Analytical Homogenization Models Research Articles

Numerical estimation via remeshing and analytical modeling of nonlinear elastic composites comprising a large volume fraction of randomly distributed spherical particles or voids

We investigate numerically via finite element (FE) simulations and analytically the nonlinear behavior of a soft matrix comprising a large volume fraction (up to 55%) of monodisperse spherical inclusions, which can be either rigid particles or voids. To address the issue of severe mesh distortion at large strains, we employ a successive remeshing and solution mapping technique that allows us to achieve a large macroscopic deformation, with the maximum attained stretch being at least two to three times when compared to that without remeshing. A general algorithm allowing to read complex arbitrary particle geometries is proposed and implemented allowing to remesh a finitely strained domain with randomly distributed inclusions of arbitrary shape. The present simulation results for a neo-Hookean elastic matrix filled with rigid particles show that the nonlinear stress–stretch uniaxial tension response can be well approximated by a neo-Hookean material with an effective shear modulus that depends on the volume fraction of the inclusions. Additionally, a study of the number of particles demonstrates that sixty four monodisperse spheres is a good compromise to reach accurate results for large deformations and maintain a reasonable CPU-time. The FE data serve as a powerful assessment tool for analytical homogenization models. Newly developed and existing models for small and large strains are proposed and assessed, respectively, for both rigid particles and voids. These results fill the gap for large volume fractions of inclusions and serve as a reference for the homogenization of the microstructures of interest.

Influence of interface transition zone on effective elastic property of heterogeneous materials with an artificial neural network study

AbstractIn this study, the influence of interface transition zone (ITZ) on the elastic properties of concrete and rock like heterogeneous materials is investigated. Direct simulations of a representative volume element are realized by using a fast Fourier transform based method and the obtained results are used as the reference solutions. Some widely used analytical homogenization models are evaluated by comparing the theoretical predictions with the reference solutions. Based on this evaluation, an artificial neural network (ANN) model is constructed in order to improve the analytical models. The proposed ANN model is trained, tested and validated against the reference solution. Its efficiency and good accuracy are clearly demonstrated.

Study of effective elastic properties of heterogeneous materials with an artificial neural network model

This paper is devoted to the estimation of macroscopic elastic properties of composites containing inclusions and pores. Some representative analytical homogenization models are first recalled and their weaknesses are highlighted by comparisons with the reference numerical results obtained from direct simulations by a Fast Fourier Transform (FFT) method. A large data set is then constructed also based on numerical results obtained from FFT simulations for different types of micro-structures and for a large range of elastic properties of constituents. An artificial neural network (ANN)-based model containing two hidden layers is constructed and trained by using this data set. Different types of validation tests of the ANN model are presented. It is found that the ANN-based model can estimate the macroscopic elastic properties of strongly heterogeneous materials with a very good accuracy, for different types of micro-structures and a large range of porosity and inclusion volume fraction.

In situ micromechanical analysis of discontinuous fiber-reinforced composite material based on DVC strain and fiber orientation fields

The influence of fiber orientation and its distribution on the global and local mechanical behavior of a short fiber-reinforced thermoplastic (SFT) composite is studied based on an in situ tensile test using synchrotron X-ray tomography. The in situ test data are utilized to compute three-dimensional (3D) strain fields inside the material at various loading steps through an in-house digital volume correlation (DVC) code. The DVC analysis finds that, although the global strain fields seem relatively uniform, the internal strain fields are locally non-uniform because of the complex reinforcement architecture. The local non-uniformity is further investigated at a single fiber level to find a correlation between the orientation of fibers, fiber volume fraction, and strain field. The investigation is performed on micro-scale subvolumes, in which all the fibers are individually segmented from the 3D tomographic data. It is found that the fibers aligned in the direction transverse to loading impede transversal shrinkage, resulting in a locally small compressive strain in that direction. The in situ test has revealed that the average fiber orientation does not change noticeably even after the global strain reaches 18%. This small change implies that initially-determined fiber orientation is crucial to the locally anisotropic mechanical behavior of a SFT-like material even undergoing large deformation. On the other hand, the loading-directional strain is more affected by the local volume fraction than the fiber orientation. The local volume fraction also mainly determines the stiffness of the subvolume that can be estimated from an analytical homogenization model. The effect of the local volume fraction on the subvolume stiffness is identified from multivariate analysis.

Experimental Study of the Mechanical Behaviour of Brick Waste Concrete and Analytical Prediction of its Elastic Modulus as a Three-Phase Composite Material

This paper focuses on the characterization of the mechanical behaviour of concrete incorporating different percentages of brick waste aggregates (BWA). Compressive strength, splitting tensile strength and elastic modulus of this material were measured based on standard laboratory tests and its microstructure was characterized based on scanning electron microscope (SEM) observations. A decrease in these properties was observed with the increase of BWA substitution ratio. However, this decrease remains moderate up to a substitution percentage of 30% (about 12% for compressive strength and elastic modulus and 8% for splitting-tensile strength). In addition, an increase in the concrete porosity was observed with the increase of BWA substitution ratio, which can explain the decrease observed in the measured mechanical characteristics. SEM views on concrete incorporating 100% of BWA showed that the interfacial transition zone (ITZ) and the cement paste present a higher porosity when compared to those of the reference concrete made with natural aggregates.Finally, a micromechanical analytical homogenization model predicting the elastic modulus of brick waste concrete (BWC) according to its composition is proposed where BWC is modelled as a three-phase composite. A good agreement was found between analytical predictions and experimental results proving that BWC mechanical characteristics are mainly governed by BWA mechanical properties and their volume fraction within concrete.

Assessing the permittivity of an unsaturated sand by combining a Lattice Boltzmann Method simulation, electromagnetic homogenization models and measurements

Dielectric permittivity is a prevailing property used for the estimation of water content in heterogeneous materials like soils and concrete. Troubles are frequently encountered when searching for the relationship between a material's permittivity and its degree of saturation. Building an electromagnetic model of the material is thus of critical importance. This paper focuses on the study of an unsaturated sand. Analytical and numerical electromagnetic homogenization procedures are confronted. Dielectric permittivity of the sand is measured at different saturation degrees within the frequency range [200 MHz; 1 GHz] thanks to a large open-ended coaxial probe. A numerical procedure is then built to calculate the effective permittivity of the medium. First, a representative elementary volume of the dry sand is created. Pore domain is then separated into water and gas phases through the use of a Lattice Boltzmann Method algorithm. The effective permittivity is then estimated using numerical homogenization. Results obtained at different water contents were eventually compared with analytical homogenization models like the Bruggeman and Maxwell-Garnett equations. A close match is obtained between measurements, numerically derived results and a Bruggeman model based on a spheroidal description of the air and water phases.

Microstructurally-guided explicit continuum models for isotropic magnetorheological elastomers with iron particles

This work provides a family of explicit phenomenological models both in the F−H and F−B variable space. These models are derived directly from an analytical implicit homogenization model for isotropic magnetorheological elastomers (MREs), which, in turn, is assessed via full-field numerical simulations. The proposed phenomenological models are constructed so that they recover the same purely mechanical, initial and saturation magnetization and initial magnetostriction response of the analytical homogenization model for all sets of material parameters, such as the particle volume fraction and the material properties of the constituents (e.g., the matrix shear modulus, the magnetic susceptibility and magnetization saturation of the particles). The functional form of the proposed phenomenological models is based on simple energy functions with small number of calibration parameters thus allowing for the description of magnetoelastic solids more generally such as anisotropic (with particle-chains) ones, polymers comprising ferrofluid particles or particle clusters. This, in turn, makes them suitable to probe a large set of experimental or numerical results. The models of the present study show that in isotropic MREs, the entire magnetization response is insensitive to the shear modulus of the matrix material even when the latter ranges between 0.003-0.3MPa, while the magnetostriction response is extremely sensitive to the mechanical properties of the matrix material.

Piezocomposite transducer design and performance for high resolution ultrasound imaging transducers

Piezocomposite design for dedicated ultrasonic imaging applications requires precise homogenization models for predicting the electromechanical characteristics of the new material. Thus, several homogenization models have been developed. As part of this work, we applied several analytical homogenization models for piezocomposite of 2–2 and 1–3 connectivities. To validate these analytical models, a comparative study was made between various models and experimental measurements. As a result, these homogenized electromechanical properties are effectively used for the calculation and comparison of electroacoustic response for typical transducers aimed at ultrasound imaging applications. An optimal design of transducer aimed at ultrasound imaging applications is proposed as a dedicated imaging performance index, elaborated through a trade-off between sensitivity and bandwidth.

Hybrid Effect on Hybrid Composite Reinforced Carbon Fibers and Flax Fibers

In this paper, hybrid composite made of carbon woven fibers and flax woven fibers is studied. This hybrid composite structure takes advantages of high resistance, high stiffness of carbon fibers and high damping and low density of flax fibers. Different structures of flax woven composites, carbon woven composites and hybrid composites were fabricated and tested experimentally. With aim of predicting the properties of the hybrid composite, a homogenization model of the composite is established. The homogenization model is based on the rule-of-mixture and iso-strain assumption. The results of the analytical homogenization model (AHM) are then compared with the results of experimental tests. The results show a good agreement between the AHM and the experimental results at the homogenization level of the woven composite. However, at the hybrid composite homogenization level, the experimental results present considerably higher stiffness than analytical results that is explained by hybrid effect on the hybrid composite.

Improved analytical homogenization of the piezoelectric macro-fiber composite: active layer embedded among passive layers

Predicting the effective electromechanical properties for piezoelectric fiber composite patches is crucial for analysis of smart structures equipped with such patches. In particular, analytical homogenization models are attractive but are limited by the complexity of the microstructure and the multiphysics involved. The macro-fiber composite (MFC) has emerged as the most popular piezoelectric fiber composite. But typical reports of its effective properties provide layerwise homogenization with low-order mechanical plate theories. This approach discards most of the out-of-plane properties and may introduce significant error from the assumptions about the plate kinematics. Moreover, the effective permittivities of the transducer are not available analytically. In an attempt to overcome these limitations, a new analytical scheme for homogenizing all layers in the MFC patch is proposed. Based on the newly discovered mechanics of structure genome, we avoid all kinematical assumptions and obtain an exact analytical solution for a periodic, multi-layered, linear-piezoelectric material. In doing so, we prove that in any such multi-layered composite, the in-plane strains and the transverse stresses are equal in each layer and the in-plane electric fields and transverse electric displacement are constant between the electrodes. Using this knowledge, a hybrid rule of mixtures is developed to homogenize all the MFC layers to obtain the complete set of properties of the full MFC. In doing the new analytical framework can homogenize any layered piezoelectric medium. Finally, since we were able to avoid the plate-kinematics assumptions, we present studies on their effect on the properties they obtain. We find that the in-plane elastic properties are not affected but the transverse shear moduli are overpredicted from 100% to 300%. What is more, studies are presented on how using the ubiquitous plane-stress-like assumptions to homogenize the fiber layers causes an 8% underprediction of the lateral elastic modulus of the MFC.

Modeling quasi-3D needle-punched C/C composites using a linear simplification representative volume element model

In this paper, a linear simplification model is proposed for predicting the flexibility of quasi-3D needle-punched C/C composites. The proposed model is based on the representative volume element, which we determined via three steps consisting of two modules: (1) a unidirectional fiber composite layer, which was composed of a homogenous matrix with random porosity throughout the material and randomly distributed short-cut fiber felt; (2) a needle-punched fiber composite layer, which possessed a random porous nature and a uniform distribution of needle-punched fibers in the matrix. We then utilized the analytical homogenization model, the Halpin–Tsai formula, and the Gox formula to predict the elasticity of the composites. The predicted elasticity properties were compared to experimental data reported in the literature to evaluate the reliability and precision of the proposed linear simplification model. The predictions were in good agreement with the experimental data. Furthermore, a series of parametric numerical calculations were performed to examine the effects of the micro-factor and integral thickness factors on the elastic properties of the composites.

Multiscale modeling of skeletal muscle tissues based on analytical and numerical homogenization

A novel multiscale modeling framework for skeletal muscles based on analytical and numerical homogenization methods is presented to study the mechanical muscle response at finite strains under three-dimensional loading conditions. First an analytical microstructure-based constitutive model is developed and numerically implemented in a general purpose finite element program. The analytical model takes into account explicitly the volume fractions, the material properties, and the spatial distribution of muscle's constituents by using homogenization techniques to bridge the different length scales of the muscle structure. Next, a numerical homogenization model is developed using periodic eroded Voronoi tessellation to virtually represent skeletal muscle microstructures. The eroded Voronoi unit cells are then resolved by finite element simulations and are used to assess the analytical homogenization model. The material parameters of the analytical model are identified successfully by use of available experimental data. The analytical model is found to be in very good agreement with the numerical model for the full range of loadings, and a wide range of different volume fractions and heterogeneity contrasts between muscle's constituents. A qualitative application of the model on fusiform and pennate muscle structures shows its efficiency to examine the effect of muscle fiber concentration variations in an organ-scale model simulation.

Evaluation of Tension, Bending and Twisting Rigidities of Single-Layer Graphene Sheets by an Analytical Asymptotic Homogenization Model

In the present study, the method of asymptotic homogenization is used to estimate elastic properties as well as tension, bending and twisting rigidities of single-layer graphene sheets (SLGSs). To this end, asymptotic homogenization of a reinforced composite is developed for modeling of SLGS by assuming that the covalent bond between carbon atoms can be represented by reinforcements. Applicable formulas are obtained for the elastic properties and rigidities of SLGS directly from the interatomic interactions through three types of potentials. It is proved that force field constants significantly affect the elastic properties of SLGS. Herein, the elastic moduli are obtained based on different types of atomic force constants. The results of the present analytical model are in close agreements with the similar theoretical results and experimental measurements. This approach can be developed to represent mechanical properties of nanotubes, nanocomposites and other nanostructures. DOI: http://dx.doi.org/10.5755/j01.mech.24.2.17822

Microstructure-guided numerical simulations to predict the thermal performance of a hierarchical cement-based composite material

This paper presents a microstructure-guided numerical homogenization technique to predict the effective thermal conductivity of a hierarchical cement-based material containing phase change material (PCM)-impregnated lightweight aggregates (LWA). Porous inclusions such as LWAs embedded in a cementitious matrix are filled with multiple fluid phases including PCM to obtain desirable thermal properties for building and infrastructure applications. Simulations are carried out on realistic three-dimensional microstructures generated using pore structure information. An inverse analysis procedure is used to extract the intrinsic thermal properties of those microstructural components for which data is not available. The homogenized heat flux is predicted for an imposed temperature gradient from which the effective composite thermal conductivity is computed. The simulated effective composite thermal conductivities are found to correlate very well with experimental measurements for a family of LWA-PCM composites considered in the paper. Comparisons with commonly used analytical homogenization models show that the microstructure-guided simulation approach provides superior results for composites exhibiting large property contrast between phases. By linking the microstructure and thermal properties of hierarchical materials, an efficient framework is available for optimizing the material design to improve thermal efficiency of a wide variety of heterogeneous materials.

Modeling of honeycombs with laminated composite cell walls

Honeycombs are versatile structures. They have been widely employed in industries where the characteristics of high stiffness, high buckling resistance, large shock absorption and light weight are required. To explore the potential of honeycombs in various mechanical applications, this paper proposes a novel honeycomb with composite laminate cell walls in order to provide wider selection of constituent materials, improved specific stiffness and distinct cell wall surfaces. Analytical homogenization model of this special type of honeycombs is established by modeling the locally heterogeneous honeycomb as a homogeneous orthotropic bulk. Both full-detailed and homogenized models are built and tested using finite element analysis, and the results showed that the analytical model has excellent accuracy in property prediction at a relatively small computational cost. Parametric studies are also conducted to investigate the effect of thickness and elastic moduli of the cell wall plies on the structure’s overall mechanical response. Based on the results, suggestions on property optimizations are discussed.

A Review of Analytical Micromechanics Models on Composite Elastoplastic Behaviour

Elasto-plastic behavior of composite structures is a key problem in light-weight application, which has been investigated vastly for decades. Most approaches for composites elasto-plastic simulation can be split into three categories: 1) numerical methods also known as computational micromechanics methods, 2) variational approaches for upper and lower bounds, and 3) analytical homogenization models. In this study only focuses on the analytical models. Two steps are essential to establish an analytical model, i.e. choosing a suitable linear elastic homogenization models and a rational linearization theory so that linear elastic models can be extended to nonlinear elasto-plastic regime. Besides, considering the situation that isotropic materials may become anisotropic due to plastic deformation, three approaches are found in the literature to deal with the anisotropic properties and corresponding Eshelby's tensor. In summary, a comparative study is made among nine different models to evaluate their capabilities on elasto-plastic behavior simulation. Experiment data of a series of unidirectional composites under different loading cases are used for the purpose of comparison.

An analytic homogenization model in traction and bending for orthotropic composite plates with the type of double corrugated cardboard

In this paper, an analytic homogenization model in traction and flexion for the double corrugated cardboard plates is presented. The proposed analytical homogenization model allows modelling the 3D double corrugated cardboard with a 2D homogenized plate. This model is essentially based on the theory of stratification and then improved by using the theory of sandwich. This model was validated by comparing the results of Abaqus-3D and H-2D model using "user's subroutine" "UGENS". The homogenization model can be used not only for corrugated cardboard plates, but also for naval and aeronautic composite structures

Effect of temperature on the failure modes of a triaxially braided polymer matrix composite

The temperature effect on the tensile behavior of a triaxially braided composite was investigated in two different loading directions. The damage evolution was monitored and the results were explained with the help of experimental, analytical and numerical methods. The evolution in the constituent elastic properties was responsible for stress redistribution within the composite explaining the differences in crack initiation and Ultimate Tensile Strength for both directions and testing temperatures. The study also revealed the potential of quick and efficient analytical homogenization models to explain failure paths in textile composites.

Analytical homogenization for in-plane shear, torsion and transverse shear of honeycomb core with skin and thickness effects

The homogenization of the honeycomb sandwich core enables to obtain an equivalent homogeneous solid and its elastic moduli thus to make numerical modeling very efficient. In the present paper, the in-plane shear, torsion, in plane-shear coupled with torsion and transverse shear are studied with skin effect: the honeycomb core’s deformation is constrained by two fairly rigid skins, so the core’s stiffness is largely increased. An analytic homogenization method, using trigonometric function series is proposed to study the influence of the honeycomb thickness on the elastic properties, and the upper and lower bounds of their equivalent elastic moduli can be given directly by the analytical homogenization models (H-models). A very good agreement has been achieved between numerical results issued from the H-models and 3D FE modeling.

Evaluation of homogenized effective properties for corrugated composite panels

Corrugated panels are widely used as structural elements in engineering fields because of their high stiffness and light weight. Moreover, corrugated composite laminates have attracted much attention as a candidate for morphing aircraft wing that need to have different behaviors in different directions due to their extremely anisotropic behavior. The simple model is required to investigate the mechanical behavior of these structures in design process. On this demand, the equivalent homogenization method of corrugated panels has been used during last decades. This homogenization model is usually treated a corrugated panel as an orthotropic panel that has different material properties in two perpendicular directions. This paper presents an analytical homogenization model for corrugated composite laminates that can be applied easily to any corrugation geometry. This paper gives explicit expressions to calculate not only the effective extensional and bending stiffness but also the effective transverse shear stiffness for a composite corrugated panel. The effective stiffness of the trapezoidal corrugation shape which is the most common corrugation geometry is evaluated and compared with those of the flat composites. To validate the proposed approach, the obtained results are compared with the previously published results.