Boundary integral equations for 2D elasticity and its application in discrete dislocation simulation in finite body: 2. Numerical implementation

In the previous paper by Yu and Diab (2013), several sets of boundary integral equations are derived for general anisotropic materials and corresponding equations for materials with different classes of symmetry are deduced. The work presented herein implements two sets of boundary element schemes to numerically solve the stress field. The integration on the element that has the singular point of the kernel is bounded and can be evaluated analytically. Four benchmark elastic problems are solved numerically to show the advantage of the two schemes over the conventional boundary element formulation in eliminating the boundary layer effect. The one with the weaker singularity has better convergence and gives more accurate results. The presented formulation also provides a direct approach to solve for stress field in a finite solid body in the presence of dislocations. Combined with discrete dislocations dynamics, boundary value problems with dislocations in finite bodies can be solved. Two examples, bending of a single crystal beam and pure shearing of a polycrystalline solid, are simulated by discrete dislocation dynamics using the scheme that has the weaker singularity. The comparisons with the published results using the well-established superposition technique validate the proposed formulation and show its quick convergence.     

Effect of solutes on dislocation motion —a phase-field simulation

Based on recent advances in phase-field models for integrating phase and defect microstructures as well as dislocation dynamics, the interactions between diffusional solutes and moving dislocations under applied stresses are studied in three dimensions. A new functional form for describing the eigenstrains of dislocations is proposed, eliminating the dependence of the magnitude of the dislocation Burgers vector on the applied stress and providing correct stress fields of dislocations. A relationship between the velocity of the dislocation and the applied stress is obtained by theoretical analysis and numerical simulations. The operation of Frank–Read sources in the presence of diffusional solutes, the effect of chemical interactions in solid solution on the equilibrium distribution of Cottrell atmosphere, and the drag effect of Cottrell atmosphere on dislocation motion are examined. The results demonstrate that the phase-field model correctly describes the long-range elastic interactions and short-range chemical interactions that control dislocation motion.     

A multiscale model for the ductile fracture of crystalline materials

In this paper, a multiscale model that combines both macroscopic and microscopic analyses is presented for describing the ductile fracture process of crystalline materials. In the macroscopic fracture analysis, the recently developed strain gradient plasticity theory is used to describe the fracture toughness, the shielding effects of plastic deformation on the crack growth, and the crack tip field through the use of an elastic core model. The crack tip field resulting from the macroscopic analysis using the strain gradient plasticity theory displayes the 1/2 singularity of stress within the strain gradient dominated region. In the microscopic fracture analysis, the discrete dislocation theory is used to describe the shielding effects of discrete dislocations on the crack growth. The result of the macroscopic analysis near the crack tip, i.e. a new K-field, is taken as the boundary condition for the microscopic fracture analysis. The equilibrium locations of the discrete dislocations around the crack and the shielding effects of the discrete dislocations on the crack growth at the microscale are calculated. The macroscopic fracture analysis and the microscopic fracture analysis are connected based on the elastic core model. Through a comparison of the shielding effects from plastic deformation and the discrete dislocations, the elastic core size is determined.     

Similarity solution of a penny-shaped fluid-driven fracture in a zero-toughness linear elastic solid

This note deals with the problem of a penny-shaped hydraulic fracture propagating in an impermeable elastic solid. Growth of the fracture is driven by injection of an incompressible Newtonian fluid at the center of the fracture. The solution is restricted to the so-called viscosity-dominated regime where it can be assumed that the solid has zero toughness. The paper describes the construction of a semi-analytical similarity solution, which incorporates the known singularity of the fluid pressure at the center of the fracture and at the tip and which is based on series expansions of the fracture opening and fluid pressure in terms of Jacobi polynomials.     

The state of stress induced by ring dislocations in a semi-infinite stepped shaft

The state of stress induced in an axi-symmetric solid formed from a half-space and a bonded semi-infinite rod, by a family of ring dislocations of arbitrary Burgers vector is found. Particular care is given to the interaction between the Cauchy singularity near the dislocation core and the geometric singularity at the rod/half-space junction.     

Propagation of a plane-strain hydraulic fracture with a fluid lag: Early-time solution

This paper studies the propagation of a plane-strain fluid-driven fracture with a fluid lag in an elastic solid. The fracture is driven by a constant rate of injection of an incompressible viscous fluid at the fracture inlet. The leak-off of the fracturing fluid into the host solid is considered negligible. The viscous fluid flow is lagging behind an advancing fracture tip, and the resulting tip cavity is assumed to be filled at some specified low pressure with either fluid vapor (impermeable host solid) or pore-fluids infiltrating from the permeable host solid. The scaling analysis allows to reduce problem parametric space to two lumped dimensionless parameters with the meaning of the solid toughness and of the tip underpressure (difference between the specified pressure in the tip cavity and the far field confining stress). A constant lumped toughness parameter uniquely defines solution trajectory in the parametric space, while time-varying lumped tip underpressure parameter describes evolution along the trajectory. Further analysis identifies the early and large time asymptotic states of the fracture evolution as corresponding to the small and large tip underpressure solutions, respectively. The former solution is obtained numerically herein and is characterized by a maximum fluid lag (as a fraction of the crack length), while the latter corresponds to the zero-lag solution of Spence and Sharp [Spence, D.A., Sharp, P.W., 1985. Self-similar solution for elastohydrodynamic cavity flow. Proc. Roy. Soc. London, Ser. A (400), 289–313]. The self-similarity at small/large tip underpressure implies that the solution for crack length, crack opening and net fluid pressure in the fluid-filled part of the crack is a given power-law of time, while the fluid lag is a constant fraction of the increasing fracture length. Evolution of a fluid-driven fracture between the two limit states corresponds to gradual expansion of the fluid-filled region and disappearance of the fluid lag. For small solid toughness and small tip underpressure, the fracture is practically devoid of fluid, which is localized into a narrow region near the fracture inlet. Corresponding asymptotic solution on the fracture lengthscale corresponds to that of a crack loaded by a pair of point forces which magnitude is determined from the coupled hydromechanical solution in the fluid-filled region near the crack inlet. For large solid toughness, the fluid lag is vanishingly small at any underpressure and the solution is adequately approximated by the zero-lag self-similar large toughness solution at any stage of fracture evolution. The small underpressure asymptotic solutions obtained in this work are sought to provide initial condition for the propagation of fractures which are initially under plane-strain conditions.     

Micromechanics analyses of interface crack

A new mechanics model based on Peierls concept is presented in this paper, which can clearly characterize the intrinsic features near a tip of an interfacial crack. The stress and displacement fields are calculated under general combined tensile and shear loadings. The near tip stress fields show some oscillatory behaviors but without any singularity and the crack faces open completely without any overlapping when remote tensile loading is comparable with remote shear loading. A fracture criterion for predicting interface toughness has been also proposed, which takes into account for the shielding effects of emitted dislocations. The theoretical toughness curve gives excellent prediction, as compared with the existing experiment data. The project supported by the National Natural Science Foundation of China     

非牛顿流体固粒悬浮流的若干问题

非牛顿流体固粒悬浮流具有广泛的应用背景,其特殊的流动属性使其成为一些新兴技术领域的核心突破点.同时,该流动又比较复杂,即便是在低固粒浓度的情况下,非牛顿流体特性也会对整个系统的微结构产生重要的影响,从而进一步影响固粒的运动.本文给出了非牛顿流体方程、固粒运动方程和非牛顿流体固粒悬浮流的特征参数,分析了这些参数的作用;阐述了单个固粒在管道中的径向移动、多固粒的相互作用和聚集、多固粒形成的链状结构以及非圆球固粒运动等方面的研究成果、结果分析以及尚未解决的问题,并对以上问题进行了总结和展望,给出了需要深入研究的具体问题和内容,旨在为进一步的研究提供参考和依据.     

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.     

On Cavitation, Configurational Forces and Implications for Fracture in a Nonlinearly Elastic Material

Experiments on polymers indicate that large tensile stress can induce cavitation, that is, the appearance of voids that were not previously evident in the material. This phenomenon can be viewed as either the growth of pre-existing infinitesimal holes in the material or, alternatively, as the spontaneous creation of new holes in an initially perfect body. In this paper our approach is to adopt both views concurrently within the framework of the variational theory of nonlinear elasticity. We model an elastomer on a macroscale as a void-free material and, on a microscale, as a material containing certain defects that are the only points at which hole formation can occur. Mathematically, this is accomplished by the use of deformations whose point singularities are constrained. One consequence of this viewpoint is that cavitation may then take place at a point that is not energetically optimal. We show that this disparity will generate configurational forces, a type of force identified previously in dislocations in crystals, in phase transitions in solids, in solidification, and in fracture mechanics. As an application of this approach we study the energetically optimal point for a solitary hole to form in a homogeneous and isotropic elastic ball subject to radial boundary displacements. We show, in particular, that the center of the ball is the unique optimal point. Finally, we speculate that the configurational force generated by cavitation at a non-optimal material point may be sufficient to result in the onset of fracture. The analysis utilizes the energy-momentum tensor, the asymptotics of an equilibrium solution with an isolated singularity, and the linear theory of elasticity at the stressed configuration that the body occupies immediately prior to cavitation. This revised version was published online in June 2006 with corrections to the Cover Date.     

刃型位错和圆弧形刚性线夹杂的干涉效应

位错和夹杂的干涉效应对于理解材料的强化和韧化机理具有十分重要的意义。文中研究了晶体材料中刃型位错和多条共圆弧刚性线夹杂的干涉作用。利用Riemann—Schwarz反照原理和复势函数的奇性主部分析技术，得到了问题的一般解答；对于只含一条刚性线夹杂的情况，给出了复势函数的封闭形式解。由Peach-Koehler公式求出了作用在刃型位错上的位错力，并讨论了圆弧形刚性线夹杂对位错力的影响规律，发现弧形刚性线对刃型位错有很强的排斥作用。本文解答不但可作为格林函数获得任意分布位错的相应解答，而且可以用于研究刚性线夹杂和任意形状裂纹的干涉效应问题。     

混凝土断裂过程区的虚拟裂纹粘聚力奇异性

混凝土断裂过程区视为具有粘聚阻力作用的虚拟裂纹，其非线性断裂和尺寸效应特性是与该虚拟裂纹粘聚力分布规律密切相关的。通过得到的粘聚应力分布函数解析结果，对该粘聚力分布特征的分析得知，在基于断裂过程区之外用线弹性场的力学模型上，该粘聚力随距离虚拟裂纹尖点的靠近，仍具有平方根奇异性。从而本文提出一个能够反映裂纹发展状态的粘聚应力奇异性强度参数，它是无粘聚力的线弹性裂纹应力强度因子和表征裂纹张开位移分布多项式参数的函数；因此，该参数可以作为混凝土非线性断裂的一个参量。文中就已有断裂试验测试结果进行了算例分析和相应的讨论。     

Temperature dependant polycrystal model application to bainitic steel behavior under tri-axial loading in the ductile–brittle transition

A polycrystal finite element (FE) model describing the temperature evolution of low carbon steel is proposed in order to forecast the local mechanical fields as a function of temperature, for bainitic microstructure submitted to tri-axial loading. The model is designed for finite strains, large lattice rotations and temperatures ranging into the brittle–ductile transition domain. The dislocation densities are the internal variables. At low temperature in Body Centred Cubic (BCC) materials, plasticity is governed by double kink nucleation of screw dislocations, whereas at high temperature, plasticity depends on interactions between mobile dislocations and the forest dislocations. In this paper, the constitutive law and the evolution of the dislocation densities are written as a function of temperature and describe low and high temperature mechanisms. The studied aggregates are built from Electron Back Scattering Diffraction (EBSD) images of real bainitic steel. The aggregate is submitted to a tri-axial loading in order to describe the material at a crack tip. Mechanical parameters are deduced from mechanical tests. The local strain and stress fields, computed for different applied loadings, present local variations which depend on temperature and on tri-axial ratio. The distribution curves of the maximal principal stresses show that heterogeneities respectively increase with temperature and decrease with tri-axial ratio. A direct application of this model provides the evaluation of the rupture probability within the aggregate, which is treated as the elementary volume in the weak link theory. A comparison with the Beremin criterion calibrated on experimental data, shows that the computed fracture probability dispersion induced by the stress heterogeneities is of the same order than the measured dispersion. Temperature and stress tri-axiality ratio effects are also investigated. It is shown that these two parameters have a strong effect on fracture owing to their influence on the heterogeneous plastic strain. These inhomogeneities can initiate cleavage fracture.     

Plane-Strain Fracture with Curvature-Dependent Surface Tension: Mixed-Mode Loading

There have been a number of recent papers by various authors addressing static fracture in the setting of the linearized theory of elasticity in the bulk augmented by a model for surface mechanics on fracture surfaces with the goal of developing a fracture theory in which stresses and strains remain bounded at crack-tips without recourse to the introduction of a crack-tip cohesive-zone or process-zone. In this context, surface mechanics refers to viewing interfaces separating distinct material phases as dividing surfaces, in the sense of Gibbs, endowed with excess physical properties such as internal energy, entropy and stress. One model for the mechanics of fracture surfaces that has received much recent attention is based upon the Gurtin-Murdoch surface elasticity model. However, it has been shown recently that while this model removes the strong (square-root) crack-tip stress/strain singularity, it replaces it with a weak (logarithmic) one. A simpler model for surface stress assumes that the surface stress tensor is Eulerian, consisting only of surface tension. If surface tension is assumed to be a material constant and the classical fracture boundary condition is replaced by the jump momentum balance relations on crack surfaces, it has been shown that the classical strong (square-root) crack-tip stress/strain singularity is removed and replaced by a weak, logarithmic singularity. If, in addition, surface tension is assumed to have a (linearized) dependence upon the crack-surface mean-curvature, it has been shown for pure mode I (opening mode), the logarithmic stress/strain singularity is removed leaving bounded crack-tip stresses and strains. However, it has been shown that curvature-dependent surface tension is insufficient for removing the logarithmic singularity for mixed mode (mode I, mode II) cracks. The purpose of this note is to demonstrate that a simple modification of the curvature-dependent surface tension model leads to bounded crack-tip stresses and strains under mixed mode I and mode II loading.     

各向异性复合材料中界面屈服/脱粘对裂纹扩展的影响

本研究针对层状复合材料中正交于层合面的裂纹,研究裂纹前方层状界面发生屈服或脱粘现象对该裂纹前沿应力场的扰动。通过利用叠加原理,借用滑移型位错密度表征界面的屈服或脱粘。利用Chebyshev数值积分法求解相应的位错密度的奇异积分方程,得到沿界面屈服/脱粘区域的位错密度分布及裂端区应力场。结果表明,若层状复合材料界面为发生屈服或脱粘,将减弱独立层裂尖的应力奇异性,进而抑制独立层中裂纹的扩展。     

In-plane stress analysis of an orthotropic plane containing multiple defects

The solution of Volterra type climb and glide edge dislocations is utilized to formulate integral equations for an orthotropic homogeneous infinite plane weakened by multiple smooth cracks and/or cavities. Cavities are considered as closed curved cracks without singularity. The integral equations are of Cauchy singular type which are converted to hypersingular integral equations. These equations are then solved numerically to determine stress intensity factors for cracks and hoop stress on the cavities. The results for isotropic and orthotropic planes are compared with available solutions in literature and excellent agreement is observed. The formulation allows stress analysis of orthotropic planes with several arbitrarily oriented cracks and cavities.     

Analysis of the stress singularity field at a vertex in 3D-bonded structures having a slanted side surface

The stress singularity that occurs at a vertex in a joint with a slanted side surface is investigated. The orders of stress singularity at a vertex and at a point on stress singularity lines for various material properties are determined using eigenanalysis. The stress distribution on an interface and the intensity of stress singularity at the vertex are investigated using BEM. It is shown that the order of stress singularity at the vertex in the joints can be reduced by slanting a side surface so as to decrease the angle between the interface and the side surface. The results of BEM analysis reveal that the distribution of stress on the interface is influenced by the slanted side surface. Finally, the 3D intensities of the singularity for stress components which are continuous at the interface are newly defined and determined for various material combinations.     

Plastic deformation of freestanding thin films: Experiments and modeling

Experimental measurements and computational results for the evolution of plastic deformation in freestanding thin films are compared. In the experiments, the stress-strain response of two sets of Cu films is determined in the plane-strain bulge test. One set of samples consists of electroplated Cu films, while the other set is sputter-deposited. Unpassivated films, films passivated on one side and films passivated on both sides are considered. The calculations are carried out within a two-dimensional plane strain framework with the dislocations modeled as line singularities in an isotropic elastic solid. The film is modeled by a unit cell consisting of eight grains, each of which has three slip systems. The film is initially free of dislocations which then nucleate from a specified distribution of Frank-Read sources. The grain boundaries and any film-passivation layer interfaces are taken to be impenetrable to dislocations. Both the experiments and the computations show: (i) a flow strength for the passivated films that is greater than for the unpassivated films and (ii) hysteresis and a Bauschinger effect that increases with increasing pre-strain for passivated films, while for unpassivated films hysteresis and a Bauschinger effect are small or absent. Furthermore, the experimental measurements and computational results for the 0.2% offset yield strength stress, and the evolution of hysteresis and of the Bauschinger effect are in good quantitative agreement.     

Motion of a slender body in a fluid under a floating plate

This paper studies the three-dimensional unsteady problem of the hydroelastic behavior of a floating infinite plate under the impact of waves generated by horizontal rectilinear motion of a slender solid in a fluid of infinite depth. An analytic solution of the problem is found based on the known solutions for the unsteady motion of a point source of mass in a fluid of infinite depth under a floating plate. Asymptotic formulas are obtained which model the motion of a solid slender body in a fluid by replacing the body with a source-sink system. These formulas are used to numerically analyze the effect of plate thickness, depth of the body, its dimensions and the velocity of rectilinear motion on the amplitude of deflection of the floating plate. The motion of a submarine under a nonbreakable plate was modeled experimentally. Theoretical and experimental data are in good agreement.     

Theoretical study of the generation of screw dislocations for Rayleigh and Lamb waves in isotropic solids

Phase singularities are generic structures which occur in all wave fields, and they are characterised by an inability to assign a value to the phase. Screw dislocations are a particular kind of phase singularity where the phase possesses a helical structure, with the singularity at the centre of the helix. In this paper we show that it is possible to generate screw dislocations on the surface of elastic isotropic solids by means of the interference of three Rayleigh waves or three Lamb waves. The dispersive character of Lamb waves leads to more complicated behaviour, which may in turn result in greater potential for applications.