Representative Volume Element Size Research Articles

Investigation on mechanical properties and deformation behavior of copper-based three-phase metal matrix composite: Experimental and micro-macro-mechanical finite element analysis

The present study involved development of copper-based metal matrix composite, reinforced with waste EN 31 steel chips and TiB2 ceramic particles. Waste EN 31 steel chips and TiB2 ceramic particles were ball-milled for 100 h to obtain a single entity. The composite material was produced with a stir-casting technique, followed by a squeeze pressure process. The addition of Cu + 10 wt% of waste steel chips + 5 wt% of TiB2 improved the tensile strength of the copper matrix by about 68.35%. Furthermore, the addition of Cu + 5 wt% of waste steel chips + 10 wt% of TiB2 and Cu + 12.5 wt% of waste steel chips + 2.5 wt% of TiB2 increased the hardness and toughness of the copper matrix by about 133.33% and 28.57%, respectively. The addition of Cu + 10 wt% waste steel chips + 5 wt% of TiB2 ensured minimal corrosion weight loss in the metal matrix composite as a result of low porosity and a strong bond between the molecules. Further, representative volume element (size: 225 × 225 × 225 nm)-based finite element analysis was done to explain the micro-mechanical deformation, interfacial strength of matrix-particle interaction and damage behavior of Cu + 10 wt% of waste steel chips + 5 wt% of TiB2 metal matrix composite. A user material sub-routine model was also written and implemented with the help of FORTRAN subroutines to simulate the macro-mechanical tension test process of Cu + 10 wt% waste steel chips + 5 wt%TiB2 metal matrix composite. The results revealed a good agreement between the micro-mechanical and macro-mechanical finite element analysis models on the one hand and the experimental results on the other. Further, the representative volume element (with matrix and particles) showed about 59% and 66.5% higher tensile strength compared to the matrix–particle interface and the matrix (without particles), respectively. The percentage difference between the micro-mechanical finite element analysis and the experiments as well as the macro-mechanical finite element analysis and the experiments was found to be 5.58% and 9.64%, respectively. The finite element analysis results established that the waste steel chip powder particles exhibited greater stress than the TiB2 powder particles.

General Statistics-Based Methodology for the Determination of the Geometrical and Mechanical Representative Elementary Volumes of Fractured Media

The upscaling of mechanical properties of fractured media requires the definition of an appropriate size for the Representative Elementary Volume (REV). Because of the stochastic nature of the fracture networks, the REV size is not deterministic and should be defined based on the variability of the equivalent properties. This work presents a new general methodology to define the size of the REV for the geometrical and elastic moduli of fractured media. Following previous works on heterogeneous materials, the decision criterion is based on the precision error that arises from the statistical theory of samples. The proposed methodology also relies on the use of the Central Limit Theorem (CLT) to assess the REV of fractured rocks. The CLT is shown to theoretically apply to both the geometrical and the elastic equivalent properties. From that observation, a general equation is drawn to predict the variance of an equivalent property for any REV candidate size, provided that the variance for one size only is known. These concepts are tested using numerous finite element simulations to obtain the distribution of the equivalent elastic moduli of two-dimensional samples containing two fracture networks previously studied for their elastic properties. These properties are confirmed to tend to a normal distribution, as stated by the CLT. Also, the standard deviations associated with the tested REV sizes were predicted with accuracy from the standard deviation obtained in the numerical simulations of only one proper reference volume. The mechanical REV was compared with the geometrical REV, which is based on the first invariant of the fracture tensor. In addition, to reduce computational costs, a procedure to reduce the number of simulations of the reference volume was proposed. A preliminary verification of the applicability of the methodology to non-elastic problems was made. Proper predictions were obtained for the standard deviation of the compression strength calculated in two studies that considered, altogether, both two-dimensional and three-dimensional samples, as well as plastic and damage models.

Representative elementary volume analysis of hardened cement paste during hydration using X-ray Computed Tomography

Representative elementary volume (REV) is an important property of cementitious materials. However, it is hard to experimentally study the REV of cement paste in micrometer scale. Here the REV size of hardened cement paste (HCP) was experimentally obtained by analyzing the microstructure information from high resolution X-Ray Computed Tomography (XCT). Results show that the selected microstructure information, the applied resolution of XCT, and the hydration ages have significant influences on REV size of HCP. With the improvement of XCT resolution, the REV size decreases. From the decreasing trend, the REV size will be close to a stable value at very high resolution. Along with the increased hydration ages, the REV size increases firstly and then decreases, while the decreasing trend slows down with higher resolution. Thus, at extremely high resolution, the REV size is expected to be constant at later hydration ages.

Numerical evaluation of the representative volume element for random composites

The Representative Volume Element (RVE) plays a central role in the homogenization of random heterogeneous microstructures, especially for composite and porous materials, with a view to predicting their effective properties. A quantitative evaluation of its size is proposed in this work in linear elasticity and linear thermal conductivity of random heterogeneous materials. A RVE can be associated with different physical and statistical properties of microstructures. The methodology is applied to specific two–phase microstructure–based random sets. Statistical parameters are introduced to study the variation in the RVE size versus volume fractions of components and the contrast in their properties. The key notion of the integral range is introduced to determine these variations. For a given desired precision, we can provide a minimal volume size for the computation of effective mechanical and thermal properties. Numerical simulations are performed to demonstrate that a volume exists which is statistically representative of random microstructures. This finding is an important component for homogenization–based multiscale modeling of materials.

Investigation of geometrical representative elementary volumes based on sampling directions and fracture sets

This paper reports that the geometrical representative elementary volume (REV) of fractured rock masses is anisotropic, and the geometric shape of an REV is of a complex, irregular geometry. A 3D fracture network model of 98 m × 80 m × 80 m was created using the field information collected from tunnel PD254 at the Maji dam site in China. One hundred random cubes of various sizes with the same sampling direction were selected to assess the volumetric fracture intensity (P32). Changing the sampling direction along the dip direction and dip angle at a 10° increment, the REV size of each sampling direction was inferred using the Wald–Wolfowitz runs test and P32 values. The REV sizes of entire rock masses and each fracture set were calculated, and the results indicated that the REV sizes varied with the different sampling directions. The anisotropy ratio and standard deviation were employed to quantify the degree of the anisotropy of REVs. Based on the results, the anisotropy of REVs for fracture set 2 is the most significant, and that of fracture set 3 is the worst. The relationship between the anisotropy of REVs and fracture sets was discussed; one fracture set, two fracture sets, and three fracture sets were selected randomly from these simulated rock masses to evaluate the characteristics of the anisotropy of REVs. The mean value and the anisotropy of REVs increased with the number of fracture sets. The maximum REV size with a sampling direction was selected as an REV for each fracture set. Four cubes with different directions and different sizes were assembled into an irregular geometry. This geometry can represent the geometric shape of the REV of entire rock masses.

Effect of 3D Representative Volume Element (RVE) Thickness on Stress and Strain Partitioning in Crystal Plasticity Simulations of Multi-Phase Materials

Crystal plasticity simulations help to understand the local deformation behavior of multi-phase materials based on the microstructural attributes. The results of such simulations are mainly dependent on the Representative Volume Element (RVE) size and composition. The effect of RVE thickness on the changing global and local stress and strain is analyzed in this work for a test case of dual-phase steels in order to identify the minimal RVE thickness for obtaining consistent results. 100×100×100 voxel representative volume elements are constructed by varying grain size and random orientation distribution in DREAM-3D. The constructed RVEs are sliced in depth up to 1, 5, 10, 15, 20, 25, 30, 40, and 50 layers to construct different geometries with increasing thickness. Crystal plasticity model parameters for ferrite and martensite are taken from already published data and assigned to respective phases. Although the global stress/strain behavior of different RVEs is similar (<5% divergence), the local stress/strain partitioning in RVEs with varying thickness and grain size shows a considerable variation when statistically compared. It is concluded that two-dimensional (2D) RVEs can be used for crystal plasticity simulations when global deformation behavior is of interest. Whereas, it is necessary to consider three-dimensional (3D) RVEs, which have a specific thickness and number of grains for determining stabilized and more accurate local deformation behavior. This estimation will help researchers in optimizing the computation time for accurate mesoscale simulations.

Yield criterion and finite strain behavior of random porous isotropic materials

The mechanical response of isotropic elastoplastic materials containing random distributions of initially spherical voids is investigated computationally based on Fast Fourier Transform simulations. Numerical limit-analysis simulations at constant stress triaxiality allow to determine the yield surfaces, leading in particular to the determination of a Representative Volume Element size for the onset of coalescence/inhomogeneous yielding. Moreover, two different coalescence regimes are observed that differ by the presence of shearing. The yield surfaces are found to be consistent with the combination of two models proposed in the literature, a GTN-type model calibrated for homogeneous yielding of random porous materials and an inhomogeneous yielding model accounting for both coalescence with or without shear. Finite strain simulations performed for different hardening moduli and stress triaxialities under axisymmetric loading conditions confirm the existence of a RVE up to the onset of inhomogeneous yielding. Coalescence strains are found to be significantly smaller for random porous materials than for periodic distribution of voids. A homogenized model is finally proposed that reproduces quantitatively the behavior of isotropic elastoplastic materials containing random distributions of voids under finite strains.

Gravitational and Finite‐Size Effects On Pressure Saturation Curves During Drainage

AbstractWe experimentally and numerically study the influence of gravity and finite‐size effects on the pressure‐saturation relationship in a given porous medium during slow drainage. The effect of gravity is systematically varied by tilting the system relative to the horizontal configuration. The use of a quasi two‐dimensional porous media allows for direct spatial monitoring of the saturation. Exploiting the fractal nature of the invasion structure, we obtain a relationship between the final saturation and the Bond number using percolation theory. Moreover, the saturation, pressure, and Bond number are functionally related, allowing for pressure‐saturation curves to collapse onto a single master curve, parameterized by the representative elementary volume size and by the Bond and capillary numbers. This allows to upscale the pressure‐saturation curves measured in a laboratory to large representative elementary volumes used in reservoir simulations. The large‐scale behavior of these curves follows a simple relationship, depending on Bond and capillary numbers, and on the flow direction. The size distribution of trapped defending fluid clusters is also shown to contain information on past fluid flow and can be used as a marker of past flow speed and direction.

Estimation of REV for Danba schist based on 3D synthetic rock mass technique

This paper aims to provide a case study of determining the structure and mechanical Representative Elementary Volume (REV) simultaneously for a specific jointed rock mass. The schist of Danba HPP project is taken as the study case here. A Synthetic Rock Mass model (SRM) is firstly constructed based on sophisticated geological loggings. With which the number of joint centers per unit volume (P30), the area of joint per unit volume (P32), and the Young’s modulus (E) are examined for various sample sizes. It is found that the SRM is a powerful tool to address those problems. The P30 reduces gradually with the increase of rock mass scale. As for P32 and E, maximum and minimum value and the variance decrease rapidly with the increase of rock mass size. A comprehensive REV size of 50~60m is then estimated based on the above 3 indexes. According to the SRM simulation, young’s modulus of the REV scale is around 6.8GPa, which is a little bit higher than the recommended value by geologists. This difference is believed to be reasonable as the recommended value usually been significantly undervalued. The procedures and findings of this paper may provide a certain reference for further studies on rock mass scale effects.

A study of the appropriateness, geometry, size, and fillers boundary restraint on representative volume element in electrical simulation for carbon-based nanocomposites

In three-dimensional Monte Carlo simulation for carbon-based conductive nanocomposites, representative volume element (RVE) establishes a relationship between the microstructure and electrical properties of nanocomposites. Generating RVE is the first step in simulation, but the lack of systematic research on RVE results in low simulation accuracy and reliability. In this work, a study of the appropriateness, geometry, size, and fillers boundary restraint on RVE for conductive nanocomposites filled with single filler of CNTs, CB or GNPs is studied. The results show that it is reasonable to use RVE in the simulation. The cube is suggested as the standard shape of RVE and the size of RVE can determine by the Chi-square test results obtained from percolation threshold. It is also concluded that periodic and aperiodic boundary restraint are superior to truncated boundary restraint. This study can provide the theoretical and technique foundation for reasonably setting RVE parameters and accurately predicting the electrical properties for nanocomposites filled with different dimension fillers.

Fractal discrete fracture network model for the analysis of radon migration in fractured media

A novel model combined with fractal theory and the discrete fracture network (DFN) model is proposed to simulate radon migration in fractured media. In the novel model, the fracture center and length distribution are determined by a first-order model, and the fracture orientations are modeled with the von Mises-Fisher distribution. The model is further developed into a computer software that can calculate the radon diffusion coefficient and exhalation rate of fractured media. An outcrop map is used to validate the ability of the model to describe complex fractures, and radon migration in natural fractures at an outcrop site is predicted using the model. Results show that the model has good reliability and robustness. Evaluation of the representative elementary volume (REV) indicates the existence of REV during the modeling of radon migration in fracture networks. Following an exponential law, REV size decreases with the increase in the length exponent.

Multiscale study on determining representative volume elements size for mechanical behaviours of complex polymer nanocomposites with nanoparticulate agglomerations

Representative volume elements (RVE) have been widely employed to describe and predict the various effective properties of heterogeneous composite systems. In this present study with an interphase property degradation model, a multiscale analysis is performed to investigate the mechanical behaviours of polymer nanocomposites containing nanoparticulate agglomerations. Using the Fourier transform approach on the structural property variation at given cell lengths, the smallest RVE size (the minimum allowable length for an approximate description of the material system) is determined based on the coefficient of variation. It is reasonable to select an RVE size that is equal to a quarter of the most dominant wavelength as the minimum RVE size. A novel user-defined factor-based approach similar to the concept of margin of error is proposed to determine the optimal sampling number at a specific RVE size. The precision approach can provide accurate estimation of the mechanical properties of the nanocomposites with reduced computational cost.

On Representative Elementary Volumes of Grayscale Micro‐CT Images of Porous Media

AbstractThe concept of linking pore‐scale data to continuum‐scale characteristics of porous media relies on the existence of a representative elementary volume (REV). The current techniques for estimating REVs require access to segmented micro‐computed tomographic (micro‐CT) images and computations of petrophysical properties which are computationally intensive and time‐consuming. Herein, a texture characterization method called the Gray‐Level Size Zone Matrix (GLSZM) is applied directly to raw grayscale micro‐CT images. GLSZM representations of 3D micro‐CT images capture information regarding the connectivity of gray‐level intensities, termed as “size‐zones.” Statistical descriptors of pore space are calculated based on GLSZM to understand the connectivity of low gray‐level intensities. These GLSZM statistics capture microstructural fluctuations and offer insights into the impact of grayscale heterogeneity on REV size. This approach allows REV sizes to be estimated directly using grayscale micro‐CT images, in a reproducible, less time‐consuming and computationally efficient manner.

INVESTIGATIONS ON REPRESENTATIVE ELEMENTARY VOLUME AND DIRECTIONAL PERMEABILITY OF FRACTAL-BASED FRACTURE NETWORKS USING POLYGON SUB-MODELS

Since the directional permeability of fractured rock masses is significantly dependent on the geometric properties of fractures, in this work, a numerical study was performed to analyze the relationships between them, in which fracture length follows a fractal distribution. A method to estimate the representative elementary volume (REV) size and directional permeability ([Formula: see text] by extracting regular polygon sub-models with different orientation angles ([Formula: see text] and side lengths ([Formula: see text] from an original discrete fracture network (DFN) model was developed. The results show that the fracture number has a power law relationship with the fracture length and the evolution of the exponent agrees well with that reported in previous studies, which confirms the reliability of the proposed fractal length distribution and stochastically generated DFN models. The [Formula: see text] varies significantly due to the influence of random numbers utilized to generate fracture location, orientation and length when [Formula: see text] is small. When [Formula: see text] exceeds some certain values, [Formula: see text] holds a constant value despite of [Formula: see text], in which the model scale is regarded as the REV size and the corresponding area of DFN model is represented by [Formula: see text] (in 2D). The directional permeability contours for DFN models plotted in the polar coordinate system approximate to circles when the model size is greater than the REV size. The [Formula: see text] decreases with the increment of fractal dimension of fracture length distribution ([Formula: see text]. However, the decreasing rate of [Formula: see text] (79.5%) when [Formula: see text] increases from 1.4 to 1.5 changes more significantly than that (34.8%) when [Formula: see text] increases from 1.5 to 1.6 for regular hexagon sub-models. This indicates that the small non-persistent fractures dominate the preferential flow paths; thereafter, the flow rate distribution becomes more homogeneous when [Formula: see text] exceeds a certain value (i.e. 1.5). A larger [Formula: see text] results in a denser fracture network and a stronger conductivity.

Geometrical and Topological Analysis of Pore Space in Sandstones Based on X-ray Computed Tomography

The pore geometry and topology properties of pore space in rocks are significant for a better understanding of the complex hydrologic and elastic properties. However, geometry and topology information about the sandstone pore structures is not fully available. In this study, we obtained the topological and geometrical pore parameters from a representative elementary volume (REV) for fluid flow in sandstone samples. For comparison, eight types of sandstones with various porosities were studied based on the X-ray micro-computed tomography technique. In this study, the REV size was selected based on the parameters from the respective pore network models (PNM), not just the porosity. Our analysis indicates that despite different porosity, all the sandstone samples have highly triangular-shaped pores and a high degree of pore structural isotropy. The high porosity group sandstones exhibit wider ranges of pore sizes than the low porosity group sandstones. Compared to the high porosity group sandstones, the low porosity group sandstones samples showing a higher global aspect ratio, indicating some pores exist in the form of bottlenecks. The pore topological properties of different sandstones show a high dependence of the porosity. The high porosity group sandstones obtain large coordination numbers, large connectivity densities and low tortuosities. The results from this study will help better understand the complex pore structure and the fluid flow in sandstone.

Representative volume element generation and its size determination for discontinuous composites made from chopped prepregs

In the present study, the composites made from chopping prepregs in rectangular chips, called discontinuous composites derived from chopped prepreg (DCCP) were modeled for stiffness prediction. Standard mechanics approach with displacement boundary conditions is used for stiffness prediction. The minimum size of the representative volume element (RVE) for this material is determined. An algorithm based on the random sequential adsorption technique (RSA) has been exploited to generate the 3D RVEs. RVEs of five different aspect ratios of the cuboid-shaped inclusions are developed using this technique. The chips are completely randomly oriented with a constant volume fraction. A detailed study was carried out in each case by further generating the RVEs with different increasing sizes, and the smallest size of RVE with geometric periodicity was identified. Further, the characteristics of these RVEs have been investigated for different aspect ratios, varying the chip volume fraction for the influence of volume fraction on RVE size and the material properties. The efficacy of the developed approach is demonstrated through repetitiveness of the effective properties prediction for various scenarios studied. The resulting material showed isotropy in the plane where chips are distributed randomly while overall isotropy about 80% is seen in most of the RVEs studied.

Efficient FFT-based upscaling of the permeability of porous media discretized on uniform grids with estimation of RVE size

An efficient numerical scheme at the crossroad of Fourier based methods and boundary element methods is derived to upscale the permeability of porous media from Stokes flow within the pore space. The method relies on a variational framework which involves trial force fields and the Green function. The discretization of the trial force fields is carried out on a uniform grid, which allows for the use of fast Fourier transforms to evaluate the convolution product arising from the Green function. A discretization of the Green function consistent with the variational framework is introduced to ensure the upper bound status of the homogenized permeability. The energy consistent discretization is compared to six alternative discretizations arising from truncation of Fourier series or from various finite difference schemes, in terms of accuracy of the homogenized permeability and of the local velocity fields as well as on the number of iterations required for convergence of the iterative solver. The method is combined to a strategy to estimate the Representative Volume Element (RVE) size at a target accuracy level from a single simulation on a large sample to study the permeability and associated RVE size for the Voronoi cell model, over the whole range of porosity.

Optimal X-ray micro-CT image based methods for porosity and permeability quantification in heterogeneous sandstones

Summary 3-D X-ray micro-CT (XCT) is a non-destructive 3-D imaging method, increasingly used for a wide range of applications in Earth Science. An optimal XCT image-processing workflow is derived here for accurate quantification of porosity and absolute permeability of heterogeneous sandstone samples using an assessment of key image acquisition and processing parameters: image resolution, segmentation method, representative elementary volume (REV) size and fluid-simulation method. XCT image-based calculations obtained for heterogeneous sandstones are compared to two homogeneous standards (Berea sandstone and a sphere pack), as well as to the results from physical laboratory measurements. An optimal XCT methodology obtains porosity and permeability results within ±2 per cent and vary by one order of magnitude around the direct physical measurements, respectively, achieved by incorporating the clay fraction and cement matrix (porous, impermeable components) to the pore-phase for porosity calculations and into the solid-phase for permeability calculations. Two stokes-flow finite element modelling (FEM) simulation methods, using a voxelized grid (Avizo) and tetrahedral mesh (Comsol) produce comparable results, and similarly show that a lower resolution scan (∼5 µm) is unable to resolve the smallest intergranular pores, causing an underestimation of porosity by ∼3.5 per cent. Downsampling the image-resolution post-segmentation (numerical coarsening) and pore network modelling both allow achieving of a REV size, whilst significantly reducing fluid simulation memory requirements. For the heterogeneous sandstones, REV size for permeability (≥1 mm3) is larger than for porosity (≥0.5 mm3) due to tortuosity of the fluid paths. This highlights that porosity should not be used as a reference REV for permeability calculations. The findings suggest that distinct image processing workflows for porosity and permeability would significantly enhance the accurate quantification of the two properties from XCT.

Numerical prediction of effective properties for heterogeneous viscoelastic materials via a temporally recursive adaptive quadtree SBFEM

A new numerical method is presented for evaluating the effective properties of heterogeneous viscoelastic materials, and an efficient and high-fidelity Direct Numerical Simulation (DNS) is addressed by integrating the advantages of the quadtree Scaled Boundary Finite Element Method (SBFEM) and a temporally recursive adaptive algorithm. The quadtree technique that can be directly implemented with input images is utilized for the mesh generation of the Representative Volume Element (RVE) with complex constitutes, and the inconvenience caused by ‘hanging nodes’ and the interface conformity requirement in the mesh generation can be removed using a polygonal Scaled Boundary Finite Element (SBFE). The computing accuracy in the time domain can be stably achieved via a recursive adaptive process when the step sizes vary. By virtue of the proposed DNS, an inverse analysis based procedure is designed to estimate the effective viscoelastic properties, and two indicators are proposed to evaluate the effectiveness of estimation. Numerical verification is provided to demonstrate the effectiveness of the proposed approach, and the impacts of the distributions and shapes of inclusions, RVE sizes, and volume fractions are taken into account.

Failure Analysis of Unidirectional Ceramic Matrix Composite Lamina and Cross-Ply Laminate under Fiber Direction Uniaxial Tensile Load: Cohesive Zone Modeling and Brittle Fracture Mechanics Approach

The current research work presents the computational micromechanical analysis of the room temperature tensile failure behavior of unidirectional (UD) and cross-ply (0/90) ceramic matrix composites (CMCs). For computational micromechanical analysis, three-dimensional (3D) representative volume element (RVE) and multi-fiber multilayer RVE (M2 RVE) models are generated that are representative of the lamina and the laminate under investigation. The RVE and M2 RVE models are generated by replicating the fiber distribution, and the placement of the fibers observed in a microscopic image of an actual CMC laminate. The generated RVE models consist of the discrete representation of individual constituent phases of the CMC such as fibers, interphase, matrix, and the fiber–interphase interface region. Under the applied external tensile load, the fiber–interphase interface interactions are modeled using the cohesive elements that follow the bilinear traction separation law. The matrix, fiber, and interphase materials failure behavior is captured using a brittle cracking model. In order to validate the proposed numerical methodology, the predicted average stress–strain curve at the UD laminate level is compared to the experimental stress–strain curve reported in the literature. In addition, the observed different phases in the predicted stress–strain curve are validated with the literature data. Using the proposed numerical methodology, a detailed local stress–strain and damage analysis leads to an observation that the so-called ductile stress–strain behavior (kink in the stress–strain curve) of a CMC UD laminate under uniaxial fiber direction tensile loads is mainly caused by the matrix damage initiation. Apart from the SiC material properties such as strength and fracture energy, it is also observed that the RVE size influences the average strength and failure strain predictions using computational micromechanics.