Representative Volume Element Size Research Articles

A precise modeling method of three‐dimensional discrete fracture network based on rectangular joint model

AbstractAs the weak structure and main seepage channel of rock mass, the discrete fracture network (DFN) has a significant influence on the physical and mechanical properties of rock mass. In this paper, based on the rectangular joint model, two algorithms are proposed to realize the precise modeling of the DFN in any polyhedron. First, the equivalence of DFN modeling with the rectangular joint model and the elliptical joint model is theoretically deduced. On this basis, the chain splicing algorithm suitable for the rectangular joint model is proposed, which can generate the DFN model according to the actual distribution of joint spacing, trace, and bridge length. Aiming at the concave boundary in the model domain, the segmentation‐stitching operation is put forward, and the continuity of the joint when passing through the interface is ensured. Finally, the effectiveness of the above algorithm is verified by several simple cases, and the relationship between the size of the geometrical representative elementary volume (REV) and discontinuity parameters is discussed. The results show that when the mean values of the joint distribution parameters are the same, even if the distribution rules are different, the number of joints generated in the same spatial region is approximately equal. The size of the geometrical REV is independent of the spacing and bridge length and is approximately three to five times the mean of the trace. The relevant algorithms and conclusions in this paper can be used as the basis for subsequent rock mass stability analysis and seepage calculation.

Determining Digital Representation and Representative Elementary Volume Size of Broken Rock Mass Using the Discrete Fracture Network–Discrete Element Method Coupling Technique

Obtaining the digital characterization and representative elementary volume (REV) of broken rock masses is an important foundation for simulating their mechanical properties and behavior. In this study, utilizing the broken surrounding rock of the main powerhouse at the Liyang pumped storage power station as an engineering background, a three-dimensional fracture network generation program is first developed based on the theories of discrete fracture network (DFN) and discrete element method (DEM). The program is then integrated with a distinct element modelling platform to generate equivalent rock mass models for broken rock masses based on the DFN–DEM coupling technique. Numerical compression tests are conducted on cylindrical rock specimens produced using the proposed modelling approach, aiming to determining the REV size of the target rock masses at the Liyang power station. A comparative validation is also performed to examine the REV result obtained from the proposed approach, which adopted a REV measuring scale index (RMSI) to determine the REV size. Results indicate that the organic integration of DFN simulation techniques and DEM platforms can effectively construct numerical models for actual broken rock masses, with structural surface distributions statistically similar to the real ones. The results also show that the REV size of the investigated rock masses determined by the cylindrical rock models is 5 m × 10 m, which aligns with the size determined by the cubic rock models, as the target cubes show the same height as the cylindrical specimens. This study provides a model and parameter basis for the numerical calculation of the mechanical behavior of broken rock mass.

REPRESENTATIVE VOLUME AND EFFECTIVE MATERIAL CHARACTERISTICS OF PERIODIC AND STATISTICALLY UNIFORMLY REINFORCED FIBER COMPOSITES

In the deformable solid mechanics, there are concepts associated with continuum points (displacements, relative elongations, shifts) and a set of continuum points – an elementary volume (mass, energy, stresses). The role of such volume in the mechanics of composite materials is played by the representative volume element (RVE).This concept was first introduced by R. Hill (1963). Modern authors use the W.J. Drugan, J.R. Willis (1996) formulation. Based on the analysis of the RVE concept, we formulated its essential features: RVE is the minimum possible sample for numerical tests to determine the effective material parameters of the composite; under any RVE loading, its macroscopic stress-strain state is uniform. Its significance for the mechanics of composite materials is revealed: the existence of RVE for a composite is a criterion for applying the effective modulus theory to the analysis of its stress-strain state; the dehomogenization of a stressed-state composite material at a point is a solution to the micromechanics problem of the RVE stress-strain state determination; the characteristic size of RVE limits the size of the sampling grid in the numerical study. An iterative algorithm for constructing a representative volume of a periodic structure composite and its effective material thermoelastic characteristics is proposed. It is shown that the geometric shape of such a composition is a rectangular parallelepiped. The RVE construction algorithm for periodic compositions is extended to the composites statistically uniformly reinforced with continuous fibers. A method for modeling such materials with a following regular structure is suggested described: in the section perpendicular to the fibers, fiber centers should be located at the vertices of regular triangles. Examples of constructing RVE and thermoelastic material characteristics of specific compositions are given. The calculation results are compared with the data obtained using certified software products.

Multiscale formulation for saturated porous media preserving the representative volume element size objectivity

SummaryA multiscale model for saturated porous media is proposed, based on the concept of representative volume element (RVE). The physics between macro and micro‐scales is linked in terms of virtual power measures given by the general theory of poromechanics. Then, applying the so‐called Principle of Multiscale Virtual Power, together with suitable admissible constraints on micro‐scale displacement and pore pressure fields, a well‐established variational framework is obtained. This setting allows deriving the weak form of micro‐scale balance equations as well as the homogenization rules for the macro‐scale stress‐like variables and body forces. Whenever the micro‐scale mechanical constitutive functionals admit, as input arguments, the full‐order expansion of the micro‐scale pore pressure field, a size effect is inherited on the macro‐scale material response. The current literature attributes this issue to the so‐called “dynamical” or “second‐order” term of the homogenized flux velocity. It has been commonly suggested that the influence of this term is negligible by assuming infinitely small micro‐scale dimensions. However, such an idea compromises the fundamental notion of the existence of RVE for highly heterogeneous media. In this work, we show that the micro‐scale size dependence can be consistently eliminated by a simple constitutive‐like assumption. Accordingly, slight and selective redefinitions in the input arguments of micro‐scale material laws are proposed, leading to a constitutive formulation that allows the combination of micro‐scale variables with different orders of expansion. Just at this specific (constitutive) level, a reduced‐order expansion is selectively adopted for the micro‐scale pore pressure field. Thus, the RVE notion is restored while still retaining the major effects of the “dynamical” component of the homogenized flux velocity. The proposed formulation is implemented within a finite element squared (FE) environment. Some numerical experiments are presented showing the viability of the methodology, including comparisons against analytical, mono‐scale and DNS solutions.

Study on the influence of statistical geometrical characteristics on the permeability tensor of fractured rock masses by the equivalent pipe network method

The permeability of the host rock of the nuclear waste underground repository is one of the most keys for estimating the potential risk for nuclide waste to escape. Therefore, a three-dimensional discrete fracture network (DFN) model using the Monte Carlo method is generated based on the statistical geometrical characteristics of fracture network in Beishan I area which is selected as the site for constructing the nuclear waste repository in China. And the permeability of Beishan I fractured rock masses is calculated through the equivalent pipe network (EPN) method and representative elementary volume (REV) of it is determined. A rotating algorithm is developed to calculate different direction permeability. Furthermore, the influence of fracture statistical geometrical characteristics on REV size is explored by creating a series of DFN models with varying log-normal distribution of side lengths. The results show that the REV size of the Beishan Ⅰ area is 1.66 × 105 m3 and the corresponding principal permeability direction is 111.29°/75.36°. The REV size and permeability both increase with the increment of the mean and variance of the log-normal distribution increases. And with the increase of mean and variance, the growth rate gradually accelerated.

A hybrid algorithm of particle swarm optimization and finite element method to identify local mesoscopic damage of concrete-like materials

Meso-scale material modeling and parameter identification are still unsettled issues in the current multi-scale methods for the field of computational mechanics. In this investigation, a hybrid algorithm of particle swarm optimization and finite element method coupled with continuum damage mechanics is proposed to forecast local mesoscopic elasticity modulus and damage of concrete-like materials based on the macroscopic stress-strain relationship data curve. The algorithm establishes a bridge between mesoscopic mechanical analysis and macroscopic mechanical analysis, which can also guarantee the consistency of analysis in different scales. A representative numerical case is conducted to support the algorithm. Additionally, the effect of different divisions of representative volume element is analyzed based on the algorithm, and the recommended size of representative volume element is 5 mm. The prediction results match well with the corresponding macroscopic experimental results and mesoscopic experimental results based on the chosen recommended size of representative volume element. The effectiveness of the algorithm is verified, which can support an effective numerical tool to optimize and identify mesoscopic mechanical parameters and damage for multi-scale material modeling and analysis.

Homogenization assumptions for the two-scale analysis of first-order shear deformable shells

This contribution presents a multiscale approach for the analysis of shell structures using Reissner–Mindlin kinematics. A distinctive feature is that the thickness of the representative volume element (RVE) corresponds to the shell thickness. The main focus of this paper is on the choice of correct boundary conditions for the RVE. Three different types of boundary conditions, which fulfil the Hill–Mandel condition, are presented to bridge the two scales. A common feature is the application of zero-traction boundary conditions at the top and bottom surfaces of the RVE. Furthermore, an internal constraint is used to reduce the dependency of the stiffness components on the RVE size. The introduced boundary conditions differ mainly in the application of shear strains and their symmetry requirements on the RVE. The characteristic features are compared by means of linear-elastic benchmark tests. It is shown that the stress resultants and tangent stiffness components are obtained correctly. Moreover, the presented approach is verified using different macroscopic shell structures and different mesostructures. Both, linear and nonlinear small strain examples are compared to analytical values or full-scale solutions and demonstrate a wide applicability of the present formulation.

Research on the RVE of the Calculated Strength of River Ice at the Mesoscale in the Freezing Period of the Yellow River

Microscopic fabric changes are the main reason for the complex physical and material properties of Yellow River ice at the macroscale. To study the physical and material properties of Yellow River ice, Yellow River ice was taken as the research object, and definition and determination methods for the representative volume element (RVE) of the Yellow River ice based on its computed strength at the microscale were proposed. A micromechanical numerical model for Yellow River ice was built, the corresponding macromechanical properties were simulated, and the RVE size of the macromechanical strength of the Yellow River ice was determined to be 250 mm. The uniaxial compressive strength of river ice in different working conditions was simulated and analyzed, and the accuracy and effectiveness of the RVE of the calculated strength of river ice were verified. The research results provide a reference for analyzing the damage process of Yellow River ice at the microscopic level, providing a theoretical basis for studying the mechanism of Yellow River ice.

The asymptotic homogenization evaluation of effective mechanical property of the C/C-ZrC composite by the reconstruction technique

In this framework, we formulate a procedure to reconstruct the structure of general random 3D braid heterogeneous media based on transformation of Gaussian random fields, where the correlation of the underlying Gaussian field is determined explicitly rather than through iterative methods. The reconstructed media can meet the key microstructural information contained in the micrographs and two-point correlation function of the reference media. A robust and efficient algorithm is the operation for accurate determination the effective physical properties from a local heterogeneous material based on the methods of asymptotic homogenization and the finite element. Comparison of the present result to the experimental data from the literature shows a good agreement. The correlation length, Gaussian complexity, representative volume element size and mesh size are investigated and selected to highlight the importance of morphologically realistic microstructures. The effective properties of 3D braid composites are evaluated numerically and the respective effect of the volume fraction and relative material parameters of the reinforcement is assessed numerically. This breakthrough is the possibility to design, predict properties, and even manufacture materials not only with the help of experiments but also with computer science.

Thermo-viscoelastic multiscale homogenization of additively manufactured short fiber reinforced polymers from direct microstructure characterization

Short fiber reinforced polymers (SFRPs) processed by Fused Filament Fabrication (FFF) show great promise for applications requiring high weight-specific mechanical properties, such as aerospace. However, the unique morphology created by the printing process and the relationship between processing parameters and part behavior is not sufficiently well understood, making structural analysis a largely empirical task. The limited ability to predict the effective viscoelastic properties and their thermal dependence is a problem from a mechanical design standpoint and is an obstacle to the widespread adoption of SFRPs made by FFF. In this work, those properties are evaluated using high-resolution imaging of physical samples, enabling the accurate prediction of time-dependent stress–strain relations across different temperatures. Multiscale homogenization is performed considering individual components (fibers, pores, matrix) at the microscale, and the regular pattern created by the printing process at the mesoscale. The efficiency of the Fast Fourier Transform (FFT) based homogenization procedure is leveraged to allow for large Representative Volume Element (RVE) sizes, which are required due to the complexity of real microstructures. The predicted thermo-viscoelastic behavior is validated against creep-recovery data of specimens printed from 30 wt% short carbon fiber reinforced poly-ether-ether ketone (PEEK), and the experimental data can be predicted within 3% mean relative error for axial loading, both at 21°and 120 °C. Imaging data and pre-processed RVEs as well as original source code are made publicly available.

An orientation corrected shaking method for the microstructure generation of short fiber-reinforced composites with almost planar fiber orientation

We present an algorithm for generating short fiber-reinforced microstructures with almost planar fiber orientation. The orientation corrected shaking (OCS) method achieves a high accuracy regarding the volume fraction, fiber length distribution and fiber orientation state. Additionally, the algorithm is capable of generating microstructures for industrial materials, e.g., for a PA66GF35 material with a volume fraction of 19.3% and an aspect ratio of 33. For typical manufacturing processes, short fiber-reinforced composites show a mainly planar fiber arrangement. Therefore, we extend the two-step shaking algorithm of Li et al. [J. Ind. Text. 51(1), pp. 506–530, 2022] for a user-selected rectangular size of the unit cell and periodic boundary conditions. Additionally, the hidden closure structure of the algorithm is uncovered and a precise realization of the fiber orientation state achieved. We examine the representative volume element size for the OCS method, observing representative errors below 2% even for unit cells with edge lengths smaller than the mean fiber length. Additionally, the influence of different closure approximations on the stiffness is investigated. When applied to an industrial PA66GF35 material with sandwich structure, the OCS method demonstrates differences below 2% and 9% for the computed directional Young’s moduli E1 and E2 compared to experimental data.

Representative volume element: Existence and extent in cracked heterogeneous medium

Acknowledging the ever-increasing demand for composites in the engineering industry, this paper focuses on the failure of composites at the microscale and augmenting the use of multiscale modelling techniques to make them better for various applications. This work aims to increase the representativeness of the volume element by attenuating the mesh and size sensitivities in representative volume element (RVE) modelling. A technique to alleviate mesh sensitivity in RVE modelling is proposed, which equalises the fracture energy observed from computational analysis with the real phenomenon, thereby keeping the response independent of the bandwidth of strain localisation. Based on the hypothesis that ensuring periodicity of strain, in addition to displacement periodicity across the domain boundary and supplementing the capability of periodic boundary conditions (PBCs) to attenuate the size dependency in RVE modelling, a set of modified PBCs (MPBCs) are formulated. One thousand two hundred RVE samples falling into combinations of five fibre volume fractions and four RVE sizes are analysed under transverse loading, and the ability of MPBCs to attenuate the effect of RVE size on the precision of material response, particularly in the softening regime, is verified. This work also focuses on various factors affecting damage initiation in 2D composite RVEs. The arrangement of a pair of fibres with their members placed close to each other, such that the angle between the direction of loading and an imaginary line drawn between their centres is less, is observed to make the region between them more favourable to damage.

Estimation of REV size of 2-D DFN models in nonlinear flow: Considering the fracture length-aperture correlation

Determining the representative elementary volume (REV) of fractured rock masses is the foundation for studying the properties of such rock masses using the equivalent continuum model. This paper focuses on estimating the REV sizes of a fractured rock mass, considering the linked trace length and fracture aperture in both linear and nonlinear flow regimes. To achieve this, Monte Carlo simulations were employed to create discrete fracture network (DFN) models based on the geometric characteristics of fractures. The linear element method, modified by the Forchheimer law, was used to calculate the equivalent permeability coefficient tensor K and the equivalent non-Darcy coefficient tensor β. These calculations were performed to determine the REV size of DFNs under nonlinear flow conditions. The findings of this study demonstrate a general consistency in the computed K values for the two flow regimes. It was observed that the REV size, determined by β value, exceeded that obtained using K for the identical fracture network. Furthermore, when considering the same flow regime, the REV size was found to be smallest when the fracture aperture remained constant, followed by cases where it was independent of the trace length and finally when it was correlated with the trace length.

Efficient generation of random fiber distributions in fiber reinforced composites

This paper proposes a new algorithm to generate representative volume elements (RVEs) with random fiber distribution in fiber reinforced composites (FRC). The proposed algorithm is straightforward and easy to implement based on judging the maximum and minimum distances between a new fiber and existing fibers. The generation results demonstrate that the maximum fiber volume fraction gradually increases and oscillates violently before reaching 78.54% as the fiber radius rises. Moreover, with the increase of RVE size, the maximum fiber volume fraction changes gently when the fiber radius does not exceed 6.5 μm, but it changes dramatically at other fiber radii. Then, the fiber distributions of the generated RVEs are evaluated using the nearest neighbor distance, Ripley’s K function, and pair distribution function. The evaluation results indicate that the fiber distributions present randomness. Lastly, the effective elastic properties of the Carbon/Epoxy unidirectional FRC are predicted using the RVEs generated by the proposed algorithm, the RVEs generated by regularization, and the Mori–Tanaka method. It is found that the prediction using the RVEs generated by the proposed algorithm is more accurate than the regularization, compared with the Mori–Tanaka and experiment results. The proposed algorithm contributes to microstructure modeling in computational micromechanics.

Spatial and temporal averaging in the homogenisation of the elastodynamic response of periodic laminates

This paper suggests a methodology to obtain homogenised material properties from a transient dynamic numerical model. The standard Hill–Mandel Principle, based on spatial averages, is extended with time averaging. Thus, in addition to a sufficiently large Representative Volume Element (RVE) to carry out the averaging in space, a sufficiently large time window is required to carry out the time averaging. The space–time averaging procedure is validated for a periodic laminate bar subjected to a variety of boundary conditions, impedance contrasts and loading conditions. The homogenised results converge to the analytical solutions and confirm that having a higher impedance contrast between laminate components requires not only larger RVE sizes but also longer time averaging windows. The most efficient macroscopic approximation is obtained by a balanced increase in RVE size and time averaging window.

Using numerical homogenization to determine the representative volume element size of paper

AbstractPaper has always been of great importance as a printing and packaging material. Due to its potential for recyclability and reusability, which result in a low environmental impact, it is expected that this importance will soar over the coming years. Nevertheless, even though paper has been in use for decades, many research questions are still open, especially in the field of mechanics. One of these questions is the appropriate size of a representative volume element (RVE) of the microstructure of paper. Therefore, a finite element network model of interconnected fibers was developed which represents the microstructure of paper. In order to create the network model realistically, geometry and material quantities were taken from experimental investigations. Additionally, single fibers and fiber bonds were investigated beforehand in order to incorporate their proper behavior into the network. Based on this, the networks were generated and subsequently loaded by applying mixed displacement and traction boundary conditions. The resulting effective stresses were compared to determine the necessary RVE size. Two approaches were considered for modeling the RVE: (i) one single “large” network, and (ii) a certain number of smaller networks whose average yielded a statistically representative effective response. As a result, the second approach turned out to be less computationally intensive due to smaller models and the possibility to run calculations simultaneously. Concluding, it was possible to determine the size of the RVE which will be used to investigate various individual microstructural effects on the macrostructure in the future.

Averaging techniques for microstructures with localization bands due to damage progression

AbstractIn multiscale analysis, homogenization methods are needed to up‐scale the micromechanical response obtained from investigating the underlying microstructure to the next higher scale. The standard homogenization schemes are based on volume averaging over the entire microstructure following Hill's approach, which requires that the virtual energies generated on the two involved scales equalize. However, these standard homogenization schemes are not applicable to softening phenomena due to localization, and representativeness of the considered microscale volume is lost. One way to overcome these drawbacks is to perform the volume averaging only within the localizing failure zone. Thereby, representative results can be achieved even in the softening region. In this paper, we apply the failure zone homogenization approach to both, mode I and mode II loading scenarios, as well as mixed‐mode loading of long fiber reinforced plastics. For an accurate description of material failure within the epoxy matrix, a scalar damage model at large strains with gradient enhancement is used, such that the obtained results are mesh‐independent. As a result, we show that for all considered cases representative volume element (RVE) sizes can be determined by using the failure zone homogenization scheme. Nevertheless, the energy distributions of all involved mechanisms have to be considered carefully in order to allow generalizations.

Second-order computational homogenisation enhanced with non-uniform body forces for non-linear cellular materials and metamaterials

Although “classical” multi-scale methods can capture the behaviour of cellular, including lattice, materials, when considering lattices or metamaterial local instabilities, corresponding to a change of the micro-structure morphology, classical computational homogenisation methods fail. On the one hand, first order computational homogenisation, which considers a classical continuum at the macro-scale cannot capture localisation bands inherent to cell buckling propagation. On the other hand, second-order computational homogenisation, which considers a higher order continuum at the macro-scale, introduces a size effect with respect to the Representative Volume Element (RVE) size, which is problematic when the RVE has to consider several cells to recover periodicity during local instability. In this paper we reformulate in a finite-strain setting the second-order computational homogenisation using the idea of equivalent homogenised volume. From this equivalence, arises at the micro-scale a non-uniform body force that acts as a supplementary volume term over the RVE. In the presented method, this non-uniform body-force term arises from the equivalence of energy, i.e. the Hill–Mandel condition, between the micro- and macroscopic volumes and depends mainly on the relation between the micro-scale and macro-scale deformation gradient. We show by considering elastic and elasto-plastic metamaterials and cellular materials that this approach reduces the RVE size dependency on the homogenised response.

Microstructural Analysis of the Transverse and Shear Behavior of Additively Manufactured CFRP Composite RVEs Based on the Phase-Field Fracture Theory

Due to the versatility of its implementation, additive manufacturing has become the enabling technology in the research and development of innovative engineering components. However, many experimental studies have shown inconsistent results and have highlighted multiple defects in the materials’ structure thus bringing the adoption of the additive manufacturing method in practical engineering applications into question, yet limited work has been carried out in the material modelling of such cases. In order to account for the effects of the accumulated defects, a micromechanical analysis based on the representative volume element has been considered, and phase-field modelling has been adopted to model the effects of inter-fiber cracking. The 3D models of representative volume elements were developed in the Abaqus environment based on the fiber dimensions and content acquired using machine learning algorithms, while fulfilling both geometric and material periodicity. Furthermore, the periodic boundary conditions were assumed for each of the representative volume elements in transversal and in-plane shear test cases,. The analysis was conducted by adopting an open-source UMAT subroutine, where the phase-field balance equation was related to the readily available heat transfer equation from Abaqus, avoiding the necessity for a dedicated user-defined element thus enabling the adoption of the standard elements and features available in the Abaqus CAE environment. The model was tested on three representative volume element sizes and the interface properties were calibrated according to the experimentally acquired results for continuous carbon-fiber-reinforced composites subjected to transverse tensile and shear loads. This investigation confirmed the consistency between the experimental results and the numerical solutions acquired using a phase-field fracture approach for the transverse tensile and shear behavior of additively manufactured continuous-fiber-reinforced composites, while showing dependence on the representative volume element type for distinctive load cases.

Computational thermal conductivity in polyurethane mixed cell foam: Numerical boundary effects and hybrid model

The effective thermal conductivity is the most useful characteristic property to distinguish between two or more insulation materials. It is always the same by using different boundary conditions or different representative elementary volume sizes. The objective of this paper is to set variational formulations of different types of boundary conditions Dirichelet boundary condition (EBC), Neumann boundary condition (NBC), Mixed boundary condition (MBC) and Periodic boundary condition (PBC). Then effective thermal conductivity are investigated by scale effect study of the representative elementary volume size of the different categories of Polyurethane foam, closed cell foam, open cell foam and mixed cell foam. The apparent conductivity remain the same for MBC, PBC and EBC in the case of closed and open unit cell foam. The effective thermal conductivity for the different categories of PU foam converge as the REVs sizes increase. A comparative study with numerical, analytical and experimental thermal conductivity is performed in order to validate the results. A hybrid model is proposed in order to overcome the computational cost of the investigation of the effective thermal conductivity of mixed cell foam.