Circular Nano-inhomogeneity Research Articles

Suppressing thermal stress in the vicinity of a circular nano-inhomogeneity via the mechanism of size effects

Thermal stress is one of the main factors threatening the reliability of nanocomposites. In this paper, we propose a set of interfacial conditions to simulate the interfacial phonon scattering and interfacial elasticity in the nanocomposite, and then obtain closed-form solutions for the thermal-elastic fields around the circular nano-inhomogeneity. The results show that the interfacial thermal stress within matrix is more sensitive to interfacial phonon scattering than that in nano-inhomogeneity, while the size change of inhomogeneity has greater impact on the stress of nano-inhomogeneity than that of the matrix. In view of the significant effects of thermal conductivities and thermal expansion coefficients on the thermal stress around inhomogeneity, the maximum von Mises stress is reduced up to three orders of magnitude by adjusting these parameters. More importantly, the thermal stresses within matrix and inhomogeneity can be, respectively, designed as “zero” even though the nanocomposite is subjected to a large thermal loading, which is viewed as an update mechanism of “neutral inhomogeneity.” Our work provides guidance for the design of nanocomposites and has important potential applications.

Analytic solution for a circular nano-inhomogeneity in a thermoelectric material considering surface elasticity

Analytic solution for a circular nano-inhomogeneity in a thermoelectric material considering surface elasticity

Design of non-circular nanoinhomogeneities with uniform heat flux in two-dimensional heat conduction

It is known that the heat flux inside a circular nanoinhomogeneity remains uniform when taking interface phonon scattering into account. We use a combination of complex variable methods and numerical techniques to design a non-circular nanoinhomogeneity with internal uniform heat flux embedded in an infinite plate subjected to uniform remote heat loading. Our results clearly demonstrate that the shape of the nanoinhomogeneity remains independent of certain specific remote loadings but depends firmly on the corresponding thermal conductivities, interface parameter and the size of the inhomogeneity itself. Moreover, we find that the magnitude and direction of the uniform heat flux within the nanoinhomogeneity can be controlled independently by adjusting the shape of the inhomogeneity. Central to our analysis is the use of a conformal mapping function obtained via Newton-Raphson iteration. The accuracy of this conformal mapping function is verified using the corresponding actual heat flux around the nanoinhomogeneity. Our results illustrate that the error in our iterative procedure to obtain the mapping function is negligible in most cases of practical interest.

Conversion efficiency and effective properties of particulate-reinforced thermoelectric composites

It is well known that nanoparticles have the ability to enhance the performance of thermoelectric materials. A comprehensive analytical model exploring the mechanism by which this enhancement takes place remains largely absent from the literature. We address this deficiency by introducing a simple model of a nanoparticle as a circular nanoinhomogeneity and analyze its effect on thermal–electric conversion efficiency and the effective properties of thermoelectric composites. Taking interface phonon scattering into consideration, the effective material parameters around a nanoinhomogeneity are derived explicitly, while the effective properties and optimal conversion efficiency are discussed in detail via numerical analysis. Our results show that the nanoinhomogeneity equally effects both the figure of merit and the conversion efficiency. A carefully selected inhomogeneity not only improves the thermoelectric power factor but also suppresses the effective thermal conductivity, and thus greatly improves the optimal conversion efficiency. Specifically, a nanoinhomogeneity with higher electric and thermal conductivities generates a higher thermoelectric figure of merit regardless of the magnitude of its Seebeck coefficient. Noting that the thermoelectric performance improved by the presence of the inhomogeneity is restricted by the intensity of interface phonon scattering, we further calculate the expressions for the effective figure of merit in certain extreme cases. Furthermore, we see that a smaller inhomogeneity has a stronger ability to improve the conversion efficiency for a given doping proportion, mainly because the interface phonon scattering of the smaller inhomogeneity has a relatively greater effect on thermal conduction.

Thermal conduction around a circular nanoinhomogeneity

Taking interface phonon scattering into consideration, we present a set of thermal conduction interface conditions to simulate the interfacial temperature jump in nanocomposites and consequently obtain closed-form solutions describing thermal fields in the vicinity of a circular nanoinhomogeneity. Our results show that the heat flux inside the inhomogeneity is uniform and that the heat flux in the surrounding matrix can also be made uniform by designing the size or thermal conductivity of the inhomogeneity in the case when the thermal conductivity of the inhomogeneity is higher than that of the matrix. Using our theoretical results, we derive an explicit expression for the effective thermal conductivity of a rectangular region containing a circular nanoinhomogeneity. Furthermore, we undertake a numerical study to further explore the effects of interface phonon scattering on thermal conduction. Our results indicate that interface phonon scattering has the ability to suppress thermal conduction around a nanoinhomogeneity and has thus important applications in improving the performance of a range of functional materials.

Analytic solution for a circular nano-inhomogeneity in a finite matrix

Abstract The Gurtin-Murdoch model has found wide applications in analyzing the mechanical behaviors of nanocomposites with surface/interface effect. In the existing literature, the matrix is usually assumed to be infinite and the surface/interface effect is considered only at the inhomogeneity-matrix interface. This assumption is indeed valid as the matrix is usually at macroscale rather than nanoscale. However, if the size of the matrix decreases to the nanoscale too, the surface/interface effect will have to be considered at the outer boundary of the matrix. In this paper, the plane deformation of a circular nano-inhomogeneity embedded inside a finite circular matrix (which implies the matrix is also at nanoscale) is investigated. The stress boundary conditions are given at the inhomogeneity-matrix interface and the outer boundary of the matrix by the G-M model. The analytic solution for the stress field is finally obtained through the complex variable method. The results show that the stress field inside the inhomogeneity is still uniform (size-dependent) when the surface/interface effect is considered. In addition, the stress field inside the bulk (including the inhomogeneity and the matrix) can be influenced not only by the size and elastic constant of the inhomogeneity, but also by those of the matrix.

Analytic solution for a circular nano-inhomogeneity with interface stretching and bending resistance in plane strain deformations

In the mechanical analysis of composites containing nano-inhomogeneities, it is customary to consider only the stretching resistance of the inhomogeneity-matrix interface but neglect the bending resistance of the interface. In this paper, we consider a circular nano-inhomogeneity in an infinite elastic plane subjected to an arbitrary uniform remote in-plane loading with both stretching and bending resistance incorporated on the interface. Analytic solutions are obtained for the stress field both inside and outside the inhomogeneity by using an integral-type boundary condition representing the jump in traction across the interface. We show that the presence of interface bending resistance has no influence on the average of the mean stress in the inhomogeneity, and for certain interface stretching and bending rigidities the stress field inside the inhomogeneity can remain uniform regardless of the specific uniform remote loading. Numerical examples are presented to examine the influence of the interface bending resistance on the interfacial tractions imposed on the inhomogeneity and matrix for a uniform remote uniaxial loading. It is found that the introduction of interface bending resistance perturbs the (interfacial) tractions imposed on the inhomogeneity only slightly whether the inhomogeneity is softer or harder than the matrix, while it may influence the (interfacial) tractions imposed on the matrix significantly when the inhomogeneity is much softer than the matrix. Moreover, it is shown that the peak of the interface bending resistance-induced jump in traction across the interface initially increases and then decreases as the inhomogeneity becomes harder (from an initial state in which the inhomogeneity is softer than the matrix).

Interaction between a screw dislocation and a circular nano-inhomogeneity with a bimaterial interface

The problem of a screw dislocation interacting with a circular nano-inhomogeneity near a bimaterial interface is investigated. The stress boundary condition at the interface between the inhomogeneity and the matrix is modified by incorporating surface/interface stress. The analytical solutions to the problem in explicit series are obtained by an efficient complex variable method associated with the conformal mapping function. The image force exerted on the screw dislocation is also derived using the generalized Peach–Koehler formula. The results indicate that the elastic interference of the screw dislocation and the nano-inhomogeneity is strongly affected by a combination of material elastic dissimilarity, the radius of the inclusion, the distance from the center of inclusion to the bimaterial interface, and the surface/interface stress between the inclusion and the matrix. Additionally, it is found that when the inclusion and Material 3 are both harder than the matrix (μ1 > μ2 and μ3 > μ2), a new stable equilibrium position for the screw dislocation in the matrix appears near the bimaterial interface; when the inclusion and Material 3 are both softer than the matrix (μ1 < μ2 and μ3 < μ2), a new unstable equilibrium position exists close to the bimaterial interface.

Closed-form solution for a circular nano-inhomogeneity with interface effects in an elastic plane under uniform remote heat flux

We study the uncoupled steady-state thermo-elastic problem of a circular nano-inhomogeneity embedded in an elastic plane subjected to a uniform remote heat flux. Nanoscale influences are included in the continuum-based model of deformation by incorporating interface effects arising from both heat conduction and elasticity (in the absence of surface tension) on the material interface. Complex variable methods are used to derive closed-form solutions for the corresponding temperature and thermal stress fields. Numerical examples are presented to examine how each of the heat or elastic interface effects influence the thermal stress field. In fact, we show that the contribution of heat related interface effects to the thermal stress field decays in both the matrix and the inhomogeneity as the heat conductivity of the inhomogeneity increases. On the other hand, the contribution of elastic interface effects to the thermal stress field decays in the matrix but increases for the inhomogeneity as the inhomogeneity becomes harder.

Uniqueness of Neutral Elastic Circular Nano-Inhomogeneities in Antiplane Shear and Plane Deformations

In elasticity theory, a neutral inhomogeneity is defined as a foreign body which can be introduced into a host solid without disturbing the stress field in the solid. The existence of circular neutral elastic nano-inhomogeneities has been established for both antiplane shear and plane deformations when the interface effect is described by constant interface parameters, and the surrounding matrix is subjected to uniform external loading. It is of interest to determine whether noncircular neutral nano-inhomogeneities can be constructed under the same conditions. In fact, we prove that only the circular elastic nano-inhomogeneity can achieve neutrality under these conditions with the radius of the inhomogeneity determined by the corresponding (constant) interface parameters and bulk elastic constants. In particular, in the case of plane deformations, the (uniform) external loading imposed on the matrix must be hydrostatic in order for the corresponding circular nano-inhomogeneity to achieve neutrality. Moreover, we find that, even when we relax the interface condition to allow for a nonuniform interface effect (described by variable interface parameters), in the case of plane deformations, only the elliptical nano-inhomogeneity can achieve neutrality.

The interaction between a screw dislocation and a nano-inhomogeneity with a semi-infinite wedge crack penetrating the interface

The interaction between a screw dislocation and a circular nano-inhomogeneity with a semi-infinite wedge crack penetrating the interface is investigated. By using Riemann-Schwartz’s symmetry principle integrated with the analysis of singularity of complex functions and the conformal mapping technique, the analytical expressions of the stress field in both the circular nano-inhomogeneity and the infinite matrix, the image force acting on the screw dislocation and the stress intensity factor at the crack tip are obtained. The influence of elastic mismatch of materials, inhomogeneity size, interface stress, wedge crack opening angle and the relative location of dislocation on the image force and on the equilibrium position of the screw dislocation and the shielding effect of the screw dislocation are discussed in detail. The results show that interface stress has a significant impact on the movement of dislocations near the interface, and the effect of interface stress enhances when the inhomogeneity radius decreases. With the decrease in the wedge crack opening angle, the influence of interface stress on the movement of the screw dislocation and on the SIF enhances. With the increment of the relative shear modulus, the influence of interface stress weakens with the screw dislocation locating in the inhomogeneity and strengthens with the screw dislocation locating in the matrix. When the screw dislocation is located in the inhomogeneity, the positive (negative) interface stress increases (decreases) the shielding effect, while this phenomenon is opposite when the screw dislocation locates in the matrix.

Stress analysis of a wedge disclination dipole interacting with a circular nanoinhomogeneity

Abstract This paper investigates the interaction between a wedge disclination dipole and a circular nanoinhomogeneity embedded in an infinite matrix. The Gurtin–Murdoch surface/interface elasticity model is applied to account for the interface stress effect of the nanoinhomogeneity. A closed form solution for the stress fields inside the inhomogeneity and matrix is derived with the complex variable method of Muskhelishvili. The influences of the interfacial and bulk material properties as well as the geometric parameters on the material force of the wedge disclination dipole are systematically discussed. It is found that the interface stress effect may influence the material force of the wedge disclination dipole significantly.

A screw dislocation near a circular nano-inhomogeneity in gradient elasticity

A screw dislocation outside an infinite cylindrical nano-inhomogeneity of circular cross section is considered within the isotropic theory of gradient elasticity. Fields of total displacements, elastic and plastic distortions, elastic strains and stresses are derived and analyzed in detail. In contrast with the case of classical elasticity, the gradient solutions are shown to possess no singularities at the dislocation line. Moreover, all stress components are continuous and smooth at the interface unlike the classical solution. As a result, the image force exerted on the dislocation due to the differences in elastic and gradient constants of the matrix and inhomogeneity, remains finite when the dislocation approaches the interface. The gradient solution demonstrates a non-classical size-effect in such a way that the stress level inside the inhomogeneity decreases with its size. The gradient and classical solutions coincide when the distances from the dislocation line and the interface exceed several atomic spacings.

Analysis of a screw dislocation interacting with an elliptical nano inhomogeneity

The interaction between a screw dislocation and an elliptical nano inhomogeneity embedded in an infinite matrix is investigated. The interface stress effects of the nano inhomogeneity are accounted for with the Gurtin–Murdoch model. The stress fields inside the inhomogeneity and matrix are then solved with the complex variable and conformal mapping method. The solution is of semi-analytical nature and is verified by studying a degenerated case wherein a screw dislocation interacts with a circular nano inhomogeneity. The image force on the screw dislocation is then calculated. The influences of the elastic mismatch between the inhomogeneity and matrix, the interfacial properties, the aspect ratio of the elliptic nano inhomogeneity and the position of the screw dislocation on the image force are systematically discussed.

On the anti-plane shear of an elliptic nano inhomogeneity

Abstract The elastic field of an elliptic nano inhomogeneity embedded in an infinite matrix under anti-plane shear is studied with the complex variable method. The interface stress effects of the nano inhomogeneity are accounted for with the Gurtin–Murdoch model. The conformal mapping method is then applied to solve the formulated boundary value problem. The obtained numerical results are compared with the existing closed form solutions for a circular nano inhomogeneity and a traditional elliptic inhomogeneity under anti-plane. It shows that the proposed semi-analytic method is effective and accurate. The stress fields inside the inhomogeneity and matrix are then systematically studied for different interfacial and geometrical parameters. It is found that the stress field inside the elliptic nano inhomogeneity is no longer uniform due to the interface effects. The shear stress distributions inside the inhomogeneity and matrix are size dependent when the size of the inhomogeneity is on the order of nanometers. The numerical results also show that the interface effects are highly influenced by the local curvature of the interface. The elastic field around an elliptic nano hole is also investigated in this paper. It is found that the traction free boundary condition breaks down at the elliptic nano hole surface. As the aspect ratio of the elliptic hole increases, it can be seen as a Mode-III blunt crack. Even for long blunt cracks, the surface effects can still be significant around the blunt crack tip. Finally, the equivalence between the uniform eigenstrain inside the inhomogeneity and the remote loading is discussed.

Size-dependent elastic interaction of a screw dislocation with a circular nano-inhomogeneity incorporating interface stress

Effect of interface stress upon the interaction between a screw dislocation and a nano-inhomogeneity is investigated. By using the complex variable method, the image force acting on the screw dislocation is obtained. The results indicate that the influence of interface stress on the motion of the dislocation near the inhomogeneity becomes significant when the radius of the inhomogeneity is reduced to nanometer scale, leading to that the normalized image force depends on the inhomogeneity size which differs from the size-independent classical solution.

Integral equation methods in plane-strain elasticity with boundary reinforcement

In this paper, a linear theory of elastic boundary reinforcement of an elastic solid is developed for plane–strain deformations. The reinforcement consists of a thin elastic coating bonded to part of the boundary of the solid. The elastic properties of the coating incorporate both extensibility and bending rigidity. Interior and exterior mixed–boundary problems are formulated and solved using integral equation methods. The boundary value problems are reduced to systems of singular integro–differential equations to which Noether–type theorems are shown to apply. The case corresponding to a coating which has only extensibility properties and no bending rigidity is particularly interesting from a mathematical point of view and is given special attention. Finally, existence and uniqueness results are presented for both interior and exterior reinforcement problems of plane–strain.