Concept Of Representative Volume Element Research Articles

Numerical prediction of thermal conductivity and thermal expansion coefficient of glass fiber-reinforced polymer hybrid composites filled with hollow spheres

The optimal performance of composites enriched with hollow spheres has been reported in contemporary literature, whereas their thermal properties have received less attention. In this regard, a finite element method (FEM)-based micromechanical model has been developed systematically to investigate the role of intra-matrix embedding of hollow spheres on the thermal conductivity and coefficient of thermal expansion (CTE) of unidirectional fiber-reinforced hybrid composites. In so doing, the concept of representative volume element (RVE) considers microstructures comprising an epoxy matrix, E-glass fiber, and E-glass hollow spheres, assuming perfect bonding (ideal interface) between the components and modified approximate periodic boundary conditions. By computing the longitudinal and transverse temperature gradients generated due to the application of uniform heat flux as well as the geometrical variation in RVE owing to temperature enhancement, thermal conductivity and CTE have been respectively determined. Comprehensive evaluations have been conducted to examine the effects of microstructural-level features, including fiber volume content and orientation, plus volume content and thickness of hollow spheres, on the effective thermal conductivity and CTE of pseudo-porous ternary E-glass/epoxy composites.

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.

Homogenized damage model for brittle materials

This work presents a homogenization damage approach for brittle composites based on the concept of Representative Volume Element (RVE) performing complex numerical analysis of the mechanical behavior of materials with individual characteristics of constituent phases. Focusing on concrete material, cement paste and transition zone are modeled as damaged materials using the Mazars’ model, while the aggregates are modeled as linear elastic material. A based homogenization damage model technique is obtained and it is applied to describe the damage in the macrostructure. Results for concrete microstructures are presented and the homogenized responses demonstrate the expected behavior, including comparison with experimental test.

Second-order multi-scale modelling of natural and architected materials in the presence of voids: Formulation and numerical implementation

This contribution proposes a second-order computational homogenisation formulation for natural and architected materials in the presence of voids. The macro-scale is described by a second gradient continuum theory in the finite strain regime, and the micro-scale is modelled by the concept of representative volume element (RVE) within the classical first-order continuum mechanics. The Method of Multi-scale Virtual Power is employed to link the two scales, ensuring a variationally consistent scale transition. Unlike the multi-scale model proposed by Rodrigues Lopes and Andrade Pires (2022), the developed formulation allows the consideration of voids in the RVE domain and boundary, which is critical for modelling cellular materials, lattice structures, and metamaterials, among others. This is achieved by defining the kinematical quantities only in the solid domain of the RVE and postulating a new homogenisation relation for the second-order gradient. The kinematic constraints are imposed on the RVE by the Lagrange multiplier method and particularised for minimal (lower bound), periodic and direct (upper bound) conditions. It is demonstrated that the homogenised macroscopic stress tensors can be expressed in terms of the Lagrange multipliers. The finite element method is adopted for the numerical solution of the micro and macro equilibrium. The Newton–Raphson scheme is employed to solve the non-linear systems of equations at both scales and the consistent macroscopic tangents required for the FE2 framework are derived. Several numerical examples of porous solids, lattice structures and metamaterials illustrate the consistency and applicability of the formulation for two and three-dimensional problems.

Modeling the evolution of representative dislocation structures under high thermo-mechanical conditions during Additive Manufacturing of Alloys

Mesoscale simulations of discrete defects in metals provide an ideal framework to investigate the micro-scale mechanisms governing the plastic deformation under high thermal and mechanical loading conditions. To bridge size and time-scale while limiting computational effort, typically the concept of representative volume elements (RVEs) is employed. This approach considers the microstructure evolution in a volume that is representative of the overall material behavior. However, in settings with complex thermal and mechanical loading histories careful consideration of the impact of modeling constraints in terms of time scale and simulation domain on predicted results is required. We address the representation of heterogeneous dislocation structure formation in simulation volumes using the example of residual stress formation during cool-down of laser powder-bed fusion (LPBF) of AISI 316L stainless steel. This is achieved by a series of large-scale three-dimensional discrete dislocation dynamics (DDD) simulations assisted by thermo-mechanical finite element modeling of the LPBF process. Our results show that insufficient size of periodic simulation domains can result in dislocation patterns that reflect the boundaries of the primary cell. More pronounced dislocation interaction observed for larger domains highlight the significance of simulation domain constraints for predicting mechanical properties. We formulate criteria that characterize representative volume elements by capturing the conformity of the dislocation structure to the bulk material. This work provides a basis for future investigations of heterogeneous microstructure formation in mesoscale simulations of bulk material behavior.

Effect of Surface Stresses on the Antiplane Stress-Strain State of a Thin Ribbon-Like Interface Inclusion

On the basis of the theory of functions of complex variable and the method of jump functions, we propose a procedure of taking into account the additional influence of surface stresses in the problem of thin interface inclusion in the bimaterial. In this problem, we take into account the possibility of imperfect contact between the inclusion and the matrix and, in particular, the possibility of contact with surface tension. This significantly extends the field of applicability of the results of simulation within the framework of the concept of representative volume element in micro- and macromechanics. We propose a generalized model of thin inclusion with arbitrary mechanical properties. The analysis of test problems reveals high accuracy and efficiency of the proposed approach. We present the results of numerical analyses of the stress fields in the case of interaction of the inclusion with concentrated forces and screw dislocations.

Isotropic yield surfaces for porous ductile materials: complete geometric representation by a computational homogenisation procedure

PurposeThe present paper aims to explore a computational homogenisation procedure to investigate the full geometric representation of yield surfaces for isotropic porous ductile media. The effects of cell morphology and imposed boundary conditions are assessed. The sensitivity of the yield surfaces to the Lode angle is also investigated in detail.Design/methodology/approachThe microscale of the material is modelled by the concept of Representative Volume Element (RVE) or unit cell, which is numerically simulated through three-dimensional finite element analyses. Numerous loading conditions are considered to create complete yield surfaces encompassing high, intermediate and low triaxialities. The influence of cell morphology on the yield surfaces is assessed considering a spherical cell with spherical void and a cubic RVE with spherical void, both under uniform strain boundary condition. The use of spherical cell is interesting as preferential directions in the effective behaviour are avoided. The periodic boundary condition, which favours strain localization, is imposed on the cubic RVE to compare the results. Small strains are assumed and the cell matrix is considered as a perfect elasto-plastic material following the von Mises yield criterion.FindingsDifferent morphologies for the cell imply in different yield conditions for the same load situations. The yield surfaces in correspondence to periodic boundary condition show significant differences compared to those obtained by imposing uniform strain boundary condition. The stress Lode angle has a strong influence on the geometry of the yield surfaces considering low and intermediate triaxialities.Originality/valueThe exhaustive computational study of the effects of cell morphologies and imposed boundary conditions fills a gap in the full representation of the flow surfaces. The homogenisation-based strategy allows us to further investigate the influence of the Lode angle on the yield surfaces.

Simulation of heterogeneous breakage in sand based on full-field X-ray tomography measurements

This work investigates onset and progression of spatially distributed particle breakage in sand by taking into account heterogeneities induced by material properties and boundary conditions. The role of porosity and mean particle size on compressive yielding are taken into account within the framework of Breakage Mechanics. In addition, initial spatially distributed heterogeneities of these state variables are replicated through a Computerized-Tomography to Finite-Element (CT-FE) mapping scheme. The concept of representative elementary volume (REV) is utilized to extract local material state variables from digital images and assign them to corresponding Gauss integration points. Analysis of CT images and constitutive parameter calibration indicates that the mechanical response and breakage evolution are sensitive to porosity fluctuation, while being nearly unaffected by changes in mean particle size within the considered range of variability for this parameter. Full-field FE numerical simulations accounting for friction at sample-platen and sample-ring interfaces are conducted to assess their influence on the distribution of breakage sites. The results show that use of both spatial heterogeneities and boundary friction provide a satisfactory match between data and simulations. It was also shown that the concentration of breakage near the loading platen is promoted by arching processes induced by ring-sample friction.

Determination of representative volume element size for a magnetorheological elastomer

Smart composite materials have been an active field of research in the last few decades. Magnetorheological elastomers (MREs) are examples of such smart composites and they show a coupling between magnetism and elasticity. MREs are heterogeneous materials and they consist of magnetic particles and a silicone-based elastomer. To describe and predict the behaviour of the macroscopic continuum accurately, microstructural information needs to be taken into account in the analysis of heterogeneous materials. Therefore, a combined approach called multi-scale analysis is used to consider various scales of observation simultaneously. The concept of Representative Volume Element (RVE) is typically employed by multi-scale approaches to describe the micro scale, and thus the size of RVE becomes a model parameter in such techniques. This has motivated the determination of the RVE size and the derivation of magnetoelastic constitutive relations in terms of the RVE sizes in our paper. The finite element method and a statistical analysis based on the coefficient of variation have been used to determine the RVE size of MREs. The results show that it is possible to determine a lower bound of the RVE size for an MRE. Furthermore, a parametric study has been conducted to examine the sensitivity of the RVE size on the different material properties of the constituents. It was found that the RVE size is primarily set by the contrast of the different material properties, i.e. the stiffness, permeability and magnetoelastic coupling coefficients.

Effects of Dissolution Fingering on Mass Transfer Rate in Three‐Dimensional Porous Media

AbstractDissolution mass transfer from the trapped phase to the flowing phase in porous media occurs in various hydrogeology processes. One of the important phenomena is dissolution fingering and its effect on dissolution mass transfer. In this work, dissolution mass transfer in porous media with various particle sizes and distribution was investigated in pore scale and meso‐scale by using X‐ray microcomputed tomography. The specific interfacial area and mass transfer coefficient were measured by following the representative elementary volume concept, and the effect of dissolution fingering was evaluated. Linear instability theory was used for the prediction of dissolution finger width () at the onset of dissolution fingering. Although dissolution fingering tends to develop when the porous media container diameter () is smaller than half of , its effect on mass transfer coefficient starts to occur when is larger than the whole . Starting this point, dissolution fingering decreases the mass transfer coefficient by three times. For validation, an additional experiment with a larger porous media container size was performed, and three dissolution fingers were successfully observed. Their size was found to be comparable with the theoretical prediction based on linear instability theory.

Anisotropic Multiscale Modelling in SAE-AISI 1524 Gas Tungsten Arc Welded Joints

A transient non-linear multiscale finite element heat flow-mechanical model to determine micro residual stresses (type III) and micro plastic strains in SAE-AISI 1524 gas tungsten arc welded joints is developed. To include anisotropy by preferred crystallographic orientation or texture, the global domain was decomposed into small subdomains based on the concept of representative volume elements (RVEs). A three-dimensional numerical procedure was developed by using the coupling DREAM.3D-ABAQUS. The macro scale temperature gradient information as prescribed driven (load) boundary conditions was used to calculate the meso thermal cycles, and the meso scale temperature gradient information was used to calculate the micro thermal cycles needed in the subsequent mechanical analysis. Anisotropy was included by randomly entering in each grain of the RVE specimen either the maximum Young’s modulus (Emax) in the stiffest direction <111>, or the minimum Young’s modulus (Emin) in the least stiff direction <100>. Under this assumption, the averaging of the grain orientations over all grains in the textured polycrystal with greater number of grains ocurred, and the strength was diluted by the spread of orientations present. Higher Mises stresses evolved in the sample with coarse grain size (16 µm), which indicates that the strong dependence of residual micro stresses on grain size was reversed. The influence of the grain size on the response of the aggregates is clearly observed.

Numerical modelling of the elastic properties of 3D-printed specimens of thermoplastic matrix reinforced with continuous fibres

Numerical modelling of 3D-printed Fused Filament Fabrication (FFF) specimens with thermoplastic matrices reinforced with continuous fibres is still in its infancy. The existing numerical work is mostly related to the adhesion between printed filaments and not to the mechanical properties. However, the latter are one of the most important parameters that define the structural behaviour of the material.A numerical three-step multi-scale model is introduced in the present study, concentrating on the fundamental aspect of the elastic properties. The most encountered reinforced Nylon FFF structures with continuous fibres of glass, carbon and, Kevlar, are examined. The concept of Representative Volume Element (RVE) is utilized for the combination of matrix and fibres at the micro-scale and the addition of voids at the meso-scale. Finally, at the macro-scale, tension simulations are performed for specimens with various lay-ups.The results of the numerical model are validated using analytical models and experimental data. A new analytical micro-mechanical model is developed as an alternative to the more cumbersome Mori–Tanaka model. A series of experimental testing is performed for Kevlar-reinforced Nylon specimens to accompany the limited existing data and aid the validation process. The comparison reveals a good correlation with the analytical models for the micro- and meso-scale and to the experimental data for the macro-scale, leading to a robust conclusion for its validity and efficiency.

How pore structure non‐stationarity compromises flow properties representativity (REV) for soil samples: Pore‐scale modelling and stationarity analysis

AbstractClassic soil physics relies heavily on the concept of representative elementary volume (REV), which is necessary to perform upscaling from the studied soil samples and parameterize continuum scale hydrological models (e.g., based on Richards or Darcy equations). In this paper, we explore the boundaries of the classic REV concept and conventional representativity studies that claim REV for any given physical property, and that its values converge to a steady value with increasing sample volume. We chose two undisturbed soil samples of standard size from Ah and B horizons, subcropped two subvolumes within each of them and performed pore‐scale flow simulations using binarized X‐ray microtomography scans as input data. The volume of the simulation domains was 9003 voxels, with a physical volume within two orders of magnitude of the whole soil core. Based on the 3D pore geometry images and the resulting flow velocity and pressure fields, we performed REV analysis for the saturated hydraulic conductivity and porosity. Although in general subvolumes showed classical REV behaviour (convergence of porosity and saturated hydraulic conductance (Ksat)to plateau‐like behaviour), their flow properties converged to different REV values. We also evaluated the stationarity of pore structures by computing the directional correlation functions to explain the observed non‐unique behaviour. We concluded that neither of the studied samples can be considered to be representative due to their structural non‐stationarity. We extensively discussed the implications of these results for the upscaling and parameterization of continuum‐scale flow models. We argued that the plateau in Ksat (or any other physical property) is a necessary, but insufficient, condition for the REV, which requires (at least) pore structure stationarity as an additional criterion. REV as a concept is much broader than previously anticipated and the possibility of establishing REVs in structured soils will require an analysis of tensorial flow properties with correct boundary conditions, multiscale soil structure imaging with stationarity analysis, and pore‐scale simulations on fused multiscale images.Highlights Ksat analysis within subvolumes showed classical REV behaviour but resulted in non‐unique REVs for the soil samples studied Based on the stationarity analysis we argued that the conventional soil core samples studied were not REVs Plateau‐like behaviour in physical properties is a necessary but insufficient condition for REV Soil structure information is necessary to properly establish REV or upscale flow properties

Thermal Stress Analysis of Environmental Barrier Coatings Considering Interfacial Roughness

A numerical analysis of the effect of roughness interface on the thermal stress in the environmental barrier coatings for ceramic matrix composites was performed. Based on the concept of representative volume elements, a micromechanical finite element model of the coated composites was established. The rough interfaces between the coating layers were described using sine curves. The cooling process after preparation and the typical service conditions for the CMCs component were simulated, respectively. The results show that the rough interface has little effect on the temperature distribution along the depth direction for the studied T/EBC coatings for SiC/SiC composites. The stress concentration occurs at the rough EBC/BC interface, which is prone to cause delamination cracking. Under typical service conditions, the high temperature can eliminate part of the thermal residual stress. Meanwhile, the thermal gradient will cause large thermal stress in the TBC layer and the stress will result in surface cracks. The stress concentrations appear at the peaks and valleys of rough interfaces. The variation range of thermal stress increases with the roughness amplitude and decreases with the wavelength.

Micromechanical modeling of bamboo short fiber reinforced polypropylene composites

In recent years, there has been a growing interest in injection moulded Fiber Reinforced Polymer (FRP) composites, particularly with the natural fiber reinforcement due to their renewability and eco friendliness. In many of the applications, stiffness and strength are of prime importance in designing of injection moulded engineering parts. Determination of elastic modulus will give the designer flexibility to design the part geometry depending on stiffness requirement for particular application. This paper discusses micromechanical modeling of bamboo short fiber reinforced polypropylene composites. Bamboo polypropylene composites (BPC) were prepared by reinforcing polypropylene homopolymer matrix with short bamboo fibers. Composites with five different reinforcement percentages (10, 20, 30, 40, and 50 wt%) were prepared and tensile test samples were injection moulded. Tensile elastic modulus and strengths of bamboo composites were measured according to ASTM standard. Various micromechanical theoretical models were studied to find best possible model that correlates well with the experimental results. Elastic Modulus predicted by “Halpin-Tsai” model was in close agreement with the experimentally measured values. Strength predicted by “Kelly-Tyson” model also matched very well with the experimental values. Micromechanic-based semi analytical model “Mori–Tanaka” in Digimat-MF (mean-field homogenization) and numerical finite-element modeling (FEM) of BPC using the concept of representative volume element (RVE) have also been studied and results are compared with experimentally measured values.

Simulating spatial heterogeneity through a CT-FE mapping scheme discloses boundary effects on emerging compaction bands

Deformation experiments on porous rocks conducted at sufficiently high confining pressure may result into localized compaction. While strain localization is expected to be triggered by material heterogeneities, evidence of compaction zones propagating from the specimen boundaries is often observed, thus indicating that sample-platen friction plays a relevant role on the compaction banding patterns. This paper aims to examine this hypothesis and quantify the relative role played by field heterogeneities and boundary effects by using a numerical replica of specimens for which compaction localization was tracked through X-ray micro-tomography. For this purpose, finite element simulations are performed, where spatially-distributed heterogeneities of the porosity field are replicated through a computerized-tomography to finite-element (CT-FE) mapping scheme. The concept of representative elementary volume is used to extract material state variables directly from the CT scans, eventually associating each integration point to a corresponding volume in the digital image. Full-field numerical analyses conducted on the virtual replicas show that frictionless specimens promote the onset of compaction localization in zones characterized by higher porosity (i.e., lower strength). By contrast, the presence of platen-specimen friction nearly eliminates the impact of material heterogeneities, such that compaction localization is forced to occur at the specimen boundaries. Furthermore, the analyses reveals the existence of an intermediate range of platen-specimen friction coefficient which favors the emergence of a transitional regime of strain localization at which material heterogeneity and boundary effects concurrently control the compaction banding patterns. Numerical simulations based on quasi-synthetic spatial fields have also revealed a marked dependence of the three zones on the degree of heterogeneity, showing that increasing values of boundary friction are necessary to offset the effect of material weaknesses in strongly heterogeneous samples.

Verification of Continuum Mechanics Predictions with Experimental Mechanics.

The general goal of the study is to connect theoretical predictions of continuum mechanics with actual experimental observations that support these predictions. The representative volume element (RVE) bridges the theoretical concept of continuum with the actual discontinuous structure of matter. This paper presents an experimental verification of the RVE concept. Foundations of continuum kinematics as well as mathematical functions relating displacement vectorial fields to the recording of these fields by a light sensor in the form of gray-level scalar fields are reviewed. The Eulerian derivative field tensors are related to the deformation of the continuum: the Euler–Almansi tensor is extracted, and its properties are discussed. The compatibility between the Euler–Almansi tensor and the Cauchy stress tensor is analyzed. In order to verify the concept of the RVE, a multiscale analysis of an Al–SiC composite material is carried out. Furthermore, it is proven that the Euler–Almansi strain tensor and the Cauchy stress tensor are conjugate in the Hill–Mandel sense by solving an identification problem of the constitutive model of urethane rubber.

Upscaling of Elastic Properties in Carbonates: A Modeling Approach Based on a Multiscale Geophysical Data Set

AbstractLinking ultrasonic measurements made on samples, with sonic logs and seismic subsurface data, is a key challenge for the understanding of carbonate reservoirs. To deal with this problem, we investigate the elastic properties of dry lacustrine carbonates. At one study site, we perform a seismic refraction survey (100 Hz), as well as “sonic” (54 kHz) and ultrasonic (250 kHz) measurements directly on outcrop and ultrasonic measurements on samples (500 kHz). By comparing the median of each data set, we show that the P wave velocity decreases from laboratory to seismic scale. Nevertheless, the median of the sonic measurements acquired on outcrop surfaces seems to fit with the seismic data, meaning that sonic acquisition may be representative of seismic scale. To explain the variations due to upscaling, we relate the concept of representative elementary volume with the wavelength of each scale of study. Indeed, with upscaling, the wavelength varies from millimetric to pluri‐metric. This change of scale allows us to conclude that the behavior of P wave velocity is due to different geological features (matrix porosity, cracks, and fractures) related to the different wavelengths used. Based on effective medium theory, we quantify the pore aspect ratio at sample scale and the crack/fracture density at outcrop and seismic scales using a multiscale representative elementary volume concept. Results show that the matrix porosity that controls the ultrasonic P wave velocities is progressively lost with upscaling, implying that crack and fracture porosity impacts sonic and seismic P wave velocities, a result of paramount importance for seismic interpretation based on deterministic approaches.

Analytical Pore-Network Approach (APNA): A novel method for rapid prediction of capillary pressure-saturation relationship in porous media

The relationship between capillary pressure and wetting fluid saturation is one of the most important functions required for the modelling of immiscible displacement within porous media. Commonly, time consuming laboratory experiments or computationally expensive pore-scale numerical simulations are performed to estimate capillary pressure− saturation relationships for a given porous media. In this research, we introduce Analytical Pore-Network Approach (APNA): a new method to forecast capillary pressure-saturation relationships in porous media. APNA is a fully coupled analytical model, that is derived using the concept of Representative Elementary Volume (REV), and underpinned by geometrical analysis of the studied pore network using pore-scale images collected by a range of imaging modalities (e.g. thin section photomicrograph, x-ray micro computed tomography, confocal microscopy, etc.). In comparison to conventional laboratory measurements and numerical simulation techniques, APNA is trivial to implement, and is capable of providing rapid estimates of capillary pressure− saturation relationships for a broad range of porous materials. We validated APNA against empirical capillary pressure− saturation data published in the literature, revealing satisfactory agreement between APNA and the experimental studies.

Hexagrid-Voronoi transition in structural patterns for tall buildings

In this paper, a first insight into the role that non-conventional structural patterns might play in the design of tall buildings is presented. The idea is to explore the mechanical properties of selected non-conventional structural patterns, in the form of both regular (Hexagrid) and irregular (Voronoi tessellation inspired) arrays, in order to assess their actual applicability in tall building design. For this aim, the concept of Representative Volume Element (RVE) and a classical homogenization-based micromechanical approach are employed for identifying the pattern units and deriving the relevant generalized stress-strain relationships. In the case of irregular patterns based on Voronoi diagrams, obtained by perturbing prescribed key geometrical features of hexagrids, a statistically significant sample of RVEs is defined on the basis of sensitivity analyses, and the related mechanical characterization is developed in statistical terms. Finally, a preliminary stiffness-based design procedure is proposed and applied to a tall building model with Voronoi exoskeleton. In conclusion, a discussion on the effectiveness of the design procedure and on the structural efficiency of the Voronoi patterns for tall buildings is presented