Representative Volume Element Size Research Articles

Rock plasticity from microtomography and upscaling

We present a workflow for upscaling of rock properties using microtomography and percolation theory. In this paper we focus on a pilot study for assessing the plastic strength of rocks from a digital rock image. Firstly, we determine the size of mechanical representative volume element (RVE) by using upper/lower bound dissipation computations in accordance with thermodynamics. Then the mechanical RVE is used to simulate the rock failure at micro-scale using FEM. Two cases of different pressures of linear Drucker-Prager plasticity of rocks are computed to compute the macroscopic cohesion and the angle of internal friction of the rock. We also detect the critical exponents of yield stress for scaling laws from a series of derivative models that are created by a shrinking/expanding algorithm. We use microtomographic data sets of two carbonate samples and compare the results with previous results. The results show that natural rock samples with irregular structures may have the critical exponent of yield stress different from random models. This unexpected result could have significant ramifications for assessing the stability of solid materials with internal structure. Therefore our pilot study needs to be extended to investigate the scaling laws of strength of many more natural rocks with irregular microstructure.

A statistical meso-damage mechanical method for modeling trans-scale progressive failure process of rock

Starting from the concept of Representative Volume Element (RVE) at the mesoscopic scale, a statistical meso-damage mechanical method (SMDMM) is developed to model the trans-scale progressive failure process of rock, based on the statistical and continuum damage mechanics theory and the finite element method (FEM). The proposed mesoscopic constitutive law of RVE is established within the framework of elastic–brittle-damage theory in which the double damage functions correspond to a tensile and compressive damage surface. A statistical approach is employed to describe the mesoscopic heterogeneity of rock material. The damage evolution and accumulation of mesoscopic RVEs is used to reflect the macroscopic failure characteristics of rock. The global stress and strain fields are solved by the FEM. An element represents a RVE, the initiation and propagation of meso-macroscopic trans-scale cracks and their interaction are manifested by removing the failed elements. Numerical analyses are carried out on a few groups of laboratory-scale rock specimens and the effects of RVE size, material homogeneity and quasi-static loading step length are investigated. Finally, a full-scale Atomic Energy of Canada Limited (AECL) Mine-by test tunnel is simulated. The proposed SMDMM is calibrated and validated for its trans-scale modeling capability to reproduce the shape and size of excavation damage zone profile around the tunnel. Accordingly, the simulation results are compared with experimental observations and numerical results predicted by other models. It is shown that the SMDMM has good performance for modeling the rock failure process from meso- to engineering/field-scale.

Morphologic Interpretation of Rock Failure Mechanisms Under Uniaxial Compression Based on 3D Multiscale High-resolution Numerical Modeling

Multiscale continuous lab oratory observation of the progressive failure process has become a powerful means to reveal the complex failure mechanism of rock. Correspondingly, the representative volume element (RVE)-based models, which are capable of micro/meso- to macro-scale simulations, have been proposed, for instance, the rock failure process analysis (RFPA) program. Limited by the computational bottleneck due to the RVE size, multiscale high-resolution modeling of rock failure process can hardly be implemented, especially for three-dimensional (3D) problems. In this paper, the self-developed parallel RFPA3D code is employed to investigate the failure mechanisms and various fracture morphology of laboratory-scale rectangular prism rock specimens under unconfined uniaxial compression. The specimens consist of either heterogeneous rock with low strength or relatively homogeneous rock with high strength. The numerical simulations, such as the macroscopic fracture pattern and stress–strain responses, can reproduce the well-known phenomena of physical experiments. In particular, the 3D multiscale continuum modeling is carried out to gain new insight into the morphologic interpretation of brittle failure mechanisms, which is calibrated and validated by comparing the actual laboratory experiments and field evidence. The advantages of 3D multiscale high-resolution modeling are demonstrated by comparing the failure modes against 2D numerical predictions by other models. The parallel RVE-based modeling tool in this paper can provide an alternative way to investigate the complicated failure mechanisms of rock.

Effect of reinforcement shape on physical properties and representative volume element of particles-reinforced composites: Statistical and numerical approaches

We are considering the effect of the particle shape on the effective elastic properties and representative volume element of two-phase composites. In this work, microstructures with ellipsoidal and spherical particles are considered. Three dimensional particles are randomly generated for different volume fractions without any contact between neighbouring spheres according to Poisson process. The used analysis is based on numerical homogenization approaches by finite element method and morphological characterization of microstructures by random fields. The effects of various parameters such as particles volume fraction, contrast in phase properties and particles shape were identified. The statistical methods are introduced and then coupled with numerical simulations. The notion of statistically representative volume, named the deterministic representative volume element (RVE), is used. The variation of this RVE as a function of particles volume fraction for heterogeneous materials is presented. These variations suggest us to identify the morphological distribution of particles which is more anisotropic and which needs a large RVE size.

Predicting the RVE size of micro-particle composite for electrical purpose using particles dispersion developed

Representative volume element (RVE) is a technique to determine and predict the mechanical and physical effective properties of any random homogeneous or heterogeneous composite material models. An optimal size of RVE model became a major objective in order to be statistically equivalent to the real composite material. In this paper, possible approaches to solving the RVE size problem is by developing a particles dispersion model in implementing microstructure of silver (Ag) micro-particle composite for electrical purpose. In this technique, a micro scale of scanning electronic microscope condition was observed for verification on Ag morphology in an epoxy matrix. The morphological analysis is used to guide the particles dispersion model that can be considered to repeat the whole macro-scale of the composite. In the RVE size study, the various composite model sizes were analyzed on the electrical conductivity for convergence study. The model size was determined by a ratio between width of RVE model and the particle size which given in ? = W/D. The numbers of RVE size (?) were set in a size of 3, 4, 5, 6, 7, 8 and 9. The efficiency of size that represents the precision of the estimation of the electrical purpose has been successfully determined by a particle dispersion model developed at model size of ? = 7. The predicted conductivity at model size of ? = 7 shows a good agreement with experimental data with the results presented at 5.06 × 10?1 and 2.69 × 10?1 S/cm for modeling and experiments, respectively.

Micro-XCT-based finite element method for prediction of elastic modulus of plane woven carbon fiber-reinforced ceramic matrix composites

In this study, an X-ray computer tomography-based finite element method was developed to predict the elastic modulus of a 2D plane woven carbon fiber-reinforced carborundum matrix composite. Micro-X-ray computer tomography was used to construct a geometrical model of the material. Then, the elastic parameters of each element were determined by the grey level of pixels in the images captured. A standard finite element model process was performed to calculate all of the elastic moduli of the material. In addition, a volume average method for determining the proper representative volume element size was developed to reduce the computational time without losing accuracy.

Numerical determination of strength and deformability of fractured rock mass by FEM modeling

The strength and deformability of fractured rock masses are important factors in the design and construction of civil and mining structures built in or on rock masses. The finite element method (FEM) was applied to study the mechanical behavior of fractured rock masses. A damage-softening statistical constitutive model of intact rock was implemented in FEM code in order to model the strain-softening behavior of rock masses. The FEM model was verified against a theoretical solution and resulted in good agreement. The FEM model was then applied to investigate the progressive failure process, scale effect and anisotropic characteristics of uniaxial compressive strength (UCS), and the deformation modulus of fractured rock masses. The numerical results showed that the representative elementary volume (REV) for strength and deformation modulus is 12 m with an acceptable variation of 10%, and the complete stress–strain curves of models greater than REV size were very similar to each other. A linear relationship was also found between the UCS and the deformation modulus. This result seems to support the idea that a rock mass has a critical strain corresponding to the UCS, regardless of the deformation modulus, scale and direction.

A Multi-Scale Scheme for Modelling Fracture under Dynamic Loading Conditions

Fracture in heterogeneous materials under dynamic loading is modelled using a multi-scale method. Computational homogenization is considered, in which the overall properties at the global-scale are obtained by solving a boundary value problem for a representative volume element (RVE) assigned to each material point of the global-scale model. In order to overcome the problems with upscaling of localized deformations, a non-standard failure zone averaging scheme is used. Discontinuous cohesive macro-cracking is modelled using the XFEM and a gradient-enhanced damage model is used to model diffuse damage at the local-scale. A continuous-discontinuous computational homogenization method is employed to obtain the traction-separation law for macro-cracks using averaged properties calculated over the damaged zone in the RVE. In the multi-scale model, a dynamic analysis is performed for the global-scale model and the local-scale model is solved as a quasi-static problem. Dispersion effects are then captured by accounting for the inertia forces at the local-scale model via a so-called dispersion tensor which depends on the heterogeneity of the RVE. Numerical examples are presented and the multi-scale model results are compared to direct numerical simulation results. Objectivity of the multi-scale scheme with respect to the RVE size is examined.

Scale-dependent homogenization of random composites as micropolar continua

Abstract A multitude of composite materials ranging from polycrystals to rocks, concrete, and masonry overwhelmingly display random morphologies. While it is known that a Cosserat (micropolar) medium model of such materials is superior to a Cauchy model, the size of the Representative Volume Element (RVE) of the effective homogeneous Cosserat continuum has so far been unknown. Moreover, the determination of RVE properties has always been based on the periodic cell concept. This study presents a homogenization procedure for disordered Cosserat-type materials without assuming any spatial periodicity of the microstructures. The setting is one of linear elasticity of statistically homogeneous and ergodic two-phase (matrix-inclusion) random microstructures. The homogenization is carried out according to a generalized Hill–Mandel type condition applied on mesoscales, accounting for non-symmetric strain and stress as well as couple-stress and curvature tensors. In the setting of a two-dimensional elastic medium made of a base matrix and a random distribution of disk-shaped inclusions of given density, using Dirichlet-type and Neumann-type loadings, two hierarchies of scale-dependent bounds on classical and micropolar elastic moduli are obtained. The characteristic length scales of approximating micropolar continua are then determined. Two material cases of inclusions, either stiffer or softer than the matrix, are studied and it is found that, independent of the contrast in moduli, the RVE size for the bending micropolar moduli is smaller than that obtained for the classical moduli. The results point to the need of accounting for: spatial randomness of the medium, the presence of inclusions intersecting the edges of test windows, and the importance of additional degrees of freedom of the Cosserat continuum.

Evaluación de propiedades elásticas de la fundición nodular empleando micromecánica computacional

Nodular cast iron is a Fe-C-Si metallic alloy whose microstructure consists of a matrix, composed in general of ferrite and pearlite, with graphite nodules embedded in it. Owing to its microscopical heterogeneity, the material response is governed by the properties, morphology and typology of the phases involved. This work reports on the evaluation of the elastic properties, i.e., Young's modulus and Poisson's ratio, of an equivalent homogeneous material that characterizes the macroscopic response of a nodular cast iron. Asymptotic homogenization is used to this end. This approach is applied to both 3D and 2D multiparticle cells simulated via the finite element method. Two different physically and geometrically-based criteria are considered to estimate the representative volume element (RVE), where the size of the RVE is found to be sensitive to the chosen criterion. The main microstructural features are obtained from a computational simulation of the solidification process of the material. The numerical predictions computed for the 3D and 2D cases are compared and discussed in terms of the resulting elastic properties. It is observed that the models employing 3D multiparticle cells require lower RVE sizes than the corresponding 2D models, where the latter present a stiffer elastic response.

Microstructure measures and the minimum size of a representative volume element: 2D numerical study

In the paper, a numerical study of the size of a representative volume element (RVE) for both the heat flow as well as the linear elasticity problems is presented. A particular two-phase random microstructure is studied and the method is applied to the digital image of reconstructed 2D realization of random media. The minimum size of RVE is determined by the investigation of the convergence of apparent properties as the size of RVE is increasing. Then, two estimates of the minimum RVE size are proposed and it is shown that the estimates are in a good agreement with the results determined by the investigation of the convergence of apparent properties. The minimum size of RVE can be successfully predicted based only on the microstructure morphology. The statistical measures used in this work are: the two-point probability and the linealpath functions.

A dispersive multi-scale crack model for quasi-brittle heterogeneous materials under impact loading

A dispersive multi-scale model is presented to model failure in heterogeneous quasi-brittle materials under high frequency loading conditions. In the dispersive multi-scale model, the heterogeneous model undergoing localized failure is replaced by a homogeneous macro-scale model with a cohesive crack and a meso-scale model with diffuse damage. Each material point of the macro-scale model is linked to a heterogeneous meso-scale model. The macro-crack is modeled as a strong discontinuity and the gradient-enhanced damage model is used to model diffuse damage in the meso-scale model. The constitutive law for the bulk material is obtained from the meso-scale model analysis using a standard computational homogenization scheme. The cohesive law for the macro-crack is obtained using a continuous–discontinuous homogenization scheme which is based on a failure zone averaging technique. In the dispersive multi-scale model, at the macro-scale, a dynamic analysis is performed and the meso-scale model is solved as a quasi-static problem. The meso-scale inertia forces are taken into account via a dispersion tensor which only depends on the meso-scale model material properties and the heterogeneity of the material. The meso-scale inertia effects appear as additional body forces in the macro-scale model and cause dispersion of the propagating wave. The effect of dispersion on the cohesive cracking is captured via a rate dependent cohesive law. The dispersive multi-scale model is verified against a direct numerical simulation and the objectivity of the scheme with respect to the representative volume element size is shown.

Numerical and statistical analysis of elastic modulus of concrete as a three-phase heterogeneous composite

This paper investigates the elastic modulus of concrete as a three-phase heterogeneous composite. Theoretically, the influence of aggregate and interfacial transition zone (ITZ) can be well characterized by the Ramesh model. Through generation of random aggregate structure, numerical samples of concrete are established and analyzed to calculate elastic modulus. It is shown that soft ITZ can greatly cancel the enhancement of aggregate while stiff ITZ cannot help much to strengthen concrete. The size of representative volume element of concrete is suggested in terms of expected error employing numerical–statistical approach. The required sample size with certain realization number is also calculated.

3D numerical simulation for the elastic properties of random fiber composites with a wide range of fiber aspect ratios

A new automatic searching & coupling (ASC) technique is proposed to generate the 3D representative volume element (RVE) to analyze the random fiber composite (RaFC) with a wide range of fiber aspect ratios (FAR), particularly the RaFC with a high FAR. Compared with the conventional model, the present model is easier to generate and more time-saving as it eliminates the drawback of free meshing. In addition, the ASC technique can remove the additional stiffness introduced by the embedded element technique (EET), and hence can improve precision and convergence. Moreover, our technique facilitates the direct application of the 3D periodic boundary conditions (PBC) to the RVE. Using the finite element (FE) method, the elastic properties of RVE are calculated, and the critical RVE size is determined. Furthermore, the 3D elastic properties of RaFC are predicted, which accord well with experimental data. Besides, the RVE is employed to explore the influences of FAR, fiber volume fraction (FVF) and the fiber incline angle on the elastic properties of RaFC, and a comparison with theoretical results is also presented.

Homogenization of the average thermo-elastoplastic properties of particle reinforced metal matrix composites: The minimum representative volume element size

The average thermo-elastoplastic properties of particle reinforced metal matrix composites (PRMMC) including the average coefficient of thermal expansion (CTE), Young’s modulus, Poisson’s ratio and isotropic hardening function are investigated. Computational homogenization method based on 3D realistic microstructures (RMs) is employed. Unit cell microstructure (UCM) based model and analytical models are also employed for comparison. As an illustration, 17vol.%SiCp (3μm)/2124Al composite is studied. Compared to RMs, UCM underestimates the average CTE and Poisson’s ratio, while it overestimates the average Young’s modulus and isotropic hardening function. The minimum representative volume element (RVE) size for determining the average CTE, Young’s modulus and Poisson’ ratio is δ=15, δ=20 and δ=20, respectively, where δ is the size ratio of microstructure model, which is defined by the ratio of the side length of the RVE to the nominal mean radius of reinforcement. The minimum size of RVE for estimating average isotropic hardening function of plastic deformation is dependent on both the temperature and the plastic deformation condition.

Local porosity distribution of cement paste characterized by X-ray micro-tomography

Porosity is one of the most important parameters for cement-based materials, which influences the mechanical property, transport property, and durability. The spatial and frequency distributions of local porosity of cement pastes are characterized using X-ray micro-tomography data and treating methods. The 3D spatial distributions for three cement paste specimens with different water cement (w/c) ratios show reasonable heterogeneity. The probability analysis also reveals this heterogeneity: the representative volume element (RVE) size based on porosity maps decreases with w/c ratio firstly, then increases with w/c ratio; and the heterogeneity on the characterized probe size or on the RVE size increases with w/c ratio. Average porosities obtained using the CT method are further compared with those by traditional methods.

Internal Deformation Measurement of Polymer Bonded Sugar in Compression by Digital Volume Correlation of In-situ Tomography

A polymer bonded sugar (PBS) cylindrical specimen, in which caster sugar grains are embedded in hydroxylterminated polybutadiene (HTPB) binder matrix, was compressed up to −31.5 % of compressive strain without confinement, while its internal microstructures were determined by X-ray micro-computed tomography. The reconstructed volumetric images were analyzed to determine three-dimensional (3D) morphologies and to investigate the damage and failure mechanisms. The image grayscale was filtered by threshold values to identify individual material constituents, and determine grain volume fraction. An incremental digital volume correlation (DVC) technique was developed to determine the internal deformations and to track the movement of individual sugar grains. The evolution of the internal 3D deformation of PBS is correlated to its microstructures, as well as debonding and void formation. The side length of a cubic representative volume element (RVE) for the PBS is determined as 10 times of the average grain size, based on the analysis of sugar volume fraction; it is consistent with the RVE size determined from convergence analysis of the average and the standard deviation of strain distribution. Image analysis indicates that void ratio can be used as an indicator to quantify the extent of damage. Debonding occurs at first in the inner core region, and then propagates to the outer annular region. Debonding is also determined as the primary mechanism for damage formation and evolution. Grain fracture was not observed during the uniaxial compression of this PBS specimen under a nominal strain of −31.5 % without confinement.

Micromechanical analysis of deformation of snow using X-ray tomography

Snow exhibits an elastic response followed by a softening behavior under compression at a strain rate of 10−4s−1 and higher. The strain softening behavior is postulated due to initiation and growth of damage in the ice matrix. A deformation controlled compression experiment at a strain rate of 2×10−4s−1 was conducted on a round grained snow sample. To investigate the link between behavior of snow and ice, X-ray tomographic imaging of the sample was performed and the size of representative volume element (RVE) with respect to ice volume fraction VϕiRVE was estimated. From the set of scanned images 112 sub-volumes of sizes equal to and larger than VϕiRVE were selected. The ice matrix formed by these images was meshed with finite elements (FE) to simulate the stress–strain curve of snow under deformation controlled compression. An elasto-plastic constitutive law for ice with provision for degradation of elastic modulus due to damage was used to simulate the stress–strain response including strain softening as observed in the experiment. The statistical representativeness of the RVE with respect to ultimate strength (S) and elastic modulus (E) of snow was further analyzed in terms of the precision of the numerical estimates of the effective properties. It was found that the standard deviation in the ultimate strength & elastic modulus is reduced by 50% for sub-volume of size 8VϕiRVE as compared to the sub-volume of size VϕiRVE. The sensitivity of overall stress–strain response to the finite size effects is also analyzed and the average coefficient of variation of the simulated stress–response with respect to the experimental response reduces from 44% for 5.745mm3 to 18% for 45.96mm3.

Microstructure level modelling for properties prediction of WC–Co cemented carbides

A random distribution micromodel was developed in order to predict the properties of WC–Co cemented carbides. MATLAB and VC++ were hybrid programmed to extract the necessary information for modelling the microstructure from scanning electron microscopy images. By analysing the distribution regularity of the microstructural parameters, the randomness characterisation of the microstructure of WC–Co cemented carbides was accomplished using probability density function. The parameterised model based on ‘random method’ was developed, in which the random distribution characteristics of microstructural topology parameters, such as average grain diameter, major axis, minor axis, centroid and grain orientation, were considered, with Co volume fraction freely controlled. The representative volume element (RVE) size was determined using the moving window method, with average grain diameter distribution as the evaluation criterion. It was found that the RVE should contain ∼120 grains. The RVE was directly imported in the finite element software ABAQUS, and the finite element simulation of microindentation tests and uniaxial tensile tests were accomplished to calculate the hardness and elastic properties of cemented carbides. The predicted results are in good agreement with the experimental dates, which reveals that the microstructure level finite element model can effectively predict the properties of WC–Co cemented carbides.

Numerical experiments for the appropriate representative volume element size selection on thin texturized metals under mechanical loads

AbstractIn this work, we analyze a strategy for the selection of the adequate RVE size for elastic problems on thin sheets with strong texture. The selection strategy includes the construction of the geometric descriptor as periodic tesselation of the sheets and the stochastic assignment of stiffness matrices for the different material grains inside the RVE. The procedure is illustrated with an example on DC01 steel sheets with thickness 50 µm. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)