The dynamic stress intensity factor for a crack due to arbitrary rectilinear screw dislocation motion

The dynamic stress intensity factor for a stationary semi-infinite crack in an elastic plane due to the rectilinear motion of a screw dislocation is obtained analytically. The intensity factor is studied for its dependence on the (initial) dislocation position, orientation and speed. The speed is subsonic and possibly non-uniform. The position and orientation are arbitrary, so that crack-dislocation intersection is considered. It is assumed that a dislocation traveling toward the crack surface arrests upon arrival. It is found that, in general, dislocation motion initiation and arrest cause discontinuities in the intensity factor. In the latter instance, the factor takes on a constant value and, in the case of arrest on the crack surface, this value depends only on the initial dislocation position.     

Multiscale simulation of material removal processes at the nanoscale

A dynamic multiscale simulation method has been used to study the nanoscale material removal processes for single crystals. The model simultaneously captures the atomistic mechanisms during material removal from the free surface and the long-range mobility of dislocations and their interactions without the computational cost of full atomistic simulations. The method also permits the simulation of system sizes that are approaching experimentally accessibly systems, albeit in 2D. Simulations are performed on single crystal aluminum to study the atomistic details of material removal, chip formation, surface evolution, and generation and propagation of dislocations for a wide range of tool speeds (20-800 m/s) at room temperature. The results from these simulations demonstrate the power of the developed method in capturing both long-range dislocation plasticity and short-range atomistic phenomena during tool advance. In addition, we have investigated the effect of the scratching depth during the material removal process. Fluctuations of scratching tangential force are related to dislocation generation events during the material removal process. A transition from dislocation generation and glides at lower tool speeds to localized amorphization at high tool speeds is found to give rise to an optimal tool speed for low cutting forces.     

The material force acting on a screw dislocation in the presence of a multi-layered circular inclusion

In this paper, we study the interaction of a screw dislocation with a multi-layered interphase between a circularly cylindrical inclusion and a matrix. The layers are coaxial cylinders of annular cross-sections with arbitrary radii and different shear moduli. The number of layers may also be arbitrary. Continuity of traction and displacement across all interfaces is assumed. We extend Honein et al.’s solution of circularly cylindrical layered media in anti-plane elastostatics to the case where all the singularities reside inside the inclusion core. The solution to this heterogeneous problem is given explicitly, for arbitrary singularities, as a rapidly convergent Laurent series, whose coefficients are expressed in terms of those of the complex potential of a corresponding homogeneous problem with the same singularities. We then consider the two particular cases of a screw dislocation, where, in the first instance, the dislocation resides inside the matrix, while, in the second instance, it is located in the inclusion core. In both instances, the Peach–Koehler force acting on the dislocation is calculated explicitly as a rapidly convergent series. We present several examples, where the effect of the layers on the material force is examined.     

Surface waves in a deformed isotropic hyperelastic material subject to an isotropic internal constraint

An isotropic elastic half-space is prestrained so that two of the principal axes of strain lie in the bounding plane, which itself remains free of traction. The material is subject to an isotropic constraint of arbitrary nature. A surface wave is propagated sinusoidally along the bounding surface in the direction of a principal axis of strain and decays away from the surface. The exact secular equation is derived by a direct method for such a principal surface wave; it is cubic in a quantity whose square is linearly related to the squared wave speed. For the prestrained material, replacing the squared wave speed by zero gives an explicit bifurcation, or stability, criterion. Conditions on the existence and uniqueness of surface waves are given. The bifurcation criterion is derived for specific strain energies in the case of four isotropic constraints: those of incompressibility, Bell, constant area, and Ericksen. In each case investigated, the bifurcation criterion is found to be of a universal nature in that it depends only on the principal stretches, not on the material constants. Some results related to the surface stability of arterial wall mechanics are also presented.     

Elastic fields of dislocation loops in three-dimensional anisotropic bimaterials

By applying semi-analytical point-force Green's functions obtained via the Stroh formulism, we derive simple line integrals to calculate the elastic displacement and stress fields for a three-dimensional dislocation loop in an anisotropic bimaterial system. The solutions for the case of anisotropy are more convenient for treating an arbitrary dislocation loop compared with traditional area integration. With this new formulation, we numerically examine the displacement, stress, and energy due to the interaction between a dislocation loop and the bimaterial interface in an Al–Cu system. The interactive image energy due to the elastic moduli mismatch across the interface is then numerically evaluated. The result shows that a dislocation loop is subjected to an attractive force by the interface when it lies in the stiff material, and a repulsive force when it lies in the soft material. Moreover, the dependence of the interactive image energy of a dislocation loop on the position and size of the dislocation loop are also demonstrated and discussed. Significantly, it is found that the interactive image energy for a dislocation loop depends only on the ratio d/a, where a is the loop diameter and d is its distance to the interface. The examples studied provide benchmark solutions for anisotropic bimaterial dislocation problems.     

Interaction energy of interface dislocation loops in piezoelectric bi-crystals

Interface dislocations may dramatically change the electric properties,such as polarization,of the piezoelectric crystals.In this paper,we study the linear interactions of two interface dislocation loops with arbitrary shape in generally anisotropic piezoelectric bi-crystals.A simple formula for calculating the interaction energy of the interface dislocation loops is derived and given by a double line integral along two closed dislocation curves.Particularly,interactions between two straight segments of the interface dislocations are solved analytically,which can be applied to approximate any curved loop so that an analytical solution can be also achieved.Numerical results show the influence of the bi-crystal interface as well as the material orientation on the interaction of interface dislocation loops.     

Electroelastic interaction between piezoelectric screw dislocation and circularly layered inclusion with imperfect interfaces

The interaction between a piezoelectric screw dislocation and an interphase layer in piezoelectric solids is theoretically investigated.Here,the dislocation located at arbitrary points inside either the matrix or the inclusion and the interfaces of the interphase layer are imperfect.By the complex variable method,the explicit solutions to the complex potentials are given,and the electroelastic fields can be derived from them.The image force acting on the dislocation can be obtained by the generalized PeachKoehler formula.The motion of the piezoelectric screw dislocation and its equilibrium positions are discussed for variable parameters.The important results show that,if the inner interface of the interphase layer is imperfect and the magnitude of degree of the interface imperfection reaches the certain value,two equilibrium positions of the piezoelectric screw dislocation in the matrix near the interface are found for the certain material combination which has never been observed in the previous studies(without considering the interface imperfection).     

Interaction of an edge dislocation with a coated elliptic inclusion

This paper presents an analytical solution for plane elasticity problems of an elliptically cylindrical layered media subject to an arbitrary edge dislocation. Based on the technique of conformal mapping and the method of analytical continuation in conjunction with the alternating technique, the general expressions of the displacements and stresses, where an edge dislocation is located in matrix, coating layer and inclusion are obtained. The numerical results of image forces exerted on a generalized edge dislocation are carried out by using the generalized Peach–Koehler equation. As a numerical illustration, both the image forces and equilibrium positions are presented for different material combinations and relative thickness of a coating layer. The result shows that the thickness and the shear modulus of the coating layer have a strong influence on the stability of dislocation.     

On XFEM applications to dislocations and interfaces

A method for modelling dislocations in systems with arbitrary materials interfaces is described. The method is based on the extended finite element method (XFEM) where dislocations are modelled in the manner of the Volterra dislocation model. A method for calculating the Peach–Koehler force by J-integrals in this framework is studied. The method is compared to closed form solutions for interface problems and excellent accuracy is obtained. The convergence and accuracy of the method is studied in two problems where analytical solutions are available: an edge dislocation interacting with a free-surface and an edge dislocation interacting with a bimaterial interface. The applicability of the method to more complicated problems is illustrated by the modelling of slip misorientation of an edge dislocation with a glide plane intersecting a material interface and dislocations in a multi-material domain with non-parallel interfaces.     

Distributional and regularized radiation fields of non-uniformly moving straight dislocations,and elastodynamic Tamm problem

This work introduces original explicit solutions for the elastic fields radiated by non-uniformly moving, straight, screw or edge dislocations in an isotropic medium, in the form of time-integral representations in which acceleration-dependent contributions are explicitly separated out. These solutions are obtained by applying an isotropic regularization procedure to distributional expressions of the elastodynamic fields built on the Green tensor of the Navier equation. The obtained regularized field expressions are singularity-free, and depend on the dislocation density rather than on the plastic eigenstrain. They cover non-uniform motion at arbitrary speeds, including faster-than-wave ones. A numerical method of computation is discussed, that rests on discretizing motion along an arbitrary path in the plane transverse to the dislocation, into a succession of time intervals of constant velocity vector over which time-integrated contributions can be obtained in closed form. As a simple illustration, it is applied to the elastodynamic equivalent of the Tamm problem, where fields induced by a dislocation accelerated from rest beyond the longitudinal wave speed, and thereafter put to rest again, are computed. As expected, the proposed expressions produce Mach cones, the dynamic build-up and decay of which is illustrated by means of full-field calculations.     

Thermo-elastic crack branching in general anisotropic media

Thermal fields may exist in addition to mechanical loading, for example, due to short term exposure to fire. In this paper, the branching of cracks in the presence of combined thermal and mechanical loads is investigated for general anisotropic media by employing the theory of Stroh’s dislocation formalism, extended to thermo-elasticity in matrix notation. A general solution to the thermo-elastic crack problem for an anisotropic material under arbitrary loading is obtained in a compact form. Green’s functions are also presented for a thermal dislocation (heat vortex) and a conventional dislocation (or, referred as mechanical dislocation), which are formulated considering the cuts located at an arbitrary angle with respect to the x₁ axis of the coordinate system (x₁, x₂, x₃). Using the derived compact expressions, the interaction between the crack and the dislocation is studied and a closed form solution for this interaction is obtained. The branching portion of the thermo-elastic crack is modelled as a continuous distribution of dislocations. This problem is then converted into a set of singular integral equations. Numerical results are presented to illustrate the possible effects of thermal loading on the propagation of the branched crack.     

Stress distribution in a two-dimensional infinite anisotropic medium with collinear cracks

The plane problem of an anisotropic material with cracks, whose surfaces are subject to surface tractions of a general kind, is studied. The medium considered if of infinite extent and the cracks are located on a single line. The Fourier transform method is employed to derive the stress and displacement components at an arbitrary point of the medium in terms of the dislocation densities and the stress discontinuities on the crack line.These formulae for stress and displacement components involve the roots of a quartic equation whose coefficients are the material constants. The cases of different roots and pairwise coincident roots are examined separately. An orthotropic medium is an important example for the case of different roots while an isotropic medium is that for the case of pairwise coincident roots. These examples are discussed in detail.As an illustration of the use of these formulae the problem of a single crack in an infinite anisotropic medium is examined in detail.Work supported in part by a grant from Council of Scientific and Industrial Research, New Delhi, India.     

Emission of dislocations from nanovoids under combined loading

Among all directions available for dislocation emission from the surface of a cylindrical circular void, the direction of the most likely emission is determined. It is shown that this direction is different from the direction of the maximum shear stress at the surface of the void due to the applied loading. The critical stress and the direction of the dislocation emission are determined for circular nanovoids under remote uniaxial, pure shear, and arbitrary biaxial loading. The analysis includes effects of the loading orientation relative to the discrete slip plane orientation. It is shown that dislocations are emitted more readily from larger nanovoids and that wider dislocations are emitted under lower applied stress than narrow dislocations. Different mechanisms, under much lower stress, operate for growth of the micron-size voids.     

A magnetoelectric screw dislocation interacting with a circular layered inclusion

Within the framework of the linear theory of magnetoelectroelasticity, the problem of a circular layered inclusion interacting with a generalized screw dislocation under remote anti-plane shear stress and in-plane magnetoelectric loads is investigated in this paper. The generalized dislocation can be located either in the matrix or in the circular layered inclusion. The layers are coaxial cylinders of annular cross-sections with arbitrary radii and different material properties. Using complex variable theory and the alternating technique, the solution of the present problem is expressed in terms of the solution of the corresponding homogeneous medium problem subjected to the same loading. Some numerical results are provided to investigate the influence of material combinations on the shear stress, electric field, magnetic and image force. These solutions can be used as Green's functions for the analysis of the corresponding magnetoelectric crack problem.     

The non-uniform motion of arbitrary dislocation distributions by climb and by glide along non-planar paths

As a basis for obtaining insight into both plastic flow in terms of dislocation motion and dynamic crack extension, the general problem of non-uniform motion of largely arbitrary dislocation distributions by climb and by glide along non-planar paths is considered.An exact solution is found in two forms: one form, vectorial in nature, shows that the essential distribution and path properties are contained in a symmetric tensor. The other form, consisting of complex functions, shows that the solution involves the inner product of the displacement discontinuity vector and complex vectors whose components normal and tangential to the path contour are related through tangent angle derivatives.Both forms illustrate that the solution has two components, one arising from the velocity discontinuity along the contour, the other arising from the displacement discontinuity at its edge and the edge speed.     

The effect of path cut on Somigliana ring dislocations in a half-space

This paper provides solutions for the stress and displacement fields induced in a half-space by an edge dislocation of constant magnitude, where the dislocation line is a circular ring lying in a plane parallel with the free surface. Two types of edge dislocations are considered; the prismatic form with a displacement discontinuity vector normal to the surface and cases with radial dislocation vectors. The latter are not true Volterra dislocations and produce different results for different path cuts these effects are investigated in some detail. The primary application of these results is for analysis of axisymmetric problems – the radial dislocations would be impossible in finite material.     

Stress-intensity factors for 3-D dynamic loading of a cracked half-space

A half-space containing a surface-breaking crack of uniform depth is subjected to three-dimensional dynamic loading. The elastodynamic stress-analysis problem has been decomposed into two problems, which are symmetric and antisymmetric, respectively, relative to the plane of the crack. The formulation of each problem has been reduced to a system of singular integral equations of the first kind. The symmetric problem is governed by a single integral equation for the opening-mode dislocation density. A pair of coupled integral equations for the two sliding-mode dislocation densities govern the antisymmetric problem. The systems of integral equations are solved numerically. The stress-intensity factors are obtained directly from the dislocation densities. The formulation is valid for arbitrary 3-D loading of the half-space. As an example, an applied stress field corresponding to an incident Rayleigh surface wave has been considered. The dependence of the stress-intensity factors on the frequency, and on the angle of incidence, is displayed in a set of figures.     

Contemporary perspectives in finite strain plasticity

From an internally consistent theory of finite strain plasticity in which are introduced a new stress tensor that includes rigid body rotation and a compatible finite strain tensor amenable to simple geometrical interpretation, a close correlation with experiment is obtained for stress paths of arbitrary composition and direction. Measured material constants are proportional to the elastic shear modulus, as indicated for dislocation theory. At large strain there exists a quantum mechanical structure for the continuum, in which are defined a series of relative reference configurations and second order transitions.     

Steady-State Response of a Thermoelastic Half-Space to the Rapid Motion of Surface Thermal/Mechanical Loads

Shear and normal tractions and a heat flux are applied to a largely arbitrary area that moves with constant subsonic speed over a half-space surface. The half-space is a coupled thermoelastic solid of the Jeffreys type, so that the governing steady-state equations involve three thermoelastic characteristic lengths and a dimensionless coupling constant. This constant and one of the lengths remain in the limit as the solid reduces to the standard coupled thermoelastic material. The problem is solved exactly in an integral transform space, and asymptotic expressions for the normal displacement and the temperature change induced on the half-space surface are extracted. These are in principle valid for large distances from the loading zone as measured along its line of travel but, because the scaling dimension is of O(10^-14)μ m, they are robust. Exact inversions are performed, and the results show marked dependence on both loading zone speed and thermoelastic parameters. Indeed, the role of the latter is enhanced as the speed is increased. Singular behavior is found, in particular, when the loading zone moves with the effective thermoelastic Rayleigh speed, an exact formula for which is also given. This revised version was published online in August 2006 with corrections to the Cover Date.     

The interaction of an edge dislocation with an inhomogeneity of arbitrary shape in an applied stress field

A general, approximate solution is presented for an edge dislocation interacting with an inhomogeneity of arbitrary shape under combined dislocation and applied stress fields. The solution shows that the contributions of the dislocation stress field and the applied stress field to the interaction follow a simple superposition principle. The dislocation stress field has a short range effect, while the applied stress field has a long range effect. As special cases, explicit solutions for some common inhomogeneity shapes are obtained for the interaction induced by the applied stress field.