Plane problems of multiple piezoelectric inclusions in a non-piezoelectric matrix

Based on the complex variable method, this paper addresses the plane problems of multiple piezoelectric inclusions in a non-piezoelectric matrix. The inclusions are assumed to be perfectly bounded to the matrix, which is loaded by in-plane mechanical loads while the inclusions are applied by anti-plane electric loads at infinity. The general solutions are first derived for the complex potentials both in the matrix and inside the inclusions, and then numerical results are presented to show the effects of applied electric field, inclusion arrays and material properties on the electroelastic fields around the inclusions. It is shown that the inclusion arrays have a significant influence on the stress distribution at the interface between the matrix and piezoelectric inclusions.     

Electrothermal stress in conducting particulate composites

Electrothermal–mechanical interaction plays an important role in controlling the performance of electromechanical structures and field-assisted processes. The understanding of electrothermal–mechanical behavior of a material requires the analyses of Joule heating and thermomechanical deformation. In this study, we analyze the current-induced thermal stress in a conducting composite consisting of conducting spherical inclusions at dilute concentration. Assuming that there is no interaction among conducting inclusions, we obtain closed-form solutions of local temperature and thermal stress. The thermal stress created by Joule heating is proportional to the square of electric current density (electric field intensity) and the von-Mises stress reaches the maximum value at the interface between the spherical inclusion and the matrix. Large electric current will likely cause local delamination along the interface.     

Interactions between N circular cylindrical inclusions in a piezoelectric matrix

Summary This paper studies the interactions between N randomly-distributed cylindrical inclusions in a piezoelectric matrix. The inclusions are assumed to be perfectly bounded to the matrix, which is subjected to an anti-plane shear stress and an in-plane electric field at infinity. Based on the complex variable method, the complex potentials in the matrix and inside the inclusions are first obtained in form of power series, and then approximate solutions for electroelastic fields are derived. Numerical examples are presented to discuss the influences of the inclusion array, inclusion size and inclusion properties on couple fields in the matrix and inclusions. Solutions for the case of an infinite piezoelectric matrix with N circular holes or an infinite elastic matrix containing N circular piezoelectric fibers can also be obtained as special cases of the present work. It is shown that the electroelastic field distribution in a piezoelectric material with multiple inclusions is significantly different from that in the case of a single inclusion.     

Volume integral equation method for multiple circular and elliptical inclusion problems in antiplane elastostatics

A Volume Integral Equation Method (VIEM) is introduced for the solution of elastostatic problems in an unbounded isotropic elastic solid containing interacting multiple isotropic and anisotropic circular/elliptical inclusions subject to remote antiplane shear. This method is applied to two-dimensional problems involving long parallel cylindrical inclusions. A detailed analysis of the stress field at the interface between the matrix and the central inclusion is carried out for square and hexagonal packing of isotropic and anisotropic inclusions. The effects of the number of isotropic and anisotropic inclusions and various fiber volume fractions on the stress field at the interface between the matrix and the central circular/elliptical inclusion are also investigated in detail. The accuracy of the method is validated by solving single isotropic and orthotropic circular/elliptical inclusion problems and multiple isotropic circular and elliptical inclusion problems for which solutions are available in the literature.     

Dynamic behaviour of a nanoscale-patterned phospholipid thin film on mica and silicon studied by atomic force microscopy

The temporal evolution of the morphology of nanoscale-patterned phospholipid thin films on mica and silicon surfaces has been investigated with an atomic force microscope (AFM). The AFM images reveal that nanoscale scratch lines on thin films prepared on mica contract with time and eventually form roundish holes. An elevated sample temperature accelerates this morphological evolution. We model such an evolution based on the interplay of the thin-film surface line tension and the friction between the thin film and the substrate. The results show that the temperature-dependent contraction is governed by the ratio of the friction coefficient and the surface line tension. The friction at the lipid/mica interface decreases to a seventh as the sample temperature rises from 18 to 60 °C. This model is supported by experiments on silicon surfaces, on which contraction of the scratch patterns is limited because of an expected greater interfacial friction.     

Inclusion agglomeration in electrified molten metal: thermodynamic consideration

ABSTRACT

The effect of electric current on inclusion agglomeration in molten metal has been investigated. It is found that the agglomeration is dependent on the electric current density, distance between inclusions and orientation of electric field. Electric current retards the agglomeration unless two inclusions are aligned along or closely to the current flow streamlines and the distance between inclusions is less than a critical value. The mechanism is also validated in the computation of cluster agglomeration. The numerical results provide a comprehensive indication for the current-induced inclusion removal and current-induced inclusion elongation. When the inclusions are in long-thin shape, the calculation predicts the current-induced microstructure alignment and current-induced microstructure refinement phenomena.

This paper is part of a themed issue on Materials in External Fields.     

Effects of inclusion on the creep rate of piezoelectric films

This paper presents an analytical method to study the effect of inclusions in piezoelectric materials on the creep rate. The driven force for the creep rate of piezoelectric materials with inclusion is from diffusional mass transport along the inclusion interface. The results show that the creep rate of piezoelectric materials containing the rigid inclusion with the shape parameter m = 0.8 appears at the minimum, and the effect of inclusion volume fraction on the creep rate of piezoelectric materials with soft inclusion becomes the smaller and smaller, as the stiffness of soft inclusion decreases. Thus, the effect of inclusion on creep characteristics can be improved by controlling the material property, the sizes, shapes, and volume ratio of inclusions.     

Modified Eshelby tensor for an ellipsoidal inclusion imperfectly embedded in an infinite piezoelectric medium

The modified Eshelby tensor for predicting the effective moduli of particle-reinforced piezoelectric composites is derived for the problem of an ellipsoidal inclusion which is imperfectly bonded to the matrix. A linear interface relation is adopted, which involves discontinuities of the mechanical displacements and electric potential across the interface, and assumes that the corresponding jumps are proportional to the continuous stresses and electric displacements at the interface. The piezoelectric field induced by a uniform eigenstrain given only in the inclusion is deduced analytically. As the induced piezoelectric field is no longer uniform, the average strains and electric displacements are calculated, and the modified piezoelectric Eshelby tensor is evaluated by both an iterative method and a direct method. By comparison, it is shown that the iterative method yields rapidly convergent results.     

Interaction between screw dislocations and inclusions with imperfect interfaces in fiber-reinforced composites

A three-phase composite cylinder model is utilized to study the elastic interaction between screw dislocations and embedded multiple circular cross-section inclusions (fibers) with imperfect interfaces in composites. By means of complex variable techniques, the explicit solutions of stress and displacement fields are obtained. With the aid of the Peach–Koehler formula, the explicit expressions of image forces exerted on screw dislocations are derived. The equilibrium positions of the appointed screw dislocation near one of the inclusions are discussed for variable parameters (interface imperfection, material mismatch and dislocation position) and the influence of the nearby inclusions and dislocations is also considered. The results show that, if the inclusion is stiffer than the matrix and the magnitude of the degree of interface imperfection reaches the certain value, a new equilibrium position for the screw dislocation in the matrix can always be produced in comparison with the previous solution (the perfect interface). The effect of elastic constants of the inclusion on the image force and the equilibrium position of the appointed screw dislocation is weak when the interface imperfection is strong. It is also seen that the magnitude of the image force exerted on the appointed dislocation caused by multiple inclusions is always smaller than that produced by a single inclusion. The impact of the closer dislocations on the mobility of the appointed dislocation is very significant.     

A piezoelectric screw dislocation in a three-phase composite cylinder model with an imperfect interface

The electroelastic coupling interaction between a piezoelectric screw dislocation and the embedded circular cross-section inclusions with imperfect interfaces in piezoelectric solids is investigated by using a three-phase composite cylinder model. By means of a complex variable technique, the explicit solutions of electroelastic fields are obtained. With the aid of the Peach-Koehler formula, the explicit expression for the image force exerted on the piezoelectric screw dislocation is derived. The image force on the dislocation and its equilibrium positions near one of the inclusions are discussed for variable parameters (interface imperfection and material electroelastic dissimilarity) and the influence of nearby inclusions is also considered. The results show that when compared with the previous solution (the perfect interface), more equilibrium positions of the screw dislocation in the matrix may be available due to the effect of the interface imperfection when the dislocation is close to the electroelastic stiff inclusion. It is also found that the magnitude of the image force exerted on the piezoelectric screw dislocation produced by multiply inclusions is always smaller than that produced by a single inclusion and the impact of nearby inclusions on the mobility of the screw dislocation is very important.     

Local Electroelastic Field and Effective Electroelastic Moduli of Piezoelectric Nanocomposites with Interface Effect

Due to the large ratio of surface area to volume in nanoscale objects, the property of surfaces and interfaces likely becomes a prominent factor in controlling the behavior of nano-heterogeneous materials. In this work, based on the Gurtin-Murdoch surface/interface elastic theory, a distinct expression is derived for embedded nano-inclusion in an infinite piezoelectric matrix coupled with interface effect. For the problem of a spherical inclusion in transversely isotropic piezoelectric medium, we reach a conclusion that the elastic and electric field are uniform when eigen-strain and eigen-electric field imposed on the inclusion are uniform even in the presence of the interface influence. The electroelastic fields in the inclusion are related to both interface electroelastic parameters and the radius of the inclusion. Then overall properties of the composites are estimated by using the dilute distribution model. Numerical results reveal that the effective electroelastic moduli are function of the interface parameters and the size of the nano-inhomogeneities.     

Exploiting optical properties of P3HT:PCBM films for organic solar cells with semitransparent anode

In this study, we demonstrate optical properties of multilayer system in an organic solar cell based on poly (3-hexylthiophene) (P3HT) and 6,6-phenyl C61-butyric acid methyl ester (PCBM) with semitransparent anode through thermal annealing effect. The optical absorption is enhanced via optimizing annealing treatment which further elevates near-field electric field amplitude. The electric field amplitude at the interface (active layer/semitransparent anode) is enhanced after thermal annealing corresponding to effective absorption near to semitransparent anode. Moreover, the thickness of the active layer is optimized via optical thin-film model for enhancing the organic solar cell efficiency.     

SH波入射双相介质半空间浅埋任意位置圆形夹杂的动力分析

采用Green 函数及复变函数方法研究了SH 波入射到双相介质半空间时,浅埋任意位置圆形夹杂的动力响应问题。首先,利用“镜像”法构造满足直角平面自由边界条件的散射波场解答,进而求出该文所需的Green函数;其次,利用“契合”思想,将模型沿着垂直界面剖分为两个直角域,并利用界面连续性条件及Green函数建立待解外力系的第一类Fredholm 积分方程组;最后,通过具体算例给出了圆形弹性夹杂周边的动应力集中系数。结果显示:界面、自由边界、圆形夹杂、入射波数等因素均对动应力集中系数有影响。     

An effective medium theory for multi-phase matrix-based dielectric composites with randomly oriented ellipsoidal inclusions

An effective medium approximation is formulated for multi-phase matrix-based dielectric composites with randomly oriented ellipsoidal inclusions. The main idea is based on considering a homogenized effective medium which is subjected to a uniform electric field, embedding in it a finite group of representative sub-elements of the composite, and then demanding that the dominant part of the far-field correction to the uniform field which prevailed in it vanishes. This condition results in an algebraic equation for the sought effective property. The calculation of the dominant part of the far-field correction is achieved in an approximate manner. That disturbance is assumed to be the sum of the disturbances caused individually by the each of the embedded elements that consist of a particle of the inclusion phases, surrounded first by some matrix material and then embedded separately in the effective medium. The volumetric ratio of the matrix shell surrounding a particle of the inclusion phase to the total volume of the embedded entity is determined according to the following strategy: the particle of an inclusion phase is assigned an amount of matrix, in proportion to the volume that this specific phase occupies relative to the total volume of all the inclusion phases in the actual composite. Numerical results are produced for a three-phase composite with randomly oriented ellipsoidal inclusions, and compared with the predictions of the average field approximation, the Mori-Tanaka mean field method, and the differential scheme. It is shown that the predictions of the average field approximation and the Mori-Tanaka model violate the multi-phase Hashin-Shtrikman bounds in several circumstances, whereas those of the effective medium approximation and the differential scheme obey those bounds.     

Effect of non-metallic inclusions on butterfly wing initiation,crack formation,and spall geometry in bearing steels

Non-metallic inclusions such as sulfides and oxides are byproducts of the bearing steel manufacturing process. Stress concentrations due to such inclusions can originate cracks that lead to final failure. This paper proposes a model to simulate subsurface crack formation in bearing steel from butterfly-wing origination around non-metallic inclusions until final failure. A 2D finite element model was developed to obtain the stress distribution in a domain subjected to Hertzian loading with an embedded non-metallic inclusion. Continuum Damage Mechanics (CDM) was used to introduce a new variable called Butterfly Formation Index (BFI) that manifests the dependence of wing formation on depth. The value of critical damage inside the butterfly wings was obtained experimentally and was used to simulate damage evolution. Voronoi tessellation was used to develop the FEM domains to capture the effect of microstructural randomness on butterfly wing formation, crack initiation and crack propagation. Then, the effects of different inclusion characteristics such as size, depth, and stiffness on RCF life are studied. The results show that stiffness of an inclusion and its location have a significant effect on the RCF life: stiffer inclusions and inclusions located at the depth of maximum shear stress reversal are more detrimental to the RCF life. Stress concentrations are not significantly affected by inclusion size for the cases investigated; however, a stereology study showed that larger inclusions have a higher chance to be located at the critical depth and cause failure. Crack maps were recorded and compared to spall geometries observed experimentally. The results show that crack initiation locations and final spall shapes are similar to what has been observed in failed bearings.     

Bounds on the effective thermal conductivity of composites with imperfect interface

A new model is developed to bound the effective thermal conductivity of composites with thermal contact resistance between spherical inclusions and matrix. To construct the trial temperature and heat flux fields which satisfy the necessary interface conditions, the transition layer for each spherical inclusion is introduced. For the upper bound, the trial temperature field needs to satisfy the thermal contact resistance conditions between spherical inclusions and transition layers and the continuous interface conditions between transition layers and remnant matrix. For the lower bound, the trial heat flux field needs to satisfy the continuous interface conditions between different regions. It should be pointed out that the continuous interface conditions mentioned above are absolutely necessary for the application of variational principles, and the thermal contact resistance conditions between spherical inclusions and transition layers are suggested by the author. According to the principles of minimum potential energy and minimum complementary energy, the bounds of the effective thermal conductivity of composites with imperfect interfaces are rigorously derived. The effects of the size and distribution of spherical inclusions on the bounds of the effective thermal conductivity of composites are analyzed. It should be shown that the present method is simple and does not need to calculate the complex integrals of multi-point correlation functions. Meanwhile, the present method provides an entirely different way to bound the effective thermal conductivity of composites with imperfect interface, which can be developed to obtain a series of bounds by taking different trial temperature and heat flux fields. In addition, the present upper and lower bounds are finite when the thermal conductivity of spherical inclusions tends to ∞ and 0, respectively.     

An irregular-shaped inclusion with imperfect interface in antiplane elasticity

Despite extensive studies of inclusions with simple shape, little effort has been devoted to inclusions of irregular shape. In this study, we consider an inclusion of irregular shape embedded within an infinite isotropic elastic matrix subject to antiplane shear deformations. The inclusion–matrix interface is assumed to be imperfect characterized by a single, non-negative, and constant interface parameter. Using complex variable techniques, the analytic function that is defined within the irregular-shaped inclusion is expanded into a Faber series, and in conjunction with the Fourier series, a set of linear algebraic equations for a finite number of unknown coefficients is determined. With this approach and without imposing any constraints on the stress distribution, a semi-analytical solution is derived for the elastic fields within the irregular-shaped inclusion and the surrounding matrix. The method is illustrated using three examples and verified, when possible, with existing solutions. The results from the calculations reveal that the stress distribution within the inclusion is highly non-uniform and depends on the inclusion shape and the weak mechanical contact at the inclusion/matrix boundary. In fact, the results illustrate that the imperfect interface parameter significantly influences the stress distribution.     

Null-field approach for piezoelectricity problems with arbitrary circular inclusions

In this paper, we derive the null-field integral equation for piezoelectricity problems with arbitrary piezoelectric circular inclusions under remote anti-plane shears and in-plane electric fields. Separable expressions of fundamental solutions and Fourier series for boundary densities are adopted to solve the piezoelectric problem with circular inclusions. Four gains are obtained: (1) well-posed model, (2) singularity free, (3) boundary-layer effect free and (4) exponential convergence. The solution is formulated in a manner of semi-analytical form since error purely attributes to the truncation of Fourier series. Two piezoelectric problems with two piezoelectric circular inclusions are revisited and compared with the Chao and Chang's solutions to demonstrate the validity of our method. The limiting case that the two inclusions separate far away leads to the Pak's solution of a single inclusion. Stress and electric field concentrations are calculated and are dependent on the distance between the two inclusions, the mismatch in the material constants and the magnitude of mechanical and electromechanical loadings. The results for the shear and electric loadings in two directions are also compared well with the Wang and Shen's results.     

Contribution to critical shear stress of nanocomposites produced by interaction of screw dislocation with nanoscale inclusion

The contribution to the critical shear stress of nanocomposites caused by the interaction between screw dislocations and nanoscale circular cylindrical inclusions with interface stresses is derived by means of the Mott and Nabarro's model. The effect of the radius of the nanoscale inclusion and the volume fraction of inclusions as well as the interface stress on the critical shear stress is examined. The most important finding is that, when the negative interface stress is considered, a critical value of the radius of the nanoscale inclusion or the volume fraction of inclusions may exist to gain the best strengthening effects for the second-phase strengthened composites which differs from the classical solution without considering the interface stress under same external conditions.     

Electroelastic fields and effective moduli of a medium containing cavities or rigid inclusions of arbitrary shape under anti-plane mechanical and in-plane electric fields

Summary We examine the two-dimensional problem of an infinite piezoelectric medium containing a solitary cavity or rigid inclusion of arbitrary shape, subjected to a coupled anti-plane mechanical and in-plane electric load at the remote boundary of the matrix. Conformal mapping techniques are employed to analyze the boundary value problems. Specific results are given for elliptical, polygonal and star-shape inclusions. Local fields of this type are used to estimate the overall moduli of a medium containing voids or rigid inclusions. This is accomplished with the help of an extension of Eshelby's formula which evaluates the total electric enthalpy by a particular line integral. Explicit estimates of the effective moduli are derived for dilute as well as for moderate area fractions of inclusions. The formulae depend solely on the cross area of the inclusion, area fraction and one particular coefficient of the mapping function. In addition, the stress and electric displacement singularities around the sharp corners of the inclusion are examined. The existence of uniform fields inside the inclusion is also envisaged. The present results, with appropriate modifications, apply equally well to those of thermoelectric and magnetoelectric effects.