Showing papers on "Representative elementary volume published in 1996"

A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites

Abstract: A variational formulation is employed to derive a micromechanics-based, explicit nonlocal constitutive equation relating the ensemble averages of stress and strain for a class of random linear elastic composite materials. For two-phase composites with any isotropic and statistically uniform distribution of phases (which themselves may have arbitrary shape and anisotropy), we show that the leading-order correction to a macroscopically homogeneous constitutive equation involves a term proportional to the second gradient of the ensemble average of strain. This nonlocal constitutive equation is derived in explicit closed form for isotropic material in the one case in which there exists a well-founded physical and mathematical basis for describing the material's statistics: a matrix reinforced (or weakened) by a random dispersion of nonoverlapping identical spheres. By assessing, when the applied loading is spatially-varying, the magnitude of the nonlocal term in this constitutive equation compared to the portion of the equation that relates ensemble average stresses and strains through a constant “overall” modulus tensor, we derive quantitative estimates for the minimum representative volume element (RVE) size, defined here as that over which the usual macroscopically homogeneous “effective modulus” constitutive models for composites can be expected to apply. Remarkably, for a maximum error of 5% of the constant “overall” modulus term, we show that the minimum RVE size is at most twice the reinforcement diameter for any reinforcement concentration level, for several sets of matrix and reinforcement moduli characterizing large classes of important structural materials. Such estimates seem essential for determining the minimum structural component size that can be treated by macroscopically homogeneous composite material constitutive representations, and also for the development of a fundamentally-based macroscopic fracture mechanics theory for composites. Finally, we relate our nonlocal constitutive equation explicitly to the ensemble average strain energy, and show how it is consistent with the stationary energy principle.

Micromechanics and homogenization of inelastic composite materials with growing cracks

Abstract: A homogenization scheme is employed to derive the effective constitutive equations of an elastoplastic composite system with growing damage. The homogenization procedure followed herein is based on the thermodynamics of dissipative media. It is shown that when damage consists of sharps microcracks the macroscopic constitutive behavior is that of a so-called generalized standard material. The latter is a general dissipative medium whose constitutive equations are completely characterized by a single scalar convex potential function of the chosen state variables and whose evolution is completely characterized by a single convex dissipation potential function of the thermodynamic forces conjugate to the chosen internal state variables. The analysis presented is valid under the assumption that the evolution of the representative volume element at hand is unique and stable. The results of the theoretical analysis are then employed for formulating an approximate method for practically deriving the macroscopic constitutive equations. Computer software development for the application of said method is currently ongoing. A simple example of the numerical results obtained so far is presented.

1 – Unified Cyclic Viscoplastic Constitutive Equations: Development, Capabilities, and Thermodynamic Framework

Abstract: This chapter focuses on the metallic materials which are developed with the objective of the inelastic analysis of structural components. They are based on the concepts of continuum mechanics, where a particular representative volume element of material can be considered as submitted to a macroscopically uniform stress, neglecting the microstress microstrain inhomogeneities at the microscale. The application domains are limited to the quasistatic deformation of metallic materials (strain rate between 10 -10 and 10 -1 ), especially under cyclic loading conditions. The constitutive equations are written in their small strain form. Also, high-temperature conditions will be considered, as well as loading under varying temperatures. Unified viscoplastic constitutive equations means the nonseparation of the plastic (rate-independent) and creep (rate-dependent) parts of the inelastic strain. Moreover, the viscoplastic equations are based on a general framework consistent both with classical plasticity (elastic domain, yield surface, loading/unloading condition) and with thermoviscoplasticity without an elastic domain. Then rate-independent conditions will be obtained consistently as a limit case of the general viscoplastic scheme.

Thermo plasticity theory for bidirectionally functionally graded materials

Abstract: A recently developed higher order theory for the thermoelastic response of composite materials with microstructures characterized by arbitrarily nonuniform reinforcement spacing in two directions (i.e., bidirectionalfy functionally graded materials) is further extended to accommodate the effect of an inelastic response of the constituent phases. 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 the imposition of boundary and interfacial conditions in an average sense between the subvolumes used to characterize the composite's functionally graded microstructure. Examples are presented that illustrate how the presence of plasticity and microstructure affect...

Microstructural design of composites using the homogenization method and digital images

Abstract: A microstructural design method is developed by utilizing the homogenization method and digital image-based (DIB) geometric modeling technique. The localization capability of the asymptotic homogenization method enables the evaluation of the microscopic deformation in the microstructure, i.e., representative volume element (RVE). This distinctive feature allows us to use the microscopic variables in the functional forms of constraints and objective function. In estimating the micromechanical responses, the systematic modeling method in terms of the pixel information of images and their processing make the construction of unit cell geometry easy and the homogenization results accurate. Therefore, while the specific microstructural geometry is taken into account in evaluating the micro and macroscopic variables, the optimal microstructural configurations can be determined depending on our objectives

Modeling transport phenomena in porous media

Abstract: The paper1 reviews the continuum approach to modelling the transport of mass, momentum and energy, of phases and of their components in a porous medium domain. The review begins with the definition of a porous medium, making use of the concept of a Representative Elementary Volume (REV) as a tool for overcoming the effect of the microscopic heterogeneity resulting from the presence of a solid matrix and a void space. The microscopic and macroscopic levels of description are defined. By averaging the description of a transport phenomenon at the microscopic level over an REV, using certain ‘averaging rules’, the macroscopic or continuum description of the same phenomenon is obtained. This methodology is first introduced in general terms for any extensive quantity, and then demonstrated for the transport of mass, momentum and energy.

3-D Microscopic Measurement and Analysis of Chemical Flow and Transport in Porous Media

Abstract: Chemical flow and transport have been studied at the pore-scale in an experimental porous medium. Measurements have been taken using a novel nonintrusive fluorescence imaging technique. The experimental setup consists of a cylindrical column carved out of a clear plastic block, packed with clear beads of the same material. A refractive index-matched fluid was pumped under laminar, slow-flow conditions through the column. The fluid was seeded with tracer particles or a solute organic dye for flow and chemical transport measurements, respectively. The system is automated to image through the porous medium for collecting microscopic values of velocity, concentration, and pore geometry at high-accuracy and high-resolution. Various geometric, flow, and transport quantities have been obtained in a full three-dimensional volume within the porous medium. These include microscopic (pore-scale) medium geometry, velocity and concentration fields, dispersive solute fluxes, and reasonable estimates ofa representative elementary volume (REV) for the porous medium.

Effective Thermoelastic Moduli of a Unidirectional Fiber Composite Containing Interfacial Arc Microcracks

Abstract: In this work, we have employed a variational model to examine the effect of fiber-matrix debonding on the thermoelastic response of a unidirectional composite. The model is designed to represent the concentric cylinder model of a composite representative volume element and it contains the capability to enhance the accuracy of a given numerical solution. The effect of the extent of debonding as well as fiber volume fraction on all the effective moduli of the unidirectional composite has been examined. Numerical results reported in the literature are compared with the results of the model developed in the present study to examine the quality of the model.

Computational simulation of continuous fiber reinforced ceramic Matrix composites behavior

Abstract: This report describes a methodology which predicts the behavior of ceramic matrix composites and has been incorporated in the computational tool CEMCAN (CEramic Matrix Composite ANalyzer). The approach combines micromechanics with a unique fiber substructuring concept. In this new concept, the conventional unit cell (the smallest representative volume element of the composite) of the micromechanics approach is modified by substructuring it into several slices and developing the micromechanics-based equations at the slice level. The methodology also takes into account nonlinear ceramic matrix composite (CMC) behavior due to temperature and the fracture initiation and progression. Important features of the approach and its effectiveness are described by using selected examples. Comparisons of predictions and limited experimental data are also provided.

An approximate representation of fiber-matrix debonding in nonperiodic metal matrix composites

Abstract: This paper presents a fiber-matrix debonding model for metal matrix composites based on a modified Needleman (1987) type cohesive zone model In this model the fiber-matrix interface is fully described by its strength and ductility under normal and shear loading Debonding initiates when a quadratic interaction of the interfacial tractions attains a critical value (the interfacial strength) Complete interfacial separation occurs when the magnitude of the resultant interfacial displacement exceeds the ductility of the interface The debonding model is implemented in the method of cells micromechanical model of Aboudi (1987) The response of composites having nonperiodic microstructures is predicted by taking an alternate view of the representative volume element That is, the interface must be representative of the conglomerate of interfaces throughout the composite Nonperiodicity is described by a single parameter that accounts for the distribution of stress concentrations at various interfaces throughout the composite The effects of nonperiodicity are illustrated and predictions presented for transverse tensile and axial shear loadings, then compared with experimental results The model is observed to accurately predict the 3-stage deformation response typical of silicon carbide/titanium composites subjected to transverse tensile loading

A Thermoviscoplastic Theory for Composite Materials by Using a Matrix--Partitioned Unmixing-Mixing Scheme

Abstract: A thermo-elasto-viscoplastic constitutive theory for fiber-reinforced composite materials is suggested by extending the existing unmixing-mixing scheme which is based upon some micromechanical observations. A new technique, which is called the "matrix-partition method,' is introduced in the model development. By this method, deformation states in the matrix phase can be represented with a set of mechanical variables for the respective parts of the partitioned matrix. As material parameters, which explain the three-dimensional microstructural effects resulting from kinematic compatibilities in the boundaries of the fibers and the matrix, the stress variation factors and the strain contribution factors are defined. The strain component of thermal expansion due to temperature changes is also included for the mathematical formulation. To verify the derived constitutive equations, the representative volume element of a unidirectionally fiber-reinforced lamina is discretized and analyzed by the developed finite...

Representative Volumes of Composite Materials

Abstract: Representative volume (RV) of a composite material is important not only for experimental determination of materials properties, but also for theoretical analysis of various properties of the materials. RV is a random variable; this is especially true for the composites with heterogeneous microstructures. In the present study, RV is redefined as a relationship among three parameters: the volume under consideration, the measured or predicted value of a property corresponding to the volume, and the scattering of the property from the real effective property. There are two steps involved to determine RV. The first is to determine the relationship among the effective property, the properties of each constituents, and volume fractions, which is the topic in composite mechanics. The second is determination of the local variation of the heterogeneity of the composite because, with increasing size of the local volume, the statistics from the local volume asymptotically approaches the value from the global average. The main purpose of this study is to develop a theoretical model to determine local variation of heterogeneity, which is obviously related to the morphological features of the composites. The model accounts for coarseness of grain sizes, volume fractions, spatial arrangement of the constituents, and properties of each constituents. By combining the model with available composite models for effective properties of composites, RV for various materials properties can be developed. As an example, RV for elastic moduli of cementitious materials is demonstrated.

Simulation of crack propagation resistance in brittle media of high porosity

Abstract: The crack propagation resistance through a porous or microstructurally heterogeneous brittle solid with local variability in strength and stiffness has been simulated. Specifically, the simulation probes the behavior of porous brittle materials in the range of porosity less than those of cellular materials and greater than those of microstructures that are in the category of dilute porosity. The simulation plane consists of a triangular network of points interacting with each other through both linear central force springs and bond angle springs, incorporating an appropriate element of a noncentral force contribution. Explicit microstructural details were incorporated into the model and the simulation was first carried out under conditions of uniaxial tensile strain in order to investigate the mechanisms of subcritical damage evolution, leading to quasi-homogeneous fracture. In order to investigate material strength and stiffness variability on the scale of a representative volume element for coherent fracture events in a crack tip stress gradient, the explicit microstructural results were incorporated into a simulation with boundary conditions characteristic of the displacement field of an infinite Mode I crack. To impart some 3D realism to the primarily 2D simulations a special 2D super-element was devised, which incorporated variability information as might be sampled by a crack front in three dimensions. For a given porosity, in general, only small differences were found between nominally diverse microstructures in terms of their tensile toughness, maximum strength and elastic moduli. The strongest dependence of the overall fracture toughness was found to come from the average porosity. The variability in local element strength and stiffness on the scale of the porosity produced highly tortuous crack paths, roughly on the scale of the chosen representative volume element. The tortuosity of the crack was largest where local variability of strength and stiffness was uncorrelated. Examples of microcrack toughening and crack bridging were observed.

Dynamic Yield Loci of A Porous Visco-plastic Material by Using A lower Bound Approach

Abstract: Based on the study of a representative volume element (RVE) of porous viscoplastic material, a void growth model is proposed in this paper. The constitutive relation of the matrix material is in the over stress form given by previous authors. A lower bound approach by constructing a statically admissible stress field in the RVE is employed, and the corresponding dynamic yield loci of this porous material are obtained. The dynamic yield loci derived from both the lower bound approach in the present paper and the upper bound approach in Ref. [5] are compared for different values of porosity f and strain rate parameter m. They are rather close.

Analysis of Crack Initiation in Structures

Abstract: The fundamental reason for studing damage mechanics is to understand why and how materials break. Together with physics, metallurgy and chemistry this knowledge allows us to improve the mechanical properties of materials and to design new multimaterials. The practical reason for studing damage mechanics is to predict when materials, as they are currently made, will break upon submission to mechanical and thermal loading. This involves the analysis of real components of structures in real or presumed situations.

Towards Scale-Dependent Constitutive Laws for Plasticity and Fracture of Random Heterogeneous Materials

Abstract: The conventional aim of micromechanics is to derive macroscopic effective constitutive laws for a Representative Volume Element (RVE) ΔV on the scales that are infinitely larger than the microscale. This is tantamount to an assumption that the macroscopic field quantities vary on scales that are much larger than the scale of ΔV However, in many problems of mechanics this is not possible — a typical example is provided by stochastic finite elements, where, due to the finite size L of an element (or window) relative to the microscale d, its stiffness matrix may display statistical fluctuations, Fig. 1. Consequently, one has to set up a stochastic stiffness matrix in relation to the actual microstructure, whereby several features have been established in the setting of elastic materials [1, 2]: i) effective stiffness tensor, and hence stiffnes matrix, on a scale δ = L/d depends on the type of boundary conditions — essential or natural, ii) this effective stiffness tensor is, in general, anisotropic, iii) only triangular finite elements are consistent with a micromechanics formulation, iv) the global response can be bounded by solving two finite element problems: one based on the displacement approach, and another based on the force approach.

Digital image-based modeling applied to the homogenization analysis of intermetallic composites

Abstract: Image processing technology is utilized for numerical analysis by the homogenization method for intermetallic composites. While the asymptotic homogenization method characterizes thermo-mechanical properties of composites and localizes the deformation to the microscopic level, the digital image-based (DIB) modeling method appropriately defines a finite element (FE) model of a representative volume element (RVE), i.e., a unit cell, through the use of digitized images. The results in linear and nonlinear analyses reflect the effects of the geometric configuration of the microstructure.