Elastic interaction between inclusions and tunable periodicity of superlattice structure in nanowires

The elastic stress distribution and the variation of the elastic energy with spacing between two inclusions of arbitrary sizes in an infinite isotropic cylindrical rod are obtained by an analytical approach and the phase field microelasticity(PFM) simulation. The results show a near-attraction and far-repulsion elastic interaction between two inclusions with hydrostatic dilatation. The critical spacing, at which the interaction changes from attraction to repulsion, is on the order of the radius ...     

The determination of the elastic field of a polygonal star shaped inclusion

Recently we found that the elastic field is uniform in a pentagonal star (five-pointed star inclusion) [1], and in a triangular inclusion [2], when an eigenstrain is distributed uniformly in these inclusions. This result is similar to the famous result of Eshelby (1957) that the elastic field is uniform in an ellipsoidal inclusion in an infinitely body when an eigenstrain is distributed uniformly in the ellipsoidal inclusion. We also found that for a Jewish star (Star of David or six points star) or a rectangular inclusion subjected to a uniform eigenstrain, the stress field is not uniform in these inclusions. These results also hold for two dimensional plane strain cases. Furthermore these analytical results are confirmed experimentally by photoelasticity method. In this paper, we investigate a more general inclusion of an m-pointed polygonal inclusion subjected to the uniform eigenstrain. We conclude that the stress field is uniform when m is odd number. This conclusion agrees with the speculation made by B. Boley after the author's talk at Shizuoka [2].     

Thickness effect of a thin film on the stress field due to the eigenstrain of an ellipsoidal inclusion

Solutions of the stress field due to the eigenstrain of an ellipsoidal inclusion in the film/substrate half-space are obtained via the Fourier transforms and Stroh eigenrelation equations. Based on the acquired solutions, the effect of a thin film’s thickness on the stress field is investigated with two types of ellipsoidal inclusions considered. The results in this paper show that if the thickness of the thin film increases, its effect on the stress field will become weaker, and can even be neglected. In the end, a guide rule is introduced to simplify the calculation of similar problems in engineering.     

基于等效特征应变原理的复合材料有效模量细观力学分析

基于等效特征应变原理,提出了一种新的复合材料有效模量细观力学分析方法。首先,在等效特征应变原理基础上提出平均等效特征应变原理,它可用于解决有限体下任意形状(无论是凸或凹形)的单个夹杂或多个夹杂的弹性变形问题。其次,将平均等效特征应变原理与细观力学直接均匀法相结合,来分析确定复合材料的有效模量。最后利用复合材料纤维与基体的力学性能参数及纤维的体分比,借助MATLAB编程方法,预测其有效模量。通过将理论预测值与已有的的试验值、其它理论预测值进行对比,验证了新分析方法的合理性和分析精度。     

The principle of equivalent eigenstrain for inhomogeneous inclusion problems

In this paper, based on the principle of virtual work, we formulate the equivalent eigenstrain approach for inhomogeneous inclusions. It allows calculating the elastic deformation of an arbitrarily connected and shaped inhomogeneous inclusion, by replacing it with an equivalent homogeneous inclusion problem, whose eigenstrain distribution is determined by an integral equation. The equivalent homogeneous inclusion problem has an explicit solution in terms of a definite integral. The approach allows solving the problems about inclusions of arbitrary shape, multiple inclusion problems, and lends itself to residual stress analysis in non-uniform, heterogeneous media. The fundamental formulation introduced here will find application in the mechanics of composites, inclusions, phase transformation analysis, plasticity, fracture mechanics, etc.     

Variational formulation of the equivalent eigenstrain method with an application to a problem with radial eigenstrains

In this paper a variational formulation of the equivalent eigenstrain method is established. A functional of the Hashin–Shtrikman type is proposed such that the solution of the equivalent eigenstrain equation is a unique minimizer of the functional. Moreover, it is also shown that the equivalent eigenstrain equation is the Euler–Lagrange equation of the potential energy of the inclusions. An approximate solution of the equivalent eigenstrain equation is then found as a minimizer of the functional on a finite dimensional span of basic eigenstrains. Special attention is paid to possible symmetries of the problem. The variational formulation is illustrated by determination of effective linear elastic properties. In particular, material with a simple cubic microstructure is considered in detail. A solution for the polynomial radial basic eigenstrains approximation is found. In particular, for the homogeneous eigenstrain approximation, the effective moduli are derived in an exact closed form.     

Continuum Dyson's equation and defect Green's function in a heterogeneous anisotropic solid

A continuum Dyson's equation and a defect Green's function (GF) in a heterogeneous, anisotropic and linearly elastic solid under homogeneous boundary conditions have been introduced. The continuum Dyson's equation relates the point-force Green's responses of two systems of identical geometry and boundary conditions but of different media. Given the GF of either system (i.e., a reference), the GF of the other (i.e., a defect system with “defect” change of materials property relative to the reference) can be obtained by solving the Dyson's equation. The defect GF is applied to solve the eigenstrain problem of a heterogeneous solid. In particular, the problem of slightly inhomogeneous inclusions is examined in detail. Based on the Dyson's equation, approximate schemes are proposed to efficiently evaluate the elastic field. Numerical results are reported for inhomogeneous inclusions in a semi-infinite substrate with a traction-free surface to demonstrate the validity of the present formulation.     

Elastic fields generated by inhomogeneities: Far-field asymptotics,its shape dependence and relation to the effective elastic properties

We discuss connections between the effective elastic properties of a solid with inhomogeneities and the far-field asymptotics of the elastic fields generated by them. We focus on the dependence of the far-field asymptotics on the inhomogeneity shape. This shape dependence in the inhomogeneity problem is in contrast with shape independence of the far field in the eigenstrain problem. For the latter, the far field applies to inclusions of any shape. We show that the external fields in the eigenstrain – and the inhomogeneity problems are interrelated by the same tensor that characterizes the compliance contribution of an inhomogeneity.     

New Green’s function for stress field and a note of its application in quantum-wire structures

The elastic field caused by the lattice mismatch between the quantum wires and the host matrix can be modeled by a corresponding two-dimensional hydrostatic inclusion subjected to plane strain conditions. The stresses in such a hydrostatic inclusion can be effectively calculated by employing the Green’s functions developed by Downes and Faux, which tend to be more efficient than the conventional method based on the Green’s function for the displacement field. In this study, Downes and Faux’s paper is extended to plane inclusions subjected to arbitrarily distributed eigenstrains: an explicit Green’s function solution, which evaluates the stress field due to the excitation of a point eigenstrain source in an infinite plane directly, is obtained in a closed-form. Here it is demonstrated that both the interior and exterior stress fields to an inclusion of any shape and with arbitrarily distributed eigenstrains are represented in a unified area integral form by employing the derived Green’s functions. In the case of uniform eigenstrain, the formulae may be simplified to contour integrals by Green’s theorem. However, special care is required when Green’s theorem is applied for the interior field. The proposed Green’s function is particularly advantageous in dealing numerically or analytically with the exterior stress field and the non-uniform eigenstrain. Two examples concerning circular inclusions are investigated. A linearly distributed eigenstrain is attempted in the first example, resulting in a linear interior stress field. The second example solves a circular thermal inclusion, where both the interior and exterior stress fields are obtained simultaneously.     

Buckling and postbuckling of magnetoelastic flat plates carrying an electric current

The basic equations of a fully nonlinear theory of electromagnetically conducting flat plates carrying an electric current and exposed to a magnetic field of arbitrary orientation are derived. The relevant equations have been obtained by considering that both the elastic and electromagnetic media are homogeneous and isotropic. The geometrical nonlinearities are considered in the von-Kármán sense, and the soft ferromagnetic material of the plate is assumed to feature negligible hysteretic losses. Based on the electromagnetic and elastokinetic field equations, by using the standard averaging methods, the 3-D coupled problem is reduced to an equivalent 2-D one, appropriate to the theory of plates. Having in view that the elastic structures carrying an electric current are prone to buckling, by using the presently developed theory, the associated problems of buckling and postbuckling are investigated. In this context, the problem of the electrical current inducing the buckling instability of the plate, and its influence on the postbuckling behavior are analyzed. In the same context, the problem of the natural frequency–electrical current interaction of flat plates, as influenced by a magnetic field is also addressed.     

Buckling and postbuckling of size-dependent cracked microbeams based on a modified couple stress theory

The elastic buckling analysis and the static postbuckling response of the Euler–Bernoulli microbeams containing an open edge crack are studied based on a modified couple stress theory. The cracked section is modeled by a massless elastic rotational spring. This model contains a material length scale parameter and can capture the size effect. The von Kármán nonlinearity is applied to display the postbuckling behavior. Analytical solutions of a critical buckling load and the postbuckling response are presented for simply supported cracked microbeams. This parametric study indicates the effects of the crack location, crack severity, and length scale parameter on the buckling and postbuckling behaviors of cracked microbeams.     

Representations of elastic fields of circular dislocation and disclination loops in terms of spherical harmonics and their application to various problems of the theory of defects

Elastic fields of circular dislocation and disclination loops are represented in explicit form in terms of spherical harmonics, i.e. via series with Legendre and associated Legendre polynomials. Representations are obtained by expanding Lipschitz-Hankel integrals with two Bessel functions into Legendre series. Found representations are then applied to the solutions of elasticity boundary-value problems of the theory of defects and to the calculation of elastic fields of segmented spherical inclusions. In the framework of virtual circular dislocation–disclination loops technique, a general scheme to solving axisymmetric elasticity problems with boundary conditions specified on a sphere is given. New solutions for elastic fields of a twist disclination loop in a spherical particle and near a spherical pore are demonstrated. The easy and straightforward way for calculations of elastic fields of segmented spherical inclusion with uniaxial eigenstrain is shown.     

Nonlinear analysis of compressed elastic thin films on elastic substrates: From wrinkling to buckle-delamination

Nonlinear buckling of elastic thin films on compliant substrates is studied by modeling and simulations to reveal the roles of pre-strain, elastic modulus ratio, and interfacial properties in morphological transition from wrinkles to buckle-delamination blisters. The model integrates an interfacial cohesive zone model with the Föppl–von Kármán plate theory and Green function method within the general framework of energy minimization. A kinetics approach is developed for numerical simulations. Subject to a uniaxial pre-strain, the numerical simulations confirm the analytically predicted critical conditions for onset of wrinkling and wrinkle-induced delamination, with which a phase diagram is constructed. It is found that, with increasing pre-strain, the equilibrium configuration evolves from flat to wrinkles, to concomitant wrinkles and buckle-delamination, and to an array of parallel straight blisters. The height and width of the buckle-delamination blisters can be approximately described by a set of scaling laws with respect to the pre-strain and interfacial toughness. Subject to an equi-biaxial pre-strain, the critical conditions are determined numerically to construct a similar phase diagram for the buckling modes. Moreover, by varying the pre-strain, modulus ratio, and interfacial toughness, a rich variety of equilibrium configurations are simulated, including straight blisters, and network blisters with or without wrinkles. These results provide considerable insight into diverse surface patterns in layered material systems as a result of the mechanical interactions between the film and the substrate through their interface, which suggests potential control parameters for designing specific surface patterns.     

Buckling and postbuckling of a compressed thin film bonded on a soft elastic layer: a three-dimensional analysis

The wrinkling of a stiff thin film bonded on a soft elastic layer and subjected to an applied or residual compressive stress is investigated in the present paper. A three-dimensional theoretical model is presented to predict the buckling and postbuckling behavior of the film. We obtained the analytical solutions for the critical buckling condition and the postbuckling morphology of the film. The effects of the thicknesses and elastic properties of the film and the soft layer on the characteristic wrinkling wavelength are examined. It is found that the critical wrinkling condition of the thin film is sensitive to the compressibility and thickness of the soft layer, and its wrinkling amplitude depends on the magnitude of the applied or residual in-plane stress. The bonding condition between the soft layer and the rigid substrate has a considerable influence on the buckling of the thin film, and the relative sliding at the interface tends to destabilize the system.     

The torsional buckling of a cruciform column under compressive load with a vertex plasticity model

The torsional buckling of a plastically deforming cruciform column under compressive load is investigated. The problem is solved analytically based on the von Kármán shallow shell theory and the virtual work principle. Solutions found in the literature are extended for path-dependent incremental behaviour as typically found in the presence of the vertex effect that is present in metallic polycrystals.At the critical load for buckling the direction of straining changes by an additional shear component. It is shown that the incremental elastic–plastic moduli are spatially nonuniform for such situations, contrary to the classical J₂ flow and deformation theories. The critical shear modulus that governs the buckling equation is obtained as a weighted average of the incremental elastic–plastic moduli over the cross-section of the cruciform.Using a plasticity model proposed by the authors, that includes the vertex effect, the buckling-critical load is computed for a aluminium column both with the analytical model and a FEM-based eigenvalue buckling analysis. The stable post-buckling path is determined by the energy criterion of path-stability. A comparison with the experimentally obtained classical results by Gerard and Becker (1957) shows good agreement without relying on artificial imperfections as necessary in the classical J₂ flow theory.     

Singularity analysis at the vertex of polygonal quantum wire inclusions

Singular behavior near the vertex/corner of polygonal inclusions in anisotropic piezoelectric planes is identified and investigated. Using a recently derived exact closed-form solution, the elastic and electric fields associated with regular polygon-shaped quantum wires are analyzed in detail, revealing the effect of factors such as material orientation, type of eigenstrain, and number of sides of polygons on the singular behavior. Our study shows that by adjusting these factors properly, one could avoid the singular behavior or reduce the level of singularity, which could be of significance in the design and fabrication of the semiconductor devices.     

弱界面异质球形夹杂本征应变问题的解析解

本文研究了具有线弹簧弱界面的异质球形夹杂的本征应变问题，所采用的线弹簧界面模型既能界面的切线方向滑动，又能考虑界面的法线方向张开，根据叠加原理、原问题的弹性场可分成三部分；二部分由真实均匀本征应变所引起，另一部分由等效的非均匀本征应变所引起，后一部分则由虚拟的Ｓｏｍｉｇｌｉａｎａ位错场所产生。本文求得了等效非均匀本征应变和虚拟位错场的Ｂｕｒｇｅｒ矢量的解析表达式，进而确定的问题的弹性场。     

平面弹性中双圆柱夹杂问题的格林函数

通过复势法并结合共形映射、解析开拓、奇点分析、Cauchy积分公式、圆环域上的Laurent级数展开等技术的联合运用,给出了含有双圆柱异相夹杂的无限大弹性基体这种三相复合系统在受到面内集中力和刃型位错作用时的级数形式的精确解答。该奇点可以作用于基体上或两夹杂内部,当一刃型位错作用于基体上时,也推导了该位错与双圆柱夹杂的交互作用能以及作用于位错上的广义。所给出的算例验证了该弹性解答的正确性并直观显示出奇点与双圆柱夹杂的相互作用。     

Interaction between dislocation and two circular cylindrical inclusions

In this paper, the elastic field of the infinite homogeneous medium with two circular cylindrical inclusions under the action of a screw dislocation is investigated and the corresponding analytical solution is obtained. Here, the conformal mapping and the theorem of analytical continuation are used. From the results obtained, it can be seen that the elastic field depends on the shear moduli of individual phases, the geometric parameters of the system, and the position and relative slip of the screw dislocation. In addition, the corresponding specific cases are also considered in this paper when two circular cylindrical inclusions are tangent to each other and they are holes and/or rigid inclusions. Finally, numerical results are illustrated to show the interaction between the screw dislocation and two circular cylindrical inclusions.