Generalized Self-consistent Scheme Research Articles

Multi-scale strategy to estimate the mechanical and diffusive properties of cementitious materials prepared with CEM II/C-M

A procedure to estimate the mechanical and diffusive properties of cement pastes, mortars and concretes made with a novel CEM II/C-M (S-LL) binder that has been developed within the European EnDurCrete project, is presented in this paper. The objective is to set up a strategy able to determine the materials composition at different scales and in various configurations, and to provide reliable estimates of the properties. The phase assemblages of the cement pastes upon hydration, and further exposed to carbonation, leaching with water or chloride containing solutions, are first simulated using thermodynamic modeling. Different homogenization schemes are then applied, to estimate the mechanical and diffusive properties of the hydrated and degraded materials at different scales. The interfacial transition zone (ITZ) in mortar and concrete is accounted for by means of specific hypotheses regarding its thickness and the respective volume fraction of hydrates and capillary pores. The application of the differential scheme (DIF) at the cement paste scale, and at higher scales of the generalized self-consistent scheme (GSCS) or a procedure accounting for ITZ modeled as an interface combined with the DIF scheme, show a good consistency with the measured Young's modulus. In parallel, simulations are performed on 3D specimens composed of spherical particles surrounded by an ITZ layer to provide additional data for comparison. Numerical results and analytical estimations are found to be in excellent agreement. Finally, a specific study focusing on the effects of the ITZ on both mechanical and diffusive properties shows a significant impact of its thickness and its composition.

Prediction on Permeability of Engineered Cementitious Composites

Permeability of concrete is regarded as a basic indicator of its durability. This paper proposed a simple model to predict the permeability of engineered cementitious composites (ECC), which are fiber reinforced cementitious composites with extremely high ductility and toughness. The permeability of cement paste in ECC was firstly determined based on the general effective media theory. The needed microstructure information of cement paste was obtained from a simulated microstructure. Porosity of the interfacial transition zone (ITZ) was obtained with an ITZ porosity model, and then used to calculate the permeability of ITZ. The permeability of the matrix was determined according to the general self-consistent scheme, and the influence of fiber was simplified with its volume fraction. The calculated permeability of ECC was verified with results from water permeability tests and the accuracy of the model was acceptable for cement-based materials.

On the effective viscosity of fresh concrete: A homogenization approach

Two new homogenization schemes together with the classical generalized self-consistent scheme (GSC) and its extension are proposed to deal with the effective viscosity of fresh concrete. All these models exploit the composite sphere morphology of suspension made of a viscous fluid and spherical particles. They differ from each other by the ways the extra fluid zone (EFZ) located in between the composite sphere are treated. The comparison with experimental data shows that the GSC provides very good result for a well arranged particle size scale that allows mapping the whole medium by composite spheres of different size scales. However, the GSC cannot be used for a suspension with a non negligible volume of the EFZ. For such a case, extensions of the GSC those take into account the contribution of the EFZ to the overall viscous behavior of the system is necessary to fit with experimental data. Two of them work very well for non-cohesive particles and the other can be employed for suspension with cohesive particles such as the case of fresh cement paste.

Anti-plane inclusion problem in the second gradient electroelasticity theory

In this paper, we compare direct and energy based homogenization schemes in the framework of the second gradient electroelasticity theory. We consider a special type of the model with simplified constitutive equations that allow to derive an exact analytical solution for the effective properties of fiber-reinforced piezoelectric composite under anti-plane loading. To find the solution we use an approach that was proposed by Benveniste for similar problem in classical electroelasticity. Based on the transformations of the field variables, we construct a solutions for the coupled anti-plane problem of gradient electroelasticity by using known solutions for the uncoupled problem of the strain gradient elasticity and similar problem of gradient electrostatics. Based on the derived solution, we demonstrate the main effects that arise in the composite armed with thin piezoelectric fibers, which diameter is of the order of the characteristic length of material’s microstructure. We show that such composites may demonstrate an unusual behavior: With the decrease of the fiber diameter its effective properties increase and its apparent behavior becomes similar to the homogeneous material. We carry out a comparison between predictions for effective properties, which were provided by the Mori–Tanaka scheme (MTS) and generalized self-consistent scheme (GSCS). We show that these methods give significantly different predictions, which can deviate from each other by hundreds of percent, especially for the effective piezoelectric constants.

Viscoelastic characterization of aged asphalt mastics using typical performance grading tests and rheological-micromechanical models

The performance grading (PG) tests use aged asphalt specimens to simulate the in situ performance of asphalt binder. The main purpose of this study is to examine the existing micromechanical models, such as the inverse rule of mixtures, Hashin-Shtrikman, generalized self-consistent scheme (GSCS), and 4-phase, to determine the dynamic shear modulus of aged asphalt mastics. The Wicket and the vGP (van Gurp-Palmen) plots were advisable tools for checking the PG test results consistency and Time-Temperature superposition principle validation on three different asphalt mastics. The generalized logistic sigmoidal model (GLSM) yielded well-fitted master curves of modulus and phase angle based on the models’ predictions for full linear viscoelastic (LVE) characterization. The 4-phase model had the best modulus predictability even better than the well-known 2S2P1D (2 Spring-2 Parabolic element-1 Dashpot) model that followed by the inverse rule of mixtures, Hashin-Shtrikman, and GSCS models. All of the models had an excellent phase angle predictability that was comparable with 2S2P1D model predictions. It is recommended to combine the typical PG tests of asphalts, a well-defined micromechanical model like 4-phase, and a well-known rheological master curve model such as GLSM for estimating the LVE properties of asphalt mastics promisingly.

Isogeometric boundary element method for calculating effective property of steady state thermal conduction in 2D heterogeneities with a homogeneous interphase

Based on the generalized self-consistent scheme (GSCS), the isogeometric boundary element method (IGABEM) is used to calculate the effective thermal conductivity of two dimensional (2D) steady state heat conduction heterogeneities with a homogeneous interphase. The heat energy formulation and the boundary integral equations adopted in this paper only contain the temperatures from the inclusion–interphase and interphase–matrix interfaces, respectively, so that the effective thermal conductivity from the generalized self-consistent model with a homogeneous interphase can be conveniently calculated. In numerical implementation, the Non-Uniform Rational B-Splines (NURBS) are employed not only to construct the exact interface shapes but also to approximate the interface temperatures. The results show that the conductivity and the thickness of the homogeneous interphase have great influence on the effective thermal conductivity. Higher interphase thermal conductivity enhances the effective thermal conductivity, whereas thinner thickness of the interphase results in the reduction of the effective thermal conductivity. Numerical examples show that the proposed method works well compared with the conventional quadratic isoparametric boundary element method.

An adaptive isogeometric boundary element method for predicting the effective thermal conductivity of steady state heterogeneity

This work presents an adaptive isogeometric boundary element method (IGBEM) for the calculation of effective thermal conductivity of steady state heterogeneities. Based on the generalized self-consistent scheme (GSCS), some integral equation formulations which only contain the unknown temperatures on the interface are used to calculate the effective thermal conductivity of steady state composites. In our approach, the geometries of the inclusion and original matrix are described using NURBS basis functions. The advantage over currently used methods is that no geometrical errors exist in the analysis process. And the geometry data in the isogeometric GSCS model can be taken directly from CAD programs. In addition, based on the upper bound of the relative error of the Gaussian quadrature formula, an adaptive integration method is used to compute the boundary integrals, which makes the computation of the integrals easier and more efficient at optimal computational cost. The comparisons between the results obtained by the present method and the existing counterparts are carried out and the good agreement can be observed.

A composite sphere assemblage model for porous oolitic rocks: Application to thermal conductivity

The present work is devoted to the determination of linear effective thermal conductivity of porous rocks characterized by an assemblage of grains (oolites) coated by a matrix. Two distinct classes of pores, i.e. micropores or intra oolitic pores (oolite porosity) and mesopores or inter oolitic pores (inter oolite porosity), are taken into account. The overall porosity is supposed to be connected and decomposed into oolite porosity and matrix porosity. Within the framework of Hashin composite sphere assemblage (CSA) models, a two-step homogenization method is developed. At the first homogenization step, pores are assembled into two layers by using self-consistent scheme (SCS). At the second step, the two porous layers constituting the oolites and the matrix are assembled by using generalized self-consistent scheme (GSCS) and referred to as three-phase model. Numerical results are presented for data representative of a porous oolitic limestone. It is shown that the influence of porosity on the overall thermal conductivity of such materials may be significant.

Multiscale modelling of transport phenomena for materials with n-layered embedded fibres. Part I: Analytical and numerical-based approaches

• The overall diffusivity tensor of a multi-layered fibre-reinforced material is given. • Diffusivity is derived from a Morphological Representative Pattern-based approach. • Closed-form analytical relations are proposed depending on morphological parameters. • Diffusivity effects of interphase areas are investigated. Numerical and analytical modelling approaches are often competing to describe a desired phenomenon. However, there is no reason to look for a single method and complementarities between distinct modelling ways are always profitable. This paper, the first of a set of two consecutive papers published in this volume, provides such a modelling methods reconciliation in the case of transport phenomena acting in transversely isotropic multi-phased materials. Microstructural Finite Element Modelling (FEM) helps supplying the analytical implementation for a better description of interactions between the constituents. As an application example, the case of molecular diffusion within a unidirectional composite medium is investigated without any restriction for considering any other transport mechanism. Based on a rational microstructure description, this study proposes a model able to predict the overall diffusion tensor of the composite and investigates more particularly diffusivity effects of interphase areas and fibre packings. For this purpose, a “ n -phase” Generalized Self-Consistent Scheme (GSCS) coupled with a Morphologically Representative Pattern (MRP) approach has been developed. This first part is focused on the modelling strategy and the concept of “transfer matrices” has been used. This approach is applied in Part II (Joannès and Hervé-Luanco, 2016) to study the influence of fibre packings on the effective behaviour of composite materials made of insulated fibres embedded in a diffusive matrix.

An interface integral formulation of heat energy calculation of steady state heat conduction in heterogeneous media

This paper presents a new boundary integral formulation of the heat energy calculation of steady state heat conduction in heterogeneous media using two methods, i.e. one which is based on some formulations from Hatta and Taya (1986) who investigated equivalent inclusion method, and another one which is closely similar to the method from Christensen (1979) who studied elastic inclusion problem. For the generalized self-consistent scheme (GSCS) model, the corresponding heat energy formulation is also given. Based on the obtained interface heat energy formulation, the boundary element method together with the Maxwell homogeneous scheme are used to calculate the effective heat conduction of 2D and 3D heterogeneous steady state heat conduction media.

Micromechanical contribution for the prediction of the viscoelastic properties of high rate recycled asphalt and influence of the level blending

This research deals with an experimental and micromechanical study of high rate recycled asphalt with 70% of RAP. Rheological measurements of shear complex modulus of virgin, RAP and blended binders were performed at different temperatures and frequencies. Then, a micromechanical model based on the generalized self-consistent scheme (GSCS) was suggested for the prediction of the effective mechanical properties of the recycled asphalt. The GSCS-based approach aims to homogenize the heterogeneous material taking into account the intergranular porosity, on the one hand, and the possible interactions between its phases on the other one. The suggested method was compared to a step-by-step (successive) homogenization approach and literature data in elastic case were used for this purpose. The results have shown that the GSCS-based approach presents a good agreement with the data especially for higher volume fractions of aggregates. Furthermore, the significant influence of the blend homogeneity level on the estimation of the effective mechanical properties of the recycled asphalt was highlighted.

Boundary integral equation formulations for steady state thermal conduction and their applications in heterogeneities

This paper presents some boundary integral equations of steady state thermal conduction for homogenization of composites based on the generalized self-consistent scheme (GSCS). The proposed formulations only contain the temperature of the interfaces which are easily numerically implemented. A formulation for calculating the heat energy only containing interface integration is also used to determinate the effective thermal conductivity of the composite. The comparison between the results obtained by the present method and the existing counterparts are carried out and the good agreement can be observed. The present method developed here is applicable to various cases of ellipsoidal, spherical and other shapes׳ inhomogeneities.

Two-scale homogenization of elastic layered composites with interfaces oscillating in two directions

In a variety of situations of practical interest, the interface between two phases in a composite cannot be reasonably assumed to be smooth but has to be taken as being rough at the microscopic scale. How to determine the effective properties of such a composite remains a largely open problem in micromechanics. The present work is concerned with layered composites in which the interface between two neighboring layers oscillates quickly and periodically along two directions in the plane normal to the layering direction. In this case, the classical homogenization theory of layered composites is no longer applicable, since the interfacial oscillations prevent the layered composite in question from being homogeneous in the plane perpendicular to the layering direction. To overcome this difficulty, a two-scale homogenization method is proposed in the present work. First, at the mesoscopic scale, each zone in which an interface oscillates is homogenized as an interphase by an asymptotic analysis. The effective elastic properties of this interphase are determined by using a numerical method based on the fast Fourier transform (FFT) or estimated by applying the generalized self-consistent scheme (GSCS). Then, at the macroscopic scale, the effective elastic moduli of the composite made of the resulting plane layers and interphases are calculated with the help of the classical homogenization theory of layered composites. Finally, numerical examples are provided to illustrate the results for the effective elastic moduli of a layered composite obtained by the two-scale homogenization method proposed and to compare them with the corresponding numerical results given by the finite element method (FEM).

The Effective Conductivity of Multiphase Composites with Imperfect Thermal Contact at Constituent Interfaces

This paper studies the effective thermal conductivity of multiphase composite in which a thermal boundary resistance exists at constituent interfaces. Based on the theoretical framework of conductivity for binary system composites in the presence of a thermal contact resistance between matrix and inclusion given by Y. Benveniste and T. Miloh (1986), the fundamental concept is generalized for the case of multiphase composites with imperfect contact which permits a temperature discontinuity between matrix and inclusions of different phases. A micromechanics model, the “generalized self-consistent scheme (GSCS)” based on a particle-matrix embedding in the effective medium, is generalized to evaluate the effective conductivity of multiphase medium with imperfect thermal contact at constituent interfaces. Numerical results are given for three-phase particulate composites with spherical particles to illustrate the effect of imperfect interfaces on the effective thermal conductivity of multiphase composites.

Rigidity of diamond reinforced metals featuring high particle contents

Diamond/metal composites with diamond contents between 57 and 72 vol% have been produced by gas pressure assisted liquid metal infiltration using Ag–3 wt% Si and Al–2 wt% Cu as matrix. The experimental data cover a range of Young’s moduli from 300 to 425 GPa and 245 to 370 GPa for the Ag–3Si and the Al–2Cu-based composites, respectively. Experimental Young’s moduli are compared to the Mori–Tanaka mean field scheme (MTM), the generalized self-consistent scheme (GSCS), the bimodal hard sphere model (TBHS), and the differential effective medium scheme (DEM). At the lower end of volume fractions investigated, the predictions by the GSCS, the TBHS, and the DEM are very close to each other and to the experimental results while the MTM is clearly lower. With increasing volume fraction the differences between the models accentuate and the data up to 72 vol% of diamond are best described by the DEM.

Revisiting the generalized self-consistent scheme in composites: Clarification of some aspects and a new formulation

The determination of an effective property in composite materials necessitates the knowledge of some averaged field quantities in the constituents (like the average heat intensity or average strain) of a composite sample, which is subjected to homogeneous boundary conditions. In the generalized self-consistent scheme (GSCS) which is today a classical micromechanics model suited for the determination of the effective properties of matrix-based composites, those average quantities are estimated by using an auxiliary configuration in which a particulate phase is first surrounded by some matrix material and then embedded in the effective medium. In the present study, we revisit the GSCS both for two- and multi-phase matrix-based composites containing spherical particles, and clarify aspects related to the volume fractions of the particle core and matrix shell within the composite element which is embedded in the effective medium. The contribution of this study is believed to be mainly on the conceptual side and resides in a new formulation of the method in which the embedding volume fractions are determined in the course of the analysis by means of some fundamental relations on the averaged fields. The study is carried out in thermal conduction and elasticity and contains new results on the effective shear modulus of multi-phase composites.

Prediction of properties of Al2O3-C refractory based on microstructure by an improved generalized self-consistent scheme

An improved generalized self-consistent scheme (GSCS), which has a simplified computation process, was proposed to predict the properties, based on microstructure, of multiphase materials. The results of the prediction of the elastic modulus, Poisson’s ratio, coefficient of thermal expansion, and coefficient of thermal conductivity of an Al2O3-C refractory using a multiscale homogenization method with this improved GSCS agree with the experimental results. It is found that understanding the microstructure and choosing a suitable matrix for the prediction are very important for prediction accuracy.

Inelastic electron tunneling spectroscopy in molecular junctions: Peaks and dips

We study inelastic electron tunneling through a molecular junction using the nonequilibrium Green's function formalism. The effect of the mutual influence between the phonon and the electron subsystems on the electron tunneling process is considered within a general self-consistent scheme. Results of this calculation are compared to those obtained from the simpler Born approximation and the simplest perturbation theory approaches, and some shortcomings of the latter are pointed out. The self-consistent calculation allows also for evaluating other related quantities such as the power loss during electron conduction. Regarding the inelastic spectrum, two types of inelastic contributions are discussed. Features associated with real and virtual energy transfer to phonons are usually observed in the second derivative of the current I with respect to the voltage Phi when plotted against Phi. Signatures of resonant tunneling driven by an intermediate molecular ion appear as peaks in the first derivative dI/dPhi and may show phonon sidebands. The dependence of the observed vibrationally induced lineshapes on the junction characteristics, and the linewidth associated with these features are also discussed.

Micromechanics applied to the thermal shock behavior of refractory ceramics

Oxides–carbon refractories are ceramics with a complex microstructure due to the variety of components present in the composition: namely, oxide aggregates and metallic additions, microcracks, porosity and a continuous carbon phase. The main aim of this work is to highlight the role of each component of a refractory subjected to severe thermal loading through the use of micromechanics. First, a multi-scale generalized self-consistent scheme (GSCS) is used to predict the macroscopic thermoelastic behavior of the refractory. Experiments on two-component specimens are performed to determine, through an inverse analysis, the properties of some components that were difficult to collect or to measure directly. The micromechanical model is validated by comparison with experimental data. Second, a microthermomechanical analysis is used to study the behavior of brittle two-phase ceramics subjected to thermal shock. A macroscopic fracture criterion is proposed that accounts for local failure mechanisms. This approach allows one to improve the classical thermal shock criteria for fracture initiation by accounting for some mechanisms of local failure in tension and the heterogeneity of the microstructure of the material. Third, this approach is used to predict the material degradation of a submerged nozzle subjected to high thermal loads. Finite element computations permit us to understand the contribution of the heterogeneity of the material to the resistance to damage of the part and to compare this approach with the prediction of classical thermal shock criteria.

An inverse method to determine the elastic properties of the interphase between the aggregate and the cement paste

Concrete is a three-phase material consisting of cement paste matrix, discrete inclusions of rock (aggregate), and an interfacial transition zone (ITZ) between the matrix and the inclusions. We model the material as a composite formed by a matrix with embedded spherical particles; each surrounded by a concentric spherical shell. Effective elastic moduli of this composite are evaluated on the basis of the generalized self-consistent scheme (GSCS). This formulation is used to solve the inverse problem of determining the elastic moduli of the ITZ from experimentally known elastic properties of the composite.