self-Consistent Methods for Composite Materials. 7.5 Self-Consistent Methods for Polycrystalline Materials. 7.6 Differential Schemes. 7.7 Comparison of Different Methods. Problems. Suggested Readings. 8 Determination of the Effective Moduli-Multiinclusion Approaches. 8.1 Composite-Sphere Model. 8.2 Three-Phase Model. 8.3 Four-Phase Model. 8.4 Multicoated Inclusion Problem. Problems. Appendix 8.A. Appendix 8.B. Appendix 8.C. References. Suggested Readings. 9 Effective Properties of Fiber-Reinforced Composite Laminates. 9.1 Unidirectional Fiber-Reinforced Composites. 9.2 Effective Properties of Multilayer Composites. 9.3 Effective Properties of a Lamina. 9.4 Effective Properties of a Laminated Composite Plate. Problems. Appendix 9.A. References. Suggested Readings. 10 Brittle Damage and Failure of Engineering Composites. 10.1 Imperfect Interfaces. 10.2 Fiber Bridging. 10.3 Transverse Matrix Cracks. Problems. Appendix 10.A. References. Suggested Readings. 11 Mean Field Theory for Nonlinear Behavior. 11.1 Eshelby's Solution and Kro..ner's Model. 11.2 Applications. 11.3 Time-Dependent Behavior of Polycrystalline Materials: Secant Approach. Problems. References. 12 Nonlinear Properties of Composites Materials: Thermodynamic Approaches. 12.1 Nonlinear Behavior of Constituents. 12.2 Effective Potentials. 12.3 The Secant Approach. Problems. Suggested Readings. 13 Micromechanics of Martensitic Transformation in Solids. 13.1 Phase Transformation Mechanisms at Different Scales. 13.2 Application: Thermodynamic Forces and Constitutive Equations for Single Crystals. 13.3 Overall Behavior of Polycrystalline Materials with Phase Transformation. Problems. References. Suggested Readings. Index.

Statistically Equivalent Representative Volume Elements for Unidirectional Composite Microstructures: Part I - Without Damage

Abstract: The representative volume element (RVE) of a material microstructure plays an important role in the analysis of heterogeneous materials, such as composites. Effective material properties in a composite material depend on the microstructural concentration and dispersion of different phases in the RVE. In this study, a combination of statistical and computational tools are proposed for identifying the statistically equivalent RVE (SERVE) for an elastic composite with nonuniform dispersion of inclusions. Numerical tests are conducted with various statistical functions of geometry, stresses, and strains to examine the validity of potential alternative definitions of the SERVE. In the first of this two-part article, methods are addressed for undamaged composite microstructures with continuous interfaces between the fiber and the matrix. The sequel article deals with the evolution of SERVE caused by damage due to interfacial debonding.

Scale‐related topology optimization of cellular materials and structures

Abstract: The integrated optimization of lightweight cellular materials and structures are discussed in this paper. By analysing the basic features of such a two-scale problem, it is shown that the optimal solution strongly depends upon the scale effect modelling of the periodic microstructure of material unit cell (MUC), i.e. the so-called representative volume element (RVE). However, with the asymptotic homogenization method used widely in actual topology optimization procedure, effective material properties predicted can give rise to limit values depending upon only volume fractions of solid phases, properties and spatial distribution of constituents in the microstructure regardless of scale effect. From this consideration, we propose the design element (DE) concept being able to deal with conventional designs of materials and structures in a unified way. By changing the scale and aspect ratio of the DE, scale-related effects of materials and structures are well revealed and distinguished in the final results of optimal design patterns. To illustrate the proposed approach, numerical design problems of 2D layered structures with cellular core are investigated.

A comprehensive closed form micromechanics model for estimating the elastic modulus of nanotube-reinforced composites

Abstract: This paper describes an approximate, yet comprehensive, closed form micromechanics model for estimating the effective elastic modulus of carbon nanotube-reinforced composites. The model incorporates the typically observed nanotube curvature, the nanotube length, and both 1D and 3D random arrangement of the nanotubes. The analytical results obtained from the closed form micromechanics model for nanoscale representative volume elements and results from an equivalent finite element model for effective reinforcing modulus of the nanotube reveal that the reinforcing modulus is strongly dependent on the waviness, wherein, even a slight change in the nanotube curvature can induce a prominent change in the effective reinforcement provided. The micromechanics model is also seen to produce reasonable agreement with experimental data for the effective tensile modulus of composites reinforced with multi-walled nanotubes (MWNTs) and having different MWNT volume fractions.

On the Size of RVE in Finite Elasticity of Random Composites

Abstract: This paper presents a quantitative study of the size of representative volume element (RVE) of random matrix-inclusion composites based on a scale-dependent homogenization method. In particular, mesoscale bounds defined under essential or natural boundary conditions are computed for several nonlinear elastic, planar composites, in which the matrix and inclusions differ not only in their material parameters but also in their strain energy function representations. Various combinations of matrix and inclusion phases described by either neo-Hookean or Ogden function are examined, and these are compared to those of linear elastic types.

Mechanics-based statistics of failure risk of quasibrittle structures and size effect on safety factors

Abstract: In mechanical design as well as protection from various natural hazards, one must ensure an extremely low failure probability such as 10−6. How to achieve that goal is adequately understood only for the limiting cases of brittle or ductile structures. Here we present a theory to do that for the transitional class of quasibrittle structures, having brittle constituents and characterized by nonnegligible size of material inhomogeneities. We show that the probability distribution of strength of the representative volume element of material is governed by the Maxwell–Boltzmann distribution of atomic energies and the stress dependence of activation energy barriers; that it is statistically modeled by a hierarchy of series and parallel couplings; and that it consists of a broad Gaussian core having a grafted far-left power-law tail with zero threshold and amplitude depending on temperature and load duration. With increasing structure size, the Gaussian core shrinks and Weibull tail expands according to the weakest-link model for a finite chain of representative volume elements. The model captures experimentally observed deviations of the strength distribution from Weibull distribution and of the mean strength scaling law from a power law. These deviations can be exploited for verification and calibration. The proposed theory will increase the safety of concrete structures, composite parts of aircraft or ships, microelectronic components, microelectromechanical systems, prosthetic devices, etc. It also will improve protection against hazards such as landslides, avalanches, ice breaks, and rock or soil failures.

Measurement of meso-scale shear deformations for modelling textile composites

Abstract: Textile preforms undergo complex in-plane and out-of-plane deformations during forming processes. In-plane shear is the most significant mode of deformation during draping or forming around double-curvature surfaces. This paper reports an improved method for measuring the constitutive shear properties of textile preforms—a wide strip bias extension test has been found to give significantly better results in comparison to conventional narrow strip bias test. Shear angles, computed directly from global strains, were in close agreement to those measured using an optical method. Constitutive shear stress–strain relation, computed from the bias extension test data, was in good agreement with KES (Kawabata Evaluation System) test results. Meso-scale tow deformations have been measured using two techniques: an inexpensive flatbed scanner-based full-field strain measurement technique for measuring in-plane tow deformations and a strain-freezing technique for measuring through-thickness tow deformations. With this data, a 3D Representative Volume Element (RVE) can be constructed for each stage of shear deformation. Beyond the geometric shear limit, tow deformations during bias extension appear to be somewhat different from those normally expected from pure shear—a reduction in tow thickness and tow cross-sectional area and a corresponding increase in packing factor.

Representative volume element of polycrystalline silicon for MEMS

Abstract: A nanoscale mechanical deformation measurement method was employed to obtain the Young’s modulus and Poisson’s ratio of polycrystalline silicon for Microelectromechanical Systems (MEMS) from different facilities, and to assess the scale at which these effective properties are valid in MEMS design. The method, based on in situ Atomic Force Microscope (AFM) imaging and Digital Image Correlation (DIC) analysis, employed 2–2.5 μm thick freestanding specimens with surface measurement areas varying between 1×2 and 5×15 μm2. The effective mechanical properties were quite invariant with respect to the fabrication facility: the Poisson’s ratio of polycrystalline silicon from the Multi-user MEMS Processes (MUMPs) and from Sandia’s Ultra planar four layer Multilevel MEMS Technology (SUMMiT-IV) was 0.22±0.02, while the elastic moduli for MUMPs and SUMMiT-IV polysilicon were 164±7 and 155±6 GPa, respectively. The AFM/DIC method was used to determine the size of the material domain whose mechanical behavior could be described by the isotropic constants. For SUMMiT polysilicon with columnar grains and 650 nm average grain size, it was found that a 10×10-μm2 specimen area, on average containing 15×15 columnar grains, was a representative volume element. However, the axial displacement fields in 4×4 or 2×2 μm2 areas could be highly inhomogeneous and the effective behavior of these specimen domains could deviate significantly from that described by isotropy. As a consequence, the isotropic material constants are applicable to MEMS components comprised of 15×15 or more grains, corresponding to specimen areas equal to 10×10 μm2 for SUMMiT and 5×5 μm2 for MUMPs, and do not provide an accurate description of the mechanics of smaller MEMS components.

Multi-scale modeling in damage mechanics of composite materials

Abstract: This paper addresses the multi-scale modeling aspects of damage in composite materials. The multiplicity of the scales of the operating mechanisms is discussed and clarified by taking examples of damage in a unidirectional ceramic matrix composite and in a cross ply polymer matrix composite laminate. Two multi-scale modeling strategies––the hierarchical and the synergistic––are reviewed in the context of deformational response. Finally, the “big picture” as it relates to the cost-effective manufacturing of composite structures intended for long-term performance is outlined and desired future direction in multi-scale modeling is discussed.

Statistically Equivalent Representative Volume Elements for Unidirectional Composite Microstructures: Part II - With Interfacial Debonding

Abstract: In this sequel to the study on microstructures without damage, methods for evaluating the statistically equivalent representative volume element (SERVE) are proposed for fiber-reinforced microstructures undergoing initiation and propagation of damage in the form of interfacial debonding. The microstructural analysis is executed using the Voronoi cell finite element model (VCFEM) in which the interface is modeled using a bilinear cohesive zone law. As introduced in the first article, a combination of statistical and computational tools is proposed to capture the evolving nature of the SERVE with increased loading. The effectiveness of alternative definitions and methods of characterizing the damaging microstructure is examined through numerical simulations.

Calculating the effective permeability of sandstone with multiscale lattice Boltzmann/finite element simulations

Abstract: The lattice Boltzmann (LB) method is an efficient technique for simulating fluid flow through individual pores of complex porous media. The ease with which the LB method handles complex boundary conditions, combined with the algorithm’s inherent parallelism, makes it an elegant approach to solving flow problems at the sub-continuum scale. However, the realities of current computational resources can limit the size and resolution of these simulations. A major research focus is developing methodologies for upscaling microscale techniques for use in macroscale problems of engineering interest. In this paper, we propose a hybrid, multiscale framework for simulating diffusion through porous media. We use the finite element (FE) method to solve the continuum boundary-value problem at the macroscale. Each finite element is treated as a sub-cell and assigned permeabilities calculated from subcontinuum simulations using the LB method. This framework allows us to efficiently find a macroscale solution while still maintaining information about microscale heterogeneities. As input to these simulations, we use synchrotron-computed 3D microtomographic images of a sandstone, with sample resolution of 3.34 μm. We discuss the predictive ability of these simulations, as well as implementation issues. We also quantify the lower limit of the continuum (Darcy) scale, as well as identify the optimal representative elementary volume for the hybrid LB–FE simulations.

Magneto-elastic modeling of composites containing chain-structured magnetostrictive particles

Abstract: Magneto-elastic behavior is investigated for two-phase composites containing chain-structured magnetostrictive particles under both magnetic and mechanical loading. To derive the local magnetic and elastic fields, three modified Green's functions are derived and explicitly integrated for the infinite domain containing a spherical inclusion with a prescribed magnetization, body force, and eigenstrain. A representative volume element containing a chain of infinite particles is introduced to solve averaged magnetic and elastic fields in the particles and the matrix. Effective magnetostriction of composites is derived by considering the particle's magnetostriction and the magnetic interaction force. It is shown that there exists an optimal choice of the Young's modulus of the matrix and the volume fraction of the particles to achieve the maximum effective magnetostriction. A transversely isotropic effective elasticity is derived at the infinitesimal deformation. Disregarding the interaction term, this model provides the same effective elasticity as Mori–Tanaka's model. Comparisons of model results with the experimental data and other models show the efficacy of the model and suggest that the particle interactions have a considerable effect on the effective magneto-elastic properties of composites even for a low particle volume fraction.

Quantification of stochastically stable representative volumes for random heterogeneous materials

Abstract: This paper details a procedure to determine lower bounds on the size of representative volume elements (RVEs) by which the size of the RVE can be quantified objectively for random heterogeneous materials. Here, attention is focused on granular materials with various distributions of inclusion size and volume fraction of inclusions. An extensive analysis of the RVE size dependence on the various parameters is performed. Both deterministic and stochastic parameters are analysed. Also, the effects of loading mode and the parameter of interest are studied. As the RVE size is a function of the material, some material properties such as Young's modulus and Poisson's ratio are analysed as factors that influence the RVE size. The lower bound of RVE size is found as a function of the stochastically distributed volume fraction of inclusions; thus the stochastic stability of the obtained results is assessed. To this end a newly defined concept of stochastic stability (DH-stability) is introduced by which stochastic effects can be included in the stability considerations. DH-stability can be seen as an extension of classical Lyapunov stability. As is shown, DH-stability provides an objective tool to establish the lower bound nature of RVEs for fluctuations in stochastic parameters.

On the size of representative volume element for Darcy law in random media

Abstract: Most studies of effective properties of random heterogeneous materials are based on the assumption of the existence of a representative volume element (RVE), without quantitatively specifying its size L relative to that of the micro-heterogeneity d . In this paper, we study the finite-size scaling trend to RVE of the Darcy law for Stokesian flow in random porous media, without invoking any periodic structure assumptions, but only assuming the microstructure9s statistics to be spatially homogeneous and ergodic. By analogy to the existing methodology in thermomechanics of random materials, we first formulate a Hill–Mandel condition for the Darcy flow velocity and pressure gradient fields. This dictates uniform Neumann and Dirichlet boundary conditions, which, with the help of two variational principles, lead to scale-dependent hierarchies on effective (RVE level) permeability. To quantitatively assess the scaling trend towards the RVE, these hierarchies are computed for various porosities of random disc systems, where the disc centres are generated by a planar hard-core Poisson point field. Overall, it turns out that the higher is the density of random discs—or, equivalently, the narrower are the micro-channels in the system—the smaller is the size of RVE pertaining to the Darcy law.

Representative volumes and multi-scale modelling of quasi-brittle materials

Abstract: Several different approaches are available in order to describe material behaviour. Considering material on the higher (macro) level of observation constitutes the macroscopic approach. However, the key to understand a macro materials behaviour lies in its mesostructure. As such the mesoscopic approach can be used which is based on the detailed material description of the lower (meso) observational level. The main focus of this dissertation is the combination of the two above techniques -- the multi-scale approach. The idea is, by means of a hierarchical multi-scale procedure, to bring the homogenised information of the detailed mesostructural description to the macro-level in the form of effective properties. Thus, the homogeneous macrostructural behaviour is driven by the heterogeneous mesostructure. Traditionally, the size of a Representative Volume Element (RVE) of the material on the meso-level is chosen as a model parameter within the multi-scale framework. Two questions arise: what should this size be and how stable is this multi-scale model based on an RVE? As an answer to the first question, a unique procedure to determine the RVE size is proposed in the dissertation. An extensive study of this size sensitivity to different test and material parameters, both deterministic and stochastic, has been discussed. With knowledge of the RVE size, the multi-scale procedure can be introduced, in which the meso-level RVE plays the role of a macro-level length-scale parameter. However, the answer to the second question is not always positive. As an example the material behaviour due to mechanical loading can be considered. Although the results are stable and reliable in the linear-elastic and hardening regimes, the picture changes in softening. This is caused by the material developing strain localisation and as a consequence losing its statistical homogeneity. For such a material a Representative Volume cannot be found and as an inference cannot be used in the multi-scale framework. A conceptually new so-called coupled-volume multi-scale approach is introduced, based on abandoning the separation of scales principle. This approach does not require an RVE be a model parameter. The idea of the approach is to uniquely link the size of the mesostructural unit cell and element size of the discretised macrostructure. The results of this coupled-volume approach show stable and reliable behaviour in all mechanical regimes.