Effective Field Method Research Articles

Efficient homogenization techniques for elastic composites: Maxwell scheme vs. effective field method

The work is devoted to the comparative analysis of four homogenization techniques (the original and generalized Maxwell schemes and the one-particle and multi-particle effective field methods) in application to the problem of calculation of the effective elastic stiffness tensor of matrix composites. The generalized Maxwell scheme and multi-particle effective field method reduce the problem to calculation of elastic fields in a finite volume of the composite embedded in the infinite homogeneous matrix medium and subjected to a constant external stress or strain field. For the solution of this problem, an efficient numerical method based on Gaussian approximating functions and fast Fourier transform technique is used. The results of the methods are compared for composites with ellipsoidal inclusions which elastic moduli are much smaller or much larger than the moduli of the matrix. The case of hybrid composites with two different families of ellipsoidal inclusion is also considered. It is shown that for various shapes and elastic properties of inclusions, the generalized Maxwell scheme and multi-particle effective field method give close results. But in the case of hard inclusions of large volume fractions the results of these methods deviate substantially from the original Maxwell scheme that does not take into account interactions between inclusions.

Method of uniform effective field in structure-dynamic approach of nanoionics

In the structure-dynamic approach of nanoionics, the method of a uniform effective field \( {F}_{\mathrm{eff}}^{j,k} \) of a crystallographic planeXjhas been substantiated for solid electrolyte nanostructures. The \( {F}_{\mathrm{eff}}^{j,k} \)is defined as an approximation of a non-uniform field \( {F}_{\mathrm{dis}}^j \)of Xj with a discrete- random distribution of excess point charges. The parameters of \( {F}_{\mathrm{eff}}^{j,k} \)are calculated by correction of the uniform Gauss field \( {F}_{\mathrm{G}}^j \) of Xj. The change in an average frequency of ionic jumps Xk → Xk + 1 between adjacent planes of nanostructure is determined by the sum of field additives to the barrier heights ηk,k + 1, and for \( {F}_{\mathrm{G}}^j \) and \( {F}_{\mathrm{dis}}^j \), these sums are the same decimal order of magnitude. For nanostructures with length ~4 nm, the application of \( {F}_{\mathrm{G}}^j \) (as \( {F}_{\mathrm{eff}}^{j,k} \)) gives the accuracy ~20 % in calculations of ion transport characteristics. The computer explorations of the “universal” dynamic response (Reσ∗ ∝ ωn) show an approximately the same power n < ≈1 for\( {F}_{\mathrm{G}}^j \) and \( {F}_{\mathrm{eff}}^{j,k} \).

Interacting cracks 3D analysis using boundary integral equation method

This paper presents a modification of the method of boundary integral equations suitable for the efficient solution of 3D problems on the arbitrarily oriented plane cracks interaction with the influence of body surface. The hypersingular boundary integral equations on the crack-surface are transformed into new form, where the solution behavior near the crack front is accounted implicitly. This modification allows the direct determination of the stress intensity factors (SIF) in the crack vicinity after solution of equations by the collocation technique. We also propose the approach based on the determination of the effective stress field formed in the vicinity of a fixed crack by neighboring cracks interacting with this crack and with boundary surface. Numerical examples concern an asymmetric problem for interacting penny-shaped plane cracks in the unlimited and limited bodies. The reliability of the results obtained by the method of effective stress field is checked by comparing with the exact solution of the problem of interaction of two penny-shaped cracks.

Investigation of the mechanical properties of nanocomposite SWCNTS/epoxy by micromechanics methods and experimental works

In the present paper, the stiffening effect of carbon nanotubes is quantitatively investigated by micromechanics methods. The Mori-Tanaka effective-field method is employed to calculate the effective elastic moduli of composites with aligned or randomly oriented straight nanotubes. In addition, the epoxy resin is modified experimentally by adding SWCNT with different ratio i.e 0, 0.1, 0.3, 0.5 and 0.7 wt.-%. A comparison between the results for SWCNT/epoxy nanocomposite which obtained analytically and experimentally is done.In the experimental work the epoxy resin is modified by adding SWCNT with different ratio i.e 0, 0.1, 0.3, 0.5 and 0.7 wt.-%. The materials are characterized in tension to obtain the mechanical properties of SWCNT/epoxy nanocomposite experimentally. The results of micromechanics methods indicated that the CNTs are highly anisotropic, with Young’s modulus in the tube direction two orders of magnitude higher than that normal to the tube. The results shows a nanotube volume fraction of 0.3%of SWCNT improve all mechanical properties such as the tensile strength, modulus of elasticity and the toughness. Avoid the volume fraction greater than 0.5% SWCNT. The optimal value achieved experimentally, (at 0.3% SWCNT) lies between the analytical values (that achieved parallel to the CNT and the randomly orientated straight CNTs).

Effective hamiltonian crystal field: Present status and applications to iron compounds

We review the basics of the Effective Hamiltonian Crystal Field (EHCF) method originally targeted for calculations of the intra‐shell excitations in the d‐shells of coordination compounds of the first row transition metal. The formalism employs in the concerted way the McWeeny's group‐function approximation and the Lowdin partition technique. It is needed for description of the transition metal complexes with partially filled d‐shells where the (static) electronic correlations are manifested. These features are particularly important for electron fillings close to “half shell” ones occurring, for example, in the Fe2+ and Fe3+ ions. Recently we extended this methodology to polynuclear coordination compounds to describe magnetic interactions of the effective spins residing in several open d‐shells. This improves the accuracy from about 1000 cm−1 to that of about 100 cm−1, that is, eventually by an order of magnitude. This approach implemented in the MagAixTic package is applied here to a series of binuclear Fe(III) complexes featuring μ‐oxygen superexchange pathways. The results of calculations are in a reasonable agreement with available experimental data and other theoretical studies of protonated bridges. Further we discuss the application of the EHCF to analysis of Mosbauer experiments performed on two organometallic solids: FeNCN and Fe(HNCN)2 and conjecture a new thermal effect in the latter material. © 2015 Wiley Periodicals, Inc.

Prediction and validation of total and regional skeletal muscle volume using B-mode ultrasonography in Japanese prepubertal children.

Very few effective field methods are available for accurate, non-invasive estimation of skeletal muscle volume (SMV) and mass in children. We aimed to develop regression-based prediction equations for SMV, using ultrasonography, in Japanese prepubertal children, and to assess the validity of these equations. In total, 145 healthy Japanese prepubertal children aged 6-12 years were randomly divided into two groups: the model development group (sixty boys, thirty-seven girls) and the validation group (twenty-nine boys, nineteen girls). Reference data in the form of contiguous MRI with 1-cm slice thickness were obtained from the first cervical vertebra to the ankle joints. The SMV was calculated by the summation of digitised cross-sectional areas. Muscle thickness was measured using B-mode ultrasonography at nine sites in different regions. In the model development group, strong, statistically significant correlations were observed between the site-matched SMV (total, arms, trunk, thigh and lower legs) measured by MRI and the muscle thickness×height measures obtained by ultrasonography, for both boys and girls. When these SMV prediction equations were applied to the validation groups, the measured total and regional SMV were also very similar to the values predicted for boys and girls, respectively. With the exception of the trunk region in girls, the Bland-Altman analysis for the validation group did not indicate any bias for either boys or girls. These results suggest that ultrasonography-derived prediction equations for boys and girls are useful for the estimation of total and regional SMV.

Acoustic field generated by an axisymmetric source embedded in a fluid-filled cylindrical hole located in a micro-cracked medium

We have calculated acoustic field generated by an impulse acoustic source in a fluid-filled cylindrical cavity located in a transversely isotropic medium. It is assumed that the anisotropy is caused by presence of crack-like inclusions. We derive effective elastic constants of the surrounding medium using the effective field method. We have shown that in the case of spheroidal crack-like inclusions with normals parallel to the axis of symmetry of the cavity the crack density can be obtained in terms of velocities of zero-order guided wave (the Stoneley wave) and shear wave propagating along the cavity (the SV-wave in the seismic terminology).

Prediction of the elastic modulus of SWCNT/epoxy composite based on the micromechanics

Abstract Due to their superior mechanical and physical properties, carbon nanotubes seem to hold a great promise as an ideal reinforcing material for composites of high strength and low density. In most of the experimental results up to date, however, only modest improvements in the strength and stiffness have been achieved by incorporating carbon nanotubes in polymers. In the present paper, the stiffening effect of carbon nanotubes is quantitatively investigated by micromechanics methods. The Mori-Tanaka effective field method has been adopted to calculate the effective elastic moduli of composites with aligned or randomly oriented straight nanotubes. The rule-of-mixtures is used to calculate the modulus of elasticity for nanocomposite. The results of micromechanics methods indicated that the CNTs are highly anisotropic, with Young's modulus in the tube direction two orders of magnitude higher than that normal to the tube. The results of micromechanics methods were compared by those obtained from the rule-of-mixtures and good agreement was also achieved when the efficiency parameter Φ = 1 and typical results were achieved with Φ = 0.25. To predict the mechanical properties of the composite materials, it is worth considering the conventional rule-of-mixtures using exact value of the efficiency parameter Φ. To predict the elastic modulus of nanocomposite reinforced by SWCNT using the conventional rule-of-mixtures, the exact value of the efficiency parameter Φ is equal to 0.25 when using nanotubes with chirality (8,3) for determination the elastic modulus of SWCNT. However, for zigzag orientation and chirality (8,0) the efficiency parameter Φ is equal to 1. The conventional rule-of-mixtures is a powerful tool and easy method compared to the micromechanics methods.

A Phenomenological Description of the MPS Signal Using a Model for the Field Dependence of the Effective Relaxation Time

In this paper, we present a new approach to describe the magnetic particle spectroscopy (MPS) signal, which is based on Shliomis' effective field method. Contrary to other approaches, we do not model single particle dynamics via the Landau-Lifshitz-Gilbert equation, but use measurable quantities to minimize the influence of unknown model parameters. The necessary material values are gained from static magnetization and magnetorelaxometry measurements. The model contains two free parameters to describe the field-dependent relaxation time. To obtain these parameters, we currently still need measured MPS data to find the relaxation parameters that match the experimental results, but we are looking into methods to avoid this matching. We present the equations to calculate the harmonic spectrum of an magnetic particle imaging tracer. We also present first results of this method obtained on FeraSpin R and other tracer formulations and compare the calculated spectrum to the measured harmonics. For all formulations, we found a coefficient of determination R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> >0.99 between our model and the measured spectra.

Size-dependent effective properties of anisotropic piezoelectric composites with piezoelectric nano-particles

Based on the electro-elastic surface/interface theory, the size-dependent effective piezoelectric and dielectric coefficients of anisotropic piezoelectric composites that consist of spherically piezoelectric inclusions under a uniform electric field are investigated, and the analytical solutions for the elastic displacement and electric potentials are derived. With consideration of the coupling effects of elasticity, permittivity and piezoelectricity, the effective field method is introduced to derive the effective dielectric and piezoelectric responses in the dilute limit. The numerical examples show that the effective dielectric constant exhibits a significant variation due to the surface/interface effect. The dielectric property of the surface/interface displays greater effect than the piezoelectric property, and the elastic property shows little effect. A comparison with the existing results validates the present approach.

Effective Hamiltonian crystal fields: Present status and applicability to magnetic interactions in polynuclear transition metal complexes

The fundamentals of the Effective Hamiltonian Crystal Field (EHCF) method, used originally to calculate intra-shell excitations in the d-shells of coordination compounds of the first row transition metals, are reviewed. The formalism of effective operators is applied to derive an explicit form of the effective operator for a dipole moment in d-shell electronic subspace, allowing us to calculate the oscillator strengths of optical d-d transitions, which are otherwise forbidden when treated in the standard EHCF approach. EHCF methodology is also extended to describing magnetic interactions of the effective spin in several open d-shells of polynuclear coordination compounds. The challenging task of improving a precision of ∼1000 cm−1 (describing the excitation energies of single d-shells by EHCF) to one of ∼100 cm−1 for the energies required to reorient spins by an order of magnitude is considered within the same paradigm as EHCF: the targeted use of McWeeny’s group function approximation and the Lowdin partition technique. These are applied to develop an effective description of a d-system. This approach is tested on a series of binuclear complexes [{(NH3)5M}2O]4+ of trivalent cations featuring oxygen super-exchange paths in order to confirm the reproducibility of the trends in the series of exchange constants values for compounds that differ in the nature of their metal ions. The results from calculations are in reasonable agreement with the available experimental data and other theoretical methods.

Dynamic effective elastic modulus of polymer matrix composites with dense piezoelectric nano-fibers considering surface/interface effect

Based on effective field method, the dynamic effective elastic modulus of polymer matrix composites embedded with dense piezoelectric nano-fibers is obtained, and the interacting effect of piezoelectric surfaces/interfaces around the nano-fibers is considered. The multiple scattering effects of harmonic anti-plane shear waves between the piezoelectric nano-fibers with surface/interface are averaged by effective field method. To analyze the interacting results among the random nano-fibers, the problem of two typical piezoelectric nano-fibers is introduced by employing the addition theorem of Bessel functions. Through numerical calculations, the influence of the distance between the randomly distributed piezoelectric nano-fibers under different surface/interface parameters is analyzed. The effect of piezoelectric property of surface/interface on the effective shear modulus under different volume fractions is also examined. Comparison with the simplified cases is given to validate this dynamic electro-elastic model.

An efficient homogenization method for elastic media with multiple cracks

An efficient numerical method for calculation of the effective stiffness tensor of 3D-elastic media with multiple interacting cracks is developed. A representative volume element (RVE) of the medium with a finite number of cracks is embedded into the background homogeneous medium. The presence of cracks outside of the RVE is accounted by placing the RVE into an effective external stress field. This field does not coincide with the external stress field applied to the cracked medium, and for construction of this field, a version of a self-consistent effective field method is proposed. The problem is reduced to a system of 2D-integral equations for the crack opening vectors inside the RVE. Discretization of these equations is performed by Gaussian approximation functions centered at a set of nodes uniformly distributed on crack surfaces. For such functions, construction of the matrix of the discretized problem is reduced to calculation of five standard 1-D integrals that can be tabulated. As a result, numerical integration is not necessary, and the matrix elements are calculated fast. Examples of various arrangements of planar cracks in isotropic elastic media are considered. The results are compared with numerical solutions and experimental data available in the literature.

Dynamic effective property of piezoelectric composites with coated piezoelectric nano-fibers

Abstract This paper addresses the size-dependent dynamic effective electro-elastic properties of piezoelectric composites embedded with coated piezoelectric nano-fibers. By using the Effective Field Method of electric–elastic coupling wave, the randomly distributed coated nano-fibers in the piezoelectric media are reduced into a typical piezoelectric coated nano-fiber. To analyze the size-dependent effect, the coupling surface/interface theory is introduced to analyze the interface effect around the coating layers. Wave function expansion method is used to express the coupling wave field in the effective coupling field. The piezoelectrically stiffened elastic modulus and effective coupling wave number are obtained. Through analysis, it is found that the coupling surface/interface effect on the piezoelectrically stiffened elastic modulus show significant variation with the material properties of coating layer and the incident wave frequency. The effect of piezoelectric property of surfaces/interfaces on the effective shear modulus is also examined. Comparison with the existing results is given to validate this dynamic electro-elastic model.

Effective shear modulus of piezoelectric film embedded with square nano-fibers under anti-plane shear waves

For piezoelectric films, the surface/interface energy around the nano-fibers plays an important role in predicting the overall properties. Square nano-fibers are very popular in piezoelectric films. This paper presents the size-dependent dynamic effective elastic modulus of piezoelectric films under anti-plane shear waves. The average electro-elastic field resulting from the randomly distributed nano-fibers is solved by applying the effective field method. To express the coupling fields in the effective field, the mapping method and wave function expansion method are introduced. The closed-form solution of piezoelectrically stiffened shear modulus with respect to the shape and volume fraction of fibers and surface/interface properties is obtained. The interacting effect of surface/interface and square shape of nano-fibers is examined.

Micromechanical analysis of the effect of void swelling on elastic and electric properties of irradiated steel

The paper focuses on changes in mechanical and electrical properties of austenitic steel due to radiation swelling. Various micromechanical models for the electrical conductivity and elastic moduli are assessed based on the experimental data available in literature. We show that, in the considered case, the effective field method provides the best prediction for both elastic and conductive properties. Comparing the two sets of experimental data, the cross-property connections that relate changes in the elastic and electrical conductivity of a material due to radiation swelling has been verified. It is shown that analytical cross-property connection is in a good agreement with experimental data and can be used to estimate changes in electrical resistivity of a material from the changes in Young’s modulus or vice versa.

On some controversial issues in effective field approaches to the problem of the overall elastic properties

Several versions of the effective field method of finding the overall elastic properties of matrix composites are examined and compared. We focus on difficulties and uncertainties encountered by these methods, in particular in cases of anisotropic multiphase composites. It is demonstrated that the schemes are best formulated in terms of the compliance/stiffness contributions tensors: such formulation exposes roots of various inconsistencies, and clarifies relations between different versions of the method. Particular attention is paid to Maxwell’s scheme which is shown to represent yet another version of the effective field method. The discussion can be extended to physical properties other than the elastic ones.

Effective Hamiltonian Crystal Field As Applied to Magnetic Exchange Parameters in μ-Oxo-Bridged Cr(III) Dimers

The calculation of the 3d-intrashell excitations in coordination compounds by means of the Effective Hamiltonian Crystal Field (EHCF) method is generalized to their polynuclear analogues to properly describe several open d-shells and their magnetic interactions. This challenge requires improving the precision of ca. 1000 cm(-1) to ca. 100 cm(-1) characteristic for the spin-reorientation energies. The method follows the successful EHCF paradigm, namely, the concerted usage of McWeeny's group-function approximation and Löwdin's partitioning technique, for an effective description of the multicenter d-systems. The novel approach is implemented in the MagAîxTic package and validated against a series of binuclear complexes of Cr(III) featuring μ-oxygen superexchange paths. The trends in the compound series in terms of exchange constants are correctly reproduced, despite differing details of composition and structure, and the numerical results agree by order of magnitude with available experimental data and other theoretical methods.

Effect of interface energy on effective dynamic properties of piezoelectric medium with randomly distributed piezoelectric nano-fibers

In the framework of coupling surface/interface theory, a self-consistent scheme of effective field method is proposed to solve the dynamic electro-elastic properties of piezoelectric nanocomposites. The piezoelectric matrix with randomly distributed piezoelectric nano-fibers is reduced to the problem of an effective piezoelectric medium with a typical piezoelectric nano-fiber. The effective wave number and piezoelectrically stiffened elastic modulus are obtained on the basis of the derived imperfect interface conditions around the piezoelectric nano-fiber. Analyses show that the energy at the surface/interface shows significant effect on the dynamic electro-elastic properties, and the effect under different incident frequencies is discussed in details.

Ferrohydrodynamic evaluation of rotational viscosity and relaxation in certain ferrofluids

A significant effect of aggregation dynamics for aqueous ferrofluid (AF) and kerosene based ferrofluid (KF) using magnetic field dependent capillary viscosity and magneto-optical relaxation measurements is studied. For better comparison parameters of AF and KF are kept similar. Ferrohydrodynamic equations of chain forming ferrofluids, dilute ferrofluids, and Brownian dynamic simulations are compared. It is observed that the rotational viscosity of AF is larger than that of KF due to field induced aggregates in it and strong dipolar interactions. It is also observed that at Ωτ ~ 0.04 both AF and KF viscosity becomes almost similar, suggesting similar behavior at that shear rate. The magneto-optical relaxation in AF exhibits nonexponential behavior when relaxed from higher magnetic field and follows irreversible thermodynamics, whereas for KF the relaxation is exponential and follows the effective field method. This discrepancy is explained based on aggregation dynamics of magnetic particles. Results are well described by the corresponding theoretical models.