Mean-field Homogenization Method Research Articles

Beyond effective stiffness: A modified differential Mori-Tanaka-Voigt homogenization for predicting stresses in individual inclusions

Mean field homogenization (MFH) methods are widely employed for homogenizing heterogeneous materials. However, they are limited to predicting effective properties and phase-averaged stresses, failing to capture stresses within individual inclusions. This paper introduces a novel homogenization approach, termed MDMT-Voigt, aimed at addressing this lacuna. The proposed model is validated extensively using finite element analysis (FEA), encompassing virtual Representative Volume Elements (RVEs) with a range of aspect ratios, volume fractions, and orientation distributions. Furthermore, validation is conducted using RVEs derived from experimentally determined microstructures via micro-computed tomography. Across all models considered, the FEA results yield a range of stresses for inclusions with same orientation and aspect ratio which is captured well by the proposed MDMT-Voigt model. Prediction of stresses in individual inclusions represents a significant advancement over conventional MFH methods, offering substantial potential for enhanced micromechanics modelling comparable to full finite element approaches, but at a computational efficiency order of magnitude lower. The paper ends with a demonstration confirming improved micromechanics using the Modified Coulomb criteria.

Heavy Metal Rejection Performance and Mechanical Performance of Cellulose-Nanofibril-Reinforced Cellulose Acetate Membranes.

In this research, cellulose acetate (CA) and CA nanocomposite membranes, reinforced with mass fractions of cellulose nanofibrils (CNF), are prepared using the phase separation technique. The membranes are extensively characterized using several techniques: Fourier Transform Infrared (FTIR) spectroscopy confirms the chemical structures, while Scanning Electron Microscopy (SEM) reveals their surface morphology. Mechanical characterization is conducted to explore the mechanical behavior of the membranes under wet and dry conditions through tensile testing. The mechanical properties of CA and CA-CNF membranes are also estimated using the Mori-Tanaka mean-field homogenization method and compared to experimental findings. The flux performance for pure and dam water, assessed at 3 bar, demonstrates that CNF reinforcement notably enhances the CA membrane's performance, particularly in flux rate and fouling resistance. The CA membrane shows high efficiency in removing Fe2+, Ba2+, and Al3+ from dam water, while CA-CNF membranes exhibit a varied range of removal efficiencies for the same ions, with the 0.5 wt % CNF variant showing superior resistance to surface fouling. Additionally, while CNF increases tensile strength and stiffness, it leads to earlier failure under smaller deformations, especially at higher concentrations. This research provides a detailed assessment of CA and CA-CNF membranes, examining their chemical, structural, and mechanical properties alongside their effectiveness in water treatment applications.

Accounting for spatial distribution in mean-field homogenization of particulate composites

Several analytical mean-field homogenization methods, which take into account the particle volume fraction, shape and orientation are readily available to estimate the effective properties of particulate composites. Models have also been proposed to account for the spatial distribution of the particles. The classical Ponte-Castañeda and Willis (PCW) model is based on a parametrization of the statistical distribution law, while the Interaction Direct Derivative (IDD) model associates a matrix cell to each inclusion, representative of close interactions. In the literature, the use of the IDD is commonly reduced to the particular case of the classical Mori and Tanaka (MT) scheme or to the aforementioned PCW model. The present study introduces an original approach to calibrate the IDD model, for 2D linear conductivity, based on representative images of the microstructure. The links between the models and the range of validity of the IDD model are discussed. Besides, an “IDD-based” PCW model and a two-step scheme are proposed for situations where the IDD estimate is inconsistent (lack of major symmetry). Finally, an image analysis method using Voronoï diagrams is implemented to define the cells associated to each inclusion and supply the models. The method is validated by comparisons between the obtained IDD and PCW estimates, the Mori–Tanaka (MT) model and benchmark full-field numerical simulations. Accounting for the inclusion distribution is seen to lead to better estimates, both qualitatively (by capturing anisotropic behaviors due to the sole distribution) and quantitatively. Possible extensions to elastic composites are discussed.

Multiscale numerical modeling of large-format additive manufacturing processes using carbon fiber reinforced polymer for digital twin applications

Large Format Additive Manufacturing (LFAM) has gained prominence in the aerospace and automotive industries, where topology optimization has become crucial. LFAM facilitates the layer-by-layer production of sizeable industrial components in carbon fiber (CF) reinforced polymers, however 3D printing at large scales results in warpage generation. Printed components are deformed as residual stresses generated due to thermal gradients between adjacent layers. This paper tackles the problem at two different scales: the micro and macroscale. Initially, the microstructure characterization of the thermoplastic ABS matrix composite material enriched with 20% short CF is used in the development of numerical models to understand the mechanical behavior of the studied material. Numerical modeling is performed simultaneously by means of Mean-Field (MF) homogenization methods and Finite Element Analysis (FEA). Outcomes validated with corrected experimental mechanical testing results show a discrepancy in the elastic modulus of 7.8% with respect to FE multi-layer analysis. Micro-level results are coupled with the a macroscopic approach to reproduce the LFAM process, demonstrating the feasibility of the tool in the development of a Digital Twin (DT).

Correlation between surface-to-volume ratio of the particle shape and elastic properties of the particulate composites

This work is motivated by real-world particle shapes, observed using a scanning electron microscopy. The focus of the presented studies was to understand the influence of the particle shapes on the effective elastic properties of the two-phase composites. For this, particles with polyhedral, undulated and other shapes were numerically modeled using analytical functions. Creation of some shapes, like polyhedral, are known from the literature but Laplace’s spherical harmonics, as well as the Goursat’s surface and some others, were used for the first time to create novel particle shapes. Elastic properties of the composites with different particle shapes were calculated using the finite element analysis. The obtained results show good agreement with mean-field homogenization methods such like Mori-Tanaka and Lielens as well as other numerical results available in the literature. Further, the dependence of the effective Young’s moduli of the composite on the shape and the corresponding surface-to-volume ratio of the particles was studied. It was observed that the effective Young’s moduli increase with the surface-to-volume ratio of the particles in the case where particles are stiffer in comparison to the matrix. It was also remarked that, in the case of particles of similar shapes, the particle surface-to-volume ratio and the effective Young’s moduli differ significantly with the surface curvature and the edge sharpness of the particles.

A Mean-field Homogenization Method-based Design of Experiments for Fiber-reinforced Plastics

Plastic materials are widely used as lighter alternatives to metals in all industries. Chopped fibers are often added as stiffeners during injection molding to increase the strength and durability of the part. The mechanical properties of fiber-reinforced plastics often exhibit a strong heterogeneity and anisotropy, which largely depends on how fibers are oriented during the injection molding process. The mean-field homogenization method is a widely used multiscale approach for modeling fiber-reinforced plastics. Using this approach, we have developed an integrated workflow of sequentially coupled plastic injection-to-structural analyses, taking into account of the fiber orientation results to better characterize the mechanical properties of the composite. In this study, we conducted two design of experiments analyses on a fully parameterized mounting boss model to study the effects of part thickness and gate location on the mass and maximum stress of the part.

MICRO-MECHANICS OF THE MECHANICAL PROPERTIES OF COMPOSITES WITH FRACTAL FIBER LENGTH DISTRIBUTION

In short-fiber composites (SFCs), fiber length distribution (FLD) is complicated and has a considerable impact on the mechanical properties of SFCs. This work proposes a fractal FLD in SFCs on the basis of the fractal theory, and develops a multi-step mean-field homogenization (MSMFH) method to accurately and efficiently predict the mechanical properties of SFCs with fractal FLD. In the developed MSMFH method, SFCs are first decomposed into virtual pseudo-grains (PGs) according to fiber orientation distribution (FOD), followed the further division of the PGs into virtual sub-pseudo-grains (SPGs) according to FLD. The Mori–Tanaka or Double-Inclusion model is adopted to homogenize the mechanical properties of each SPG in the first step, and the Voigt model is implemented to homogenize the mechanical properties of all the SPGs and the PGs, respectively, in the sequential steps. Fiber length and orientation averaging algorithms for the developed MSMFH method are detailed. The developed MSMFH method and the proposed fractal FLD are validated to accurately predict the mechanical properties of SFCs by the means of the comparison with the FE method and the available experimental tests.

Quantifying the evolution of mechanical behaviours of concrete materials resting on realistic microstructural characteristics: A micro-to-macro validated combination analysis method

Understanding the physical and chemical essence of concrete material corrosion and the evolution process of mechanical properties is crucial for the study of durability. However, there is currently no suitable universal method to obtain the results of mechanical property evolution through physical and chemical essence. The purpose of this paper is to bridge durability parameters and mechanical behaviors through the micro-to-macro-scale, resting only on realistic microstructural characteristics. This model took the essential parameters of durability degradation such as phase change, porosity, and corrosion degree after corrosion as input parameters. A structure with a corroded layer wrapping the uncorroded inner core was proposed. This structure is closer to the actual corroded structure to a greater extent. This method quantifies and unravels the deterioration mechanisms of mechanical behaviour in concrete caused by external sulfuric acid and sulphate attack. The remarkable features are that: (1) corrosion products form only in outer C-S-H layers through “ink-bottle” theory; (2) considers interfacial transition zone effects and (3) combines mean-field homogenization method with the finite element analysis. Additionally, the mechanical properties obtained from the multi-scale model have been experimentally verified, demonstrating the robustness and reliability of the model.

Improved micromechanical prediction of short fibre reinforced composites using differential Mori-Tanaka homogenization

Due to the spatial distribution of inclusions in random heterogeneous media, individual inclusions are stressed differently. Mean field homogenization (MFH) methods, a popular homogenization method, cannot account for this variability, instead predicting the same mean value of stress for a particular phase. In this paper, a differential scheme is expanded to calculate the stresses in the individual inclusions, including the scatter and variation of local stresses. The accurate prediction of stress variability of stresses within a particular phase has led to more accurate and realistic modelling of inclusion matrix debonding and inclusion breakage.Extensive benchmarking of the proposed models against finite element (FE) results confirms excellent predictive abilities of scatter at three length scales (effective property, individual inclusions, and matrix-inclusion interface). The potential of modelling stress variability post-onset of damage is also demonstrated.Different realisations of the differential Mori Tanaka (MT) lead to varying predictions of scatter in the individual inclusion stresses. However, the effective properties predictions remain consistent.The prediction of individual inclusion stresses, and their scatter constitutes a significant gap in the literature and has been addressed in this paper. This new scheme could lead to the development of several intricate models of damage in various composites without the need for extensive FE modelling.

Damage constitutive model for concrete under the coupling action of freeze–thaw cycles and load based on homogenization theory

The constitutive properties of concrete in freeze–thaw (F–T) environments are the basis for evaluating whether its mechanical properties are adequate for engineering practice in cold regions. The aim of this paper is to establish a constitutive model considering F–T damage and mesoscopic structural damage during loading to effectively predict the mechanical properties of freeze-thawed concrete. Firstly, concrete is conceptualized as the structural body and the frictional band at the mesoscale, based on which the F–T damage mechanism and the load damage mechanism of concrete are analyzed. Then, the macro–mesoscopic constitutive relationship is established based on the mean-field homogenization method. The constitutive properties of the structural body and the frictional band are defined as elastic and elastoplastic, respectively. The damage evolution equation under the coupling action of F–T cycles and load is developed based on the damage mechanism of concrete. In addition, inhomogeneous deformations between the structural body and the frictional band are considered in the model. Finally, the rationality of the proposed model is demonstrated by the test results of the frozen–thawed concrete. The deterioration characteristics of F–T cycles can be well reproduced.

Non-local integral-type damage combined to mean-field homogenization methods for composites and its parallel implementation

Modern lightweight materials are often decomposed into several distinct materials, e.g. a matrix material and some reinforcements. A manufacturing process for these materials often induces eigenstrains, both reversible and non reversible. This could lead to an initial damage of a composite material, and eventually result into fracture of the structure. This contribution deals with non-local integral-type damage of composites subjected to eigenstrains. It is well known that local damage models lead to mesh sensitive simulation results. To avoid this effect, non-local integral-type regularization of equivalent strains and damage variables are exploited. Effective macroscopic properties are obtained using mean-field homogenization methods. A regularization in each material phase within the framework of an iterative Newton–Raphson procedure leads to the update of a non-local tangent stiffness matrix at each equilibrium iteration. To reduce the computational time parallel non-local assembly and averaging procedures are introduced. In this regard, the performance of different representations of dynamic sparse matrices and the scalability of the parallel implementation are investigated. Various microstructural morphologies are distinguished in representative examples to show the capability of the proposed work. Furthermore, the transition from damage to fracture is performed by aid of an element deletion method. To achieve the best compromise between memory consumption and speed over a range of non-zero entries, an array of flatmaps should be used. It is shown that the introduced parallel algorithms enable a drastic reduction of computational time. Moreover, it is illustrated that for various volume fractions of an inclusion a mesh sensitivity is clearly reduced.

Two-Scale analysis of fretting fatigue crack initiation in heterogeneous materials using Continuum Damage Mechanics

The phenomenon of fretting fatigue damage is related to tribological behaviour and commonly occurs at the contact interface between two bodies that sustain oscillation loads. Based on the complicated mechanical situation, several methods have been developed to predict the crack initiation location and life. The Continuum Damage Mechanics (CDM) approach in conjunction with Finite Element Method (FEM) is used in this paper. The fretting fatigue lifetime includes two parts, crack initiation lifetime and crack propagation lifetime, and this paper focuses on the crack initiation phase. Homogeneous materials are usually studied in fretting fatigue problems, and the influence of defects is necessary to be considered. In this study, the authors proposed several models to consider the effect of heterogeneity on crack initiation location and lifetime. The numerical model is a 2D one with plane strain assumption. To get more accurate numerical results with low computational cost, a Two-Scale Analysis approach is developed in this paper. Two macroscopic models and three microscopic models are proposed, including a Macroscopic homogeneous model (Macro-HOMO), a Multiscale Homogenization model with Direct Numerical Simulation (MH-DNS), a Microscopic homogeneous model (Micro-HOMO), a Microscopic Model with equivalent material properties by using Mean Field Homogenization method (Micro-MFH), and a microscopic model with Direct Numerical Simulation (Micro-DNS). The results obtained by using the CDM approach with different microscopic models are compared, and show that the prediction accuracy of the Micro-DNS is the best. In addition, in terms of computational cost, the Micro-DNS model shows great advantages as well comparing to the Macro-DNS model.

Of spheres and infinite cylinders: A critical relook at multi-step mean-field homogenization formulations

Mean field homogenization (MFH) methods are commonly deployed for homogenization of composite materials. Owing to physical admissibility problems and other concerns, one usually resorts to multi-step formulations of the Mori-Tanaka (MT) method involving the use of MT homogenization in more than one step. Using Representative Volume Elements (RVE) consisting of varying proportion of spheres and infinite cylinders, six variants of the Multi-step MT are implemented and benchmarked against FE and single step MT formulation. The physically admissibility conditions are checked for the different schemes, the average stresses in the inclusion and matrix phases and predicted effective modulus are compared with each other. The choice of the RVE has led to clear identification of errors associated with each of the seven mean field implementations. An important consideration for some of the multi-step MT formulations is the choice of the Poisson's ratio of the intermediate effective medium after the first step of homogenization, consequences of this choice are studied in this paper.Single step MT yields physically inadmissible values of the strain concentration tensor, while all the other multi-step MT formulations are able to circumvent this problem. Significant deviations and errors were observed for both the phase average stresses and the effective modulus with no one scheme giving acceptable results for all the cases considered in this paper. The order of homogenization in the multi-step MT formulation has significant effect on the predictions of the effective modulus as well as the phase average stresses. When the volume fraction of one of the phases tends to zero, the predictions of some multistep MT schemes do not always converge to the single step MT.

Multi-scale study on interfacial bond failure between normal concrete (NC) and ultra-high performance concrete (UHPC)

It is a promising way to strengthen and repair the damaged normal concrete (NC) structure with ultra-high performance concrete (UHPC), and the interfacial performance between NC and UHPC is a critical issue. This paper aims to reveal the debonding failure performance between UHPC (repair material) and NC (substrate material) under tensile loading and shear loading. A multi-scale analysis method is established to simulate the mechanical behavior of the NC-UHPC interface, considering the complex interface geometry on the meso-scale. The failure of NC matrix is described by the elastic-plastic damage model in form of coupled gradient-enhanced damage evolution, the interface is represented by the cohesive zone model (CZM), and the UHPC is characterized by the second-order mean-field homogenization (MFH) method. After extracting the roughness characteristic parameters of the tested interfaces, the representative volume element (RVE) corresponding to the roughness is established, the load transfer and the debonding failure behavior are investigated, and the effective traction-separation law (TSL) is obtained. Finally, the tensile and the shear separation mechanisms of the NC-UHPC interface are revealed, and the effects of the roughness and strength of the interface on the macro shear and tensile strength are evaluated. Both the interface roughness and the substrate concrete strength have positive effects on the mechanical properties of the interface.

Multi-scale and multi-step modeling of thermal conductivities of 3D braided composites

To accurately predict the Effective Thermal Conductivities (ETCs) of Three-Dimensional (3D) braided composites, this work proposes a multi-scale and multi-step Mean-Field Homogenization (MFH) method. The composites are considered as the composites with 3D multi-directional braiding yarns as inclusions at the meso-scale, and the braiding yarns are viewed as unidirectional fibers reinforced composites at the micro-scale. The ETCs of the braiding yarns are firstly determined using the MFH method at the micro-scale. According to the yarn orientation, the composites are virtually decomposed into several pseudo-grains, each of which consists of the oriented braiding yarns and matrix, and the multi-step MFH method is then developed to predict the ETCs of the pseudo-grains and the composites sequentially at the meso-scale, i.e. the MFH prediction of the ETCs of each pseudo-grain individually and the composites consisting of all the pseudo-grains in two sequential steps. The proposed multi-scale and multi-step MFH method is validated by comparing with the multi-scale FE homogenization method and the available experimental test. The modeling results show that the through-thickness and in-plane ETCs of the composites increase with the increase of fiber volume fraction and transverse ETC, and decrease and increase with the increase of interior braiding angle, respectively. The proposed multi-scale and multi-step MFH method possesses the advantages of better computational efficiency and simpler implementation compared with the multi-scale FE homogenization method.

FE-based damage modeling approach for short fiber reinforced thermoplastics under quasi-static load coupling anisotropic viscoplasticity and matrix degradation

It is challenging to predict the material degradation and crack initiation of short fiber reinforced thermoplastics (SFRT) components with high computation efficiency and low parameter identification effort. To achieve that, this work utilizes a hybrid method combining micro- and macro-mechanical approaches to describe the damage-coupled material behavior of SFRT. The Mori-Tanaka mean-field homogenization method is used to determine the effective linear elastic properties of SFRT, whereas the consideration of plasticity is based on a macro-mechanical anisotropic viscoplastic model. The effect of micro-damage in the matrix material on the macroscopic behaviors of SFRT is considered within the Continuum Damage Mechanics (CDM) framework. A nonlinear damage evolution law is implemented to account for the nonlinearity in the damage evolution. Targeting industrial applications, the proposed damage-coupled material model is implemented into the commercial FE software ANSYS with a novel stepwise damage updating process, requiring no user-defined subroutine. The model is calibrated with experimental data on flat specimens under monotonic and cyclic tensile loading. The FE simulation with the calibrated material model accurately describes the anisotropic deformation and crack initiation of the investigated SFRT material observed in experiments.

A new incremental variational micro-mechanical model for porous rocks with a pressure-dependent and compression–tension asymmetric plastic solid matrix

In this paper, we shall formulate a new micro-mechanical model for describing both local and macroscopic elastic–plastic behavior of porous rocks with a pressure-sensitive and tension–compression asymmetric solid matrix. For this purpose, a new incremental variational mean field homogenization (IVMFH) method is first established. The novelty lies in the fact that the new formulation allows considering solid matrix obeying a nonlinear plastic yield criterion rather than the linear Drucker–Prager one adopted in the previous work Zhao et al. (2019). To this end, a new optimization procedure is proposed to estimate the effective incremental potential involving a stress-dependent relation between volumetric and deviatoric parts of local plastic strain field of solid matrix. The proposed incremental variational model takes into account the effect of non-uniform distribution of local stress and strain fields in solid matrix by incorporating their first and second moments in the homogenization process. The proposed model is able to provide not only macroscopic but also local mechanical responses of porous rocks. Its efficiency is assessed by a series of comparisons with full-field finite element calculations. Application examples are also presented through comparisons with experimental data for two typical porous rocks: Vosges sandstone and Lixhe chalk.

Multiscale modelling framework for elasticity of ultra high strength concrete using nano/microscale characterization and finite element representative volume element analysis

Ultra-High Strength Concrete (UHSC) (greater than 100 MPa) is a mechanically superior material compared with the Normal Strength Concrete (NSC) due to its inherent performance characteristics. Improved modulus of elasticity is one of the key target characteristics in the development of UHSC. Macroscopic response of UHSC is a result of a multitude of phases in different spatial length scales such as mesoscale, microscale, nanoscale etc. and investigating these spatial scales can yield a better understanding about contribution of each heterogenous phases to the macroscopic behaviour of UHSC. In this paper, a new multiscale modelling method including procedures to obtain micro/nano scale properties is proposed to predict the macroscopic elastic modulus using nanoindentation experiments, hydration simulations, Scanning Electron Microscopy (SEM) and Finite Element Representative Volume Element (FE-RVE) modelling.Characterization of nanomechanical properties of the cementitious composite was carried out using nanoindentation, microstructural characterization was performed using scanning electron microscopy, and hydration simulation of the cementitious paste was carried out using Virtual Cement and Concrete Testing Laboratory (VCCTL) software. A five-level multiscale framework is proposed for UHSC and results from these experimental testing and simulations were used as inputs in the proposed framework.A novel algorithm which can model any volume fraction of different phases was developed to generate geometries for RVEs to be used in FE-RVE simulations. Upscaling of elastic modulus using FE-RVE was found to be very accurate, and this method can generate detailed variation of microfields inside the RVE. A parametric study was carried out on how varying inhomogeneities in the RVE, boundary conditions, and the shape of the inhomogeneities would affect the homogenized elastic modulus.Continuum micromechanics models such as Mori-Tanaka method and Self Consistent Scheme were used for the analytical homogenization at each scale for comparison with FE-RVE method. The results of the proposed FE-RVE analysis, the Mean Field Homogenization (MFH) method, and experiment were compared and found to be a very good fit.

Adaptive affine homogenization method for Visco-hyperelastic composites with imperfect interface

Despite intense research on homogenization methods, it is a still challenging task to predict the nonlinear mechanical responses of visco-hyperelastic particle-reinforced composites. In this work, we propose an adaptive affine method, a novel mean-field homogenization method designed to ensure the consistency of the accumulated strain state and the concentration tensor, and apply the method to predict the mechanical responses of the composites in the large strain regime under uniaxial, cyclic, and biaxial loading conditions. Our method is also extended to predict the mechanical response in the presence of interfacial imperfections described by linear cohesive traction-separation laws. The analytic predictions are validated against finite element analyses of representative volume elements. We believe that our adaptive affine method can be extended to model various nonlinear responses of load-bearing composites including the effects of (visco)plasticity and finite deformation.

Transfer learning for enhancing the homogenization-theory-based prediction of elasto-plastic response of particle/short fiber-reinforced composites

Mean-field homogenization methods relying on the solution of Eshelby’s inclusion problem do not provide accurate predictions when the interaction among reinforcements becomes significant, or in the inelastic response regime when the shape of the reinforcement has a high aspect ratio. Herein, we propose a combined theoretical and data-driven approach in which a homogenization method is followed by transfer learning to enhance the prediction of the nonlinear mechanical response of particle/short fiber-reinforced composites. We trained a deep neural network (DNN) with a massive stress-strain curve dataset of the composites subjected to uniaxial and cyclic loading in the elasto-plastic regime based on the adaptive incrementally affine homogenization method; then, we fine-tuned the pre-trained DNN via transfer learning with a relatively small dataset based on time-consuming three-dimensional (3D) finite element analyses (FEA). The transfer learning approach exhibited better predictive performance than the DNN directly trained only with the FEA dataset did, for a wide range of reinforcement volume fractions and shapes. With transfer learning, the DNN exploits the knowledge learned from the homogenization theory to perform new learning tasks on small datasets from 3D FEA. The proposed approach can be extended to other composite analyses requiring excessive time and cost for large datasets.