Collected Works of J. D. Eshelby

Zener has shown how thermoelastic effects give rise to damping of the mechanical vibrations of solid. For example, in a vibrating reed opposite sides are alternately compressed and extended. This gives rise to an alternating temperature-difference across the width of the reed, and the resulting flow of heat leads to dissipation of mechanical energy.

An expression is derived for the displacements in an isotropic elastic medlum which contains an edge dislocation moving with uniform velocity c. When c=0 the solutlon reduces to that given by Burgers for a stationary edge dislocation. The energy density in the medium becomes infinite as c approaches c 2, the velocity of shear waves in the medurn; this velocity therefore sets a limit beyond which the dislocation cannot be accelerated by applied stresses. The atomc structure of the medium is next partly taken into account, following the method already used by Peierls and Nabarro for the stationary dislocation. The solution found in this way differs from the one in which the atomic structure is neglected only within a region of width ζ which extends not more than a few atomic distances from the centre. ζ varies with c and vanishes when c=c r, the velocity of Rayleigh waves. It becomes negative when c r<c<c 2. Thus c r rather than c 2 appears to be the limiting velocity when the atomic nature of the mehum is taken into account. Since c r≃0.9c 2 the difference is not of much importance.The same method applied to a screw dislocation giws, in the purely elastic case, the expression already derived by Frank. The corresponding Peierls-Nabarro calculation shows that the width ζ is proportional to (1−c 2/c 2 2 )1/2. This “relativistic” behaviour is analogous to Frenkel and Kontorowa’s results for their one-dimensional dislocation model.

Burgers has discussed dislocations in materials with cubic symmetry for the general case in which the dislocation axis is an arbitrary curve. The results of the present paper are limited to dislocations of edge type with an infinite straight line as axis, but apply to a material with the symmetry of any of the crystal classes. The axis of the dislocation may be arbitrarily inclined to the symmetry axes of the material. Expressions are given for the elastic displacements and the energy of the dislocation. Nabarro’s calculation of the wldth of a dislocation is extended to the anisotropic case.

This paper sketches the picture which theoretical physics gives of the mechanisms of heat conduction in metals and insulators for the temperature range used in normal engineering practice.

A method is given for finding the equilibrium positions of a set of like dislocations in a common slip-plane under the influence of a given applied stress. Their positions are given by the roots of a certain set of orthogonal polynomials. The case of a set of free dislocations piled up against a fixed dislocation by a constant applied stress is discussed in detail and the resulting stress-distribution is compared with that produced by a crack with freely slipping surfaces.

The parallel between the classical theory of elasticity and the modern physical theory of the solid state is incomplete; the former has nothing analogous to the concept of the force acting on an imperfection (dislocation, foreign atom, etc.) in a stressed crystal lattice. To remedy this a general theory of the forces on singularities in a Hookean elastic continuum is developed. The singularity is taken to be any state of internal stress satisfjring the equilibrium equations but not the compatibility conditions. The force on a singularity can be given as an integral over a surface enclosing it. The integral contains the elastic field quantities which would surround the singularity in an infinite medium, multiplied by the difference between these quantities and those actually present. The expression for the force is thus of essentially the same form whether the force is due to applied surface tractions, other singularities or the presence of the free surface of the body (‘image force’). A region of inhomogeneity in the elastic constants modifies the stress field; if it is mobile one can define and calculate the force on it. The total force on the singularities and inhomogeneities inside a surface can be expressed in terms of the integral of a ‘Maxwell tensor of elasticity’ taken over the surface. Possible extensions to the dynamical case are discussed.

The stress due to screw dislocation passing normally through an infinite plate or a disc is largely confined to the neighbourhood of the dislocation line, in contrast to the case of a dislocation in an infinite medium. Two screw dislocations in a plate attract or repel one another with a short-range force in place of the inverse first power law for infinite parallel dislocations. The stress due to an edge dislocation is not essentially different in the plate and infinite body so long as the plate remains flat, but in some circumstances the stress may be largely relieved by buckling of the plate.

The general solution of the elastic equations for an arbitrary homogeneous anisotropic solid is found for the case where the elastic state is independent of one (say x 3) of the three Cartesian coordinates x 1, xx 2, x 3. Three complex variables z (l)=x 1 + p (l) x 2 (l=1, 2, 3) are introduced, the p (l) being complex parameters determined by the elastic constants. The components of the displacement (u 1, u t2, u 3) can be expressed as linear combinations of three analytic functions, one of z (1), one of s (2), and one of z (3). The particular form of solution which gives a dislocation along the x 3-axis with arbitrary Burgers vector (a 1, a 2, a 3) is found. (The solution for a uniform distribution of body force along the x 3-axis appears as a by-product.) As is well known, for isotropy we have u 3=0 for an edge dislocation and u 1=0, u 2=0 for a screw dislocation. This is not true in the anisotropic case unless the xl xa plane is a plane of symmetry. Two cases are discussed in detail, a screw dislocation running perpendicular to a symmetry plane of an otherwise arbitrary crystal, and an edge dislocation running parallel to a fourfold axis of a cubic crystal.

In connection with Galt and Herring’s observation on thin whiskers of tin, the properties of a screw dislocation in a cylinder are worked out. When all boundary conditions are taken into account, the image force tends to keep the dislocation along the axis. Only when it is displaced about half-way to the surface does the image force tend to pull it out of the rod. Generators of the cylindrical rod become helices when the dislocation is introduced. The dislocation can be ejected from the rod by twisting or bending it suitably.

The elastic field surrounding an arbitrary moving screw dislocation is found, and a useful analogy with two-dimensional electromagnetic fields is pointed out. These results are applied to a screw dislocation accelerating from rest and approaching the velocity of sound asymptotically. The applied stress needed to maintain this motion is found on the assumption that the Peierls condition is satisfied near the center of the dislocation. A general integral equation of motion is derived for a simplified dislocation model, and the kind of behavior it predicts is illustrated.

Peach’s1 very pretty explanation of the formation of metallic whiskers2 seems to be ruled out by the observation3 that they grow at the root.The growth seems to be influenced4 by the atmosphere over the surface.

The results reported by Miller and Russell1 may be illustrated by a direct calculation for a spherical crystal.

The expression u=c r/r 3, (where c is a constant) sometimes assumed for the displacement around a point imperfection (interstitial or substitutional impurity, lattice vacancy) gives a nonzero stress at the surface of the solid. The additional “image displacement” necessary to insure that this stress vanishes is usually neglected, but may be important. For example, it accounts for from 30 to 50 percent of the volume change produced by such defects. This and other effects of the image term are discussed. Miller and Russel have pointed out that for a point imperfection near the center of a sphere the apparent volume change deduced from measurements of the x-ray lattice constant is greater than the geometrical volume change. It is shown that the reverse is true when the defect is near the surface, and that for a large number of defects scattered uniformly through the sphere the geometrical and x-ray expansions are equal. It can be shown quite generally that a body of arbitrary shape is expanded uniformly by a statistically uniform distribution of point imperfections, and that the x-ray diffraction pattern is altered in the way to be expected for such an expansion. To establish this, however, it is essential to take the image terms into account.

In an isotropic material it is known that two point defects regarded as centres of dilatation interact with one another only indirectly through the modification of their elastic fields by the surface d the body. In a cubic material there is an additional direct interaction energy equal to the product d the inverse cube of their separation and a function d direction whose average over all angles is zero. This function is evaluated approximately. If in a more relined elastic model we replace each defect by a centre d dilatation plus a small region where the elastic constants differ fiom those of the matrix there is an additional interaction proportional to the inverse sixth power of the distance between them.

In a recent paper,(1) Freudenthal and Weiner have proposed that thermal stresses produced during fatigue arc severe enough to create microcracks;these stresses are due to highly looahzed “thermal flashes,” associated with repeated and reversed slip processes, which in the extreme case may raise the temperature near the slip plane by some 200°C. The heating eifcot of dislocations moving along slip planes is in fact of importance in many aspects of plastic deformation. In this note the heating effect is calculated in a different manner, and some of the assumptions of Freudenthal and Weiner are discussed.

The conditions governins the motion of a screw dislocationat velocities above and below the velocity of shear waves are contrasted. For supersonic motion the Peierls-Nabarro equation becomes a differential equation, in contrast to the subsonic integral equation. It has a solution representing a moving restoration of fit along a slip plane across which there was originally complete misfit.In a dispersive medium where the sound velocity decreases with decreasing wavelength a dislocation moving between the maximum and minimum sound velocities suffers a retarding force, which may be calculated if the shape of the dislocation is supposed to be known.

Among the imperfections to which a crystal is subject,1 some (interstitial and impurity atoms, vacant lattice sites, dislocations . . .) are relatively permanent. The introduction of one of them generally alters the position of every lattice point. Obviously in calculation we cannot take every lattice point into account explicitly in a crystal of any size, and must be content to treat the greater part of the crystal as a continuum. In favorable cases the exact behavior in the regions where the continuum approximation is inappropriate is unimportant and can be taken into account by giving suitable values to certain parameters appearing in the continuum solution.

It is supposed that a region within an isotropic elastic solid undergoes a spontaneous change of form which, if the surrounding material were absent, would be some prescribed homogeneous deformation. Because of the presence of the surrounding material stresses will be present both inside and outside the region. The resulting elastic field may be found very simply with the help of a sequence of imaginary cutting, straining and welding operations. In particular, if the region is an ellipsoid the strain inside it is uniform and may be expressed in terms of tabulated elliptic integrals. In this case a further problem may be solved. An ellipsoidal region in an infinite medium has elastic constants diierent from those of the rest of the material; how does the presence of this inhomogeneity disturb an applied stress-field uniform at large distances? It is shown that to answer several questions of physical or engineering interest it is necessary to know only the relatively simple elastic field inside the ellipsoid.

Eshelby(partly written): The relative rotation d0 of the ends of a whisker containing a screw dislocation when it is stretched by an amount dl can be calculated by modifying the analysis of (1951) of the simultaneous finite extension and infinitesimal torsion of a cylinder of arbitrary cross section, made of isotropic material with an arbitrary stress-strain relation.

If the energies required to form positive and negative ion vacancies in and ionic crystal are unequal, then in thermal equilibrium dislocations in the crystal will be electrically charged and surrounded by a Debye-Hückel cloud of vaccines. If the vacancy cloud is immobile a finite force is required to separate the dislocation from the cloud, and so the crystal will possess a yield stress below which plastic flow will not occur. The presence of divalent impurities modifies the magnitude of the charge on a dislocation, and may even reverse it. If precipitation of the impurity or association of impurity atoms and vacancies can occur, the concentration of impurities may be a complicated function of temperature. The yield stress of the crystal may then exhibit maxima and minima when plotted as a function of temperature. Experimental results showing this behaviour are presented and tentatively compared with the theory.

The twist due to a screw dislocation parallel to the axis of an isotropic cylinder of arbitrary cross section can be found from the solution of the ordinary torsion problem for the same cylinder. Some particular cases are worked out. The results are also valid for certain kinds of anisotropy.

It is shown that between rigid misfitting spheres in an elastic continuum there is a repulsive interaction if the spheres are bonded to the medium. The effect of other boundary conditions is discussed, and a simplified method is given for calculating the interaction between various types of defect.

Part of the increase in yield stress during the strain-ageing of a low-carbon steel develops too rapidly to be explained by long-range duffusion. The rate at which the initial rapid rise in yield stress develops and the dependence of its magnitude on the dissolved solute content are shown to be those expected from a contribution due to stress-induced ordering of the interstitial solute atoms in the stress fields of dislocations. A simple theoretical treatment gives values in reasonable agreement with experiment.

In calculations relating to defects in a crystal lattice the solid state physicist draws largely on continuum concepts familiar to the engineering. Some topics relating to the two points of view are considered. By way of the theory of continuous distributions of dislocations, microscopic internal stresses may be traced to the lattice defects which are their source. A simple method for calculating the stresses in and around precipitations and transformed regions is outlined; in some cases it may be useful in dealing with the interactions between stresses on a microscopic scale. Generally, however, such interactions have to be discussed in terms of energy changes.

The results of an earlier paper are extended. The elastic field outside an inclusion or inhomogeneity is treated in greater detail. For a general inclusion the harmonic potential of a certain surface distribution may be used in place of the biharmonic potential used previously. The elastic field outside an ellipsoidal inclusion or inhomogeneity may be expressed entirely in terms of the harmonic potential of a solid ellipsoid. The solution gives incidentally the velocity field about an ellipsoid which is deforming homogeneously in a viscous fluid. An expression given previously for the strain energy of an ellipsoidal region which has undergone a shear transformation is generalized to the case where the region has elastic constants different from those of its surroundings. The Appendix outlines a general method of calculating biharmonic potentials.

This review is concerned with the two following problems in the infinitesimal theory of elasticity, and with their inter-relation and generalization.

An expression is found for the force required to move a dislocation steadily through a visco-elastic medium, in particular Zener’s standard linear solid. It is applied to the case of a screw dislocation in iron exhibiting the Snoek effect. Schoeck and Seeger’s atomic treatment of the same problem contains an error. When this is corrected the two methods give solutions of the same form.

A kink on a dislocation in an isotropic elastic medium is treated as a ‘point defect’ with a certain mass, constrained to move along a line and subject to a radiation reaction. A value for the mass is obtained from the well known stretched-string model, and the radiation reaction is found by calculating the rate at which an oscillating kink radiates energy into the medium. It is found that the kink has a scattering cross-section for elastic waves which is proportional to the square of its width. For long waves the cross-section is independent of frequency, in contrast to the case of ordinary point defects. A kink moving through an isotropic flux of elastic waves experiences a retarding force proportional to the product of its velocity and the energy density of the waves. In connexion with a similar result for the retarding force on a dislocation moving rigidly it has been suggested that the expression for the energy density should include the zero-point energy. A formal quantum-mechanical calculation shows that this is not so in the case of a kink.

CERTAIN results of the theory of dislocations in isotropic materials are extended to the case of a dislocation lying in the basal plane of a hexagonal crystal. Espressions are found for the energy of a circular loop and for the line tension of a dislocation. Numerical results are presented for graphite.

It is shown that some problems concerning the distortion of plates and rods by dislocations can be solved very simply with the help of Colonetti’s theorem. The relation between the plastic deformation and the electrical potentials produced by the movement of charged dislocations is also examined.

The following problem, of interest in the theory of work-hardening, is considered. Under the influence of an applied stress dislocation loops are emitted from a source and pile up against an elliptical barrier. To determine their equilibrium number and distribution.

Consider two misfitting inclusions A and B bounded by surfaces S A, S B embedded in a matrix having the same elastic constants as the inclusions.

The elastic field of an edge dislocation is found in a simple manner by making use of the relation between an edge dislocation and a ‘wedge’ dislocation make by inserting or removing a narrow wedge of material.

The behaviour of a small-amplitude infinite standing or travelling sine wave on a dislocation is examined. For a certain range of frequencies and wavelengths no elastic energy is radiated, but in general a suitable applied stress is required to maintain the motion. When, however, a certain relation between frequency and wavelength is satisfied no applied forces are necessary and the motion is self-maintaining. This relation may be expressed analytically in the limit of very long wavelengths, but in general it must be evaluated numberically. Curves are given for edge, screw and 60° dislocations in materials with various Poisson ratios. In the course of the calculation a value for the line tension of a sinusoidally deformed stationary dislocation is also obtained.

A physical meaning is suggested for Li’s terminating dislocation, based on Heaviside’s interpretation of the magnetic field, which the formula of Biot and Savart gives when applied to an incomplete circuit.

This chapter summarizes the elements of the theory of elasticity, illustrated by some problems basec in fracture mechanics. It is note absolutely necessary to absorb the whole chapter to appreciate the succeeding one. However, it was hoped that readers not already familiar with the topic may find that the application of complex variable theory to elasticity is not in fact very difficult, particularly if we start with the simple case of anti-plane strain (Mode III deformation)

In fracture mechanics the interes lies in the elastic state near the crack tip. This state is characterized by three quantities K I, K II, K III, the stress intensity factors. If these quantities are the same for two crack tips, even though the crack geometries and the types of loading are different, then conditions are identical near the two tips. Closely related to the stress intensity factor is the crack extension force G which gives the amount of energy released from the system cracked-specimen-plus-loading-mechanism for unit advance of the crack front. This chapter discusses K and G and then briefly outlines the physical interpretation of fracture criteria and the plasticity around crack tips.

A modification of Craggs’ method for calculating the flow of energy into the tip of a moving crack is proposed. For a plane crack extending uniformly at both ends the flow falls to zero at the Rayleigh velocity, contrary to Craggs’ result, but in agreement with that of Broberg.

Dislocations enter the theory of fracture in two ways. First, as crystal dislocations, they play a role in the physics of fracture. Secondly, they can serve as convenient basic elements in the macroscopic treatment of fracture. This is a result of the fact that a crack is equivalent to a continuous array of dislocations. This equivalence is made use of in developing the mathematical theory of dislocation arrays and cracks. The problem of determining the equilibrium position of dislocations in a linear array is discussed for the cases of discrete dislocations and continuous distributions of dislocations. The basic problem in the theory of cracks is the determination of the way in which a crack modifies an applied stress field. The problem is first treated as one in antiplane strain (type III deformation), and it is then shown that the results obtained may easily be modified so as to apply to plane deformation (type I and type I1 deformation). The methods for handling energy changes which have been developed in fracture mechanics and dislocation theory are discussed and compared. These theoretical results are applied to a number of problems in fracture mechanics and to dislocation models of crack initiation. Among the topics reviewed are crack theories in which some attempt is made to take account of cohesive forces, as developed by Barenblatt, or of plastic relaxation at the crack tips, as developed by Bilby, Cottrell, and Swinden.

With the help of a thorem of Bateman’s an expression is found for the dynamic elastic field of a crack when one of its tips moves arbitrarily in the plane of the crack, starting from rest. The linear isotropic theory of elasticity is used, and only states of anti-plane strain (mode III deformation) are considered. The crack is initially of finite length and subject to any static anti-plane loading. The solution obtained becomes inaccurate in regions into which distrubrances reflected at the other tip have penetrated. The error is estimated for some special cases. The results are used to discuss the equation of motion of a crack tip.

In principle one could solve the problem in the way I described in [2] of the author’s paper, because the field of a point force in a transversely isotropic medium can be written down explicitly.1,2 However, I am sure this would be very cltimsy, and that Dr. Chen’ method is the right way to tackle the problem.

A theorem of Bateman’s is used to find the elastic field near the tip of an antiplane crack which starts moving in an arbitrary manner. The energy release rate is calculated and found to confirm a general expression previously proposed.

The force on a dislocation or point defect, as understood in solid-state physics, and the crack extension force of fracture mechanics are examples of quantities which measure the rate at which the total energy of a physical system varies as some kind of departure frm uniformity within it changes its configuration. One may define similarly a force acting on each element of a mobile interface (a phase boundary or martensitic interface, for example).Methods for calculating sech effective forces are reviewed for both quasi-static and dynamic processes, the latter with particular reference to the motion of crack tips. The elastic energy-momentum tensor proves to be a useful tool in such calculations.

The production of flakes of a brittle solid by pressure or impact is studied with the help of a simplified mathematical model.

The early work of Griffith and others on crack propagation has in recent years been extended and systematized to form the science of Fracture Mechanics. In dealing with stationary cracks its principal theoretical tools are the stress intensity factor which characterizes the elastic field near a crack tip and the energy release rate associated with the extension of a crack. The behaviour of fast-moving cracks presents some intriguing theoretical problems which are only beginning to be solved.

A fault plane which has undergone slip over a limited area, a thin intrusion or a crack whose faces have been caused to slide over one another or separate by the action of an applied stress are all physical realizations of a dislocation, that is, an internal surface in an elastic solid across which there is a discontinuity of displacement. Since this discontinuity varies from point to point of the internal surface it is actually a so-called Somigliana dislocation. It can, however, be built up from the more familiar dislocations of crystal physics which have a constant displacement discontinuity.

The connection between the elastic energy-momentum tensor and the surface or path-independent integral for energy release rates and kindred concepts is recalled and extended to grade 2 materials. For a completely general elastic field there are apparently only the three types of path-independent integrals derived by Günther from Noether’s theorem, but if the field is anti-plane,(mode III) there is an indefinite number of them. In plane strain the energy-momentum tensor can be re-interpreted as an ordinary compatible stress. This observation, coupled with the elastic reciprocal theorem, leads to an indefinite number of path-independent integrals for this case also. Some other possibilities are also touched on. The paper contains some specimen calculations for specific cases.

In this chapter we shall mainly be concerned with point defects (vacancies and interstitials) in pure crystals, though substitutional and interstitial impurities will receive some attention, particularly as ideas originally developed to deal with them have subsequently been applied to vacancies and self-interstitials.

Bilby, B.A., Eshelby, J.D. and Kundu, A.K., 1975. The change of shape of a viscous ellipsoidal region embedded in a slowly deforming matrix having a different viscosity. Tectonophysics, 25: 265–274.The general theory of the elastic fields round ellipsoidal inclusions and inhomogeneities is applied to solve the problem of the slow deformation of a visous material containing an ellipsoidal inhomogeneity of different viscosity. The theory is used to calculate the two-dimensional finite strain of an elliptic cylinder and of prolate and oblate spheroids under applied pure-strain rates, and attention is drawn to applications in the fields of rock deformation and in the mixing and homogenization of viscous liquids.

The application to continuum mechanics of the general methods of the classical theory of fields is advocated and illustrated by the example of the static elastic field. The non-linear theory of elasticity is set up in the most convenient form (Lagrangian coordinates and stress tensor). The appropriate energy-momentum tensor is derived, and it is shown that the integral of its normal component over a closed surface gives the force (as the term is used in the theory of solids) on defects and inhomogeneities within the surface. Other topics discussed are Günther’s and related integrals, symmetrization of the energy-momentum tensor, and the Eulerian formulation. Some further extensions, existing and potential, are indicated.

We are glad to see our results laid out in a handy form, although we would prefer our solution to be described as an exact one rather than one which is “more exact” than Gay’s original solution which was, in fact, incorrect.

Simple derivatioins are given for some results in the continuum theory of point defects; in particular the interaction between defects, both directly and through image (surface) effects. It is argued that these results] have a wider validity than appears at first sight. The jump frequency for particles diffusing from one minimum to another of a rigid one-dimensional potential is derived by a simple argument which shows clearly the origin of the frequency factor, which turns out not to be, in any reasonable sense, a measure of the number of attacks on the barrier per unit time, as commonly stated. The method can quite easily be extended to give the jump frequency of a point defect or vacancy in three dimensions both when the effect of the host lattice is simulated by a rigid set of potential well and also when, more realistically, it is note (Vineyard formula).

The presence of free surfaces can sometimes affect the deformation of an internally stressed solid on an almost macroscopic scale: for instance, crystal whiskers containing an axial screw dislocation exhibit a twist which increases as their cross-section decreases. Such effects can be handled with the help of the theory of elasticity. On a microscopic scale relaxation due to the presence of the free surfaces of the foil affects the image contrast in thin-film electron microscopy. Of course, in this case we are really interested in the deformation of the lattice, but usually for want of anything better we have to treat the atoms as points embedded in an imaginary elastic continuum and deforming with it.

It is argued that the Peach-Koehler force on a disclination in a nematic liquid crystal is, unlike its namesake for a dislocation in a solid, not a fictitious configuration force, but a real force which, for equilibrium, must be balanced by an external force applied to the singular line. Formally this is a consequence of the near identity of the Ericksen stress and the appropriate energy-momentum tensor.

In the physics of solids and in fracture mechanics the idea of an effective force acting on some kind of defect in an elastic solid (e.g. a lattice vacancy or other point defect, a dislocation or a crack tip) has become familiar.

Various topics in the theory of dislocations in linear elastic continua are reviewed using, wherever possible, simple physical arguments to derive the mathematical results. The elastic field of a general Somigliana dislocation is derived and specialized to the physically important case where its discontinuity vector is constant. A general account of the effect of free surfaces is illustrated by some specific examples. In discussing dislocation energetics it is emphasized that once the Peach-Koehler expression describing the interaction with an externally applied stress field has been established, its extension to the case of internal stresses and image stresses follows without further mathematics. In the final Section it is demonstrated that certain solutions for uniformly and non-uniformly moving dislocations can be derived rather simply from the corresponding static solutions.

CHEREPANOV[1] has considered the problem of a finite thin inextensible fibre embedded in an elastic solid which is stretciied by a uniform stress σ∞ parallel to the fibre.