Showing papers on "Representative elementary volume published in 2012"

Continuum damage mechanics : a continuum mechanics approach to the analysis of damage and fracture

Abstract: Preface.- List of Symbols.- PART I Foundations of Continuum Damage Mechanics.- 1. Material Damage and Continuum Damage Mechanics.- 1.1 Damage and its Microscopic Mechanisms.- 1.2 Representative Volume Element and Continuum Damage Mechanics.- 2. Mechanical Representation of Damage and Damage Variables.- 2.1 Mechanical Modeling of Damage.- 2.2 Mechanical Representation of Three-Dimensional Damage State.- 2.3 Effective Stress and Hypothesis of Mechanical Equivalence.- 2.4 Elastic Constitutive Equation and Elastic Modulus Tensor of Damaged Material.- 2.5 Procedure of Continuum Damage Mechanics and its Refinement.- 3. Thermodynamics of Damaged Material.- 3.1 Thermodynamics of Continuum.- 3.2 Thermodynamic Constitutive Theory of Inelasticity with Internal Variables.- 3.3 Extension of Thermodynamic Constitutive Theory of Inelasticity.- 4. Inelastic Constitutive Equations and Evolution Equations of Material with Isotropic Damage.- 4.1 One-Dimensional Inelastic Constitutive Equation of Material with Isotropic Damage.- 4.2 Three-Dimensional Inelastic Constitutive Equations of Material with Isotropic Damage.- 4.3 Strain Energy Release Rate and Stress Criterion for Damage Development in Elastic-Plastic Damage.- 4.4 Inelastic Damage Theory based on Hypothesis of Total Energy Equivalence.- 5. Inelastic Constitutive Equation and Damage Evolution Equation of Material with Anisotropic Damage.- 5.1 Elastic-Plastic Anisotropic Damage Theory based on Second-Order Symmetric Damage Tensor.- 5.2 Elastic-Plastic Anisotropic Damage Theory in Stress Space.- 5.3 Fourth-Order Symmetric Damage Tensor and its Application to Elastic-Plastic-Brittle Damage.- PART II Application of Continuum Damage Mechanics.- 6. Elastic-Plastic Damage.- 6.1 Constitutive and Evolution Equations of Elastic-Plastic Damage - Ductile Damage, Brittle Damage and Quasi-Brittle Damage.- 6.2 Ductile Damage and Ductile Fracture.- 6.3 Application to Metal Forming Process.- 6.4 Analysis of Sheet Forming Limit by Anisotropic Damage Theory.- 6.5 Constitutive Equations of Void-Containing Ductile Material.- 6.6 Continuum Damage Mechanics Theory with Plastic Compressibility.- 7. Fatigue Damage.- 7.1 High Cycle Fatigue .- 7.2 Low Cycle Fatigue.- 7.3 Uncoupled Numerical Analysis of Very Low Cycle Fatigue.- 8. Creep Damage and Creep-Fatigue Damage.- 8.1 Creep Damage and Phenomenological Theory of Creep Damage.- 8.2 Viscoplastic Damage Theory of Creep Damage.- 8.3 Creep-Fatigue Damage.- 8.4 Effect of Damage Field on Stress Field at a Creep Crack Tip.- 9. Elastic-Brittle Damage.- 9.1 Damage of Elastic-Brittle Material.- 9.2 Isotropic Damage Theory of Concrete.- 9.3 Anisotropic Brittle Damage Theory by Second-Order Damage Tensor.- 9.4 Anisotropic Brittle Damage Theory with Elastic Modulus Tensor as Damage Variable.- 9.5 Anisotropic Brittle Damage Theory with Compliance Tensor as Damage Variable.- 10. Continuum Damage Mechanics of Composite Material.- 10.1 Damage of Laminate Composites.- 10.2 Elastic-Brittle Damage of Ceramic Matrix Composites.- 10.3 Local Theory of Metal Matrix Composites.- 11. Local Approach to Damage and Fracture Analysis.- 11.1 Local Approach to Fracture Based on Continuum Damage Mechanics and Finite Element Method.- 11.2 Mesh-Sensitivity in Time-Independent Deformation.- 11.3 Regularization of Strain and Damage Localization in Time-Independent Materials.- 11.4 Mesh-Sensitivity in Time-Dependent Deformation.- 11.5 Causes of Mesh-Sensitivity in Time-Dependent Deformation.- APPENDIX Foundations of Tensor Analysis.- A.1 Vectors and Tensors.- A.2 Vector Product, Tensor Product and the Components of Tensors.- A.3 Orthogonal Transformation, Invariants and Eigenvalues of Tensors.- A.4 Differentiation and Integral of Tensor Fields.- A.5 Differential Calculus of Tensor Functions.- A.6 Representation Theorem for Tensor Functions.- A.7 Matrix Representation of Tensors and Tensor Relations.- Reference Books and Bibliography.- Subject Index.

Elastic–plastic behavior of non-woven fibrous mats

Abstract: Electrospinning is a novel method for creating non-woven polymer mats that have high surface area and high porosity. These attributes make them ideal candidates for multifunctional composites. Understanding the mechanical properties as a function of fiber properties and mat microstructure can aid in designing these composites. Further, a constitutive model which captures the membrane stress–strain behavior as a function of fiber properties and the geometry of the fibrous network would be a powerful design tool. Here, mats electrospun from amorphous polyamide are used as a model system. The elastic–plastic behavior of single fibers are obtained in tensile tests. Uniaxial monotonic and cyclic tensile tests are conducted on non-woven mats. The mat exhibits elastic–plastic stress–strain behavior. The transverse strain behavior provides important complementary data, showing a negligible initial Poisson's ratio followed by a transverse:axial strain ratio greater than −1:1 after an axial strain of 0.02. A triangulated framework has been developed to emulate the fibrous network structure of the mat. The micromechanically based model incorporates the elastic–plastic behavior of single fibers into a macroscopic membrane model of the mat. This representative volume element based model is shown to capture the uniaxial elastic–plastic response of the mat under monotonic and cyclic loading. The initial modulus and yield stress of the mat are governed by the fiber properties, the network geometry, and the network density. The transverse strain behavior is linked to discrete deformation mechanisms of the fibrous mat structure including fiber alignment, fiber bending, and network consolidation. The model is further validated in comparison to experiments under different constrained axial loading conditions and found to capture the constraint effect on stiffness, yield, post-yield hardening, and post-yield transverse strain behavior. Due to the direct connection between microstructure and macroscopic behavior, this model should be extendable to other electrospun systems and other two dimensional random fibrous networks.

Evaluation of generalized continuum substitution models for heterogeneous materials

Abstract: Several extensions of standard homogenization methods for composite materials have been proposed in the literature that rely on the use of polynomial boundary conditions enhancing the classical affine conditions on the unit cell. Depending on the choice of the polynomial, overall Cosserat, second gradient, or micromorphic homogeneous substitution media are obtained. They can be used to compute the response of the composite when the characteristic length associated with the variation of the applied loading conditions becomes of the order of the size of the material inhomogeneities. A significant difference between the available methods is the nature of the fluctuation field added to the polynomial expansion of the displacement field in the unit cell, which results in different definitions of the overall stress and strain measures and higher order elastic moduli. The overall higher order elastic moduli obtained from some of these methods are compared in the present contribution in the case of a specific periodic two-phase composite material. The performance of the obtained overall substitution media is evaluated for a chosen boundary value problem at the macroscopic scale for which a reference finite element solution is available. Several unsatisfactory features of the available theories are pointed out, even though some model predictions turn out to be highly relevant. Improvement of the prediction can be obtained by a precise estimation of the fluctuation at the boundary of the unit cell.

Representative volume element to estimate buckling behavior of graphene/polymer nanocomposite

Abstract: The aim of the research article is to develop a representative volume element using finite elements to study the buckling stability of graphene/polymer nanocomposites. Research work exploring the full potential of graphene as filler for nanocomposites is limited in part due to the complex processes associated with the mixing of graphene in polymer. To overcome some of these issues, a multiscale modeling technique has been proposed in this numerical work. Graphene was herein modeled in the atomistic scale, whereas the polymer deformation was analyzed as a continuum. Separate representative volume element models were developed for investigating buckling in neat polymer and graphene/polymer nanocomposites. Significant improvements in buckling strength were observed under applied compressive loading when compared with the buckling stability of neat polymer.

The influence of the representative volume element (RVE) size on the homogenized response of cured fiber composites

Abstract: The influence of the representative volume element (RVE) size (in terms of fiber packing and number of fibers for a given fiber-volume fraction) on the residual stresses created during the curing process of a continuous fiber-reinforced polymer matrix tow is investigated with the ultimate goal of finding a minimum unit cell size that can be used later for a homogenization procedure to calculate the response of woven fiber textile composites and in particular, fiber tows. A novel network curing model for the solidification of epoxy is used to model the curing process. The model takes into account heat conduction, cure kinetics and the creation of networks in a continuously shape changing body. The model is applied to the curing of a fiber/matrix RVE. The results for the minimum size of the RVE, obtained on the basis of the curing problem, are compared with a similar RVE, modeled as an elastic-plastic solid subjected to external loads, in order to compare the minimum RVE sizes obtained on the basis of different boundary value problem solutions. (Some figures may appear in colour only in the online journal)

Effect of carbon nanotube orientation on the mechanical properties of nanocomposites

Abstract: Carbon nanotubes (CNTs) have been regarded as ideal reinforcements of high-performance composites with enormous applications. In this paper, nano-structure is modeled as a linearly elastic composite medium, which consists of a homogeneous matrix having hexagonal representative volume elements (RVEs) and homogeneous cylindrical nanotubes with various inclination angles. Effects of inclined carbon nanotubes on mechanical properties are investigated for nano-composites using 3-D hexagonal representative volume element (RVE) with short and straight CNTs. The CNT is modeled as a continuum hollow cylindrical shape elastic material with different angles. The effect of the inclination of the CNT and its parameters is studied. Numerical equations are used to extract the effective material properties for the hexagonal RVE under axial as well as lateral loading conditions. The computational results indicated that elastic modulus of nano-composite is remarkably dependent on the orientation of the dispersed SWNTs. It is observed that the inclination significantly reduces the effective Young’s modulus of elasticity under an axial stretch. When compared with lateral loading case, effective reinforcement is found better in axial loading case. The effective moduli are very sensitive to the inclination and this sensitivity decreases with the increase of the waviness. In the case of short CNTs, increasing trend is observed up to a specific value of waviness index. It is also found from the simulation results that geometry of RVE does not have much significance on stiffness of nano-structures. The results obtained for straight CNTs are consistent with ERM results for hexagonal RVEs, which validate the proposed model results.

Transformation-Induced, Geometrically Necessary, Dislocation-Based Flow Curve Modeling of Dual-Phase Steels: Effect of Grain Size

Abstract: The flow behavior of dual-phase (DP) steels is modeled on the finite-element method (FEM) framework on the microscale, considering the effect of the microstructure through the representative volume element (RVE) approach. Two-dimensional RVEs were created from microstructures of experimentally obtained DP steels with various ferrite grain sizes. The flow behavior of single phases was modeled through the dislocation-based work-hardening approach. The volume change during austenite-to-martensite transformation was modeled, and the resultant prestrained areas in the ferrite were considered to be the storage place of transformation-induced, geometrically necessary dislocations (GNDs). The flow curves of DP steels with varying ferrite grain sizes, but constant martensite fractions, were obtained from the literature. The flow curves of simulations that take into account the GND are in better agreement with those of experimental flow curves compared with those of predictions without consideration of the GND. The experimental results obeyed the Hall-Petch relationship between yield stress and flow stress and the simulations predicted this as well.

Stiffness and strength of hierarchical polycrystalline materials with imperfect interfaces

Abstract: In this study we investigate the effect of imperfect (not perfectly bonded) interfaces on the stiffness and strength of hierarchical polycrystalline materials. As a case study we consider a honeycomb cellular polycrystal used for drilling and cutting tools. The conclusions of the analysis are, however, general and applicable to any material with structural hierarchy. Regarding the stiffness, generalized expressions for the Voigt and Reuss estimates of the bounds to the effective elastic modulus of heterogeneous materials are derived. The generalizations regard two aspects that are not included in the standard Reuss and Voigt estimates. The first novelty consists in considering finite thickness interfaces between the constituents undergoing damage up to final debonding. The second generalization consists of interfaces not perpendicular or parallel to the loading direction, i.e., when isostress or isostrain conditions are not satisfied. In this case, approximate expressions for the effective elastic modulus are obtained by performing a computational homogenization approach. In the second part of the paper, the homogenized response of a representative volume element (RVE) of the honeycomb cellular polycrystalline material with one or two levels of hierarchy is numerically investigated. This is put forward by using the cohesive zone model (CZM) for finite thickness interfaces recently proposed by the authors and implemented in the finite element program FEAP. From tensile tests we find that the interface nonlinearity significantly contributes to the deformability of the material. Increasing the number of hierarchical levels, the deformability increases. The RVE is tested in two different directions and, due to different orientations of the interfaces and Mixed Mode deformation, anisotropy in stiffness and strength is observed. Stiffness anisotropy is amplified by increasing the number of hierarchical levels. Finally, the interaction between interfaces at different hierarchical levels is numerically characterized. A condition for scale separation, which corresponds to the independence of the material tensile strength from the properties of the interfaces in the second level, is established. When this condition is fulfilled, the material microstructure at the second level can be efficiently replaced by an effective homogeneous continuum with a homogenized stress–strain response. From the engineering point of view, the proposed criterion of scale separation suggests how to design the optimal microstructure of a hierarchical level to maximize the material tensile strength. An interpretation of this phenomenon according to the concept of flaw tolerance is finally presented.

Prediction of material properties of biaxial and triaxial braided textile composites

Abstract: Biaxial and triaxial braided composites are modeled using finite element methods to predict effective material properties: for biaxial braids, diamond pattern (1/1), regular pattern (2/2), and Hercules pattern (3/3) are modeled, while for triaxial braids, a regular pattern (2/2) is modeled. A micromechanical approach is adopted to calculate material properties of the tows, which are critical compositions of braided composites. By applying periodical boundary conditions (PBC), four representative unit cells (RUC) in correspondence to the four types of braids are analyzed and compared. Subsequently, effective material properties are obtained with the braiding angle varying from 15° to 75° with an increment of 5°, from which the variation of the engineering constants with the braiding angle is studied. The prediction results are compared with the experimental values for two material systems, and good agreement is achieved.

Steady Couette flows of elastoviscoplastic fluids are nonunique

Abstract: The Herschel–Bulkley rheological fluid model includes terms representing viscosity and plasticity. In this classical model, below the yield stress the material is strictly rigid. Complementing this model by including elastic behavior below the yield stress leads to a description of an elastoviscoplastic (EVP) material such as an emulsion or a liquid foam. We include this modification in a completely tensorial description of cylindrical Couette shear flows. Both the EVP model parameters, at the scale of a representative volume element, and the predictions (velocity, strain and stress fields) can be readily compared with experiments. We perform a detailed study of the effect of the main parameters, especially the yield strain. We discuss the role of the curvature of the cylindrical Couette geometry in the appearance of localization; we determine the value of the localization length and provide an approximate analytical expression. We then show that, in this tensorial EVP model of cylindrical Couette shear f...

Permeability prediction of fibrous porous media with complex 3D architectures

Abstract: We present a new three-dimensional finite element technique to solve efficiently flows in a representative porous volume with fibrous microstructures, which employs a fictitious domain method to deal with immersed microstructures and a mortar-element method to satisfy rigorously the tri-periodic boundary condition for the representative volume element. Through the extensive numerical simulations for various fiber and fabric architectures, we investigate the relationship between the permeability and fiber architectures in order to establish a reasonable approximation method in estimating the permeability of such complex 3D architectures. Specifically we discuss the applicability and the limitation of the macroscopic permeability averaging rule for those purposes, using the permeability of simple structural building blocks. Finally, we present the Kozeny constants of various microstructures for a wide range of the fiber volume fraction, which may facilitate simple permeability estimation of complex 3D porous structures using the Kozeny–Carman model.

Determination of Geometrical and Structural Representative Volume Elements at the Baihetan Dam Site

Abstract: A case study at the Baihetan dam site was undertaken to obtain a representative volume element (RVE) size. Two-dimensional fracture information in an exploration tunnel was used to generate a large three-dimensional fracture network. By dividing the entire modeled rock mass into cubes, the volumetric fracture density (P32) value of each cube was determined. The size effect can be determined by changing the cube size. The RVE was determined using P32 calculation and statistical tests, including Kolmogorov-Smirnov (KS) and Wilcoxon rank-sum tests. The P32 value depends on the geometrical parameters of fracture density and size; in this study, this value is called the geometrical RVE. P32 is dependent on fracture density and size, and cannot appropriately reflect certain data such as fracture dip or dip angle. Therefore, we propose a structural RVE (SRVE) considering fracture dip direction, dip angle, density, and size, which together constitute the available information about the fracture network. Therefore, the SRVE is more applicable for use in solving geological problems than the RVE. In this analysis, the KS and Wilcoxon rank-sum tests were used to determine the SRVE size.

Analytical Estimation of Elastic Properties of Polypropylene Fiber Matrix Composite by Finite Element Analysis

Abstract: A structural composite is a material system consisting of two or more phases on a macroscopic scale, whose mechanical performance and properties are designed to be superior to those of constituent materials acting independently. Fiber reinforced composites (FRP) are slowly emerging from the realm of advanced materials and are replacing conventional materials in a variety of applications. However, the mechanics of FRPs are complex owing to their anisotropic and heterogeneous characteristics. In this paper a representative volume model has been considered and a finite element model incorporating the necessary boundary conditions is developed using available FEA package ANSYS to predict the elastic property of the composite. For verification, the numerical results of elastic properties are compared with the analytical solution and it is found that there is a good agreement between these results.