Large Representative Volume Elements Research Articles

On effective thermal conductivity of heterogeneous material: From periodic unit inclusion cell to large representative volume element

The custom design of a composite material requires the knowledge of its effective behavior which depends on several parameters such as the properties of the constituents, their distributions, shapes and sizes. In thermal conductivity, all these parameters have a direct and significant influence on the effective thermal conductivity (ETC) of composites. These parameters act differently on the behavior of the composite according to its use as much as a good conductor by improving its ETC or by reducing it by considering it as an insulator. The main goal of this work is an extensive study of all possible situations of ETC by a numerical homogenization technique based on the Finite Elements Method. For this purpose, the two main categories of composites, namely the periodic composites represented by the unit cell and the random RVE, are considered. For each category of composite, different contrasts, different particles’ volume fractions, aspect ratios, positions, and orientations are studied. In order to ensure the simulation results are effective and representative, two boundary conditions, namely Uniform Temperature Gradient Boundary Conditions (UGT) and Periodic Thermal Boundary Conditions (PBC), were imposed on all considered microstructures. The results of this study constitute a synthesis of the effective thermal conductivities of all the possible situations which can be summarized as follows: On the one hand, the reinforcements shape effect is very important irrespective of the nature of the periodic and random composites and on the other hand, the two types of composites do not have the same behavior as that for the reinforcements with a circular or almost circular shape. This study also emphasizes the situations where the ETC can be estimated analytically and when homogenization is necessary. From a practical point of view, and on the basis of the obtained results, this study made it possible to show clearly in which situation the composite becomes more conducting and in the opposite case when it becomes more insulating.

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.

Investigating the deformation behavior of hot-pressed Ti/Al/Ti laminated composite

In this paper, the deformation behavior of Ti/Al/Ti composite fabricated by the hot-pressing process was investigated by simulation and experimental analyses. Furthermore, the elastic properties of laminated composites were predicted by using different representative volume elements (RVEs) subjected to compression and tension. The SEM images displayed well-bonded straight interfaces. Also, XRD and EDS analyses revealed that TiAl3 was the only intermetallic compound that grew at the Ti/Al interfaces. The simulation results showed that stress concentration and strain localization existed at the interfaces during hot compression. In addition, the elastic behavior of simulated RVEs agreed relatively well with the experimentally-obtained values. Nevertheless, the predicted values of effective elastic constants of large RVE were consistent with the experimental values. This is due to the actual thickness, volume fraction, and stacking order of composite layers.

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.

Dynamic response of Advanced Placed Ply composites

This work investigates the high strain rate behavior of AP-PLY composites. The large representative volume elements and brittle nature of this material necessitated the use of a bespoke Split-Hopkinson bar apparatus. AP-PLY and baseline laminates were subjected to tensile loading at strain rates of 30 s−1. Results were compared with quasi-static data to evaluate whether the laminate architecture introduced any strain rate dependency. In addition, the dynamic experiments were simulated using a multiscale modeling framework, providing further insights into the micromechanisms governing material behavior. The moduli of the AP-PLY composites were found to be strain rate independent, however, strengths were found to be marginally higher than those of their baseline counterparts. At high strain rates, the strain concentrations induced by the geometry of the individual tapes at through thickness undulations and tow boundaries were less significant due to reduced out-of-plane tow straightening and delamination. As a result, no reduction in AP-PLY strength in comparison to the baseline laminates was obtained. These differences in deformation micromechanisms led to an improvement of the damage tolerance when subjected to dynamic loading.

Deep learning aided evaluation for electromechanical properties of complexly structured polymer nanocomposites

In the present work, two types of deep neural networks (DNNs) were employed to establish the structure–property relationship of polymer nanocomposites. The trained DNNs based on multiscale analysis results can not only overcome the limitations of the conventional clustering density-based model and multivariate regression models but also exhibit superior performance in evaluating the electromechanical properties of polypropylene matrix composites, wherein spherical SiC nanoparticles were randomly distributed and dispersed. A simple graph convolution network showed better capability than a complex artificial neural network, despite fewer features considered; this implies that the graph convolution network is more appropriate and user-friendly for evaluating the effect of nanoparticle distribution and agglomeration. In addition, the trained graph convolution network can effectively provide mechanical and electrical properties corresponding to large representative volume element (RVE) without a loss of accuracy. The present study demonstrates that deep learning techniques can be put to practical use for the design of next-generation polymer nanocomposite materials.

Network science approach to predict the yield strength of metals containing soft particles and pores

Local fluctuations in particle size and spatial distributions can substantially affect the yield strength of metals containing soft particles and/or porosities. The phenomenon can be studied via computational homogenization techniques, which, however, can be highly computationally demanding when large representative volume elements are needed. A more efficient approach to model the plastic deformation that controls the yield strength of metals containing pores has recently been proposed by the author and co-workers. The key idea is to treat the material as a network rather than as a continuum. In fact, plastic deformation in porous metals occurs via the formation of shear bands connecting the pores, and it can thus be modelled as a time-evolving network where the nodes represent the pores and the links represent the shear bands. So far, the applicability of this new approach has only been tested against 2D synthetic microstructures. The present work takes it a step further by applying the approach to real spheroidal graphite iron. Based on strain data collected in-situ via digital volume correlation, it is shown that the new network approach can explain the spatial variations of the plastic deformation that arise from local variations of the particle distribution throughout a tensile specimen.

From unit inclusion cell to large Representative Volume Element: Comparison of effective elastic properties

The concept of equivalent morphology has received much consideration in recent decades. The importance of this concept is reflected in the fact that an inclusion of any morphology can be replaced by a circular one in simulation. If this concept is confirmed, it will facilitate the modeling and simulation of complex configuration microstructures.To decide on this concept, an in–depth study is carried out in this work, trying to answer it in a clear and definitive way by trying to identify all the possible situations. For this the two types of composites, namely, periodic interpreted by an elementary cell and random interpreted by a Representative Volume Element (RVE) with 200 inclusions are considered. To be sure that the isotropy is provided by the RVE of the periodic microstructure, two types of elementary cells were treated: one circular and the other square. In order to cover all possible situations, the inclusion of the elementary cell is considered with several situations, centered position with different orientations at constant and random steps, random position with orientation at constant steps and random position and orientation at random steps. For each situation, the effective property is determined by the average of 20 cases are processed for elementary cells, while for the large RVE, the properties are obtained by a unit realization. To take into account the effect of contrast, two situations are considered, namely, rigid inclusion case and rigid matrix case. Several results are obtained and given in the conclusion.

An FFT-based approach for Bloch wave analysis: application to polycrystals

A method based on the Fast Fourier Transform is proposed to obtain the dispersion relation of acoustic waves in heterogeneous periodic media with arbitrary microstructures. The microstructure is explicitly considered using a voxelized Representative Volume Element (RVE). The dispersion diagram is obtained solving an eigenvalue problem for Bloch waves in Fourier space. To this aim, two linear operators representing stiffness and mass are defined through the use of differential operators in Fourier space. The smallest eigenvalues are obtained using the implicitly restarted Lanczos and the subspace iteration methods, and the required inverse of the stiffness operator is done using the conjugate gradient with a preconditioner. The method is used to study the propagation of acoustic waves in elastic polycrystals, showing the strong effect of crystal anistropy and polycrystaline texture on the propagation. It is shown that the method combines the simplicity of classical Fourier series analysis with the versatility of Finite Elements to account for complex geometries proving an efficient and general approach which allows the use of large RVEs in 3D.

Variational upscaling for modeling state of strain-dependent behavior and stress-induced crystallization in rubber-like materials

The purpose of this paper is to present a general upscaling strategy for deriving macroscopic constitutive laws for rubber-like materials from the knowledge of the network distribution and a mechanical description of the individual chains and of their free energy. The microscopic configuration is described by the position of the cross-links and is not obtained by an affine assumption but by minimizing the corresponding free energy on stochastic large representative volume elements with adequate boundary conditions. This general framework is then approximated by using a microsphere (directional) description of the network. It is presented in a global setting and is extended in order to handle situations with tube-like constraints and stress-induced crystallization.

Multi-scale reliability-based design optimisation framework for fibre-reinforced composite laminates

PurposeThe purpose of this study is to enable performing reliability-based design optimisation (RBDO) for a composite component while accounting for several multi-scale uncertainties using a large representative volume element (LRVE). This is achieved using an efficient finite element analysis (FEA)-based multi-scale reliability framework and sequential optimisation strategy.Design/methodology/approachAn efficient FEA-based multi-scale reliability framework used in this study is extended and combined with a proposed sequential optimisation strategy to produce an efficient, flexible and accurate RBDO framework for fibre-reinforced composite laminate components. The proposed RBDO strategy is demonstrated by finding the optimum design solution for a composite component under the effect of multi-scale uncertainties while meeting a specific stiffness reliability requirement. Performing this using the double-loop approach is computationally expensive because of the number of uncertainties and function evaluations required to assess the reliability. Thus, a sequential optimisation concept is proposed, which starts by finding a deterministic optimum solution, then assesses the reliability and shifts the constraint limit to a safer region. This is repeated until the desired level of reliability is reached. This is followed by a final probabilistic optimisation to reduce the mass further and meet the desired level of stiffness reliability. In addition, the proposed framework uses several surrogate models to replace expensive FE function evaluations during optimisation and reliability analysis. The numerical example is also used to investigate the effect of using different sizes of LRVEs, compared with a single RVE. In future work, other problem-dependent surrogates such as Kriging will be used to allow predicting lower probability of failures with high accuracy.FindingsThe integration of the developed multi-scale reliability framework with the sequential RBDO optimisation strategy is proven computationally feasible, and it is shown that the use of LRVEs leads to less conservative designs compared with the use of single RVE, i.e. up to 3.5% weight reduction in the case of the 1 × 1 RVE optimised component. This is because the LRVE provides a representation of the spatial variability of uncertainties in a composite material while capturing a wider range of uncertainties at each iteration.Originality/valueFibre-reinforced composite laminate components designed using reliability and optimisation have been investigated before. Still, they have not previously been combined in a comprehensive multi-scale RBDO. Therefore, this study combines the probabilistic framework with an optimisation strategy to perform multi-scale RBDO and demonstrates its feasibility and efficiency for an fibre reinforced polymer component design.

Analytical and Numerical Investigation of Hardening Behavior of Porous Media

In this study, a comparative analysis is presented between a new proposed analytical model and numerical results for macroscopic behavior of porous media with isotropic hardening in its matrix. The macroscopic behavior of a sufficiently large representative volume element (RVE), with 200 identical spherical voids, was simulated numerically using finite element method and compared with elementary volume element that contains one void. The matrix of the porous material is considered as elasto-plastic with isotropic hardening obeys exponential law for isotropic hardening. A new parameter was added with exponential law for isotropic hardening to represent the new proposed analytical model for macroscopic isotropic porous hardening. The new added parameter B depended only on the porosity. The results of the new proposed analytical model were compared with numerical results for different types of cyclic loading. Very good agreements were found between the numerical results and the proposed analytical model.

Multiscale analysis of solid, waffle, ribbed and hollowcore reinforced concrete slabs

We present a multiscale plate element for the analysis of solid, waffle and hollowcore reinforced concrete slabs based on the higher order computational continua (HC2) formulation. The salient features of the proposed formulation are: (i) the ability to consider large representative volume elements (RVE) characteristic to waffle and hollowcore slabs, (ii) versatility stemming from the ease of handling damage and prestressing, and (iii) computational efficiency resulting from model reduction, combined with the well-established damage law rescaling method that yields simulation results nearly mesh-size independent. The multiscale formulation has been validated against experimental data for solid, hollowcore and waffle reinforced concrete slabs with and without prestressing.

Computational analysis of particle reinforced viscoelastic polymer nanocomposites – statistical study of representative volume element

The concept of representative volume element (RVE) is widely used to determine the effective material properties of random heterogeneous materials. In the present work, the RVE is investigated for the viscoelastic response of particle-reinforced polymer nanocomposites in the frequency domain. The smallest RVE size and the minimum number of realizations at a given volume size for both structural and mechanical properties are determined for a given precision using the concept of margin of error. It is concluded that using the mean of many realizations of a small RVE instead of a single large RVE can retain the desired precision of a result with much lower computational cost (up to three orders of magnitude reduced computation time) for the property of interest. Both the smallest RVE size and the minimum number of realizations for a microstructure with higher volume fraction (VF) are larger compared to those of one with lower VF at the same desired precision. Similarly, a clustered structure is shown to require a larger minimum RVE size as well as a larger number of realizations at a given volume size compared to the well-dispersed microstructures.

Micromechanics of brain white matter tissue: A fiber-reinforced hyperelastic model using embedded element technique

A transverse-plane hyperelastic micromechanical model of brain white matter tissue was developed using the embedded element technique (EET). The model consisted of a histology-informed probabilistic distribution of axonal fibers embedded within an extracellular matrix, both described using the generalized Ogden hyperelastic material model. A correcting method, based on the strain energy density function, was formulated to resolve the stiffness redundancy problem of the EET in large deformation regime. The model was then used to predict the homogenized tissue behavior and the associated localized responses of the axonal fibers under quasi-static, transverse, large deformations. Results indicated that with a sufficiently large representative volume element (RVE) and fine mesh, the statistically randomized microstructure implemented in the RVE exhibits directional independency in transverse plane, and the model predictions for the overall and local tissue responses, characterized by the normalized strain energy density and Cauchy and von Mises stresses, are independent from the modeling parameters. Comparison of the responses of the probabilistic model with that of a simple uniform RVE revealed that only the first one is capable of representing the localized behavior of the tissue constituents. The validity test of the model predictions for the corona radiata against experimental data from the literature indicated a very close agreement. In comparison with the conventional direct meshing method, the model provided almost the same results after correcting the stiffness redundancy, however, with much less computational cost and facilitated geometrical modeling, meshing, and boundary conditions imposing. It was concluded that the EET can be used effectively for detailed probabilistic micromechanical modeling of the white matter in order to provide more accurate predictions for the axonal responses, which are of great importance when simulating the brain trauma or tumor growth.

Towards practical multiscale approach for analysis of reinforced concrete structures

We present a novel multiscale approach for analysis of reinforced concrete structural elements that overcomes two major hurdles in utilization of multiscale technologies in practice: (1) coupling between material and structural scales due to consideration of large representative volume elements (RVE), and (2) computational complexity of solving complex nonlinear multiscale problems. The former is accomplished using a variant of computational continua framework that accounts for sizeable reinforced concrete RVEs by adjusting the location of quadrature points. The latter is accomplished by means of reduced order homogenization customized for structural elements. The proposed multiscale approach has been verified against direct numerical simulations and validated against experimental results.

Thermal shock resistance of magnesia spinel refractories—Investigation with the concept of configurational forces

In magnesia-spinel refractories, spinel inclusions are embedded in a magnesia matrix in order to increase the thermal shock resistance by micro-crack toughening. The paper investigates damage evolution in the material during the cooling at the end of the production process by a sophisticated numerical analysis. After a stress-strain analysis on a large representative volume element, the development of the damage zone between spinel grains is simulated with the concrete damaged plasticity model of ABAQUS. Application of the concept of configurational forces allows the physically correct determination of the driving force for cracks in the elastic quasi-plastic matrix material. The results demonstrate that inherent matrix cracks are unable to grow during the cooling, but they strongly influence the initiation and development of the damage zone. The pronounced damage initiation, in combination with the low crack driving force, is the main reason for the good thermal shock resistance of magnesia spinel refractories.

Automatic three-dimensional geometry and mesh generation of periodic representative volume elements for matrix-inclusion composites

This paper introduces an efficient method to automatically generate and mesh a periodic three-dimensional microstructure for matrix-inclusion composites. Such models are of major importance in the field of computational micromechanics for homogenization purposes utilizing unit cell models. The main focus of this contribution is on the creation of cubic representative volume elements (RVEs) featuring a periodic geometry and a periodic mesh topology suitable for the application of periodic boundary conditions in the framework of finite element simulations. Our method systematically combines various meshing tools in an extremely efficient and robust algorithm. The RVE generation itself follows a straightforward random sequential absorption approach resulting in a randomized periodic microstructure. Special emphasis is placed on the discretization procedure to maintain a high quality mesh with as few elements as possible, thus, manageable for computer simulations applicable to high volume concentrations, high number of inclusions and complex inclusion geometries. Examples elucidate the ability of the proposed approach to efficiently generate large RVEs with a high number of anisotropic inclusions incorporating extreme aspect ratios but still maintaining a high quality mesh and a low number of elements.

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

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

Hyperelastic and Elastic-Plastic Approaches for Modelling Uniaxial Tensile Performance of Woven Fabrics

This article presents the findings of experimental and finite element simulation warp direction uniaxial tensile testing of plain 1/1, 2/2 twill and 8 ends satin woven fabrics with respect to a woven fabric model developed in IGES using UniverFilter. Woven fabrics have been specifically configured as a balanced weave thereby allowing systematic investigation of the effect of uniaxial tensile stress on the weave. Static automatic incrementation of large representative volume elements has enabled characterisation of the response of two-dimensional woven fabrics under uniaxial tensile stress with respect to hyperelastic and elastic-plastic material properties. Plain 1/1 and 8 ends satin woven fabrics were well-described by the hyperelastic model and the elastic-plastic model predicted extended strain percentages. The modelling indicates that satin woven fabric possesses the lowest strain distribution and compression stress in the unloaded weft direction compared to plain and twill woven fabrics.