Generalized Self-consistent Method Research Articles

Non-equilibrium phase transitions in active rank diffusions

We consider N run-and-tumble particles in one dimension interacting via a linear 1D Coulomb potential, an active version of the rank diffusion problem. It was solved previously for N = 2 leading to a stationary bound state in the attractive case. Here the evolution of the density fields is obtained in the large-N limit in terms of two coupled Burger's type equations. In the attractive case the exact stationary solution describes a non-trivial N-particle bound state, which exhibits transitions between a phase where the density is smooth with infinite support, a phase where the density has finite support and exhibits “shocks”, i.e., clusters of particles, at the edges, and a fully clustered phase. In the presence of an additional linear potential, the phase diagram, obtained for either sign of the interaction, is even richer, with additional partially expanding phases, with or without shocks. Finally, a general self-consistent method is introduced to treat more general interactions. The predictions are tested through extensive numerical simulations.

Acoustic waves in random distributions of double porosity cylinders

Effective quantities – wavenumber, bulk modulus and mass density – are sought for a random distribution of double porosity parallel circular cylinders saturated by the surrounded fluid. For this purpose, a Generalized Self Consistent Method (GSCM) is applied to Linton and Martin’s formula for the wavenumber, which accounts for multiple scattering between cylinders. Since Linton and Martin’s formula contains the Independent Scattering Approximation (ISA), the simplest case of GSCM applied to ISA is also examined. The analysis is restricted to low frequencies where the implicit equations derived from generalized self consistent schemes can be solved analytically. The study focuses especially on the dependence of the effective quantities with the P1, P2, P3 and S waves propagating in the double porosity domains of the heterogeneous medium. Influence of volume fraction of scatterers and of porosity is illustrated.

Extended mean-field homogenization of unidirectional piezoelectric nanocomposites with generalized Gurtin-Murdoch interfaces

This paper presents for the first time an extended Mori-Tanaka approach aimed at identifying the little-explored piezoelectric response of unidirectional nanoporous composites with energetic surfaces. The interface is simulated using the generalized Gurtin-Murdoch coherent interface model considering nonvanishing in-plane surface stress and surface electric displacement. The analytical solutions for Eshelby’s inhomogeneity problems are obtained by solving four systems of equations, generated from the known macroscopic electromechanical loading conditions. The dilute concentration tensors for the fiber and energetic surface are evaluated exactly for coupled electromechanical fields, which are then utilized in the extended multiphysics Mori-Tanaka homogenization scheme. The reliability and accuracy of the extended theory are demonstrated through an extensive comparison with the predictions of the composite cylinder assemblage (CCA), the generalized self-consistent method (GSCM), as well as the finite-element (FE) homogenization technique. New results generated in this work demonstrate that, except for the transverse shear modulus, the extended Mori-Tanaka, CCA, and FE provide indistinguishable results under axisymmetric, axial shear, and electric loading, even at high volume fractions and small pore sizes where the surface piezoelectric effects are significant.

Homogenization of size-dependent multiphysics behavior of nanostructured piezoelectric composites with energetic surfaces

Surface piezoelectricity considering the extended Gurtin--Murdoch coherent interface model has been incorporated into the composite cylinder assemblage (CCA), generalized self-consistent method (GSCM), as well as the multiphysics finite-element micromechanics (MFEM), for simulating the size-dependent multiphysics response of nanoporous materials wherein interface stress and electric displacement prevail. In the case of the CCA/GSCM model, the coherent interface model is implemented through the generalized Young--Laplace equations that govern the variation of the surface stress and the surface electric displacement. Three loading modes are utilized to identify the closed-form solutions for a complete set of Hill's moduli, and piezoelectric and dielectric constants. In the case of the MFEM, surface piezoelectricity is incorporated directly through additional surface energies associated with the elements that stretch along the interface. In order to assess the accuracy of the developed computational approaches, the generalized Kirsch problem under far-field transverse electric displacement loading is developed for recovering electric displacement concentration in the vicinity of the pore boundary. Homogenized properties are generated and critically examined for a broad variety of parameters and dimensions, predicted by the CCA/GSCM and MFEM methods. It is shown that all the predicted effective properties of these two families of homogenization techniques are similar except for the transverse shear moduli where they show marked differences that are reminiscent of what has been observed in the absence of surface electricity.

Predicting the chloride diffusion in concrete incorporating fly ash by a multi-scale model

A prediction model was developed in this paper to quantify the effect of fly ash on the chloride diffusion process in concrete samples. To capture the hierarchical characteristics of constituents at multiple scales, the model adopted the general self-consistent (GSC) method to calculate the effective chloride diffusivity of fly ash concrete. Moreover, the proposed model adopted the Papadakis model to estimate the amount of hydration products of fly ash pozzolanic reaction as well as cement hydration. In addition, the chloride diffusivity of fly ash concrete was corrected by introducing the filling effect of unreacted fly ash particles and the aggregate tortuosity factor. Beside the theoretical modelling, this study also carried out RCM and RCPT tests on concrete samples with different replace ratio of fly ash, whose measurements were used to validate the proposed model. A good agreement was obtained between model calculations and experimental measurements demonstrates the proposed model can successfully characterize the effect of fly ash on chloride diffusion in concrete samples, which can be an effective means to assessing the durability of fly ash concrete materials and structures in practical engineering.

Homogenization accounting for size effects in particulate composites due to general interfaces

Two analytical approaches are developed to determine the overall size-dependent response of composites embedding general interfaces. The first approach extends the composite sphere assemblage (CSA) approach and the generalized self-consistent method (GSCM) to account for the general interface model resulting in new bounds and estimates on the macroscopic properties of particulate composites. In the second approach, we develop an interface-enhanced Mori–Tanaka method that not only determines the effective properties but also provides the state of the stress and strain in each phase of the medium. The general interface model captures both elastic and cohesive interface models. Computational analysis is carried out using the finite element method to verify the analytical results. A remarkable agreement between the proposed analytical solutions and the computational results is obtained. A thorough parametric study is carried out to shed light on the role of the general interfaces in the overall behavior of composites. Motivated by the numerical and analytical findings, the material behavior is found to be bounded. Thus, two notions of ultimate bounds and size-dependent bounds are introduced and discussed.

Stress fields and effective modulus of piezoelectric fiber composite with arbitrary shaped inclusion under in-plane mechanical and anti-plane electric loadings

The piezoelectric fiber composite, which has been widely used in vibration control of aeronautic and aerospace structures, contains an active layer constructed by piezoceramic fibers embedded in polymer matrix. In this paper, an analytical framework is developed to predict the stress fields and the effective elastic and piezoelectric modulus of a piezoelectric fiber composite with arbitrary shaped inclusion under in-plane mechanical and anti-plane electric loadings. Firstly, a three-phase model is presented based on the general self-consistent method. It assumes that a representative volume element, consisting of an arbitrary shaped piezoelectric fiber and non-piezoelectric matrix, is embedded in infinite equivalent medium of piezoelectric fiber composite. Also, in order to obtain the complex potentials in different phases, a new conformal mapping method is proposed to translate two arbitrary connected domains into an annular domain. Then, the general solution of each complex potential can be expressed in series form with some coefficients. The continuity conditions of the interfaces and the homogeneous conditions are used to build up a set of equations to determine the coefficients. After the complex potentials are solved, the exact stress responses and effective modulus of piezoelectric fiber composite with arbitrary shaped inclusion are obtained. It is shown that the piezoelectric fiber shape and volume fraction have significant influence on the stress distributions around the piezoelectric fiber and effective modulus of the piezoelectric fiber composite. Furthermore, the proposed analytical framework provides a powerful instrument for solving related problems about fiber-reinforced composites with imperfect interface or with coated phase.

Comparison between the Mori-Tanaka and generalized self-consistent methods in the framework of anti-plane strain inclusion problem in strain gradient elasticity

In the present paper, two different averaging schemes are compared in the framework of the anti-plane inclusion problem of the strain gradient elasticity. Effective longitudinal shear modulus of fiber-reinforced composites was evaluated based on the Mori-Tanaka method (MTM) within the direct approach and based on the generalized self-consistent method (GSCM) within an energy approach. It is known that, in gradient theories, the non-uniform fields are realized inside the inhomogeneity even in the presence of a uniform strain or stress field prescribed at the infinity. Therefore, in MTM, an averaged concentration tensor is utilized. GSCM is used together with Eshelby's formula, which evaluates the total strain energy of the composite representative fragment by a particular surface integral. Thus, in this energy-based approach, one does not need to use averaged field quantities. The two mentioned approaches were compared and validated by finite element modeling in the framework of the simplified strain gradient elasticity theory. It was shown that both methods capture the inclusion size effects, however the quantitative predictions of these methods are significantly different for high volume fraction of small inclusions. It was shown that MTM underestimates the effective modulus compared to numerical simulations, while GSCM provides a sufficiently good solution. Thus, it seems that energy-based averaging schemes are preferable in the framework of gradient elasticity. Nevertheless, the MTM can be useful for the effective properties approximate evaluation of composites with small volume fraction of inclusions.

Equivalent inclusions in micromechanics with interface energy effect

In order to apply classical micromechanics in predicting the effective prop-erties of nanocomposites incorporating interface energy, a concept of equivalent inclusion (EI) is usually adopted. The properties of EI are obtained by embedding a single inclusion with the interface into an infinite matrix. However, whether such an EI is universal for different micromechanics-based methods is rarely discussed in the literature. In this pa-per, the interface energy theory is used to study the applicability of the above mentioned EI. It is found that some elastic properties of the EI are related only to the properties of the inclusion and the interface, whereas others are also related to the properties of the matrix. The former properties of the EI can be applied to both the classical Mori-Tanaka method (MTM) and the generalized self-consistent method (GSCM). However, the latter can be applied only to the MTM. Two kinds of new EIs are proposed for the GSCM and used to estimate the effective mechanical properties of nanocomposites.

The application of general self-consistent model on mechanical behaviour of fibre-reinforced cementitious composites

Fibre-reinforced cementitious composite (FRCC), by adding short discrete fibres randomly in cementitious composites, exhibits substantially improved mechanical properties than conventional cementitious composites due to the fibre bridging action and the existence of multiple micro-cracks. In this paper, based on the general self-consistent method a new micromechanical analytical model is developed to model the material properties of the FRCCs. A hexagonal-shaped representative volume element model with multiple micro-cracks is established based on the microstructure of the FRCC to represent the transversely isotropic material characteristics. The developed model is used to evaluate the equivalent Young’s modulus of a typical FRCC before and after cracking, and the results obtained are compared to those obtained from other analytical models and experimental data for validation. The influence of the aligned orientation of fibre and the crack density of the matrix on Young’s modulus of the composite is also studied.

Effective properties of coated fibrous piezoelectric composites with spring-type interfaces under anti-plane mechanical and in-plane electrical loads

In this paper, we investigate the effective properties of three-phase (matrix/coating/fiber) cylindrical piezoelectric composites with imperfect interfaces under anti-plane mechanical and in-plane electrical loads. By using the electromechanically coupling spring-type interface model and the generalized self-consistent method (GSM), we analytically derived the effective electroelastic moduli. The present solutions include as special cases the three-phase cylindrical piezoelectric composites with perfect interfaces as well as the two-phase (matrix/fiber) case with imperfect or perfect interfaces. Selected calculations are graphically shown to demonstrate dependence of the effective moduli on the interfacial properties. The particular size-dependent characteristic due to the interfacial imperfection is also discussed.

Rayleigh limits for effective wavenumbers of randomly distributed porous cylinders. Comparison of explicit and implicit methods

The effective wavenumber of the coherent wave propagating in a fluid containing parallel porous cylinders randomly distributed in space is derived at the Rayleigh limit for (i) explicit formulas: Independent Scattering Approximation (ISA), Waterman and Truell (WT) and Linton and Martin (LM) and (ii) for implicit formulas: Coherent Potential Approximation (CPA) and Generalized Self Consistent Method (GSCM) applied to WT and to LM. The effective mass density and bulk modulus are also derived. The validity of all the effective quantities is checked by recovering, when the porosity of the scatterers tends to zero, the case of an inhomogeneous medium of elastic cylinders.

Effective elastic moduli of fiber-reinforced composites with interfacial displacement and stress jumps

The generalized self-consistent method (GSCM) is applied to obtain the closed-form estimates for the 5 elastic moduli of a transversely isotropic composite consisting of an isotropic matrix reinforced by unidirectional isotropic fibers of circular cross-section. The interfaces between the matrix and the fibers in this composite are taken to be imperfect and characterized by the general elastic isotropic model which includes as particular cases the widely used elastic spring-layer and membrane-type imperfect interface models. The displacement, strain and stress fields in an infinite homogeneous medium containing a composite cylinder with a general imperfect interface and subjected to each of 4 elementary remote uniform loadings are specified and used in deriving the estimates for the effective elastic moduli. These results extend and bridge the special relevant results reported in the literature for fiber-reinforced composites with imperfect interfaces characterized by the spring-layer or membrane-type imperfect interface model. Numerical examples are also given to illustrate some results.

Truly self-consistent solution of Kohn-Sham equations for extended systems with inhomogeneous electron gas

The density functional approach in the Kohn-Sham approximation is widely used to study properties of many-electron systems. Due to the nonlinearity of the Kohn-Sham equations, the general self-consistent solution method for infinite systems involves iterations with alternate solutions of the Poisson and Schrödinger equations. One of problems with such an approach is that the charge distribution, updated by solving the Schrodinger equation, may be incompatible with the boundary conditions of the Poisson equation for Coulomb potential. The resulting instability or divergence manifests itself most appreciably in the case of infinitely extended systems because the corresponding boundary-value problem becomes singular. In this work the stable iterative scheme for solving the Kohn-Sham equations for infinite systems with inhomogeneous electron gas is described based on eliminating the long-range character of the Coulomb interaction, which causes the tight coupling of the charge distribution with the boundary conditions. This algorithm has been previously successfully implemented in the calculation of work function and surface energy of simple metals in the jellium model. Here it is used to calculate the energy spectrum of quasi-two-dimensional electron gas in the accumulation layer at the semiconductor surface n-InAs. The electrons in such a structure occupy states that belong to both discrete and continuous parts of the energy spectrum. This causes the problems of convergence in the usually used approaches, which do not exist in our case. Because of the narrow bandgap of InAs, it is necessary to take the nonparabolicity of the conduction band into account; this is done by means of a new effective mass method. The calculated quasi-two-dimensional energy bands correspond well to experimental data measured by the angle resolved photoelectron spectroscopy technique.

A Novel Approach 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 sand (aggregate), and an interfacial transition zone (ITZ) between the matrix and the inclusions. In order to calculate the elastic properties of the ITZ of cement mortar, We model the material as a composite formed by a matrix with embedded spherical particles; each surrounded by a concentric spherical shell. With help of the generalized self-consistent method (GSCM), equations of bulk modulus and shear modulus of ITZ are deduced, and Elastic properties of ITZ are computed by using experimentally known elastic properties of the composite. It is found that the shear modulus of ITZ is about 50% of that of the cement paste matrix.

Effective Properties of Electro-Elastic Composites with Multi-Coating Inhomogeneities

This work presents a micromechanics-based model to investigate the effective thermo-electric properties of piezoelectric composite materials. The effective thermo-electric properties are derived by considering a multi-coated ellipsoidal inhomogeneity embedded in a host material in the framework of the generalized self-consistent method (GSCM). An incremental scheme, in which the reinforcements are incrementally put in the host material, is implemented. The validation of the micromechanical model is performed with experimental data. The model proposed has a wide range of applications and can be extended to other physical properties.

On the effective electroelastic properties of microcracked generally anisotropic solids

In this study we first obtain the explicit expressions for the 15 effective reduced elastic compliances of an elastically anisotropic solid containing multiple microcracks with an arbitrary degree of alignment under two-dimensional deformations within the framework of the non-interaction approximation (NIA). Under special situations, our results can reduce to the classical ones derived by Bristow (J Appl Phys 11: 81–85, 1960), and Mauge and Kachanov (J Mech Phys Solids 42(4):561–584, 1994). Some interesting phenomena are also observed. For example, when the undamaged solid is orthotropic, the effective in-plane shear modulus is dependent on the degree of the crack alignment. The NIA method is then extended to obtain the effective electroelastic properties of an anisotropic piezoelectric solid containing two-dimensional insulat- ing, permeable or conducting microcracks with an arbitrary degree of alignment. We also derive a set of fifteen coupled nonlinear equations for the unknown effective reduced elastic compliances of a microcrac- ked, anisotropic, elastic solid by using the generalized self-consistent method (GSCM). The set of coupled nonlinear equations can be solved through iteration.

Anisotropic effective moduli of microcracked materials under antiplane loading

This study focuses on the prediction of the anisotropic effective elastic moduli of a solid containing microcracks with an arbitrary degree of alignment by using the generalized self-consistent method (GSCM). The effective elastic moduli pertaining to anti-plane shear deformation are discussed in detail. The undamaged solid can be isotropic as well as anisotropic. When the undamaged solid is isotropic, the GSCM can be realized exactly. When the undamaged solid is anisotropic it is difficult to provide an analytical solution for the crack opening displacement to be used in the GSCM, thus an approximation of the GSCM is pursued in this case. The explicit expressions of coupled nonlinear equations for the unknown effective moduli are obtained. The coupled nonlinear equations are easily solved through iteration.

Poroelastic properties of anisotropic cylindrical composite materials

The present work is devoted to the determination of the effective poroelastic properties of anisotropic materials, such as porous nanocomposites and microbiotissues. The poroelastic field of composite materials in a transversely isotropic medium is determined by using the general self-consistent method (GSCM), and the dependence of material properties of the composites on porosity is considered. A hexagonal transversely isotropic model is used and treatment processes are proposed for application to microstructures or macrostructures. Several numerical examples are presented to validate the proposed model, and the results obtained from the model are compared with those obtained from a random-array model, a hexagonal-array model using the strain energy approach (SEA), the dilute suspension of the GSCM, the self-consistent method (SCM) and the effective self-consistent method (ESCM). An explicit solution of poroelasticity is provided, and the effects of the solid matrix anisotropy and pore space on the effective poromechanical properties considered. The numerical results for the hexagonal microstructure using the GSCM agree much better with those obtained from experimental investigation in both wet and dry situations than those obtained from other methods. Finally, the influences of fatigue on micromechanical properties for wet and dry conditions are also described.

An exact solution for the three-phase thermo-electro-magneto-elastic cylinder model and its application to piezoelectric–magnetic fiber composites

A three-phase cylindrical model for analyzing fiber composite subject to in-plane mechanical load under the coupling effects of multiple physical fields (thermo, electric, magnetic and elastic) is presented. By introducing an eigenstrain corresponding to the thermo-electro-magnetic-elastic effect, the complex multi-field coupling problem can be reduced to a formal in-plane elasticity problem for which an exact closed form solution is available. The present three-phase model can be applied to fiber/interphase/matrix composites, such that a lot of interesting thermo-electro-magnetism and stress coupling phenomena induced by the interphase layer are revealed. The present model can also be applied to fiber/matrix composites, in terms of which a generalized self-consistent method (GSCM) is developed for predicting the effective properties of piezoelectric–magnetic fiber reinforced composites. The effective piezoelectric, piezomagnetic, thermoelectric and magnetoelectric moduli can be expressed in compact explicit formulae for direct references and applications. A comparison of the predictions by the GSCM with available experimental data is presented, and interesting magnification effects and peculiar product properties are discussed. As a theoretical basis for the GSCM, the equivalence of the three sets of different average field equations in predicting the effective properties are proved, and this fact provides a strong evidence of mathematical rigor and physical realism in the formulation.