Piezoelectric screw dislocations interacting with a circular inclusion with imperfect interface

The electroelastic coupling interaction between multiple screw dislocations and a circular inclusion with an imperfect interface in a piezoelectric solid is investigated. The appointed screw dislocation may be located either outside or inside the inclusion and is subjected to a line charge and a line force at the core. The analytic solutions of electroelastic fields are obtained by means of the complex-variable method. With the aid of the generalized Peach–Koehler formula, the explicit expressions of image forces exerted on the piezoelectric screw dislocations are derived. The motion and the equilibrium position of the appointed screw dislocation near the circular interface are discussed for variable parameters (interface imperfection, material electroelastic mismatch, and dislocation position), and the influence of the nearby parallel screw dislocations is also considered. It is found that the piezoelectric screw dislocation is always attracted by the electromechanical imperfect interface. When the interface imperfection is strong, the impact of material electroelastic mismatch on the image force and the equilibrium position of the dislocation becomes weak. Additionally, the effect of the nearby dislocations on the mobility of the appointed dislocation is very important.     

Dislocation core spreading at interfaces between metal films and amorphous substrates

Experiments with transmission electron microscopy have shown that in a strong electron beam the contrast of dislocations may gradually disappear at an incoherent interface between a metal thin film and an amorphous substrate. There are reasons to believe that this phenomenon is caused by radiation-induced dislocation core spreading at the interface. A quantitative model accounting for this effect will be necessary for a better understanding of dislocation structures and plastic deformation in metal thin films. As a first step toward this objective, we develop a number of mathematical solutions for dislocation core spreading at an incoherent interface. For simplicity, we consider screw dislocations, and consider the interface to be characterized by a shear adhesive strength, τ₀, below which no core spreading occurs, and above which spreading takes place in a viscous manner. We determine the final equilibrium core width and the rate of core spreading for single or planar arrays of dislocations in a homogeneous bulk material or at the interface between a thin film and a semi-infinite substrate where the film and substrate may have the same, or different, elastic constants. Some of our solutions are analytic and others are based on an implicit finite difference method with a Gauss-Chebyshev quadrature scheme. The phenomenon of dislocation core spreading is expected to have a dramatic effect on the strength of crystalline films deposited on amorphous substrates.     

Trapping and escape of dislocations in micro-crystals with external and internal barriers

We perform three-dimensional dislocation dynamics simulations of solid and annular pillars, having both free-surface boundary conditions, or strong barriers at the outer and/or inner surfaces. Both pillar geometries are observed to exhibit a size effect where smaller pillars are stronger. The scaling observed is consistent with the weakest-link activation mechanism and depends on the solid pillar diameter, or the annular pillar effective diameter, D_eff = D − D_i, where D and D_i are the external and internal diameters of the pillar, respectively. An external strong barrier is observed to dramatically increase the dislocation density by an order of magnitude due to trapping dislocations at the surface. In addition, a considerable increase in the flow strength, by up to 60%, is observed compared to simulations having free-surface boundary conditions. As the applied load increases, weak spots form on the surface of the pillar by dislocations breaking through the surface when the RSS is greater than the barrier strength. The hardening rate is also observed to increase with increasing barrier strength. With cross-slip, we observe dislocations moving to other glide planes, and sometimes double-cross-slipping, producing a thickening of the slip traces at the surface. Finally the results are in qualitative agreement with recent compression experimental results of coated and centrally-filled micropillars.     

Elastic fields due to dislocation arrays in anisotropic bimaterials

Based on the single-dislocation Green’s function, analytical solutions of the elastic fields due to dislocation arrays in an anisotropic bimaterial system are derived by virtue of the Cottrell summation formula. The singularity in the Peach–Koehler (P–K) force is removed by both rigorous mathematical approach and physical energy consideration. Numerical results for dislocation arrays in the Cu/Nb bimaterial with Kurdjumov–Sachs (K–S) orientation show that: (1) the traction continuity and periodic condition are both satisfied; (2) the maximum magnitude of the traction at the interface due to a mixed dislocation array is smaller than that due to a single mixed dislocation. In other words, the traction at the interface could be suppressed by the corresponding array with a relatively high density (L < 10 nm); however, the shear stress on the glide plane increases with increasing dislocation density; (3) the Cu/Nb interface attracts the mixed dislocation array in copper and repels the screw one there. This implies that the P–K force depends not only on the material properties, but also on the crystal orientation and the type of Burgers vector, among others.     

Size dependent mechanical behaviour of tantalum

The size dependence of deformation of Ta was studied using compression tests of focused ion beam (FIB) machined microcolumns. Columns with diameters between 0.5 and 8 μm with 〈1 1 1〉 and 〈1 0 0〉 orientations along the column axis were tested. By comparing results of bcc Ta columns with results from previous experiments on fcc metals it was found that Ta shows significantly higher normalized yield stresses in combination with a weaker sample size dependence. The differences between bcc and fcc metals can be attributed to the different dislocation behaviour of bcc metals, especially to the lower mobility of screw dislocations.     

On the interaction between a dislocation and a circular inhomogeneity with imperfect interface in antiplane shear

The solution of appropriate elasticity problems involving the interaction between inclusions and dislocations plays a fundamental role in many practical and theoretical applications, namely, it increases the understanding of material defects thereby providing valuable insight into the mechanical behavior of composite materials.Although the problem of a three-phase circular inclusion interacting with a dislocation in antiplane shear has been presented [Xiao and Chen, Mech. Mater. 32 (2000) 485], the analysis is limited to the classical perfect bonding condition. The current paper considers the solution for a homogeneous circular inclusion interacting with a dislocation under thermal loadings in antiplane shear. The bonding along the inhomogeneity–matrix interface is considered to be imperfect with the assumption that the interface imperfections are constant. It is found that when the inhomogeneity is soft, regardless of the level of interface imperfection, the inhomogeneity will always attract the dislocation. As a result, no equilibrium positions are available. Alternatively, when the inhomogeneity is hard, an unstable equilibrium position is found which depends on the imperfect interface condition and the shear moduli ratio μ₂/μ₁.     

Diffusionally assisted grain-boundary sliding and viscoelasticity of polycrystals

Motivated by recent attenuation experiments on finely grained samples, we reanalyse the Raj-Ashby model of grain-boundary sliding. Two linearly elastic layers having finite thickness and identical elastic constants are separated by an interface (grain boundary) whose location is a given periodic function of position. Dissipation is confined to that interfacial region. It is caused by two mechanisms: a slip (boundary sliding) viscosity, and grain-boundary diffusion, with corresponding Maxwell relaxation times t_v and t_d. Owing to the assumption of a given, time-independent interface, the resulting boundary-value problem (b.v.p.) is linear and time-separable. The response to time-periodic forcing depends on angular frequency ω, on the ratio M=t_v/t_d of Maxwell times, and on the characteristic interface slope. The b.v.p. is solved using a perturbation method valid for small slopes. To relate features of the mechanical loss spectrum previously studied in isolation, we first discuss the solution as a function of M. Motivated by experiments, we then emphasize the case M?1 in which the relaxation times are widely separated. The loss spectrum then always has two major features: a frequency band 1?ωt_d?M^-1 within which the loss varies relatively weakly with ω; and a loss maximum at ωt_d∼M^-1 due to the slip viscosity. If corners on the interface are sufficiently rounded, those two universal features are separated by a third feature: between them, there is a strong minimum whose location is (entirely) independent of slip viscosity. The existence of that minimum has not previously been reported. These features are likely to occur even in solutions for finite interface slopes, because they are a consequence of the separation of timescales. The precise form of the spectrum in the weakly varying band must, however, be slope-dependent because it is controlled by stress singularities occurring at corners, and the strength of those singularities depends on the angle subtended by the corner.     

Discrete dislocation plasticity analysis of the grain size dependence of the flow strength of polycrystals

The grain size dependence of the flow strength of polycrystals is analyzed using plane strain, discrete dislocation plasticity. Dislocations are modeled as line singularities in a linear elastic solid and plasticity occurs through the collective motion of large numbers of dislocations. Constitutive rules are used to model lattice resistance to dislocation motion, as well as dislocation nucleation, dislocation annihilation and the interaction with obstacles. The materials analyzed consist of micron scale grains having either one or three slip systems and two types of grain arrangements: either a checker-board pattern or randomly dispersed with a specified volume fraction. Calculations are carried out for materials with either a high density of dislocation sources or a low density of dislocation sources. In all cases, the grain boundaries are taken to be impenetrable to dislocations. A Hall–Petch type relation is predicted with Hall–Petch exponents ranging from ≈0.3 to ≈1.6 depending on the number of slip systems, the grain arrangement, the dislocation source density and the range of grain sizes to which a Hall–Petch expression is fit. The grain size dependence of the flow strength is obtained even when no slip incompatibility exists between grains suggesting that slip blocking/transmission governs the Hall–Petch effect in the simulations.     

Interaction between a screw dislocation and a viscoelastic piezoelectric bimaterial interface

The complex variable method is employed to derive analytical solutions for the interaction between a piezoelectric screw dislocation and a Kelvin-type viscoelastic piezoelectric bimaterial interface. Through analytical continuation, the original boundary value problem can be reduced to an inhomogeneous first-order partial differential equation for a single function of location z = x + iy and time t defined in the lower half-plane, which is free of the screw dislocation. Once the initial, steady-state and far-field conditions are known, the solution to the first order differential equation can be obtained. From the solved function, explicit expressions are then derived for the stresses, strains, electric fields and electric displacements induced by the piezoelectric screw dislocation. Also presented is the image force acting on the screw dislocation due to its interaction with the Kelvin-type viscoelastic interface. The derived solutions are verified by comparing with existing solutions for the simplified cases, and various interesting features are observed, particularly for those associated with the image force.     

含非完整界面涂层夹杂中螺型位错的弹性分析

研究了含非完整界面圆形涂层夹杂内部一个螺型位错在夹杂、涂层与无限大基体材料中产生的弹性场.运用复变函数函数方法,获得了三个区域复势函数的解析解答.利用求得的应力场和Peach-Koehler公式,得到了作用在螺型位错上位错力的精确表达式.主要讨论了两个非完整界面对位错力的影响规律.结果表明,涂层界面对夹杂内部螺型位错的吸引力随着界面粘结强度的弱化而变大.界面非完整程度增加削弱材料弹性失配对位错力的影响.在一定条件下,非完整界面可以改变夹杂内位错与涂层/基体系统之间的引斥干涉规律,并使位错在夹杂内部产生一个稳定或非稳定的平衡点.     

Simulation for surface self-nanocrystallization under shot peening

Driven by high frequency and multi-directional shot peens, dislocations of various orientations proliferate into the metal, and accumulate in high density in the surface layer of a shallow depth. Migration, generation and annihilation of dislocations dictate the evolution of mobile dislocation density. Simulation for the experiment of pure iron under repeated shot peen flux of 800 times per square millimeter is carried out, and a dislocation density up to 2.17×10¹¹ mm⁻² is achieved. Dislocations of such density in the surface layer are shown to be capable of forming nano-grains whose size is about 10 nm. Molecular dynamics simulation verifies the formation of nano-grained metals at such dislocation density level. The dislocations are first regrouped to form subcrystallites, then combined to form stable nanocrystallized grains after sufficiently long time of relaxation. The project supported by the National Natural Science Foundation of China (10121202)     

Terminal velocity of a Taylor drop in a vertical pipe

A scaling analysis based on the field equations for two phases and the jump conditions at the interface is carried out to deduce a balance of forces acting on a Taylor drop rising through stagnant liquid in a vertical pipe. The force balance is utilized to deduce a functional form of an empirical correlation of terminal velocity of the Taylor drop. Undetermined coefficients in the correlation are evaluated by making use of available correlations for two limiting cases, i.e. extremely high and low Reynolds number Taylor bubbles in large pipes. Terminal velocity data obtained by interface tracking simulations are also used to determine the coefficients. The proposed correlation expresses the Froude number Fr as a function of the drop Reynolds number Re_D, the Eötvös number Eo_D and the viscosity ratio μ^*. Comparisons between the correlation, simulations and experimental data confirm that the proposed correlation is applicable to Taylor drops under various conditions, i.e., 0.002 < Re_D < 4960, 4.8 < Eo_D < 228, 0 ? μ^* ? 70, 1 < N < 14700, −12 < log M < 4, and d/D < 1.6, where N is the inverse viscosity number, M the Morton number, d the sphere-volume equivalent drop diameter and D the pipe diameter.     

Dislocation based high-rate plasticity model and its application to plate-impact and ultra short electron irradiation simulations

This paper focuses on the development of a plasticity model to describe high rate deformations of metals. Modeling of target mechanical response is performed in frames of continuum mechanics. Plastic flow is described as the result of an over barrier dislocation sliding in specific slip planes. Computations of shock wave propagation in fcc, bcc and hcp metals modeling in comparison with shock wave experiments are performed to verify the model. The model predicts yield strength increase on elastic precursor in aluminum monocrystal and titanium of high purity at high temperatures.The action on a copper target of the electron beams with energy density (the total energy incident on an unit area during an irradiation pulse) 8.6 J cm⁻² and varied pulse duration has been investigated. At the considered irradiation regime the target remains in a solid state (maximal temperature is 710 K) and shear stresses can reach values of about 0.72 GPa. Depth distribution of dislocation density after irradiation has a maximum that is localized on a distance of 10 μm from the irradiated surface and the maximum dislocation density is about 6 × 10⁹ cm⁻² in the target. The shortening of the exposure time to 1 ns leads to the increase of the dislocation density. Further reduction of exposure time has a weak effect on the dislocation density because the shear stresses reach a limit.     

Crystallographic orientation and size dependence of tension-compression asymmetry in molybdenum nano-pillars

Uniaxial tension and compression experiments on [0 0 1] and [0 1 1] oriented molybdenum nano-pillars exhibit tension-compression asymmetry, a difference in attained stresses in compression vs. tension, which is found to depend on crystallographic orientation and sample size. We find that (1) flow stresses become higher at smaller diameters in both orientations and both loading directions, (2) compressive flow stresses are higher than tensile ones in [0 0 1] orientation, and visa versa in [0 1 1] orientation, and (3) this tension-compression asymmetry is in itself size dependent. We attribute these phenomena to the dependence of twinning vs. antitwinning deformation on loading direction, to the non-planarity of screw dislocation cores in Mo crystals, and to the possibly lesser role of screw dislocations in governing nano-scale plasticity compared with bulk Mo.     

Screw dislocation pile-ups in two phase materials of finite rigidity

The arrangement of discrete screw dislocations piled-up under the action of a uniform applied stress against the welded interface between different elastically isotropic half-spaces has been determined by representing the pile-up as a continuous distribution of infinitesimal dislocations. The dislocation slip plane is inclined at an arbitrary angle $\text{1}\text{2}\text{aπ}$ to the normal to the interface, assuming a to be a rational number. The singular integral equation expressing the condition for static equilibrium of the dislocations under a constant applied stress is solved by a method based on the Wiener-Hoph technique with the Mellin transform, and from this solution the mean density of dislocations and the stress field of the pile-up are determined.     

The piling-up of screw dislocations against a rigid inclusion

This paper determines the stress and displacement distributions near the tip of an array of continuously distributed screw dislocations piled-up against a rigid cylindrical inclusion; the inclusion-matrix interface near the pile-up tip is inclined at an angle β(≠ $\text{1}\text{2}$π) to the slip plane and the solid deforms in an anti-plane strain mode. The local stresses are a power function of the distance r from the pile-up tip, and both the radial and angular dependencies of the stresses are the same irrespective of whether or not there exists a shear stress σ within the interval containing the dislocations. This state of affairs contrasts markedly with that for the special case β = $\text{1}\text{2}$π discussed by E. Smith (1972), when the local stresses are independent of r if σ = 0 and have a logarithmic form when σ ≠ 0. The similarity of the model with that of two intersecting screw dislocation pile-ups in a homogeneous solid is also demonstrated.     

Polychromatic microdiffraction analysis of defect self-organization in shock deformed single crystals

A spatially resolved X-ray diffraction method – with a submicron 3D resolution together with SEM and OIM analysis are applied to understand the arrangements of voids, geometrically necessary dislocations and strain gradient distributions in samples of Al (1 2 3) and Cu (0 0 1) single crystals shocked to incipient spallation fracture. We describe how geometrically necessary dislocations and the effective strain gradient alter white beam Laue patterns of the shocked materials. Several distinct structural zones are observed at different depths under the impact surface. The density of geometrically necessary dislocations (GNDs) is extremely high near the impact and back surface of the shock recovered crystals. The spall region is characterized by a large density of mesoscale voids and GNDs. The spall region is separated from the impact and back surfaces by compressed regions with high total dislocation density but lower GNDs density. Self-organization of shear bands is observed in the shock recovered Cu single crystal.     

Macro-to-microchannel transition in two-phase flow: Part 1 - Two-phase flow patterns and film thickness measurements

The classification of macroscale, mesoscale and microscale channels with respect to two-phase processes is still an open question. The main objective of this study focuses on investigating the macro-to-microscale transition during flow boiling in small scale channels of three different sizes with three different refrigerants over a range of saturation conditions to investigate the effects of channel confinement on two-phase flow patterns and liquid film stratification in a single circular horizontal channel (Part 2 covers the flow boiling heat transfer and critical heat flux). This paper presents the experimental two-phase flow pattern transition data together with a top/bottom liquid film thickness comparison for refrigerants R134a, R236fa and R245fa during flow boiling in small channels of 1.03, 2.20 and 3.04 mm diameter. Based on this work, an improved flow pattern map has been proposed by determining the flow patterns transitions existing under different conditions including the transition to macroscale slug/plug flow at a confinement number of Co ≈ 0.3-0.4. From the top/bottom liquid film thickness comparison results, it was observed that the gravity forces are fully suppressed and overcome by the surface tension and shear forces when the confinement number approaches 1, Co ≈ 1. Thus, as a new approximate rule, the lower threshold of macroscale flow is Co = 0.3-0.4 while the upper threshold of symmetric microscale flow is Co ≈ 1 with a transition (or mesoscale) region in-between.     

Reissner-Sagoci problem for a homogeneous coating on a functionally graded half-space

This paper is concerned with the axisymmetric elastostatic problem related to the rotation of a rigid punch which is bonded to the surface of a nonhomogeneous half-space. The half-space is composed of an isotropic homogeneous coating in the form of layer, which is attached to the functionally graded half-space. The shear modulus of the FGM is assumed to vary in the direction of axis Oz normal to the boundary as μ₁(z) = μ₀(1 + αz)^β, where μ₀, α, β are positive constants. The punch undergoes rotation due to the action of the internal loads. By using Hankel's integral transforms, the mixed boundary value problem is reduced to dual integral equations, and next, to a Fredholm's integral equation of the second kind, which is solved numerically for the case of β = 2. The final results show the effect of non-homogeneity on the shear stresses and an unknown moment of punch rotation.