Mean-field Homogenization Research Articles

A general framework for mean-field homogenization of multi-layered linear elastic composites subjected to thermal and curing induced strains

The aim of this work is to derive a general three-scale framework for the homogenization of polymer composites accounting for the degree of cure, besides thermal and mechanical contributions. For this purpose an additive decomposition of the small strain tensor into an elastic, a thermal and a chemical part of the small strain tensor is considered at all three scales. The macroscale of the three-scale framework represents the homogeneous material. The mesoscale is decomposed into a polymer-based material and an inclusion, which is coated or even multi-layered. On this scale, the degree of curing varies between the different constituents and is therefore inhomogeneous. The microscale of the three scale framework represents a step-growth polymerization mechanism. On this scale the degree of cure is homogeneous. Effective thermal, chemical and elastic properties are derived in a general way based on the transformation field analysis and the mean-field homogenization framework. In a comparative study the capability of the proposed framework is shown. In detail, effective properties for a system of multiple growing spheres and for multiple growing ellipsoids are determined versus the degree of cure. Macroscopic effective properties for a fibre reinforced composite are derived versus the degree of cure and versus the matrix volume fraction. Additionally, results for a coated-fibre reinforced composite versus the degree of cure are displayed.

Modeling of geometric configuration and fiber interactions in short fiber reinforced composites via new modified Eshelby tensors and enhanced mean-field homogenization

Determination of the effective elastic properties of unidirectional short fiber reinforced composites (SFRCs) is a key fundamental requirement which is usually estimated by mean-field homogenization (MFH) methods. Available classical MFH methods only account for volume fraction and aspect ratio of the fibers and do not consider the physical cylindrical geometry of the fibers. Moreover, classical MFHs are not designed for predicting the effective properties of SFRCs with distinct packing configurations and order of fibers. In this work, new modified Eshelby tensors associated with an enhanced MFH are developed which resolve the mentioned limitations of the available MFH methods and preserve their computational efficiency. While classical Eshelby tensors used in MFHs depend only on the aspect ratio of the fictitious ellipsoidal inclusion and material properties of the matrix, the introduced modified Eshelby tensors additionally account for physical cylindrical geometry as well as interactions of the short fibers with different packing configurations. Moreover, non-uniform homogenizing eigenstrains required for accurate homogenization of the short fibers are also incorporated by the proposed analytical treatment. Predictions of the developed enhanced MFH for various packing configurations are compared with those obtained from Finite Element Method implementing periodic boundary conditions and very good agreements are observed. It is shown that packing configuration and geometrical specifications of short fibers can significantly affect the effective properties of the SFRC.

Per-phase spatial correlated damage models of UD fibre reinforced composites using mean-field homogenisation; applications to notched laminate failure and yarn failure of plain woven composites

A micro-mechanical model for fibre bundle failure is formulated following a phase-field approach and is embedded in a semi-analytical homogenisation scheme. In particular mesh-independence and consistency of energy release rate for fibre bundles embedded in a matrix phase are ensured for fibre dominated failure. Besides, the matrix cracking and fibre-matrix interface debonding are modelled through the evolution of the matrix damage variable framed in an implicit non-local form. Considering the material parameters of both fibre and epoxy matrix phases identified from manufacturer data sheets, it is shown that the failure strength of a ply loaded along the longitudinal direction is in agreement with the reported values. Finally, the multi-damage homogenisation framework is applied to model, on the one hand, the failure of a notched laminate, in which case the failure modes are observed to be in good agreement with experiments, and, on the other hand, the failure of yarns in a plain woven composite unit-cell under uni-axial tension.

Phase mean strain and stress estimates in possibly multi-continuous composites under affine elastic-like behavior

N-phase materials (perfect phase bonding, no cracks) can have one single, several or none continuously through-sample spanning (i.e. topologically continuous or connected) phases. The former phase assemblage type is known as the inclusion-matrix structures (all but one phase are discontinuous hence embedded) and the latter one is a special case of aggregates. The type when several p phases - possibly up to all - among n are arranged such as to be simultaneously (co-) continuous, to be called “multi-continuous” composites, gathers all the other assemblage possibilities, of greatest number for n≥3. Earlier works from the author and co-workers proposed specific linear “elastic-like” (i.e. including piezo-type coupling) effective property estimates for any n≥p≥2, from making use of laminate system (LS) schemes, within the Hashin-Shtrikman (HS) mean field homogenization framework. The characteristics of these LS estimates likely to represent p-phase co-continuity were identified in comparing with the related ones for inclusion-matrix structures and for random aggregates. The objective is here to examine the phase (mean) strain and stress estimates that result from the localization counterparts of the LS schemes, in broadening the framework to affine elastic-like behavior, i.e. linear elastic-like with phase eigen-strains. One makes use of the previous result firstly recalled that each phase assemblage type, regardless of complexity varieties, can be mainly characterized by a specific (n+1)th reference medium layer with null concentration. The localization characteristics are then examined from this theoretical viewpoint for n-phase composites, specializing for the two-phase LS(A,B) estimates which we compare with the HS(A,B) ones for the “A in B″ and “B in A″ phase arrangements. Specific LS(2) localization features are pointed.

A fractal model of effective mechanical properties of porous composites

In this work, a simple but effective fractal model is proposed to predict the mechanical properties of complex porous composites. By extending the Mori-Tanaka and Double Inclusion models, we develop a computationally efficient algorithm based on the mean-field homogenization, to quickly determine the macroscopic elastic properties of the random porous media by the fractal theory. The random pore sizes and locations are properly characterized by a single fractal dimension, and the fractal model agrees well with the numerical results obtained by the finite element homogenization technique and the existing experimental data from literature. Moreover, the effect of pore size distribution has been investigated, indicating that the elastic properties decrease with the increase in the fractal dimension. In particular, the computation time can be reduced from 4 h for the finite element analysis to 15 s based on the fractal model. Based on the mathematically analogous governing equations, the present fractal modeling has great potential to determine the other significant physical properties such as thermal conductivity and coefficient of thermal expansion.

Residual stresses in deep-drawn cups made of duplex stainless steel X2CrNiN23-4

Residual stress development in deep drawing processes is investigated based on cylindrical cups made of duplex stainless steel sheet. Using a two-scale approach combining finite element modelling with a mean field homogenization scheme the macro residual stresses as well as the phase-specific micro residual stresses regarding the phases ferrite and austenite are calculated for steel X2CrNiN23‑4 for various drawing depths. The simulation approach allows for the numerical efficient prediction of the macro and phase-specific micro residual stress in every integration point of the entire component. The simulation results are validated by means of X‑ray diffraction residual stress analysis applied to a deep-drawn cup manufactured using corresponding process parameters. The results clearly indicate that the fast simulation approach is well suited for the numerical prediction of residual stresses induced by deep drawing for the two-phase duplex steel; the numerical results are in good agreement with the experimental data. Regarding the investigated process, a significant influence of the drawing depth, in particular on the evolution of the residual stress distribution in drawing direction, is observed. Considering the appropriate phase-specific strain hardening, the two-scale approach is also well suited for the prediction of phase specific residual stresses on the component level.

On mean field homogenization schemes for short fiber reinforced composites: Unified formulation, application and benchmark

This paper revisits the topic of mean field homogenization for short fiber reinforced composite materials. A short glass fiber reinforced thermoplastic polyamide 6.6 with a fiber mass fraction of 35 % is used as an example material for which microstructural analyses and experimental tests were performed. We cover a set of common models: the Self-Consistent (SC), Mori–Tanaka (MT), Ponte Castañeda-Willis (PCW), Interaction Direct Derivative (IDD) and Two-Step (TS) schemes. We first recast them into a unified formulation that permits a thorough theoretical comparison, including the topics of loss of major symmetry as well as connections between the models. We are able to show that the MT, PCW and IDD schemes can be expressed in a surprisingly similar form. This extends to the equations for prediction of effective stiffness and compliance tensors as well as the strain and stress localization tensors. Secondly, we address the resolution of the material microstructure within mean field homogenization schemes, comparing classical and more recent and efficient methods. Last, we benchmark the different mean field schemes and modeling approaches, including comparisons to numerical homogenization by Fast Fourier Transformation (FFT) and experimental data from tensile tests. These show that the MT and TS schemes are both capable of accurately modeling the composite material for complex orientations and homogeneous fiber phases. For possibly inhomogeneous fiber phases only the TS scheme yields both accurate and physically reasonable results. Other models such as the IDD or PCW are of great theoretical importance, but cannot be generally applied for the given material class.

A mean-field homogenization approach to predict fracture in as-quenched microstructures of Ductibor® 500-AS steel: characterization and modelling

Ductibor® 500-AS is a hot-stamping steel that has been designed to introduce local ductile regions within hot-stamped tailor-welded blanks (TWBs) used for energy-absorbing sections of automotive structures. This study investigates the flow and fracture behavior of a range of the microstructures of this alloy, corresponding to mostly ferritic (~96% ferrite plus 4% martensite), intermediate ferritic + martensitic (~57% ferrite plus 43% martensite), and mostly martensitic (~10% ferrite plus 90% martensite), obtained by applying various quench rates after austenitization. A series of mechanical tests in various stress states, ranging from simple shear to biaxial tension, was performed to characterize fracture for the developed microstructures. A mean-field homogenization (MFH) technique was then used to predict the flow and fracture response as a function of the microstructure. An MFH scheme designated as the interpolative Samadian-Butcher-Worswick 1 (INSBW1) within the first-order secant-based linearization method was considered to predict the macroscopic hardening behavior of the ferritic-martensitic microstructures. In the MFH modelling, the flow response of the ferritic and martensitic phases was modelled based on a dislocation-based hardening model, taking into account the chemical composition, carbon partitioning between phases, and dislocation generation and annihilation. Damage accumulation and the onset of fracture were predicted using a phenomenological damage indicator defined within each constituent phase given their calculated fracture loci and instantaneous stress and strain states. The experimental results showed that the mostly-martensitic microstructure had the lowest ductility and highest strength, and ductility and strength increased and decreased, respectively, as the martensite volume fraction in the microstructure decreased. The predicted flow curves and fracture strains as well as the fracture-limit curves for all of the microstructures agreed well with the measured data and interpolated modified Mohr-Coulomb (MMC) fracture loci.

A systematic design to develop high-performance sintered particulate copper-composite as heat spreader material

A computational framework built on an effective-medium theory and mean-field homogenization is used to design particulate copper (Cu) composite for high-performance heat spreaders for electronic devices. Several potential particulate types as a second-phase are considered based on their intrinsic properties to design various Cu-composite heat-spreaders leading to desirable properties such as low coefficient of thermal expansion (CTE), low density, and improved structural response without compromising on required thermal conductivity. Based on computational predictions, the thermal interface of matrix-particulate is found to be the dominant factor for maintaining required thermal conductivity (~300 W m−1 K−1), which suggested ceramic particles, particularly Beo, SiC, and AlN, as preferred candidates when loaded up to 30 vol%. Due to very high thermal conductivity and low CTE, diamond particles demonstrated the best results provided its wettability with copper is improved as a result of the surface coating. To validate the model results, numerous Cu composites with diamond as a second-phase are sintered using the spark plasma sintering technique, and the experimental data is found in close agreement with the predictions. The Cu composites containing Ni-coated diamond particles exhibited thermal conductivity more than the value of pure Cu at extremely low volume concentration (5%). The effect of particle size is also studied and it is found that the composites sintered with sub-micron particles resulted in more porosity as compared to micron-sized particles, which displayed direct bearing on thermal and mechanical properties as demonstrated by experimental data. The measured data on densification and particle–matrix interface condition is used in calibrating the models for establishing their effect on resulting thermal and structural properties. The presented integrated computational design methodology is validated using experimental data, which is expected to help the researchers and electronic industry develop high-performance heat spreaders with tailored properties.

Tensile failure model of carbon fibre in unidirectionally reinforced epoxy composites with mean-field homogenisation

This paper presents an extension of the so-called incremental-secant mean-field homogenisation (MFH) formulation accounting for fibre bundle failure and matrix cracking in Unidirectional (UD) composites. First a model for fibre bundle failure is developed based on the strength failure probability of the carbon fibre described by a Weibull distribution. This fibre bundle failure model is then framed in a damage model of embedded bundles in a matrix by considering an exponential relation to describe the longitudinal stress build-up profile experimentally observed during failure of embedded fibre bundles. Cracking of the matrix in UD composites is accounted for through an anisotropic non-local damage model, which allows capturing the so-called 0° splits experimentally observed during the longitudinal tension of UD plies. A Mean Field Homogenisation (MFH) model is then extended to account for these damage models as component behaviours of the 2-phase composite material. A finite element multi-scale simulation of a notched laminate shows that the intra-laminar failure modes observed by an in situ experiment reported in the literature are well captured by the damage variables related to the matrix and fibre bundle failure processes. Inter-laminar failure is also captured by an extrinsic cohesive law introduced between the plies.

Experimental study and micromechanical modelling of the effective elastic properties of Fe–TiB2 composites

The aim of this paper is to investigate the effective properties of Fe–TiB2 composites obtained after hot or cold rolling. The elastic moduli of both hot and cold rolled composites are measured experimentally using several methods. Microstructure analyses based on SEM observations are performed to characterize the distribution of particles and cracks, and are then used to generate 3D representative microstructures using the RSA method. This allows the numerical determination of the overall elastic behavior of Fe–TiB2 composites using full-field FFT-based simulations. In addition, Young’s moduli of the hot rolled Fe–TiB2 composites are also determined analytically using the mean-field homogenization scheme of Mori–Tanaka. The elastic properties determined experimentally, analytically and numerically are in a good agreement. Overall, a significant improvement of the specific stiffness in comparison to standard steels is achieved irrespective of the processing conditions.

New incremental secant linearization method for mean-field homogenization approach of elasto-viscoplastic microscopic heterogeneous materials

Unmatched time scales of elastic and viscoplastic responses are not reasonably considered in linearizing the constitutive laws of constituent phases in elasto-viscoplastic microscopic heterogeneous materials, which makes the interaction among the constituent phases very difficult to be described by homogenization approaches. To address this issue, a new incremental secant linearization method is developed by solving linearized equations of stress, strain and time increments obtained from the Hooke’s law and the Taylor’s expansion of stress increment function. Subsequently, the new linearization method is implemented into the Mori-Tanaka’s (M−T) and self-consistent (SC) homogenization approaches. Finally, the stress–strain responses of elasto-viscoplastic microscopic heterogeneous materials (including the composites and polycrystalline materials) under different loading conditions are predicted by the incremental secant linearization-based M−T and SC approaches, and the predicted results are compared with the results obtained by other approaches, such as finite element, fast Fourier transform and generalized affine linearization methods. The comparison shows that the new secant linearization takes an important role in the accurate and effective simulations of the stress–strain responses of elasto-viscoplastic microscopic heterogeneous materials, and the predictions are independent of loading step size if the step size is not too large. Meanwhile, the homogenization approaches of elasto-viscoplastic and elasto-plastic microscopic heterogeneous materials are expected to be unified since the new secant linearization method provides the same mathematical structure for the linearized elasto-viscoplastic constitutive model as that for the elasto-plastic one.

Micro-mechanics and data-driven based reduced order models for multi-scale analyses of woven composites

Order reduction of woven composite materials is based on the definition of short fibre reinforced matrix material pseudo-grains completed by pure matrix parts. The former ones model the curved yarns, which are assimilated to continuous fibre reinforced matrix materials, in woven composites, and the latter ones model the matrix response. The homogenisation is achieved by recursively using micro-mechanics models, such as mean-field homogenisation and Voigt’s rule of mixtures, and the laminate theory.The pseudo-grains number and micro-structural features such as orientation, aspect ratio and volume fraction are considered as the Reduced Order Model (ROM) parameters and are identified following the approach of Deep Material Network (DMN): a set of homogenised elasticity tensors evaluated by computational homogenisation of woven unit cells is used as training data in order to identify the topological parameters of the ROM. Once the topological parameters are identified, the proposed ROM can be used to conduct nonlinear analyses of woven composites.The accuracy and efficiency of the proposed ROM have been verified by comparing the predictions with direct numerical simulations on two different woven unit cells.

Model reduction by mean-field homogenization in viscoelastic composites. III. Dual theory

The mean-field homogenization scheme for viscoelastic composites proposed by Lahellec & Suquet (2013 Int. J. Plasticity 42, 1–13 ( doi:10.1016/j.ijplas.2012.09.005 )) is revisited from the standpoint recently adopted in a companion paper (Idiart MI et al. 2020 Proc. R. Soc. A 20200407 ( doi:10.1098/rspa.2020.0407 )). It is shown that the scheme generates a reduced-order approximation wherein the microscopic kinetics of the composite are described in terms of a finite set of macroscopic forces identified with the phase averages and intraphase covariances of the various microscopic force fields, which can be evaluated by mean-field homogenization techniques. The approximation exhibits a two-potential structure with a convex complementary energy density but a non-convex force potential. The consequential properties of the approximation are exposed and their implications are discussed. The exposition is supplemented by proofs of equivalence between the present scheme and other candidate schemes proposed in the literature for composites with elementary local rheologies of Maxwellian type.

On the multiscale analysis of a two phase material: crystal plasticity versus mean field

In this paper, a comparison is made between two multiscale methods, namely crystal plasticity finite element and mean field on a material composed of two phases. Both methods are used to homogenize a given microstructure. In order to obtain macroscopic behavior, in the mean field approach, a Self-Consistent scheme is used to evaluate stress and strain partitioning among the phases. In this method, an average of the fields is estimated and local distributions cannot be captured. In parallel, crystal plasticity simulations on Representative Volume Elements (RVEs) composed of hexagonal grains are performed. In these simulations, grain orientations are attributed randomly respecting Mackenzie's distribution function in order to achieve isotropic behavior and macroscopic hardening is extracted from the simulations. The results on macroscopic hardening of both methods are compared to distinguish the extents of validity of mean field homogenization. In addition to Self- Consistent, other mean field schemes such as Voigt, Reuss and Bound-Interpolation are compared in terms of efficiency and accuracy. The comparison manifests that Self-Consistent scheme is capable of predicting material behavior well.

Micromechanics-based theoretical prediction for thermoelectric properties of anisotropic composites and porous media

By employing a mean-field homogenization theory to an anisotropic thermoelectric composite involving spherical fillers, we obtain analytical predictions of the effective Seebeck coefficient, effective electrical conductivity, and effective thermal conductivity in the presence of interfacial electrical and thermal resistances. First, we validate the thermoelectric Eshelby tensor and concentration tensor for an anisotropic matrix in the presence of interfacial resistance using finite element analysis (FEA) results and subsequently demonstrate that the predicted effective thermoelectric properties match well with the FEA up to a filler volume fraction of less than 20%. Based on the homogenization theory, we investigate the effect of filler size and filler properties on the effective thermoelectric properties of a composite. An interesting observation is that the porosity, which was known not to affect the effective Seebeck coefficient in isotropic materials, noticeably affects the effective Seebeck coefficient in anisotropic materials. Under strong anisotropy, there was a mismatch between the Seebeck coefficients of the porous medium obtained from numerical simulation and theoretical prediction. We discuss the origin of such phenomena in terms of the thermoelectric eddy current existing in an anisotropic medium: as the electric field is not conservative anymore, there is a circulating current in the porous anisotropic materials.

An Integrated Approach to Design and Develop High-Performance Polymer-Composite Thermal Interface Material.

A computational framework based on novel differential effective medium approximation and mean-field homogenization is used to design high-performance filler-laden polymer thermal interface materials (TIMs). The proposed design strategy has the capability to handle non-dilute filler concentration in the polymer matrix. The effective thermal conductivity of intended thermal interface composites can be tailored in a wide range by varying filler attributes such as size, aspect ratio, orientation, as well as filler–matrix interface with an upper limit imposed by the shear modulus. Serval potential polymers and fillers are considered at the design stage. High-density polyethylene (HDPE) and thermoplastic polyurethane (TPU) with a non-dilute concentration (~60 vol%) of ceramic fillers exhibit high thermal conductivity (4–5 W m−1 K−1) without compromising the high compliance of TIMs. The predicted thermal conductivity and coefficient of thermal expansion are in excellent agreement with measured data of various binary composite systems considering HDPE, TPU, and polypropylene (PP) loaded with Al2O3 and AlN fillers in varying sizes, shapes, and concentrations, prepared via the melt-mixing and compression-molding route. The model also validates that manipulating filler alignment and aspect ratio can significantly contribute to making heat-conducting networks in composites, which results in ultra-high thermal conductivity.

Mean-field homogenization coupled low-velocity impact analysis of nano fibre reinforced composites

Finding lightweight materials that are suitable for high-strength impact application is imperative. However, incorporating empirical methods for the same can be an exorbitant affair. This study aims to obviate these experimental methods and replace them with a numerical approach to the problem using the finite element method (FEM) in tandem with mean-field homogenization. CNT and BNNT nano-fillers were used in the polymer matrix for the sake of comparison of dynamical response. The preliminary step involves finding material properties of composite panels having different configurations and concentration of nanofillers both computationally and experimentally. Later, an explicit dynamic based finite element analysis was performed to simulate the low-velocity impact on each of the composite plate developed. Finally, temporal responses of the plate such as contact force, displacement, and velocity after the impact were illustrated and compared for both CNT and BNNT composite plates for different heterogeneities of fibre.

Effective viscoelastic behavior of polymer composites with regular periodic microstructures

Polymer matrix exhibit linear viscoelasticity during processing of composites. The viscoelastic homogenization problem in the time-domain play a vital role in the virtual process simulations. In this paper, full-field simulations using finite element (FE) are carried out for different regular periodic microstructures of unidirectional (UD) fiber reinforced polymer (FRP) (i.e., short, and long fiber) composite and particulate composites. The incremental variational based mean-field homogenization (MFH) as proposed by Lahellec and Suquet (Lahellec and Suquet, 2007a) is used here for comparing with the full-field solutions. Two different elastic bounds-based homogenization methods are coupled with this scheme. Implementation is validated with benchmark problems of particulate composites. These are then compared with the solutions of different particulate microstructures (face centered cubic (FCC), body centered cubic (BCC) and simple cubic (SC)). Mean-field response is closer to FCC arrangement. Additionally, FCC and BCC arrangement indicated nearly the same response with different local field statistics. Relaxation in the effective modulus of UD FRP composites obtained from the MFH compared well with the full-field solution as a function of direction and time for hexagonal arrangement of long fiber and staggered arrangement of ellipsoidal short fibers. The effect of different aspect ratio, volume fraction and contrast in properties on the macroscopic response is additionally investigated. Glass and carbon fiber with an isotropic approximation is used for studying the effect of contrast. Double inclusion MFH scheme coupled with the incremental variational approach indicated better comparisons with the full-field solution of UD FRP composites.

Identification methodology of a rate-sensitive constitutive law with mean field and full field modeling approaches for polycrystalline materials

The present paper deals with the consideration of the rate-sensitivity mechanical behavior of metallic materials, in the framework of mean field and full field homogenization approaches. We re-examine the possibility of describing properly this rate sensitivity with a simple and widely used power law expressed at the level of the slip system, and we propose a methodology to accelerate the identification of the global material constitutive law for Finite Element (FE) simulations. For such an aim, simulations of a tensile test are conducted, using a simple homogenization model (the Taylor one, used in a relaxed constraint form) and an FE code (Abaqus), both using the same single-crystal rate-dependent constitutive law. It is shown that, provided that the identification of this law is performed with care and well adapted to the examined case (rate-sensitive or insensitive materials, static and/or dynamic ranges), the simple power law can be used to simulate the macroscopic behavior of polycrystalline aggregates in a wide range of strain rate (including both static and dynamic regimes) and strain-rate sensitivity values (up the rate-insensitive limit).