Mean-field Homogenization Model Research Articles

An improved mean-field homogenization model for the three-dimensional elastic properties of masonry

Accurate assessment of the overall mechanical behavior of masonry, composed of bricks and mortar joints, remains challenging due to its inhomogeneous and orthotropic nature. In this study, the feasibility of various mean-field homogenization schemes for the three-dimensional orthotropic elastic properties of masonry is comprehensively investigated. Three kinds of masonry patterns are considered, including the stack bonded pattern, the running bonded pattern and the double-leaf Flemish bonded pattern that has received limited attention so far. Special attention is paid to the homogenization schemes which have not been applied to the masonry case, such as Lielens’ interpolative double inclusions (D-I) and the interaction direct derivative (IDD) schemes. After a comparison between the well-known mean-field homogenization schemes, an improved micro-mechanical model is proposed by combining the advantages of the IDD and D-I models. The validation of the proposed model is conducted through a comparison against experimental data from literature and numerical results obtained via finite element analyses (FEA). The results show that the proposed model can accurately evaluate the orthotropic elastic properties of the three masonry typologies for a wide range of stiffness ratios between brick and mortar, ranging from 1 to 1000. The proposed model also shows better performance than the classical schemes especially when the stiffness ratios between brick and mortar are higher than 10, which is of major importance for the application of mean-field homogenization based multiscale methods to the nonlinear analysis of masonry. Furthermore, the presented homogenization method can be of interest for other anisotropic materials, e.g., laminate materials.

Viscoelastic-viscoplastic polymer composites: Development and evaluation of two very dissimilar mean-field homogenization models

This paper deals with the micromechanical modeling of polymer composites with viscoelastic-viscoplastic (VE-VP) constituents. Two mean-field homogenization (MFH) models based on completely dissimilar theoretical approaches are extended from elasto-viscoplasticity (EVP) to VE-VP and assessed. The first approach is the incremental-secant method. It relies on a fictitious unloading of the composite at the beginning of each time step. Then, a thermoelastic-like Linear Comparison Composite (LCC) is constructed from the computed residual state directly in the time domain. The method provides naturally isotropic per-phase incremental-secant operators for isotropic VE-VP constituents. It takes into account both the first and the second statistical moment estimates of the equivalent stress micro-field. The second approach is the integral affine method. It starts by linearizing the rates of viscoplastic strain and internal variables. The linearized constitutive equations are then recast in a hereditary integral format to which the Laplace-Carson (L-C) transform is applied. A thermoelastic-like LCC is built in the L-C domain, where MFH is carried out. Finally, the composite’s response in the time domain is recovered by numerical inversions of L-C transforms. The method is able to overcome the issue of heterogeneous viscous stresses encountered by time domain MFH models. The two proposed MFH formulations are able to handle non-monotonic, non-proportional and multi-axial loading histories. Their accuracy was assessed against full-field finite element (FE) results for different microstructures and loadings. The computational cost of both methods is negligible compared to FE analyses. Overall, the incremental-secant approach is much simpler mathematically and numerically than the integral affine formulation, its accuracy ranges from acceptable to excellent, and important improvements can be expected in the future by controlling the virtual unloading time increment.

Investigation of thermal conductivity for liquid metal composites using the micromechanics-based mean-field homogenization theory.

For the facile use of liquid metal composites (LMCs) for soft, stretchable and thermal systems, it is crucial to understand and predict the thermal conductivity of the composites as a function of liquid metal (LM) volume fraction and applied strain. In this study, we investigated the effective thermal conductivity of LMCs based on various mean-field homogenization frameworks including Eshelby, Mori-Tanaka, differential and double inclusion methods. The double inclusion model turned out to make the prediction closest to the experimental results in a wide range of LM volume fractions. Interestingly, we found that the theoretical models based on the assumption of ideal LM dispersion and zero interfacial resistance underestimated the thermal conductivity compared to the experimental results in a low volume fraction regime. By considering the accompanied variations in the LM inclusion's aspect ratios under a typical size distribution of inclusions (∼μm), the change of effective thermal conductivity was predicted under a uniaxial 300% tensile strain. Our study will deepen the understanding of the thermal properties of LMCs and support the designs of stretchable thermal interfaces and packaging with LMCs in the future.

Fracture characterization of tailored Usibor® 1500-AS and damage modelling based on a coupled-micromechanical-phenomenological strategy

Usibor® 1500-AS steel is commonly used in the automotive industry in the die-quenched form due to its very high level of strength after hot stamping. In order to increase the ductility of this steel, in-die-heated, tailored hot stamping can be utilized to promote the formation of softer phases instead of the hard martensitic phase normally formed during die quenching. In this study, the fracture behavior of the different microstructures of this steel, including fully martensitic, 60% martensitic plus 40% bainitic, and fully bainitic, were characterized through several mechanical testing. A coupled-micromechanical-phenomenological approach was developed to predict the fracture of Usibor® 1500-AS with mixed martensitic-bainitic microstructures under different loading conditions. In this strategy, the flow behavior of the steel as well as stress and strain partitioning between phases were predicted using mean-field homogenization (MFH) schemes, given the stress-strain curves of the constituent phases. At the same time, damage accumulation in the constituents was calculated using the phenomenological Generalized Incremental Stress-State-dependent damage MOdel (GISSMO) coupled with the fracture loci of the corresponding constituent phases based on the strain- and stress-state history within each phase. Finally, the onset of material fracture was predicted to correspond to the damage parameter for any phase reaching unity. In this framework, three MFH schemes, corresponding to the interpolative Mori-Tanaka, self-consistent, and Samadian-Butcher-Worswick-1 (SBW1) approaches, were evaluated using the first-order secant-based linearization approach. To investigate the accuracy of the numerical results, the predicted flow curves and fracture strains of a two-phase microstructure were compared with the corresponding measured data. The comparisons showed that the predictions based on all of the MFH models were in good accord with the measured data. However, the fracture locus predicted based on the SBW1 scheme, especially in the case of the plane-strain-tension stress state, provided the best agreement with the interpolated Bai-Wierzbicki fracture locus obtained using the experimental data.

New mean-field homogenization schemes for the constitutive modelling of the elastic and elastoplastic deformation behavior of multi-phase materials

In this study, a new class of mean-field homogenization (MFH) models were developed based on the underlying concepts of the direct and inverse Mori-Tanaka and Lielens MFH schemes. The proposed models were employed to predict the elastic and elastoplastic constitutive responses for a range of representative multi-phase materials, including polymer- and metal-matrix composites and a dual-phase steel. To evaluate the performance of the developed MFH models in the elastic region, their predictions for the elastic constants of graphite-reinforced epoxy, Al2O3-reinforced AA7075, and TiB2-reinforced AISI 1045 steel with different ellipsoidal-inclusion shapes and volume fractions were compared with the results of unit-cell finite element method (FEM) simulations under uniaxial-tension loading. For the plastic region, the proposed MFH models were coupled with various linearization approaches, including the first-order secant-based and tangent-based schemes, to estimate the flow behavior of DP590 steel in uniaxial tension. The MFH predictions were shown to compare favorably with the results of unit-cell FEM simulations and experiment. Moreover, the predictive ability of the developed models for both elastic and plastic responses was assessed via comparisons with the approximations of the self-consistent and Mori-Tanaka MFH models. The results showed that the proposed MFH models can describe the elastic and elastoplastic behavior of heterogeneous materials with different constituent volume fractions as well as or better than conventional MFH models.

Bayesian inference of non-linear multiscale model parameters accelerated by a Deep Neural Network

We develop a Bayesian Inference (BI) of the parameters of a non-linear multiscale model and of its material constitutive laws using experimental composite coupon tests as observation data. In particular we consider non-aligned Short Fibers Reinforced Polymer (SFRP) as a composite material system and Mean-Field Homogenization (MFH) as a multiscale model. Although MFH is computationally efficient, when considering non-aligned inclusions, the evaluation cost of a non-linear response for a given set of model and material parameters remains too prohibitive to be coupled with the sampling process required by the BI. Therefore, a Neural-Network (NNW) is first trained using the MFH model, and is then used as a surrogate model during the BI process, making the identification process affordable.

Bayesian identification of Mean-Field Homogenization model parameters and uncertain matrix behavior in non-aligned short fiber composites

We present a stochastic approach combining Bayesian Inference (BI) with homogenization theories in order to identify, on the one hand, the parameters inherent to the model assumptions and, on the other hand, the composite material constituents behaviors, including their variability. In particular, we characterize the model parameters of a Mean-Field Homogenization (MFH) model and the elastic matrix behavior, including the inherent dispersion in its Young’s modulus, of non-aligned Short Fibers Reinforced Polymer (SFRP) composites. The inference is achieved by considering as observations experimental tests conducted at the SFRP composite coupons level. The inferred model and material law parameters can in turn be used in Mean-Field Homogenization (MFH)-based multi-scale simulations and can predict the confidence range of the composite material responses.

A new interpolative homogenization model for evaluation of the effective elasto-plastic responses of two-phase composites

To accurately predict the effective elasto-plastic responses of two-phase composites, an interpolative M-T/D-I mean-field homogenization model with the tangent formulation is proposed, and the numerical implementation algorithm of the proposed M-T/D-I model with the tangent formulation is presented. The comparisons between the effective elastic properties and elasto-plastic responses of two-phase composites predicted by the M-T/D-I model with the tangent formulation and the RVE based FE homogenization method show that the M-T/D-I model with the tangent formulations can accurately evaluate the effective mechanical properties of two-phase composites. The simulation results demonstrate that the elastic properties of two-phase composites predicted by the M-T/D-I model are more accurate than those predicted by the D-I and M-T models, and the effective elasto-plastic responses of two-phase composites predicted by the M-T/D-I model vary in the range of those predicted by the D-I and M-T models. It is revealed that the M-T/D-I model with the tangent formulation allows to simulate the tensile/compression and biaxial tensile/shear elasto-plastic responses of two-phase composites. Lastly, the effects of the inclusion volume fraction and elastic properties on the effective elasto-plastic responses of two-phase composites are discussed for showing the capability of the proposed interpolative M-T/D-I model with the tangent formulation.

Experimental and multi-scale numerical evaluations for effective mechanical properties of 2-D Cf/Mg composites

2-D Cf/Mg composites consisting of AZ91D magnesium alloy and carbon non-woven fabrics are prepared by using the LSEVI process and their mechanical properties are experimentally tested. The experimental results show that the 2-D Cf/Mg composites have the better mechanical properties: elastic modulus of 71.0 GPa and ultimate tensile strength of 366.0 MPa, compared to AZ91D magnesium alloy matrix with the elastic modulus of 45.0 GPa and ultimate tensile strength of 250.0 MPa. The effective elastic properties of the 2-D Cf/Mg composites are numerically predicted by using a multi-scale homogenization method: the RVE based FE homogenization method at the macro-scale coupling the D-I mean-field homogenization model at the micro-scale. The agreement between the results of the experimental tests and the multi-scale homogenization method illustrates the validation of the multi-scale homogenization method to evaluate the effective mechanical properties of the 2-D Cf/Mg composites. For improving the computational efficiency, at the macro-scale the laminated composites theory is coupled with the RVE based FE homogenization method to predict the effective mechanical properties of the 2-D Cf/Mg composites.

Micro-mechanical failure prediction of radar-absorbing structure dispersed with multi-walled carbon nanotubes considering multi-scale modeling

Conventional radar-absorbing structure is typically manufactured with high weight percentage (wt.%) of carbonaceous nano-conductive particles in the polymer matrix to tailor its microwave absorbing performance. However, these manufacturing methods have some physical limitations with regard to fabrication, due to the high viscosity in the polymer matrix and, inhomogeneous in mechanical and electrical properties. No study has been conducted with micro-mechanical failure prediction of radar-absorbing structure dispersed with multi-walled carbon nanotubes. In order to address these limitations, radar-absorbing structures dispersed with multi-walled carbon nanotubes were designed in the Ku-band (12.4–18 GHz). Additionally, to establish and verify the micro-mechanical failure analysis based on multiscale modeling, finite element analysis was carried out using the Mori–Tanaks mean-field homogenization model within the representative volume element model in the microstructure. In order to verify the Hashin criteria of radar-absorbing structure dispersed with multi-walled carbon nanotube (0.5 wt.%, 1.0 wt.% and 1.5 wt.%), mechanical tests (tensile, compressive and shear test) were conducted according to ASTM standards. In this paper, radar-absorbing structure with irregularly arranged filler and matrix with representative volume element was modeled from the micro-mechanical point of view and the results from Hashin failure criterion were verified both by simulations and experimental results of prediction strengths within the expected error range (lower than 6%). The reliability of application in micro-mechanical prediction of radar-absorbing structure was confirmed considering the multi-scale modeling.

Mean-field homogenization based approach to evaluate macroscopic coefficients of thermal expansion of composite materials

In this paper, a mean-field homogenization based approach implemented in the two steps sequentially is proposed to evaluate the macroscopic coefficients of thermal expansion (CTEs) of multi-phase composites. The adopted mean-field homogenization models are the quadratic interpolative M-T/D-I mean-field homogenization model in the first-step homogenization procedure and the simple interpolative Voigt/Reuss mean-field homogenization model in the second-step homogenization procedure, respectively. The proposed mean-field homogenization based approach is validated by comparing with the RVE based FE homogenization method. The simulation results indicate that with the increase of the inclusion volume fraction, the macroscopic CTEs of multi-phase composites generally decrease because that the inclusions possess the smaller CTEs than the matrices. However, in the case of multi-phase composites with the planarly distributed inclusions, the through-thickness CTEs firstly increase with the increase of the inclusion volume fraction and then decrease with the increase of the inclusion volume fraction. The effect of the inclusion aspect ratio on the macroscopic CTEs of multi-phase composites can be neglected, while the effect of the inclusion transversely isotropic material properties on the macroscopic CTEs of multi-phase composites must be considered.

Finite strain mean-field homogenization of composite materials with hyperelastic-plastic constituents

A finite strain mean-field homogenization (MFH) formulation is proposed for a class of composites where multiple phases of solid inclusions or cavities are embedded in a continuum matrix. Local constitutive equations of each solid phase are based on a multiplicative decomposition of the deformation gradient onto elastic and inelastic parts and hyperelastic-plastic stress-strain relations. For the special situation of hyperelastic constituents, a mixed variational formulation is presented which handles both compressible and quasi-incompressible cases within the same framework. A special emphasis is put on the proper definition of various macroscopic stress measures and tangent operators. For an extended Mori-Tanaka MFH model, numerical algorithms were developed and implemented. The MFH predictions were extensively tested against direct finite element simulations of representative volume elements or unit cells, for several heterogeneous microstructures under various loadings.

Evaluation for elastic properties of metal matrix composites with randomly distributed fibers: Two-step mean-field homogenization procedure versus FE homogenization method

The modified two-step mean-field homogenization procedure including the quadratic interpolative model on the basis of the Mori-Tanaka (M-T) and interpolative double inclusion (D-I) mean-field homogenization models in the first-step homogenization procedure and the simple interpolative model in the second-step homogenization procedure on the basis of the Voigt and Reuss mean-field homogenization models, and the direct finite element (FE) homogenization method based on the concept of the representative volume element (RVE) and the periodic boundary conditions (the RVE based FE homogenization method), are implemented to predict the effective elastic properties of metal matrix composites with the randomly distributed fibers. Compared with the results measured from the uniaxial tensile experiments, the modified two-step mean-field homogenization procedure and the RVE based FE homogenization method provide the accurate predictions on the effective elastic properties of metal matrix composites with the randomly distributed fibers. However, in the case of neglecting the detailed stress and strain fields in the metal matrix composites with the randomly distributed fibers, the modified two-step mean-field homogenization procedure gives the far better computational efficiency than the RVE based FE homogenization method.

A study of composite laminates failure using an anisotropic gradient-enhanced damage mean-field homogenization model

The failure of carbon fiber reinforced epoxy laminates is studied using an anisotropic gradient-enhanced continuum damage model embedded in a mean-field homogenization scheme.In each ply, a homogenized material law is used to capture the intra-laminar failure. The anisotropy of the homogenized material model results from the homogenization method and from the reformulation of the non-local continuum damage theory to account for the material anisotropy. As a result the damage propagation direction in each ply is predicted with accuracy as compared to the experimental results, while the problems of losing uniqueness and strain localization, which occur in classical finite element simulations when strain softening of materials is involved, can be avoided.To model the delamination process, the hybrid discontinuous Galerkin/extrinsic cohesive law method is introduced at the ply interfaces. This hybrid method avoids the need to propagate topological changes in the mesh with the propagation of the delamination while it preserves the consistency and stability in the un-cracked interfaces.As a demonstration, open-hole coupons with different stacking sequences are studied numerically and experimentally. Both the intra- and inter-laminar failure patterns are shown to be well captured by the computational framework.

Prediction of the elastic properties of single walled carbon nanotube reinforced polymers: A comparative study of several micromechanical models

This paper deals with the prediction of the elastic properties of polymer composites reinforced with single walled carbon nanotubes. Our main contribution is the investigation of several micromechanical models, while most of the papers on the subject deal with only one approach. We implemented four homogenization schemes, a sequential one and three others based on various extensions of the Mori–Tanaka (M–T) mean-field homogenization model: two-level (M–T/M–T), two-step (M–T/M–T) and two-step (M–T/Voigt). Several composite systems are studied, with various properties of the matrix and the graphene, short or long nanotubes, fully aligned or randomly oriented in 3D or 2D. Validation targets are experimental data or finite element results, either based on a 2D periodic unit cell or a 3D representative volume element. The comparative study showed that there are cases where all micromechanical models give adequate predictions, while for some composite materials and some properties, certain models fail in a rather spectacular fashion. It was found that the two-level (M–T/M–T) homogenization model gives the best predictions in most cases.