Showing papers on "Representative elementary volume published in 2009"

Micromechanics of spatially uniform heterogeneous media: A critical review and emerging approaches

Abstract: Outside of the classical microstructural detail-free estimates of effective moduli, micromechanical analyses of macroscopically uniform heterogeneous media may be grouped into two categories based on different geometric representations of material microstructure. Analysis of periodic materials is based on the repeating unit cell (RUC) concept and the associated periodic boundary conditions. This contrasts with analysis of statistically homogeneous materials based on the representative volume element (RVE) concept and the associated homogeneous boundary conditions. In this paper, using the above classification framework we provide a critical review of the various micromechanical approaches that had evolved along different paths, and outline recent emerging trends. We begin with the basic framework for the solution of micromechanics problems independent of microstructural representation, and then clarify the often confused RVE and RUC concepts. Next, we describe classical models, including the available RVE-based models, and critically examine their limitations. This is followed by discussion of models based on the concept of microstructural periodicity. In the final part, two recent unit cell-based models, which continue to evolve, are outlined. First, a homogenization technique called finite-volume direct averaging micromechanics theory is presented as a viable and easily implemented alternative to the mainstream finite-element based asymptotic homogenization of unit cells. The recent incorporation of parametric mapping into this approach has made it competitive with the finite-element method. Then, the latest work based on locally-exact solutions of unit cell problems is described. In this approach, the interior unit cell problem is solved exactly using the elasticity approach. The exterior problem is tackled with a new variational principle that successfully overcomes the non-separable nature of the overall unit cell problem.

Modeling of graphene?polymer interfacial mechanical behavior using molecular dynamics

Abstract: Carbon nanotube (CNT) polymer-matrix composites exhibit promising properties as structural materials for which appropriate constitutive models are sought, to predict their macroscale behavior. The reliability of determining the homogenized response of such materials depends upon the ability to accurately capture the interfacial behavior between the nanotubes and the polymer matrix. In this work, molecular dynamics simulations, using the Consistent Valence Force Field (CVFF) to describe the atomistic interactions, are used to study nanoscale load transfer between polyethylene and a graphene sheet, a model system chosen to characterize the force-separation behavior between CNTs and the polymer matrix. Separation studies are conducted for both opening as well as sliding modes and cohesive zone parameters such as peak traction and energy of separation are evaluated for each mode. Studies are also carried out to investigate the effect of tension and compression on sliding mode separation. Size dependence studies are conducted utilizing different sizes of the computational domain and different boundary conditions, to obtain the representative volume element and connect to continuum level properties. These results set the stage for continuum length-scale micromechanical models which may be used in determining the overall material response, incorporating interfacial phenomena.

A Micromechanics Model for the Electrical Conductivity of Nanotube-Polymer Nanocomposites

Abstract: The introduction of carbon nanotubes (CNTs) into nonconducting polymers has been observed to yield orders of magnitude increases in conductivity at very low concentrations of CNTs. These low percolation concentrations have been attributed to both the formation of conductive networks of CNTs within the polymer and to a nanoscale effect associated with the ability of electrons to transfer from one CNT to another known as electron hopping. In the present work, a micromechanics model is developed to assess the impact of the effects of electron hopping and the formation of conductive networks on the electrical conductivity of CNT-polymer nanocomposites. The micromechanics model uses the composite cylinders model as a nanoscale representative volume element where the effects of electron hopping are introduced in the form of a continuum interphase layer, resulting in a distinct percolation concentration associated with electron hopping. Changes in the aspect ratio of the nanoscale representative volume element are used to reflect the changes in nanocomposite conductivity associated with the formation of conductive networks due to the formation of nanotube bundles. The model results are compared with experimental data in the literature for both single- and multi-walled CNT nanocomposites where it is observed that the model developed is able to qualitatively explain the relative impact of electron hopping and nanotube bundling on the nanocomposite conductivity and percolation concentrations.

A semi‐discrete shell finite element for textile composite reinforcement forming simulation

Abstract: A triangular shell element for the simulation of textile composite reinforcements forming is proposed. This element is made up of unit woven cells. The internal virtual works are added on all woven cells of the element. They depend on tensions, in-plane shear and bending moments that are directly those given by the experimental tests that are specific to textile composite reinforcement. The element has only displacement degrees of freedom; the bending curvatures are obtained from the displacement of the neighbouring elements. A set of example shows the efficiency of the approach and the relative roles of the tensile, in-plane shear and bending rigidities. Especially their influence on the appearance and the development of wrinkles in draping and forming tests is analysed. Copyright © 2009 John Wiley & Sons, Ltd.

Micromechanical approach to the strength properties of frictional geomaterials

Abstract: The present paper describes a micromechanics-based approach to the strength properties of composite materials with a Drucker–Prager matrix in the situation of non-associated plasticity. The concept of limit stress states for such materials is first extended to the context of homogenization. It is shown that the macroscopic limit stress states can theoretically be obtained from the solution to a sequence of viscoplastic problems stated on the representative elementary volume. The strategy of resolution implements a non-linear homogenization technique based on the modified secant method. This procedure is applied to the determination of the macroscopic strength properties and plastic flow rule of materials reinforced by rigid inclusions, as well as for porous media. The role of the matrix dilatancy coefficient is in particular discussed in both cases. Finally, finite element solutions are derived for a porous medium and compared to the micromechanical predictions.

Variationally consistent computational homogenization of transient heat flow

Abstract: A framework for variationally consistent homogenization, combined with a generalized macrohomogeneity condition, is exploited for the analysis of non-linear transient heat conduction. Within this framework the classical approach of (model-based) first-order homogenization for stationary problems is extended to transient problems. Homogenization is then carried out in the spatial domain on representative volume elements (RVE), which are (in practice) introduced in quadrature points in standard fashion. Along with the classical averages, a higher order conservation quantity is obtained. An iterative FE2-algorithm is devised for the case of non-linear diffusion and storage coefficients, and it is applied to transient heat conduction in a strongly heterogeneous particle composite. Parametric Studies are carried Out, in particular with respect to the influence of the 'internal length' associated with the second-order conservation quantity. Copyright (C) 2009 John Wiley & Sons, Ltd.

Continuum Microviscoelasticity Model for Aging Basic Creep of Early-Age Concrete

Abstract: We propose a micromechanics model for aging basic creep of early-age concrete. Therefore, we formulate viscoelastic boundary value problems on two representative volume elements, one related to cement paste (composed of cement, water, hydrates, and air), and one related to concrete (composed of cement paste and aggregates). Homogenization of the “nonaging” elastic and viscoelastic properties of the material’s contituents involves the transformation of the aforementioned viscoelastic boundary value problems to the Laplace-Carson (LC) domain. There, formally elastic, classical self-consistent and Mori-Tanaka solutions are employed, leading to pointwisely defined LC-transformed tensorial creep and relaxation functions. Subsequently, the latter are back-transformed, by means of the Gaver-Wynn-Rho algorithm, into the time domain. Temporal derivatives of corresponding homogenized creep and relaxation tensors, evaluated for the current maturation state of the material (in terms of current volume fractions of cem...

The effect of yarn distortion on the mechanical properties of 3D four-directional braided composites

Abstract: A representative volume cell (RVC) is chosen to analyze the mechanical properties of 3D (3 dimensions) four-directional braided composites. Owing to braid yarns (an assembly of fibers) squeezing against each other in the braided composites, the braid yarns are distorted. Based on geometrical characteristics of the braided composites, cross-section of each braid yarn is supposed to be an octagon and divided into seven regions in the RVC. The distortion characteristics of yarns are considered in each region. Elastic properties of each region obtained by stochastic function theory are introduced into finite element model to calculate the mechanical properties of the RVC. The influences of yarn distortion on the stiffness and strength of the braided composites are obtained and discussed.

Influence of Martensite Mechanical Properties on Failure Mode and Ductility of Dual-Phase Steels

Abstract: The effects of the mechanical properties of the martensite phase on the failure mode and ductility of dual-phase (DP) steels are investigated using a micromechanics-based finite element method. Actual microstructures of DP steels obtained from scanning electron microscopy (SEM) are used as representative volume elements (RVEs) in the finite element calculations. Ductile failure of the RVE is predicted as plastic strain localization during the deformation process. Systematic computations are conducted on the RVE to quantitatively evaluate the influence of the martensite mechanical properties and volume fraction on the macroscopic mechanical properties of DP steels. These properties include the ultimate tensile strength (UTS), ultimate ductility, and failure modes. The computational results show that, as the strength and volume fraction of the martensite phase increase, the UTS of DP steels increases, but the UTS strain and failure strain decrease. In addition, shear-dominant failure modes usually develop for DP steels with lower martensite strengths, whereas split failure modes typically develop for DP steels with higher martensite strengths. The methodology and data presented in this article can be used to tailor DP steel design for its intended purposes and desired properties.

Effective Permeability of Media with a Dense Network of Long and Micro Fractures

Abstract: This article presents a new methodology to estimate the effective permeability of random fractured media of any anisotropy containing both microfractures and a large number of long fractures crosscutting the representative volume element. The fractures are replaced by fictitious permeable materials for which the tangential permeability is deduced from a Poiseuille flow. A self-consistent scheme is proposed to derive the macroscopic permeability. On the one hand, the contribution of long fractures to the effective permeability writes by simple superposition of the fracture tangential permeabilities. On the other hand, the contribution of microfractures needs to resort to auxiliary problems requiring the computation of second-order Hill (or Eshelby) tensors related to ellipsoids embedded in an anisotropic matrix, for which a complete procedure is detailed. The effect of the microfracture normal permeability is put in evidence in the upscaling scheme and analyzed. In particular, it is shown that it must be chosen large enough to allow the connections between families. Examples are finally developed and compared to numerical simulations in the 2D case.

A micromechanical hyperelastic modeling of brain white matter under large deformation

Abstract: A finite element based micromechanical model has been developed for analyzing and characterizing the microstructural as well as homogenized mechanical response of brain tissue under large deformation. The model takes well-organized soft tissue as a fiber-reinforced composite with nonlinear and anisotropic behavior assumption for the fiber as well as the matrix of composite matter. The procedure provides a link between the macroscopic scale and microscopic scale as brain tissue undergoes deformation. It can be used to better understand how macroscopic stresses are transferred to the microstructure or cellular structure of the brain. A repeating unit cell (RUC) is created to stand as a representative volume element (RVE) of the hyperelastic material with known properties of the constituents. The model imposes periodicity constraints on the RUC. The RUC is loaded kinematically by imposing displacements on it to create the appropriate normal and shear stresses. The homogenized response of the composite, the average stresses carried within each of the constituents, and the maximum local stresses are all obtained. For each of the normal and shear loading scenarios, the impact of geometrical variables such as the axonal fiber volume fraction and undulation of the axons are evaluated. It was found that axon undulation has significant impact on the stiffness and on how stresses were distributed between the axon and the matrix. As axon undulation increased, the maximum stress and stress in the matrix increased while the stress in the axons decreased. The axon volume fraction was found to have an impact on the tissue stiffness as higher axon volume fractions lead to higher stresses both in the composite and in the constituents. The direction of loading clearly has a large impact on how stresses are distributed amongst the constituents. This micromechanics tool provides the detailed micromechanics stresses and deformations, as well as the average homogenized behavior of the RUC, which can be efficiently used in mechanical characterization of brain tissue.

Characterisation of formability behaviour of multiphase steels by micromechanical modelling

Abstract: Multiphase steels are new Advanced High Strength Steels (AHSS), which have been developed for the automobile industry for the purpose of reducing of car body weight. These steels offer an excellent combination of high strength and large elongation due to the coexistence of harder and softer phases in their microstructure. The advantageous properties of multiphase steels can be utilised by regulating the type, amount, formation and spatial distribution of the different constituent phases. To describe the influences of the heterogeneous microstructure on the mechanical properties and the complex fracture mechanisms, an approach is presented using Representative Volume Elements (RVEs) on a micro level. Three dimensional RVE simulations were conducted under considerations of metallographic analysis for a Dual Phase (DP) steel and a Transformation-Induced Plasticity (TRIP) steel. The Gurson–Tvergaard–Needleman (GTN) damage model was applied to investigate the local crack initiation in steel sheets during various forming processes. In conclusion, a failure prediction based on microstructure is proposed for the material characterisation in sheet metal forming of multiphase steels.

Effects of inter-fibre spacing on damage evolution in unidirectional (UD) fibre-reinforced composites

Abstract: A three dimensional (3D) micromechanical study has been performed in order to investigate local damage in UD composite materials under transverse and longitudinal tensile loading. In particular, the influence of non-uniform distribution of fibres in RVEs (representative volume element) with a hexagonal packing array and the effects of thermal residual stresses has been investigated. To examine the effect of inter-fibre spacing and residual stress on failure, a study based on the Maximum Principal Stress failure criterion and a stiffness degradation technique has been used for damage analysis of the unit cell subjected to mechanical loading. Results indicate a strong dependence of damage onset and its evolution from the fibres position within the RVE. Predicted mechanical properties, damage initiation and evolution are also clearly influenced by the presence of residual stress.

Damage Evolution in Composites with a Homogenization-based Continuum Damage Mechanics Model

Abstract: This paper develops a 3D homogenization-based continuum damage mechanics (HCDM) model for fiber reinforced composites undergoing micro- mechanical damage. Micromechanical damage in the representative volume element (RVE) is explicitly incorporated in the form of fiber-matrix interfacial debonding. The model uses the evolving principal damage coordinate system as its reference in order to represent the anisotropic coefficients. This is necessary for retaining accuracy with nonproportional loading. The material constitutive law involves a fourth order orthotropic tensor with stiffness characterized as macroscopic internal variable. Damage in 3D composites is accounted for through functional forms of the fourth order damage tensor in terms of macroscopic strain components. The HCDM model parameters are calibrated by using homogenized micromechanical (HMM) solutions for the RVE for a few strain histories. The proposed model is validated by comparing the CDM results with HMM response of single and multiple fiber RVEs subjected to arbitrary loading history. Finally the HCDM model is incorporated in a macroscopic finite element code to conduct damage analysis in a structure. The effect of different microstructures on the macroscopic damage progression is examined through this study.

Influence of Martensite Distribution on the Mechanical Properties of Dual Phase Steels: Experiments and Simulation

Abstract: The application of multiphase steels in the automotive industry has been rapidly increased according to economic, environmental and safety reasons. To determine an optimal combination of high strength and good formability of multiphase steels by using the FE modelling, their complex microstructures have to be considered. Two-dimensional Representative Volume Elements (RVEs) were currently developed based on real microstructures for dual phase (DP) steels. In general, the microstructure of DP steels contains hard martensite particles and a soft ferritic matrix. The strain hardening behaviour of the individual phases was described in the model taking the microstructural constituents and the carbon partitioning during intercritical annealing into account. Two dual phase microstructures with same martensitecontent but different martensite distributions were investigated in experiment as well as in FEM simulation by means of the RVE. The resulting mechanical properties of these steels are strongly influenced by the phase distribution and interaction. As validation, calculated flow curves were compared with the experimental results from quasi-static tensile tests. In addition, the local stress and strain partitioning between both phases depending on the spatial phase distribution and morphology is discussed.

Independent mesh method-based prediction of local and volume average fields in textile composites

Abstract: An independent mesh method (IMM) for three-dimensional stress analysis in composites with complex fiber architectures is proposed. The method represents a combination of direct meshing and voxel-based methodology and allows the modeling of complex tow geometries not readily amenable to traditional finite element meshing. Each fiber tow is meshed independently, while the matrix is meshed throughout the volume of interest. The matrix approximation is then truncated by disregarding the shape functions, whose support is completely inside a tow or completely covered by more than one tow in regions such as tow intersections. The calculation of average stiffness properties of both an oblong fiber-matrix representative volume element (RVE) and a plain weave composite RVE is performed for verification and convergence evaluation purposes. The digital chain technique was used to model fiber architecture in the tri-axial braided composite with high fidelity including the effects of nesting and compaction of plies. Local deformations of the digital architecture due to relief of residual processing stress following a saw cut were predicted by using IMM. These deformations in the tri-axial braided composite were then measured experimentally using Moire interferometry. The degree of agreement between the predicted strain fields and those measured experimentally was shown to correlate with the degree of accuracy of digital architecture and varied from agreement in average behavior to practically point wise agreement across the entire field of measurement.