Surface/interface effects on elastic behavior of a screw dislocation in an eccentric core–shell nanowire

The elastic behavior of a screw dislocation which is positioned inside the shell domain of an eccentric core–shell nanowire is addressed with taking into account the surface/interface stress effect. The complex potential function method in combination with the conformal mapping function is applied to solve the governing non-classical equations. The dislocation stress field and the image force acting on the dislocation are studied in detail and compared with those obtained within the classical theory of elasticity. It is shown that near the free outer surface and the inner core–shell interface, the non-classical solution for the stress field considerably differs from the classical one, while this difference practically vanishes in the bulk regions of the nanowire. It is also demonstrated that the surface with positive (negative) shear modulus applies an extra non-classical repelling (attracting) image force to the dislocation, which can change the nature of the equilibrium positions depending on the system parameters. At the same time, the non-classical solution fails when the dislocation approaches very close to the surface/interface with negative shear modulus. The effects of the core–shell eccentricity and nanowire diameter on dislocation behavior are discussed. It is shown that the non-classical surface/interface effect has a short-range character and becomes more pronounced when the nanowire diameter is smaller than 20 nm.     

Interface effects on elastic behavior of an edge dislocation in a core–shell nanowire embedded to an infinite matrix

The elastic behavior of an edge dislocation located inside the core of a core–shell nanowire which is embedded in an infinite matrix is studied within the surface/interface elasticity theory. The corresponding boundary value problem is solved exactly by using complex potential functions. An important parameter so-called interface characteristic parameter which has the dimension of length and is a combination of the interface moduli enters the formulations. The stress field of the dislocation, image force acting on the dislocation, and the dislocation strain energy is calculated by considering the interface effect. The introduced characteristic parameter allows the examination of the core–shell size on the image forces acting on the dislocation. The repelling and attracting effects of the interface parameter on the image force are discussed. The equilibrium position of the dislocation is also studied. The dislocation strain energy in the interface elasticity framework is only slightly different from that of traditional elasticity when the dislocation is placed in the central region of the core and reaches its maximum value when it is located near the core–shell interface.     

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.     

A screw dislocation by nonlinear continuum mechanics

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Constitutive analysis of finite deformation field dislocation mechanics

Driving forces for dislocation motion and nucleation in finite-deformation field dislocation mechanics are derived. The former establishes a rigorous analog of the Peach-Koehler force of classical elastic dislocation theory in a nonlinear, nonequilibrium field-theoretic context; the latter is a prediction of the theory. The structure of the stress response and permanent distortion are also derived. Sufficient boundary and initial conditions are indicated, and invariance under superposed rigid motions is discussed. Hyperelasticity and finite-deformation elastic theory of dislocations are shown to be special cases of the framework. Owing to the nonlocal nature of the theory, the results as well as the methods used to derive them appear to be novel.     

The image force on a dislocation near an elliptic hole in nonlocal elasticity

Summary A basic solution to the problem of a screw dislocation near a circular hole in nonlocal clasticity is first presented. The nonlocal stress and nonlocal image force on the screw dislocation are derived whereby the singularities in both classical stress and classical image force are eliminated. Then, the nonlocal image force on a screw dislocation near an elliptic hole is obtained by a conformal mapping technique. The maximum of the image force and the corresponding position of hte dislocation are estimated for a dislocation very close to the holes.

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Bildkraft auf eine Versetzung neben einem elliptischen Loch bei nichtlokaler Elastizität

Übersicht Zuerst wird eine Lösung für das Problem einer Schraubenversetzung neben einem kreisförmigen Loch bei nichtlokaler Elastizität vorgestellt. Die nichtlokal Spannungen und die nichtlokalen Bildkräfte werden abgeleitet, wobei alle Singularitäten verschwinden. Dann wird die nichtlokale Bildkraft auf eine Schrauben versetzung neben einem elliptischen Loch mit Hilfe einer konformen Abbildung erhalten. Für die Versetzung sehr nahe dem Loch werden das Maximum der Bildkraft und ihre Position geschätzt.

     

梯度型非局部高阶梁理论与非局部弯曲新解法

针对文献中关于纳米结构刚度受非局部效应影响趋势的不一致预测,基于梯度型的非局部微分本构模型,利用迭代法及泰勒展开法求得了非局部高阶应力的无穷级数表达,非局部应力相当于经典弯曲应力与非局部挠度的逐阶梯度之和. 然后推导了梯度型非局部高阶梁弯曲的挠曲轴微分方程,并结合正则摄动思想,求解了非局部挠度的表达式. 最后给出数值算例,具体量化挠度受非局部尺度因子的影响大小. 研究表明:相比于其经典值,纳米结构的非局部弯曲挠度可呈现出或增大或减小或不变的趋势,考虑梯度型非局部高阶应力降低或提高或不影响纳米结构的刚度,具体结果依赖于外载和边界约束的类型. 算例显示外载形式和边界约束条件均各自独立地影响着纳米结构的非局部弯曲挠度,同时首次观察到非局部最大弯曲挠度的位置可能受非局部尺度因子的影响. 研究结论解决了非局部弹性力学应用于纳米结构的若干疑点,并为理论的发展和优化提供支持.      

A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation

In recent years there have been many papers that considered the effects of material length scales in the study of mechanics of solids at micro- and/or nano-scales. There are a number of approaches and, among them, one set of papers deals with Eringen's differential nonlocal model and another deals with the strain gradient theories. The modified couple stress theory, which also accounts for a material length scale, is a form of a strain gradient theory. The large body of literature that has come into existence in the last several years has created significant confusion among researchers about the length scales that these various theories contain. The present paper has the objective of establishing the fact that the length scales present in nonlocal elasticity and strain gradient theory describe two entirely different physical characteristics of materials and structures at nanoscale. By using two principle kernel functions, the paper further presents a theory with application examples which relates the classical nonlocal elasticity and strain gradient theory and it results in a higher-order nonlocal strain gradient theory. In this theory, a higher-order nonlocal strain gradient elasticity system which considers higher-order stress gradients and strain gradient nonlocality is proposed. It is based on the nonlocal effects of the strain field and first gradient strain field. This theory intends to generalize the classical nonlocal elasticity theory by introducing a higher-order strain tensor with nonlocality into the stored energy function. The theory is distinctive because the classical nonlocal stress theory does not include nonlocality of higher-order stresses while the common strain gradient theory only considers local higher-order strain gradients without nonlocal effects in a global sense. By establishing the constitutive relation within the thermodynamic framework, the governing equations of equilibrium and all boundary conditions are derived via the variational approach. Two additional kinds of parameters, the higher-order nonlocal parameters and the nonlocal gradient length coefficients are introduced to account for the size-dependent characteristics of nonlocal gradient materials at nanoscale. To illustrate its application values, the theory is applied for wave propagation in a nonlocal strain gradient system and the new dispersion relations derived are presented through examples for wave propagating in Euler–Bernoulli and Timoshenko nanobeams. The numerical results based on the new nonlocal strain gradient theory reveal some new findings with respect to lattice dynamics and wave propagation experiment that could not be matched by both the classical nonlocal stress model and the contemporary strain gradient theory. Thus, this higher-order nonlocal strain gradient model provides an explanation to some observations in the classical and nonlocal stress theories as well as the strain gradient theory in these aspects.     

Nonlocal interaction of a dislocation with a crack

Summary Based on the classical stress analysis, the stress field of a screw dislocation near a semi-infinite crack is investigated by a conformal mapping method. The classical image force on the dislocation due to the crack is presented by the obtained classical image stress. To eliminate the singularity of the stress field at the crack tip, the nonlocal elasticity theory is adopted. A maximum of the image stress is found near the crack tip and the nonlocal image force is then acquired by the obtained nonlocal stress field. Both the stress and the image force show much difference from the classical ones and seem to be more reasonable physically.

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Nichtlokale Wechselwirkung zwischen einer Versetzung und einem Riß

Übersicht Basierend auf der klassischen Spannungsanalyse einer Schraubenversetzung neben einem halb-unendlichen Riß mittels einer konformen Abbildung wird die klassische Bildkraft auf die Schraubenversetzung als Folge des Risses vorgestellt. Um die Spannungssingularitäten neben der Rißspitze zu beseitigen, wird die nichtlokale Elastizitätstheorie herangezogen, mittels ihrer wird das Maximum der Spannung neben dem Riß gefunden und die nichtlokale Bildkraft ermittelt. Die Spannungen und die Bildkraft der nichtlokalen Elastizitätstheorie zeigen große Unterschiede gegenüber der klassischen Analyse und sind physikalisch vernünftiger.

     

A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations

This study develops a gradient theory of single-crystal plasticity that accounts for geometrically necessary dislocations. The theory is based on classical crystalline kinematics; classical macroforces; microforces for each slip system consistent with a microforce balance; a mechanical version of the second law that includes, via the microforces, work performed during slip; a rate-independent constitutive theory that includes dependences on a tensorial measure of geometrically necessary dislocations. The microforce balances are shown to be equivalent to nonlocal yield conditions for the individual slip systems. The field equations consist of the yield conditions coupled to the standard macroscopic force balance; these are supplemented by classical macroscopic boundary conditions in conjunction with nonstandard boundary conditions associated with slip. As an aid to solution, a weak (virtual power) formulation of the nonlocal yield conditions is derived. To make contact with classical dislocation theory, the microstresses are shown to represent counterparts of the Peach-Koehler force on a single dislocation.     

Defects in gradient micropolar elasticity—II: edge dislocation and wedge disclination

A theory of gradient micropolar elasticity based on first gradients of distortion and bend-twist tensors for an isotropic micropolar medium has been proposed in Part I of this paper. Gradient micropolar elasticity is an extension of micropolar elasticity such that in addition to double stresses double couple stresses also appear. The strain energy depends on the micropolar distortion and bend-twist terms as well as on distortion and bend-twist gradients. We use a version of this gradient theory which can be connected to Eringen's nonlocal micropolar elasticity. The theory is used to study a straight-edge dislocation and a straight-wedge disclination. As one important result, we obtained nonsingular expressions for the force and couple stresses. For the edge dislocation the components of the force stress have extremum values near the dislocation line and those of the couple stress have extremum values at the dislocation line and for the wedge disclination the components of the force stress have extremum values at the disclination line and those of the couple stress have extremum values near the disclination line.     

Effect of interface stresses on the image force and stability of an edge dislocation inside a nanoscale cylindrical inclusion

Dislocation mobility and stability in inclusions can affect the mechanical behaviors of the composites. In this paper, the problem of an edge dislocation located within a nanoscale cylindrical inclusion incorporating interface stress is first considered. The explicit expression for the image force acting on the edge dislocation is obtained by means of a complex variable method. The influence of the interface effects and the size of the inclusion on the image force is evaluated. The results indicate that the impact of interface stress on the image force and the equilibrium positions of the edge dislocation inside the inclusion becomes remarkable when the radius of the inclusion is reduced to nanometer scale. The force acting on the edge dislocation produced by the interface stress will increase with the decrease of the radius of the inclusion and depends on the inclusion size which differs from the classical solution. The stability of the dislocation inside a nanoscale inclusion is also analyzed. The condition of the dislocation stability and the critical radius of the inclusion are revised for considering interface stresses.     

纵向剪切下含共焦刚性核弹性椭圆夹杂中的螺型位错干涉

摘 要 研究了无穷远纵向剪切下无限大基体中含共焦刚性核弹性椭圆夹杂内任意位置螺型位错的干涉问题。运用复变函数保角映射、解析延拓等方法,获得了基体与夹杂区域的应力场的级数形式精确解,并得出了位错像力的解析表达式,导出了纵向剪切下两椭圆界面最大应力及其比值公式。分析结果表明：夹杂内部的刚性核对位错与夹杂的干涉产生明显的扰动效应,排斥硬夹杂内位错,并使之不断趋近弹性夹杂界面。对于软夹杂,夹杂中的位错存在稳定的平衡位置,当位错位于刚性核和平衡位置之间时,位错会趋于弹性夹杂界面;当位错位于平衡位置和弹性夹杂界面之间时,位错会离开界面。结果还显示,夹杂的长轴和短轴之比对位错与夹杂的干涉也有着不可忽视的影响,尤其当位错在刚性核附近时,随着夹杂的长、短轴比值的减小,核对位错的排斥力也明显减弱。本文解答包含了多个以往文献成果。     

TWISTING STATICS AND DYNAMICS FOR CIRCULAR ELASTIC NANOSOLIDS BY NONLOCAL ELASTICITY THEORY

The torsional static and dynamic behaviors of circular nanosolids such as nanoshafts, nanorods and nanotubes are established based on a new nonlocal elastic stress field theory. Based on a new expression for strain energy with a nonlocal nanoscale parameter, new higher-order governing equations and the corresponding boundary conditions are first derived here via the variational principle because the classical equilibrium conditions and/or equations of motion can- not be directly applied to nonlocal nanostructures even if the stress and moment quantities are replaced by the corresponding nonlocal quantities. The static twist and torsional vibration of circular, nonlocal nanosolids are solved and discussed in detail. A comparison of the conventional and new nonlocal models is also presented for a fully fixed nanosolid, where a lower-order governing equation and reduced stiffness are found in the conventional model while the new model reports opposite solutions. Analytical solutions and numerical examples based on the new nonlocal stress theory demonstrate that nonlocal stress enhances stiffness of nanosolids, i.e. the angular displacement decreases with the increasing nonlocal nanoscale while the natural frequency increases with the increasing nonlocal nanoscale.     

Defects in gradient micropolar elasticity—I: screw dislocation

A gradient micropolar elasticity is proposed based on first gradients of distortion and bend-twist tensors for an isotropic micropolar medium. This theory is an extension of the theory of micropolar elasticity with couple stresses together with gradient elasticity in a way that in addition to hyper stresses, hyper couple stresses also appear. In particular, the strain energy, besides its dependence upon the distortion and bend-twist terms of a micropolar medium (Cosserat continuum), depends also on distortion and bend-twist gradients. Using a simplified but rigorous version of this gradient theory, we can connect it to Eringen's nonlocal micropolar elasticity. In addition, it is used to study a screw dislocation in gradient micropolar elasticity. One important result is that we obtained nonsingular expressions for the force and couple stresses. The components of the force stress have maximum values near the dislocation line and those of the couple stress have maximum values at the dislocation line.     

Interaction between a dislocation and a core–shell nanowire with interface effects

The problem of a screw dislocation interacting with a core–shell nanowire (coated nanowire) containing interface effects (interface stresses) is first investigated. The interaction energy and the interaction force are calculated. The interaction force and the equilibrium position of the dislocation are examined for variable parameters (interface stress and material mismatch). The influence of the core–shell nanowire and the interface stresses on the interaction between two screw dislocations is also considered. The results show that the impact of the interface stresses on the motion and the equilibrium position of the dislocation near the core–shell nanowire is very significant when the radius of the nanowire is reduced to nanometer scale.     

非局部平面应变和平面应力问题界定及其精确性讨论

在非局部弹性理论框架下对平面应变和平面应力状态重新界定.首先,分别在其相应简化假设下推导控制方程,并与经典局部情况进行比较.然后,引入变形协调条件对两类非局部平面问题的精确性进行讨论.其中,对于非局部平面应力状态,通过应变协调方程的Fourier变换形式来进行研究,使问题得以简化.通过以上分析,最终得到一些有价值的结论.     

Dynamic modeling of preloaded size-dependent nano-crystalline nano-structures

The vibration behavior of size-dependent nano-crystalline nano-beams is investigated based on nonlocal, couple stress and surface elasticity theories. A nanocrystalline nano-beam is composed of three phases which are nano-grains, nano-voids,and interface. Nano-voids or porosities inside the material have a stiffness-softening impact on the nano-beam. A Eringen's nonlocal elasticity theory is applied in the analysis of nano-crystalline nano-beams for the first time. Residual surface stresses which are usually neglected in modeling nano-crystalline nano-beams are incorporated into nonlocal elasticity to better understand the physics of the problem. Also, a modified couple stress theory is used to capture rigid rotations of grains. Applying a differential transform method(DTM) satisfying various boundary conditions, the governing equations obtained from the Hamilton's principle are solved. Reliability of the proposed approach is verified by comparing the obtained results with those provided in the literature. The effects of the nonlocal parameter, surface effect, couple stress, grain size, porosities, and interface thickness on the vibration characteristics of nano-crystalline nano-beams are explored.     

Dynamic meshless method applied to nonlocal crack problems

The dynamic meshless methods for local and nonlocal field theories are formulated in this paper. Application to two crack problems is presented. The meshless method of local theory gives solution that is in good agreement with the classical analytical crack tip solution, while the nonlocal theory yields a solution without stress singularity at the crack tip. The numerical results also show the embedded nonlocal nature of meshless methods.     

微结构非局部效应的多重性模型及在微梁弯曲中的应用

基于Eringen非局部弹性理论,直接利用逐次逼近法推导了非局部应力场的精确表达,该精确的非局部应力可具体表示为一个无穷级数的形式. 然后以微梁的横向弯曲和纯弯曲变形为例,建立平衡方程并求解及分析了挠度受非局部效应的影响. 结果表明：根据所取非局部小尺度参数大小的不同,非局部微梁的弯曲挠度可低于也可以高于经典力学下的挠度,非局部效应的增大可提高亦可降低结构的抗弯刚度. 本文结果证明了Wang以及Lim等人分别提出的两种相反的非局部模型的各自正确性. 同时首次发现,弯曲挠度随着非局部效应的增大而上下波动且存在若干跳跃点,挠度是非局部小尺度参数的非单调函数,研究同时给出了一种确定材料非局部常数的建议途径.