Two-step Homogenization Research Articles

Nonlinear, Large Deformation Finite-Element Beam/Column Formulation for the Study of the Human Spine: Investigation of the Role of Muscle on Spine Stability

A nonlinear, large deformation beam/column formulation is used to model the behavior of the human spine under compressive load. The stabilizing roles of muscles are accounted for using Patwardhan’s assumption that muscles act to direct the load along the tangent of the column. Three aspects of the spinal structure are then investigated. First, we look at the effects of two different assumptions for the action of muscles, leading to significant differences in the spine behavior. Second, the difference in mechanical properties between the vertebrae and the spinal disks is explored. Third, a nonlinear mechanical response of the spinal disk that arises from a two-step hierarchical homogenization technique is used. It is found that these factors have an important influence on the overall behavior of the spine structure. The present formulation offers a versatile model to investigate various features of the human spine, while remaining affordable computationally. It also provides an interesting framework for fu...

Multiscale homogenization models for the elastic behaviour of metal/ceramic composites with lamellar domains

This study predicts the elastic properties of an innovative metal–ceramic composite with statistically oriented domains of parallel ceramic platelets embedded in a eutectic Al–Si-alloy. For this purpose, a two-step homogenization procedure was employed by finite element- and micromechanical modelling. In a first step, the microstructure of the specimen was divided in domains with the same orientation of lamellae and the elastic properties of single domains were calculated while a precise representation of the shape of the lamellae was attempted. In a second step the elastic constants of a large specimen consisting of many domains were computed both by finite element and micromechanical modelling. The experimental Young’s modulus of such poly-domain specimens was determined by an acoustic resonance method and was lower than predicted. The differences can be explained by microcracks caused by large residual microstresses produced in these materials when they are cooled from the manufacturing temperature.

Macroscopic Constitutive Law for Mastic Asphalt Mixtures from Multiscale Modeling

A well established framework of an uncoupled hierarchical modeling approach is adopted here for the prediction of macroscopic material parameters of the Generalized Leonov (GL) constitutive model intended for the analysis of flexible pavements at both moderate and elevated temperature regimes. To that end, a recently introduced concept of a statistically equivalent periodic unit cell (SEPUC) is addressed to reflect a real microstructure of Mastic Asphalt mixtures (MAm). While mastic properties are derived from an extensive experimental program, the macroscopic properties of MAm are fitted to virtual numerical experiments performed on the basis of first order homogenization scheme. To enhance feasibility of the solution of the underlying nonlinear problem a two-step homogenization procedure is proposed. Here, the effective material properties are first found for a mortar phase, a composite consisting of a mastic matrix and a fraction of small aggregates. These properties are then introduced in place of the matrix in actual unit cells to give estimates of the model parameters on macroscale. Comparison with the Mori-Tanaka predictions is also provided suggesting limitations of classical micromechanical models.

Calculation of effective properties of textile reinforced aluminum alloy by a two-step homogenization procedure

Vehicle weight reduction is one of the main topics of modern automotive production. The use of aluminum and its alloys fulfils this requirement in an outstanding manner. Having the advantage of low density and therefore light weight, these materials still have a disadvantage in stiffness, which is lower than that of various steels. Here, textile reinforced aluminum cast parts offer excellent mechanical properties in lightweight design applications where high compression strength is needed. Such materials are inhomogeneous and their properties differ strongly from the properties of the initial aluminum alloy. To calculate the mechanical properties of the aluminum alloy, reinforced by ceramic (alumina) textile structure, homogenization based on asymptotic methods was used. On the macroscopic level, the material was considered being composed of the textile structure infiltrated by aluminum alloy. Here, the properties of the pure alumina could not be taken to describe the textile structure, because of the infiltration of the alloy in rovings, constructing the textile structure. So, on the microscopic level the rovings were considered as composed of the ceramic filaments infiltrated by aluminum alloy. In that way, calculation of the effective Young’s modulus and Poisson ratio of this material built the first level of the homogenization procedure. On the next step, these calculated values were used as properties of one of the structure components (textile) to calculate the effective mechanical behavior of the whole reinforced part.

Material Parameter Identification of Interpenetrating Metal-Ceramic Composites

A new class of metal/ceramic composites has recently been developed. A porous ceramic preform, the pore structure of which is created via a freeze-casting technique, is melt-infiltrated with metallic alloy via sqzeeze-casting. The microstructure of the composite has lamellar-like domains with geometrical characteristics which are dependent on the manufacturing parameters. The aim of our study is to find a good micromechanical model in order to deduce the mechanical properties of the single domains and of the whole material as a function of the microstructural geometry and the material parameters of the ceramics (alumina) and the alloy (Al-Si eutectic). Firstly, the statistical analysis of polarized light microscopic micrographs of the cross section of the specimen was performed. Domains with the same orientation of lamellae, so-called single domains were detected, selected and measured. The material modeling was performed by a two-step homogenisation procedure using a combination of different micromechanical models. Predicted material properties were compared with ultrasonic measurements for a single domain and for the whole microstructure.

MICROSTRUCTURAL EVOLUTION OF Al-Cu-Mg-Ag ALLOY WITH TRACE Zr ADDITION DURING TWO-STEP HOMOGENIZATION

The microstructural evolution in an Al - Cu - Mg - Ag alloy with trace Zr addition during homogenization treatment was characterized by Differential Scanning Calorimetry (DSC), Scanning Electron Microscopy (SEM) and Energy-dispersive X-ray Spectroscopy (EDS). It was shown that the low-melting-point phase segregating toward grain boundaries is Al 2 Cu , with a melting point of 523.52°C. A two-step homogenization process was employed to optimize the microstructure of the as-cast alloy, during which the alloy was first homogenized at a lower temperature, then at a higher temperature. After homogenized at 420°C for 6 h, Al 3 Zr particles were finely formed in the matrix. After that, when the alloy was homogenized at an elevated temperature for a longer time, i.e., 515°C for 24 h, most of the precipates at the grain boundaries were removed. Furthermore, the dispersive Al 3 Zr precipitates were retained, without coarsening greatly in the final homogenization step. A kinetics model is employed to predict the optimal homogenization time at a given temperature theoretically, and it confirms the result in present study, which is 420°C/6h+515°C/24h.

Homogenization of acoustic metamaterials of Helmholtz resonators in fluid

By using a two-step homogenization approach, we derive analytical formulas of effective mass density ${\ensuremath{\rho}}_{e}$ and effective bulk modulus ${B}_{e}$ for two- and three-dimensional acoustic metamaterials of Helmholtz resonators (HRs) in fluid. A negative ${B}_{e}$ is found at certain frequencies due to the monopolar resonance, leading to a low-frequency acoustic band gap. A unified picture is presented for metamaterials of HRs and three-component metamaterials of negative ${\ensuremath{\rho}}_{e}$. Our work supports recent observations in a one-dimensional array of HRs [N. Fang et al., Nat. Mater. 5, 452 (2006)] and presents important high-dimensional extensions for exploring more fascinating phenomena.

Assessment of the Thermoelastic Properties of an Injection molded short-fiber Composite: Experimental and Modelling

The accurate prediction of both the elastic properties and the thermal expansion coefficients is very important for the precise simulation of such processes as injection molding of short-fiber polymer-matrix composites. In this work, a two-step homogenization procedure is applied and compared with experimental values obtained on a polyarylamide/glass fiber composite for a broad range of temperatures. It is observed that the stiffness averaging version of the model surpasses the compliance averaging variant, especially when it is combined with a precise evaluation of the fourth-order orientation tensor. It is also demonstrated that the orthotropic closure approximations are significantly better than previous ones (linear, quadratic, and hybrid) and than a very recent one. Among the orthotropic closure approximations, the fitted ones lead to acceptable results, which are very close to those obtained with the experimentally measured fourth-order orientation tensor

Faraday magneto-optical rotation in compositionally graded films

We present a two-step homogenization method for studying the Faraday magneto-optical effect in graded metal-dielectric composite films of width W, in which the volume fraction of metal particles in a slice varies along the direction perpendicular to the film surface. First, we adopt the effective-medium theory to formulate the equivalent (local) dielectric permittivity tensor for a z slice. Second, the graded composite films are homogenized with an effective (overall) dielectric permittivity tensor including the diagonal and off-diagonal elements. Faraday rotation is studied as a function of the graded profile p(z) with the same total volume fraction. For a power-law form p(z)=a(z/W)m with different m, it is found that with increasing m, the magnitude of Faraday rotation becomes weak near the surface plasmon resonant band, accompanied with the redshift of the resonant center. Interestingly, it is possible to achieve strongly enhanced Faraday rotation in the high-frequency region, and to change the direction of rotation in the low-frequency one. Moreover, the magnitude can be further enhanced for needle-like particles. In the dilute limit, we show that Faraday rotation is indeed independent of m within Maxwell-Garnett theory.

Effect of homogenization treatment on microstructure and hot workability of high strength 7B04 aluminium alloy

The effects of homogenization treatment on microstructure, overburnt temperature and hot rolling plasticity of high strength 7B04 aluminium alloy were investigated. Under the condition of homogenization at 470 °C, the starting melting temperature of the primary eutectics in ingot of non-equilibium solidified 7B04 alloy is 478 °C. Using two-step homogenization processing at ultra-high temperature which comprises heating the ingots to 470 °C at 10 °C/h and holding for 64 h, and then heating to 500 °C at 1 °C/h and holding for 10 h, the ingots of 7B04 aluminium alloy could safely pass the sensitive overburnt zone between 480 °C and 495 °C, and the ordinary burnt phenomena of the ingots between 480 °C and 495 °C does not occur because the excess low-melting point eutectic phases in the as-cast alloy dissolve into the matrix during the two-step homogenization processing. Consequently, the hot rolling plasticity of ingot of 7B04 aluminium alloy is greatly improved.

Homogenization of Negative-Index Composite Metamaterials: A Two-Step Approach

We present a two-step homogenization method for composite metamaterials. First, each layer of wires or resonators is homogenized as a slab with negative permittivity or permeability, respectively. Second, the single negative stack which results is homogenized to form the effective medium. Comparing the predictions of the first and second step can serve as a gauge of the homogeneity of the composite. We thus take a gradual approach to homogenization, asking not whether, but to what extent a composite metamaterial approaches the sought after effective medium. Our two-step approach can also capture phenomena which otherwise may be wrongly attributed to effective medium behavior. We illustrate by qualitatively reproducing and reinterpreting a set of experimental data from the literature.

Comparison of several closure approximations for evaluating the thermoelastic properties of an injection molded short-fiber composite

The accurate prediction of both the elastic properties and the thermal expansion coefficients is very important for the precise simulation of such processes as injection molding of short-fiber polymer–matrix composites. In this work, a two-step homogenization procedure is applied and compared with experimental values obtained on a polyarylamide/glass fiber composite for a broad range of temperatures. It is observed that the stiffness averaging version of the model surpasses the compliance averaging variant, especially when it is combined with a precise evaluation of the fourth-order orientation tensor. It is also demonstrated that the orthotropic closure approximations are significantly better than previous ones (linear, quadratic, and hybrid) and than a very recent one. Among the orthotropic closure approximations, the fitted ones lead to acceptable results, which are very close to those obtained with the experimentally measured fourth-order orientation tensor.

Effective thermal conductivity of transversely isotropic media with arbitrary oriented ellipsoïdal inhomogeneities

The objective of this work is to investigate the thermal conduction phenomena in transversely isotropic geomaterials or rock-like composites with arbitrary oriented ellipsoïdal inhomogeneities of low aspect ratio. Based on the evaluation of the Green function, we provide here new expressions for the interaction tensor whose knowledge permits to obtain the concentration tensor of the polarization field used itself to evaluate the effective thermal conductivity tensor by homogenization. Some particular cases of the obtained general solution are equally presented, in order to validate the developed formalism. The obtained results are next used to study the effect of matrix anisotropy, pores systems and microstructure-related parameters on the overall effective thermal conductivity in transversely isotropic rocks. A two-step homogenization scheme is developed for the prediction of the initial anisotropy effects and to test the ability of the proposed model in the evaluation of effective thermal conductivity. With the help of an Orientation Distribution Function (ODF) the anisotropy due to the pore systems is also accounted. Numerical applications and comparisons with available experimental data are finally carried out for a partially saturated Opalinus clay and an argillite which are both composed of an argillaceous matrix and multiple solid minerals constituents.

Effective thermal conductivity of partially saturated porous rocks

The present work is concerned with the determination of the effective thermal conductivity of porous rocks or rock-like composites composed by multiple solid constituents, in partially saturated conditions. Based on microstructure observations, a two-step homogenization scheme is developed: the first step for the solid constituents only, and the second step for the (already homogenized) solid matrix and pores. Several homogenization schemes (dilute, Mori–Tanaka, the effective field method and Ponte Castañeda–Willis technique) are presented and compared in this context. Such methods are allowing: (i) to incorporate in the modellization the physical parameters (mineralogy, morphology) influencing the effective properties of the considered material, and the saturation degree of the porous phase; (ii) to account for interaction effects between matrix and inhomogeneities; (iii) to consider different spatial distributions of inclusions (spherical, ellipsoïdal). An orientation distribution function (ODF) permits simultaneously to incorporate in the modelling the transverse isotropy of pore systems. Appearing as homogeneous at the macroscopic scale, it is showed that the effective conductivity depends on the physical properties of all subsidiary phases (microscopic inhomogeneities). By considering the solution of a single ellipsoïdal inhomogeneity in the homogenization problem it is possible to observe the significant influence of the geometry, shape and spatial distribution of inhomogeneities on the effective thermal conductivity and its dependence with the saturation degree of liquid phase. The predictive capacities of the two-step homogenization method are evaluated by comparison with experimental results obtained for an argillite.

General mean-field homogenization schemes for viscoelastic composites containing multiple phases of coated inclusions

We consider matrix materials reinforced with multiple phases of coated inclusions. All materials are linear viscoelastic. We present general schemes for the prediction of the effective properties based on mean-field homogenization. There are four contributions in this work. First, we present a two-step homogenization procedure in a general setting which besides the usual assumptions of Eshelby-based models, does not suffer any restriction in terms of material properties, aspect ratio or orientation. Second, for a matrix reinforced with coated inclusions, we propose two general homogenization schemes, a two-step method and a two-level recursive scheme. We develop and compare the mathematical expressions obtained by the two schemes and a generalized Mori–Tanaka (M–T) model. Third, for a two-phase composite, either standalone or stemming from two-step or two-level schemes, we use a double-inclusion model based on a closed-form but non-trivial interpolation between M–T and inverse M–T estimates. Fourth, we conduct an extensive validation of the proposed schemes as well as others against experimental data and unit cell finite element simulations for a variety of viscoelastic composite materials. Under severe conditions, the proposed schemes perform much better than other existing homogenization methods.

Shotcrete Elasticity Revisited in the Framework of Continuum Micromechanics: From Submicron to Meter Level

Understanding the chemomechanical behavior of shotcrete (sprayed concrete) is a prerequisite for better design or failure risk assessment of civil engineering structures. Macroscopic material laws for shotcrete are preferably based on hydration degree–dependent material functions. For a given shotcrete mixture—that is, for a fixed water-cement ratio (w∕c) and aggregate-cement ratio (a∕c) —these functions are determined directly from macroscopic experiments. In a purely macroscopic context, considering changes in the w∕c ratio (as often encountered in large tunnel projects), or in the a∕c ratio would require additional experiments. As an alternative, such hydration degree–dependent material functions can be predicted by considering quantitative information at the microlevel of the cementitious material, in the framework of continuum micromechanics. In this contribution, we focus on the dependence of the elastic stiffness on the hydration degree. We employ a two-step homogenization procedure: within a repre...

Influence of hydrolysed lecithin addition on protein adsorption and heat stability of a sterilised coffee cream simulant

Abstract Within an overall goal to enhance the stability of coffee cream, the influence of hydrolysed soybean lecithin addition was investigated. To avoid the seasonal variations in milk composition a model system was used containing 5% (w/w) soybean oil and 12% (w/w) skimmed milk powder with and without lecithin addition before a two-step high-pressure homogenisation. Addition of 20% (w/v) caesium chloride enabled a nearly complete recovery of the fat in the cream layer upon centrifugation of casein-stabilised emulsions. From the fat and protein content of this cream layer, it was concluded that lecithin addition did not significantly affect the protein content of the cream layer after emulsion preparation, but significantly reduced the additional milk protein adsorption at the oil/water interface upon sterilisation. The experimental data suggest that the enhanced heat stability of the lecithin-supplemented coffee cream simulant may be explained by the fact that hydrolysed lecithin largely reduces attractive protein–protein interactions during sterilisation.

Mean-field homogenization of multi-phase thermo-elastic composites: a general framework and its validation

This paper deals with mean-field Eshelby-based homogenization techniques for multi-phase composites and focuses on three subjects which in our opinion deserved more attention than they did in the existing literature. Firstly, for two-phase composites, that is when in a given representative volume element all the inclusions have the same material properties, aspect ratio and orientation, an interpolative double inclusion model gives perhaps the best predictions to date for a wide range of volume fractions and stiffness contrasts. Secondly, for multi-phase composites (including two-phase composites with non-aligned inclusions as a special case), direct homogenization schemes might lead to a non-symmetric overall stiffness tensor, while a two-step homogenization procedure gives physically acceptable results. Thirdly, a general procedure allows to formulate the thermo-elastic version of any homogenization model defined by its isothermal strain concentration tensors. For all three subjects, the theory is presented in detail and validated against experimental data or finite element results for numerous composite systems.

Hydrolysis of the oil phase of a W/O/W emulsion by pancreatic lipase

W/O/W emulsions are expected to protect bioactive substances from degradation by pancreatic enzymes. We investigated the enzymatic hydrolysis of the oil phase and release of a marker substance from the inner-aqueous phase to the outer-aqueous phase using an artificial digestive fluid. Octanoic acid triacylglycerol (C8TG) was used as the oil phase. W/O/W emulsions were prepared by two-step homogenization and succeeding membrane filtration. When the artificial digestive fluid containing lipase and gall was added to the emulsion, release of the marker substance from the inner-phase solution, oil-phase hydrolysis, and emulsion coalescence occurred in that order. When a coarse emulsion and 0.2- and 0.8-μm membrane-filtered fine emulsions were treated with the fluid for 1 h, the degrees of C8TG hydrolysis were 3.8%, 55% and 57%, the fractions of the marker substance released from the inner-water phase were 2.7%, 89% and 72%, and the median diameters of the oil droplets were changed from 32 to 23 μm, 0.71 to 27 μm, and from 2.2 to 26 μm, respectively. These results suggested that the diameter of the oil droplets in the W/O/W emulsion significantly affected the release profile of the marker loaded in the inner-water phase of the emulsion.

Micromechanics-Based Modeling of Woven Polymer Matrix Composites

A novel approach is combined with the generalized method of cells (GMC) to predict the elastic properties of plain-weave polymer matrix composites (PMCs). The traditional one-step three-dimensional homogenization procedure that has been used in conjunction with GMC for modeling woven composites in the past is inaccurate due to the lack of shear coupling inherent in the model. However, by performing a two-step homogenization procedure in which the woven composite repeating unit cell is homogenized independently in the through-the-thickness direction prior to homogenization in the plane of the weave, GMC can now accurately model woven PMCs. This two-step procedure is outlined and implemented, and predictions are compared with results from the traditional one-step approach as well as other model and experimental results from the literature. Full coupling of this two-step technique within the recently developed Micromechanics Analysis Code with GMC software package will result in a widely applicable, efficient, and accurate tool for the design and analysis of woven composite materials and structures.