Showing papers on "Representative elementary volume published in 1999"

Higher-order theory for functionally graded materials

Abstract: This paper presents the full generalization of the Cartesian coordinate-based higher-order theory for functionally graded materials developed by the authors during the past several years. This theory circumvents the problematic use of the standard micromechanical approach, based on the concept of a representative volume element, commonly employed in the analysis of functionally graded composites by explicitly coupling the local (microstructural) and global (macrostructural) responses. The theoretical framework is based on volumetric averaging of the various field quantities, together with imposition of boundary and interfacial conditions in an average sense between the subvolumes used to characterize the composite's functionally graded microstructure. The generalization outlined herein involves extension of the theoretical framework to enable the analysis of materials characterized by spatially variable microstructures in three directions. Specialization of the generalized theoretical framework to previously published versions of the higher-order theory for materials functionally graded in one and two directions is demonstrated. In the applications part of the paper we summarize the major findings obtained with the one-directional and two-directional versions of the higher-order theory. The results illustrate both the fundamental issues related to the influence of microstructure on microscopic and macroscopic quantities governing the response of composites and the technologically important applications. A major issue addressed herein is the applicability of the classical homogenization schemes in the analysis of functionally graded materials. The technologically important applications illustrate the utility of functionally graded microstructures in tailoring the response of structural components in a variety of applications involving uniform and gradient thermomechanical loading.

A two scale damage concept applied to fatigue

Abstract: The ductile type of damage is a phenomenon now well understood. Once the fully coupled set of constitutive equations is identified, Damage Mechanics is a powerful tool to predict failure. Brittle materials do not exhibit such a damageable macroscopic behavior. Nevertheless, they still fail. On the idea that damage is localized at the microscopic scale, a scale smaller than the mesoscopic one of the Representative Volume Element (RVE), we propose a three-dimensional failure modeling for monotonic as well as for fatigue loading. We develop a two scale model of what we shall call brittle damage: at the microscopic scale, micro-cracks or micro-voids exhibit a damageable plastic-like behavior with no effect on the global (mesoscopic) elastic behavior. Microscopic failure is assumed to coincide with the RVE failure. This model turns out to represent quite well physical phenomena related to high cycle fatigue such as the mean stress effect, the nonlinear accumulation of damage, initial strain hardening or damage effect and the nonproportional loading effect for bi-axial fatigue. The model has been implemented as a post-processor computer code. A simplified identification procedure for the determination of the material properties is given.

Scale effects on the elastic behavior of periodic andhierarchical two-dimensional composites

Abstract: The apparent stiffness tensors of two-dimensional elastic composite samples smaller thanthe representative volume element (RVE) are studied as a function of system size. Numericalexperiments are used to investigate how the apparent properties of the composite converge withincreasing scale factor n, defined to be the ratio between the linear size of the composite and thelinear size of the unit cell. Under affine (Dirichlet-type) or homogeneous stress (Neumann-type) boundary conditions, the apparent elastic moduli overestimate orunderestimate, respectively, the effective elastic moduli of the infinitely periodic system. Theresults show that the difference between the Dirichlet, Neumann and the effective stiffnesstensors depends strongly on the phase stiffness contrast ratio. Dirichlet boundary conditionsprovide a more accurate estimate of the effective elastic properties of stiff matrix composites,whereas Neumann boundary conditions provide a more accurate estimate for compliant matrixstructures. It is shown that the apparent bulk and shear moduli may lie outside of theHashin–Shtrikman bounds. However, these bounds provide good upper and lower estimates forthe apparent bulk and shear moduli of structures with a scale factor n ⩾ 2. A similar approach isused to study hierarchical composites containing two distinct structural levels with a finiteseparation of length scales. It is shown, numerically, that the error associated with replacing thesmallest-scale regions by an equivalent homogeneous medium is very small, even when the ratiobetween the length scales is as low as three.

Thermal Conductivity of Fiber Reinforced Composites by the FEM

Abstract: Applicability of the finite element method (FEM) in predicting the effective transverse thermal conductivity of fiber reinforced composites is systematically studied. Four different boundary condition combinations representing the periodicity of the temperature field are employed for ideal composites having perfect bond between fiber and matrix. Both circular and square cross-section fibers are studied. Comparisons of present FEM results with available analytical and experimental results reveal that periodicity realized by prescribed temperatures yields most accurate results up to high fiber volume fractions. In composites with interfacial thermal barrier resistance the effective conductivity varies in a wide range depending on the interfacial conductance between fiber and matrix. Best fit with available experimental results is obtained for both circular and square fibers when the dimensionless interfacial conductance is about 30. By employing the modeling practice found successful in the cases for which ...

Prediction of the large-strain mechanical response of heterogeneous polymer systems : local and global deformation behaviour of a representative volume element of voided polycarbonate

Abstract: The mechanical behaviour of voided polycarbonate has been predicted by using a detailed finite element model of the microstructure and an accurate elasto–viscoplastic model for the glassy polymeric matrix material. On the microstructural level a spatially periodic plane strain matrix with irregularly distributed voids has been considered. The voids represent low-modulus non-adhering rubbery particles under negative pressure. The constitutive model for the homogeneous parts of the material reflects the typical yield and post-yield behaviour of glassy polymers: strain rate and history dependent yield, intrinsic strain softening and subsequent strain hardening. The finite element simulations show that the irregular void distribution causes a radical change in deformation behaviour. In particular the macroscopic strain softening disappears. This transformation in macroscopic behaviour originates from the arbitrary order in which local shear bands between the randomly distributed voids are formed and subsequently harden. In the averaged overall mechanical response the individual unstable yield and post-yield behaviour of the local shear bands is evened out, resulting in an overall stable macroscopic deformation behaviour. This mechanism is believed to be primarily responsible for the toughness enhancement of heterogeneous polymer systems through the addition of easily cavitating rubbery particles.

Application of shakedown analysis to the plastic design of composites

Abstract: Shakedown theory is applied to the determination of plastically admissible ranges of external loads for periodic metal-matrix composites. A link is established between local stress-fields in a representative volume element on the microlevel and admissible stresses on the macrolevel by the use of the homogenisation technique. The developed methodology is applied for illustration to the problem of the best choice of design parameters for a composite.

A microstructure based numerical method for constitutive modeling of composite and porous materials

Abstract: An elastic–plastic constitutive model that incorporates the details of material microstructure, is proposed for modeling composite and porous materials. It is based on two-scale analysis using the asymptotic homogenization method and the Voronoi cell finite element model for detailed microstructural analysis. The linear elastic behavior is represented by an orthotropic elasticity tensor, obtained by solving an elastic microstructural problem with periodicity boundary conditions followed by homogenization. In the isotropic hardening plasticity with associate flow rule representation, a pressure-dependent yield function with strain and plastic work dependent parameters is postulated. The variation of these parameters are determined from a numerical database in the strain space that is generated from an ordered sequence of microstructural RVE (representative volume element) analyses with asymptotic homogenization. Numerical examples are conducted to examine the effectiveness of the constitutive model by comparing results of macroscopic analysis using the model and those of incremental two-scale analysis with homogenization. For a wide range of microstructures and loading conditions, the proposed model is found to yield extremely accurate solutions. Considerable advantage is achieved with respect to computational efficiency.

3-D Thermoelastic Moduli and Saturation Crack Density for Cross-Ply Laminates with Transverse Cracks

Abstract: For stiffness critical structures, such as those found in space applications, the structural design and functionality may depend on several stiffness components. In this work we study the effect of transverse cracking on the effective Young's moduli, the effective Poisson ratios and the effective coefficient of thermal expansion for glass/epoxy and graphite/epoxy cross-ply laminates. In particular, included in the study are the effect of prescribed boundary conditions on the representative volume element used to describe the problem, the effect of variable crack spacing, and the interaction of transverse crack stress fields. The analysis is accomplished by the use of a model based on Reissner's variational principle that has been shown to accurately model stress fields and energy release rates in flat laminates. The present model predictions are compared to shear lag and finite element predictions from the literature and found to be in good agreement with published experimental data. A criterion to estima...

A Novel 3D Microstructural Model for Trabecular Bone: II. The Relationship Between Fabric and the Yield Surface.

Abstract: A novel 3D microstructural model was proposed and validated in part I of this publication. In part IT, the model was used to identify the yield surface of a representative volume element of human trabecular bone as a function of volume fraction and degree of anisotropy. Finite element models of open and closed cells geometries were used to calculate effective yield stresses for a variety of loading cases with periodic boundary conditions. The postyield behaviour of the trabecular tissue was assumed from data available for cortical tissue. The yield stresses defined by a 0·2% offset in the global stress-strain curve were fit to an orthotropic Hill criterion and the parameters of the surface calculated. Similarly to the previous elastic analysis, distinct but strong relationships were obtained between volume fraction, fabric and the yield surface parameters for both the open and closed cell geometries. This finding suggests that volume fraction and fabric may be used to predict the initiation of mechanical ...

Modeling of Diffusion in a Micro-Cracked Composite Laminate Using Approximate Solutions

Abstract: A set of approximate solutions for predicting through-thickness diffusivity and solvent uptake behavior of micro-cracked composite laminates is presented. In order to maintain solution tractability, Fickian diffusion is adapted as the basic framework of the model. Starting with the assumption of a uniform micro-crack array on the exposed faces of the laminate, an RVE (representative volume element) approach is employed to estimate the laminate through-thickness diffusivity as a function of crack depth and crack density. It is also established that, for the Graphite/Epoxy material system under consideration, the rate of moisture ingress into micro-cracks due to capillary imbibition does not have an appreciable long-term influence on the laminate diffusivity and uptake. The approximate solutions are verified by comparison with available test data from moisture uptake experiments and with finite element predictions.

Local analysis and global nonlinear behaviour of periodic assemblies of bodies in elastic contact

Abstract: This work attempts to capture the effects of microstructural changes on the overall response of a composite made of bodies in elastic contact, and to define numerically a homogenised constitutive relationship for the global behaviour. The analysis is restricted to small strains, plane problems and monotonic proportional loads. An important feature of this work is the quasi-static frictional contact analysis of the microstructure composed of deformable elastic bodies by means of parametric quadratic programming principle and its corresponding algorithmic implementation. The generality of the homogenisation algorithm, which, in principle, can be applied to a large variety of non-linear behaviour affecting the representative volume element, is clearly described. Numerical examples are given to demonstrate the efficiency and validity of the algorithm presented in the paper.

Texture Development of Aluminum Polycrystals Under Finite Plastic Deformations

Abstract: A procedure to generate uniformly distributed orientations from random numbers has been introduced and turns out to be a simple way to get large sets of initial orientations Subjecting the pretextured aggregate to the shear deformation results in improved texture predictions in terms of both the intensity levels and the location of certain texture components This emphasizes the fact that the texture development is strongly influenced by the initial texture For initial isotropy the resulting texture can be described in terms of the A- and B-fibres [4] When starting from textured state the E-fibre is dominant The RVE homogenization method leads to a smoothing of the density distribution within the fibres The two homogenization techniques result principally in the same volume fractions of fibres vs shear number with exception of the A-fibre for which the Taylor model predicts a larger volume fraction

Global Non-Linear Behaviour of Periodic Composite Materials

Abstract: This paper aims to define a homogenised constitutive relation for the global behaviour of periodic composite structures in the case of non-linear material components. Special emphasis is put on the description of the generality of the algorithm which, in principle, can be applied to any kind of nonlinear material behaviour affecting the representative volume element. The method is currently restricted to plane situations with monotonic proportional loading.

Mechanics of Three-Dimensional Textile Structural Composites: Performance Modeling

Abstract: This paper presents a model for the prediction of anisotropic elastic stiffness of 3-D textile structural composites based on the preform architecture and the preforming process parameters. The methodology employed in this paper uses preform modeling to represent the fiber microstructure and uses micro-mechanical analysis to determine the macroscopic anisotropic stiffness. Representative volume elements (macro-cells) are identified for three-dimensionally woven preforms and geometric descriptions of the path and interactions between the tows within the cells are generated. This geometric modeling enables the estimation of the fiber volume fraction and the directional distribution of the fiber in the composite. The macro-cell is further decomposed into simpler elements whose stiffness is determined from tow geometry and the stiffness of the fiber/matrix constituents. The macroscopic stiffness of the textile composite is obtained from the macro-cell definition using an Effective Response Comparison (ERC) technique. Numerical results are compared with data from experimental characterization of several woven composites.

A Mechanical Model for the Prediction of the Elastic Properties of Polymeric Resins Reinforced with Liquid Crystal Polymers

Abstract: The mechanical behavior of semicrystalline polymeric resins reinforced with short fiber liquid crystal polymers (LCP) under mechanical loading has been investigated. A micromechanical approach is considered and a representative unit cell model is developed and employed in conjunction with the finite element method of analysis to predict the effective elastic properties of the composite. The advantage of such an approach is that it could contain an improved reference to the microstructural elements by including their geometric description and their non-isotropic behavior. In the present work, the composite is assumed to be macroscopically homogeneous and linearly elastic with periodic fiber distribution array. The finite element method was used in the solution of the micromechanical boundary value problem. Three-dimensional models of the representative unit cell were generated. Models with varying short fiber geometry and volume fractions were developed and a number of boundary conditions and loading cases were used in the analysis. Displacement and stress fields in the composite are obtained and used in the calculation of the effective composite properties. Polypropylene composites reinforced with short fiber Vectra LCP in various volume fractions were fabricated. Tensile experiments were performed. The model predicted effective properties are in good agreement with micromechanical equations values and experimental results.

On the Modelling of Solid Phase Sintering of Hardmetal Based on Viscoplasticity

Abstract: This thesis is primarily concerned with issues related to the modelling of sintering of hardmetal, which is composed of WC-grains that are embedded in a Co-binder. The aims are to develop a constitutive model framework and computational tools by which it is possible, in a unified manner, to simulate the two subsequent steps of cold compaction and (free) sintering. Only solid phase sintering is considered, i.e. it is tacitly assumed that the sintering process takes place below the melting point of the binder material. Free sintering is carried out without any external load. The "driving force" of the process (denoted the "sintering stress") arises from the change of surface energy between constituents and pores. From a continuum point of view, the sintering stress (that brings about the densification) acts as a hydrostatic pressure. An explicit expression for the sintering stress is derived from thermodynamics and from a simplified microstructure. This stress is applied in two ways: (1) As an external load on an "Representative Volume Element" (RVE), in a mesomechanical approach. (2) As a part of the effective stress that is employed in the constitutive equations in a continuum model. An RVE-generator, that uses the initial relative density and the weight portions of Co and WC as input, is developed. The creep mechanisms of the constituents (WC,Co) are modelled using an incompressible elasto-viscoplasticity model with isotropic hardening and temperature dependent parameters. The RVE is calibrated using experimental data from free sintering and uniaxially loaded specimens. A thermo-hyperelastic-viscoplastic model with isotropic hardening and temperature dependent parameters is used as a continuum model. The particular model chosen is based on quasi-static and dynamic yield surfaces that are elliptic in the meridian planes of the stress space. This model is, after calibration and validation using experimental data, implemented into a commercial FE-code. Numerical investigations of the compaction and subsequent sintering of specimens are carried out.

Homogenisation of heterogeneous viscoplastic materials

Abstract: Heterogeneous materials have been used extensively in the past few decades, since their mechanical properties, such as strength, stiffness and toughness are being improved continuously. Experimental work has clearly demonstrated the significant influence of the micromechanical phenomena on the resulting overall macroscopic deformation behaviour of the material. Nevertheless, more fundamental studies are needed for a better understanding of the deformation behaviour of these materials. Relations between the microstructural phenomena and the macroscopic deformation behaviour are indispensable when predicting macroscopic properties from the microstructure. Homogenisation provides a way in determining this relation. The homogenisation process aims at replacing the heterogeneous material with an equivalent continuum model, for which a closed-form constitutive equation is derived. Assuming statistical homogeneity of the heterogeneous material, it is possible to identify an element whose mechanical behaviour is representative for the heterogeneous material as a whole, a so-called representative volume element (RVE). In the homogenisation method proposed in this paper, finite element calculations are performed on the microstructural level, and therefore no simplifying assumptions concerning the microstructure of the material are required. First, the proposed homogenisation strategy is validated numerically. Here, the obtained homogenised values are used to simulate various loading histories on a perforated plate. The obtained global behaviour is confronted with direct finite element calculations on the complete heterogeneous structure. From this, we have concluded that the proposed homogenisation method results in a fairly accurate and time-efficient prediction of the mechanical behaviour of the heterogeneous structure. Experiments have been used as well to evaluate the method. Thermoplastic sheets with an arbitrary distributed hole pattern, are subjected to tensile and shear loadings. First, the material is characterised by standard techniques. Then, the RVE is defined, after which an effective parameter set for this RVE is obtained. This parameter set is then used to model the mechanical behaviour of the experimentally tested perforated plate.

Numerical determination of yield surfaces of Al/SiC laminated composite plates

Abstract: In this study the elastic-plastic properties of an aluminum alloy reinforced by unidirectional SiC-fibers are computed by using nonlinear finite element simulations. In order to carry out these computations, a representative volume element assuming a regular quadratic arrangement of the fibers is used. The plastic properties of the composite are described by a quadratic flow criterion which is able to consider both direction and tension-compression anisotropy. Additionally, the flow rule takes into account plastic volume changes. The hardening of the material is defined by a mixed isotropic-kinematic hardening model. By using this modeling yield surfaces of multi-layer laminated composite plates under an in-plane loading are determined.

Numerical determination composite plates of yield

Abstract: 124 Abstract In this study the elastic-plastic properties of an aluminum alloy reinforced by unidirectional SiC-fibers are computed by using nonlinear finite element simulations. In order to carry out these computations, a representative volume element assuming a regular quadratic arrangement of the fibers is used. The plastic properties of the com- posite are described by a quadratic flow criterion which is able to consider both direction and tension-compression anisotropy. Additionally, the flow rule takes into account plastic volume changes. The hardening of the material is defined by a mixed isotropic-kinematic hardening model. By using this modeling yield surfaces of multi-layer lam- inated composite plates under an in-plane loading are determined.