## Abstract

The effect of prestress on dislocation (and inclusion) fields in nonlinear elastic solids is analysed by extending previous solutions by Eshelby and Willis. Using a plane-strain constitutive model (for incompressible incremental nonlinear elasticity) to describe the behaviour of ductile metals (*J*_{2}-deformation theory of plasticity), we show that when the level of prestress is high enough that shear band formation is approached, strongly localized strain patterns emerge, when a dislocation dipole is emitted by a source. These may explain cascade activation of dislocation clustering along slip band directions.

## 1. Introduction

The theory of dislocations (and inclusions) in solids has been thoroughly developed for elastic materials, unloaded in their natural state. We extend this theory to cover the possibility that the material is prestressed, through a generalization of solutions found by Eshelby [1–3] and Willis [4], by introducing an incremental formulation for incompressible materials, in which the nominal stress is related to the incremental displacement gradient, within a constitutive framework (which embraces Mooney–Rivlin and Ogden materials and also material models describing softening [5]) under the plane-strain constraint, even if several of the presented results remain valid within a three-dimensional context.

Anisotropy strongly influences near dislocation stress fields (as shown in figure 1; see appendix A for details), and almost all crystals are anisotropic, so that anisotropy has been the subject of an intense research effort [6–8] and has been recently advocated as a way to study dislocation core properties [9,10].

Our interest is to analyse the effect of orthotropy induced by prestress on dislocation (and inclusion) fields, within the general framework of incremental nonlinear elasticity, but with a special emphasis on a material model for metals (*J*_{2}-deformation theory [11,12]), so that our investigation is addressed to ductile metals subject to extreme strain, where the nucleation of a clustering of dislocations into a ‘super dislocation’ perturbs a material that has a low stiffness, so low that the differential equations governing the incremental equilibrium are close to the boundary of ellipticity loss.

When this boundary is approached (from the interior of the elliptic region), our solution for edge dislocations (but also, in general, for inclusions) reveals features of severely deformed metals near the shear band formation. In this situation, we show that emission of a dislocation (which can also be viewed as a ‘super dislocation’) dipole produces incremental fields strongly localized along the directions of the shear bands, formally excluded *within* the elliptic region. This may induce a cascade of dislocation clustering, which may explain the fact that the amount of slip that takes place on an active shear band is three orders of magnitude greater than could be produced by the passage of a single dislocation [13].

This paper is organized as follows. A boundary integral equation, proposed by Eshelby [2] and Willis [4] for *an inclusion in an infinite plane subject to a generic transformation strain*, is generalized in §2 to incremental, incompressible nonlinear elasticity in plane strain (for a uniformly prestressed material), and a new boundary equation is formulated for the in-plane incremental mean stress (when the prestress is set to be equal to zero, our generalization reduces for isotropic incompressible material to novel formulae because the incompressible case has never been explicitly addressed). As an example of application of the derived equations, we present the case of a circular inclusion subject to a uniform purely dilatational transformation strain. This solution, in the case of *J*_{2}-deformation theory of plasticity, shows the strong effect of prestress, particularly when the material is prestressed near the boundary of ellipticity loss. In §3, the solution for an edge dislocation incrementally deformed within a uniformly prestressed elastic material is derived and, after treatment of the boundary integral equations in view of the numerical implementation (§4), applications are presented in §5, where a dislocation dipole (different from a force dipole, see Bigoni [14] for a discussion on the differences) is emitted within the *J*_{2}-deformation theory of plasticity material homogeneously deformed near the elliptic boundary.

## 2. Inclusions in prestressed elastic materials

### (a) Material model

We refer to an incompressible nonlinear elastic material deformed under plane-strain conditions in the *x*_{1}–*x*_{2} reference system, whereas *x*_{3} represents the out-of-plane direction. Under these hypotheses, and assuming the current configuration as the reference configuration, the most general material model has been provided by Biot [15], which in the Bigoni & Dal Corso [16] notation, can be written as a linear relation between the increment in the nominal (unsymmetric) stress and the incremental displacement *v*_{i} (plus the incompressibility constraint) as
2.1where repeated indices are summed between 1 and 2,*δ*_{ij} is the Kronecker delta, is the increment in the in-plane mean stress and the non-null components of the fourth-order tensor are
2.2which are functions of the dimensionless prestress and anisotropy parameters
2.3Within this framework, *μ* and *μ*_{*} are, respectively, the incremental shear moduli parallel to, and inclined at 45^{°} to, the principal stress axes. For the Mooney–Rivlin material, these moduli depend on the maximum current stretch *λ*>1 and are expressed as [14]
2.4with *μ*_{0} the ground state shear modulus, whereas for the Ogden material, the definitions are the following:
2.5where *μ*_{i} and *β*_{i} are material parameters.

We will restrict the analysis to the elliptic regime, which corresponds to 2.6and may be further subdivided into an elliptic complex region, 2.7and an elliptic imaginary region, 2.8

A special case of the above constitutive framework is the *J*_{2}-deformation theory of plasticity, proposed by Hutchinson & Neale [11], in which
2.9where (*λ* is the in-plane maximum stretch) is the logarithmic strain and *N* is a hardening parameter ∈(0,1), so that a vanishing *N* corresponds to ideal plastic behaviour. This material touches the elliptic–hyperbolic boundary when the logarithmic strain reaches the critical value solution of
2.10so that, for instance, *N*=0.4 yields .

### (b) The inclusion problem

We follow and generalize Eshelby [2] and Willis [4] by considering an infinite elastic plane, homogeneously prestressed and incompressible and therefore obeying the incremental constitutive laws (2.1), containing an inclusion of arbitrary shape, in which a uniform incremental displacement gradient *v*^{P}_{i,j} is prescribed, which can be thought as an inelastic (e.g. plastic or thermal) deformation.

Note that the Eshelby inclusion problem in linear elasticity is formulated by prescribing an inelastic *strain*, not a displacement gradient (if a displacement gradient is assigned instead, the skew symmetric part of this, representing a rigid-body infinitesimal rotation, produces null fields outside the inclusion, meaning that the solution for the infinite body containing the inclusion consists of a pure, uniform, rigid-body rotation), a situation different from incremental nonlinear elasticity, where the effect of prestress is to alter the incremental response, even for a rigid-body rotation.

Because the inclusion is constrained by the surrounding matrix material, an elastic deformation *v*^{E}_{i,j} is produced, so that the ‘total’ incremental displacement gradient *v*_{i,j} within the inclusion can be obtained through the additive rule
2.11It is important to note that, although the material is incompressible, *the prescribed inelastic incremental displacement* *need not satisfy the incompressibility constraint*, so that because *v*^{E}_{i} does (namely *v*^{E}_{k,k}=0), it follows that *v*_{k,k}=*v*^{P}_{k,k}.

The elastic part of the incremental deformation produces the incremental nominal stress
2.12through two incremental mean stresses and , the latter being a homogeneous incremental mean stress, defined inside the inclusion and associated to the deformation *v*^{P}_{i,j} (we will show later that results will be independent of this, but it is better for the moment to keep track of a part of the stress that is related to the inclusion transformation).

Body forces are not considered, so that the incremental nominal stress has to satisfy the equilibrium equations, which for an infinite body containing a concentrated unit force can be written as
2.13where is Green's function for incremental nominal stress, in other words, the *ij*-component of the nominal stress at ** x** produced by a unit point force applied in the

*g*-direction at a point

**, and**

*y**δ*(

**−**

*x***) is the Dirac delta. This Green function, valid within the present constitutive framework, has been given by Bigoni & Capuani [17].**

*y*#### (i) The incremental displacement

We consider now the inclusion problem sketched in figure 2, where an inclusion of volume *D*_{in} and surface ∂*D*_{in} is included in an infinite region. We assume that the singularity at point ** y** is enclosed by a disc

*C*

_{ε}centred in

**, with radius**

*y**ε*and surface ∂

*C*

_{ε}. We define a closed, finite and simply connected domain

*D*

_{out}outside both the inclusion and the disc surrounding the singularity at

**, so that its boundary ∂**

*y**D*

_{out}can be regarded as the sum of the surfaces of inclusion and disc (∂

*D*

_{in}and ∂

*C*

_{ε}) and an external boundary ∂

*D*

_{ext}as follows: 2.14

On the above-defined region, *D*_{out}, we may use the Betti identity, thus yielding
2.15where the comma denotes differentiation with respect to ** x**, the same variable for which integration is performed (as noted by the symbol d

*V*

_{x}), and is the infinite-body Green function for incremental displacements [17].

Using the rule of product differentiation, equation (2.15) becomes
2.16because the quantity
2.17is equal to zero, owing to the major symmetry of and the incompressibility constraint (). On application of the divergence theorem to equation (2.16), it follows that:
2.18where *n*_{i} is the unit vector normal to the integration boundary and pointing towards the external of the domain (figure 2).

Recalling equation (2.14), the integration regions given by the domain *D*_{out} and the contour ∂*D*_{out} can be split. Because , we can write
2.19whereas
2.20is a limit that can be obtained from equation (2.13) using the delta function properties. Therefore, assuming that the incremental stress and displacement fields induced by the inclusion decay at infinity, where the outer boundary is moved, and for *ε*→0, the integral equation (2.18) becomes
2.21where now *n*_{i} is the outward unit normal to the inclusion surface ∂*D*_{in}.

A further application of the divergence theorem yields
2.22but within the inclusion the gradient of the velocity can be written as
2.23so that, using equation (2.12), we may write
2.24where is Green's incremental in-plane mean stress, defined in Bigoni & Capuani [17]. Because the field is uniform and the incremental displacement field solenoidal, the application of the divergence theorem to the first term of equation (2.24) yields an *integral equation for the incremental displacements outside the inclusion produced by the uniform inelastic field* ,
2.25Note that equation (2.25) involves both the deviatoric and the volumetric part of and that the volumetric term vanishes for a purely deviatoric inelastic incremental displacement gradient.

If we introduce a potential such that 2.26equation (2.25) can be rewritten as 2.27showing that now the velocity field is expressed only in terms of a boundary integral.

A simple way to calculate is as follows. Within the two-dimensional framework, we may introduce the coefficient as
2.28where *δ*_{ij} is the Kronecker delta, *i*,*j*=1,2 and , so that we can obtain a family of potentials depending on the arbitrary coefficient in the form
2.29where the index *i* is *not* summed.

Following Willis [4], we are now in a position to derive an expression that is alternative, but equivalent, to (2.25). This can be carried out through application of the divergence theorem, so that, collecting the derivative with respect to *x*_{l}, we obtain
2.30an expression that can be transformed using incremental equilibrium, and the major symmetry of in the form
2.31to yield
2.32

A second application of the divergence theorem allows us to obtain an *integral equation for the incremental displacements outside the inclusion produced by the uniform inelastic field* , fully equivalent to (2.25),
2.33and expressed in terms of the transformation incremental inelastic displacement .

Introducing the following notation for Green's incremental tractions along the surface of unit normal *n*_{i}:
2.34equation (2.33) becomes
2.35Note that the expressions for the components of are given both in singular and regularized forms by Bigoni *et al*. [18], and can be used to evaluate the integral equation (2.35).

The gradient of incremental displacement can be given by two expressions, one when equation (2.25) is used, 2.36and the other when equation (2.33) is used, 2.37According to the two representations (2.36) and (2.37), the second gradient can be expressed as 2.38or as 2.39because , as is homogeneous.

#### (ii) The incremental mean stress

To complete the solution, the incremental mean stress has to be calculated. For this purpose, taking into account that is homogeneous, the incremental equilibrium equations (2.1) allow us to derive the gradient of in the form
2.40where the differentiation is carried out with respect to the variable *x*_{i}.

Using equation (2.38) in equation (2.40), we obtain 2.41which, using the rate equilibrium equations (2.1), yields 2.42

Defining the function *F*(** x**−

**) as 2.43Bigoni & Capuani ([17], their appendix B) have shown that 2.44which, applied to equation (2.42), allows one to eliminate the differentiation with respect to**

*y**y*

_{i}, thus yielding an

*integral equation for the incremental mean stress outside the inclusion, produced by the uniform inelastic field*, 2.45

The earlier-mentioned procedure can be repeated using the expression (2.39) instead of (2.38), namely the second formulation for the displacement field, to derive an expression that is alternative, but equivalent, to (2.45). Equation (2.41) transforms into
2.46whereas rate equilibrium equations (2.1), taking into account that for Green's velocity field, (see [17]) yield
2.47so that equation (2.46) becomes
2.48Using equation (2.44), we may write
2.49so that the differentiation with respect to *y*_{i} can be eliminated, thus yielding an *integral equation for the incremental mean stress outside the inclusion, produced by the uniform inelastic field* ,
2.50where *n*_{i} is the outward unit normal to the inclusion surface ∂*D*_{in}.

As a conclusion, the mechanical fields outside an inclusion of arbitrary shape, embedded in a prestressed elastic incompressible infinite matrix, can be summarized as follows:

— incremental displacement field, given by equation (2.25) or equation (2.33);

— incremental mean stress field, given by equation (2.45) or equation (2.50); and

— incremental nominal stress rate field given by 2.51where equations (2.25) or (2.33) and equations (2.45) or (2.50) can be alternatively used.

#### (iii) Example: the circular inclusion

As a simple example, we consider a circular inclusion of radius *a*, subject to an inelastic purely volumetric dilatational Eulerian incremental strain, *v*^{P}_{i,j}=*βδ*_{ij}. With reference to the coordinate system sketched in figure 3, we consider a source point ** x** lying on the inclusion surface (so that ) and a generic point

**(outside the inclusion) at which we will calculate the displacement and mean stress fields; furthermore, the inclusion is centred at the origin**

*y**O*of the

*x*

_{1}–

*x*

_{2}reference system. With these assumptions and defining the distance between the points

**and**

*x***as 2.52and using equations (2.25) and (2.45), the boundary equations for**

*y**incremental displacements and mean stress around a circular inclusion in a prestressed nonlinear elastic material*become, respectively, 2.53and 2.54

Equations (2.53) and (2.54) can be rewritten using equations (2.33) and (2.50) in the fully equivalent forms 2.55and 2.56

Equation (2.53) has been used to generate the incremental solution shown in figure 4, for an isotropic elastic (with null prestress) matrix material (figure 4*a*) and for a *J*_{2}-deformation theory matrix material uniformly deformed near the boundary (but still within) the elliptic region.

The latter material, with a hardening exponent *N*=0.380, is pre-deformed at a logarithmic strain , a value close to loss of ellipticity, occurring at .

The strong effect of prestress is evident from figure 4, so that *the incremental displacement fields are completely different* and the situation near the ellipticity loss shows the emergence of strongly localized fields, focused parallel to the four shear band directions (for a *J*_{2}-material with *N*=0.380, the shear bands are inclined at ±27.37^{°} with respect to the *x*_{1}-axis).

In the simple case of null prestress, *k*=0 and *η*=0, equations (2.53) and (2.54) reduce to
2.57so that introducing isotropy, *ξ*=1, they become
2.58and the velocity can be evaluated as
2.59We can note that the last expression can be written in a polar coordinate system, which yields only a radial velocity field,
2.60where , namely the same result of linear elasticity (the so-called Lamé solution).

## 3. Edge dislocations in prestressed elastic materials

The integral equations determining the incremental displacement and mean stress for a straight edge dislocation can be obtained from equations (2.33) and (2.50) by considering a thin (thickness *h*) rectangular inclusion (without loss of generality) with one edge centred at the origin of the *x*_{1}–*x*_{2} axes, and subject to the incremental simple shear displacement field,
3.1where *n*_{k} is the unit vector orthogonal and *b*_{k} is a vector parallel to the long edges of the rectangle. Note that the modulus of ** b** is twice the maximum displacement induced by the simple shear inside the rectangle. The incremental displacement field (3.1) satisfies
3.2so that inserting equation (3.1) into equations (2.33) and (2.50) and taking the limit

*h*→0, we obtain

*the integral equations for a straight edge dislocation in a prestressed material,*3.3where

*L*is the dislocation line of unit normal

*n*

_{i}and

*b*

_{i}is the (constant) Burgers vector, defining the jump in the incremental displacement imposed across the dislocation, see figure 5.

We consider a straight edge dislocation dipole with one of the two dislocations centred at the origin of the *x*_{1}–*x*_{2} reference system, a generic source point ** x** lying on the dislocation line (so that ) and a generic point

**at which we will calculate the displacement and mean stress fields, as shown in figure 6. Representing the dislocation line with a polar coordinate system (**

*y**ρ*,

*ψ*), where

*ρ*∈[0,

*a*], the Burgers vector

**and the normal vector**

*b***become 3.4whereas the distance between the points**

*n***and**

*x***is defined as 3.5similar to the circular inclusion example.**

*y*Because ** b** is constant and orthogonal to

**, the incremental displacement and mean stress fields for**

*n**an edge dislocation dipole*can be obtained, respectively, in the following form: 3.6aand 3.6bwhere 3.7

### (a) The edge dislocation solution along the dipole line

The displacement and the mean stress fields can be explicitly evaluated along the dislocation line through equations (3.6) and the following considerations on the Green function structure. From figure 5, the point ** y**, when taken along the dislocation line, is represented by and the angle

*ϕ*is constant and equal to

*ψ*. In this case, we have that

*s*=

*z*=

*ρ*

_{y}−

*ρ*because

*ε*=0 along the dislocation line. This constraint allows us to express Green's function gradient for displacement and mean stress as 3.8where and are functions of the sole variable

*ψ*. Now the integration along

*ρ*can be performed because, in equation (3.8), the dependence on

*s*is explicit, so that the incremental displacement and mean stress fields along the dislocation take the following form: 3.9aand 3.9bNote that the incremental displacement and mean stress fields exhibit essentially different asymptotic behaviours at the dislocation tips (near both points,

*ρ*

_{y}=0 and

*ρ*

_{y}=

*a*). In fact, as one can expect, the displacement field shows a logarithmic singularity (similar to that found by Eshelby [3]), whereas the mean stress displays a 1/

*s*singularity.

### (b) Curiosity of the incompressible isotropic linear elastic solution

In the simple case of null prestress, *k*=0 and *η*=0, equations (3.6) reduce to
3.10aand
3.10bso that introducing isotropy, *ξ*=1, and considering an edge dislocation dipole aligned to the *x*_{1}-axis, *ψ*=0, they become
3.11

The integration of equations (3.11) can be performed in an explicit way, thus yielding 3.12

Equation (3.12) coincides with the linear elastic (compressible) solution for a dislocation [2], when taken with Poisson's ratio equal to 1/2.

An issue of interest is that the logarithmic behaviour near the dislocation tip is not present in the incompressibility limit, so that a singular stress field is generated by a displacement field not showing the usual logarithmic singularity.

## 4. The numerical treatment of the boundary integral equations

The numerical treatment of the boundary integral equations (3.6) involves a Cauchy-type integral, for equation (3.6a), and a hypersingular integral, for equation (3.6b). The use of these equations implies the knowledge of the gradient of Green's function for incremental displacement and for incremental in-plane mean stress; the former has been given by Bigoni & Capuani [17] and will not be repeated, while the latter can be obtained using eqns (48) and (62) given by Bigoni & Capuani [17], so that we arrive at the final expression
4.1where *ζ*_{gj}(** x**,

**,**

*y**α*) and

*Ξ*

_{gj}(

**,**

*x***,**

*y**α*) are functions (not reported for brevity) of the distance between the source point

**, the generic point**

*x***and the angle**

*y**α*, as defined by Bigoni & Capuani [17] (their fig. 1); coefficients

*γ*

_{1}and

*γ*

_{2}are also defined in Bigoni & Capuani [17] (their eqn (15)). Note also that

*δ*

_{1g},

*δ*

_{2g},

*δ*

_{1j}and

*δ*

_{2j}are all Kronecker deltas (taking the values 0 and 1). Note that the term

*ζ*

_{gj}(

**,**

*x***,**

*y**α*) is related to the gradient of Green's hydrostatic nominal stress, whereas the term

*Ξ*

_{gj}(

**,**

*x***,**

*y**α*) is related to the second gradient of Green's velocity.

The numerical evaluation of the boundary integral equation (3.6a) requires the following treatment. First, we introduce the reference system shown in figure 5, where
4.2
4.3so that
4.4where *ε* can become a small parameter,
4.5

We introduce the change of variables
4.6so that
4.7Note from figure 5 that, whereas the source point ** x** ranges along the dislocation line

*ρ*∈[0,

*a*], point

**is arbitrary. Therefore, the variable**

*y**z*(does not) vanishes for all

**whose projections lie (out-) inside the dislocation line (**

*y**ρ*

_{y}∉(0,

*a*))

*ρ*

_{y}∈(0,

*a*), so that the problem in managing equations (3.6) occurs when

*ρ*

_{y}∈(0,

*a*). In this situation,

*ε*can be made arbitrarily small, but different from zero, whereas variable

*z*can be expanded around zero.

Using (4.6), the integrals involved in (3.6a) can be written in the following form:
4.8where
4.9in which function *Δ*(*ε*,*z*,*α*) takes a complicated expression, not reported for brevity.

Therefore, a Taylor series expansion of function *Δ*(*ε*,*z*,*α*) in the variable *z* yields
4.10where
4.11so that function *G*(*ε*,*z*) can be regularized as
4.12where the derivative of function *Δ*(*ε*,*z*,*α*) in the variable *z* can be easily calculated (though it takes a complicated expression, which is not reported for conciseness).

As a conclusion, instead of with the integral (4.8), we can work with its regularized version, written as 4.13in which the singular terms have been explicitly evaluated. The integral equation for in-plane mean stress increment (3.6b) can be treated in a way similar to that used to obtain (4.13), which is used in §5 to produce numerical values for the incremental displacement fields near a dislocation dipole.

## 5. Dislocation clustering in a metal near the elliptic border

We are now in a position to explore the effect of prestress on a metal deformed near the elliptic boundary. For this purpose, we can use equation (3.6a) in the regularized version (4.13) for an edge dislocation dipole (which may be also thought of as a ‘super dislocation’, i.e. a collection of dislocations smeared out along a certain direction) of length *a*, which is assumed to be nucleated in a *J*_{2}-deformation theory material with a hardening parameter *N*=0.363 (see equations (2.9) and (2.10)). Incremental displacement fields for a unit length Burgers vector, at a prestrain (so that the material is close to the ellipticity threshold , but still within the elliptic region) are plotted in figure 6 for different inclinations *ψ* of the dipole with respect to the orthotropy axes (see the sketch in figure 6, upper part on the left).

Note that the dislocation solution depends on the parameter *η*, which has been assumed equal to 0.490. The following inclinations have been considered: *ψ*={0,*π*/6,*π*/4,*π*/3,*π*/2}.

It may be worth observing that the perturbation induced by a dislocation dipole is different from that induced by a force dipole (as considered by Bigoni & Capuani [17]). In fact, force and dislocation dipoles can produce similar effects only in the far fields and only under the assumption that the prestress is absent [14].

In all cases reported in figure 6, we observe the formation of zones of intense deformation, aligned parallel to the inclination of the shear bands (±27.37^{°} with respect to the *x*_{1}-axis), formally possible only at loss of ellipticity. The response of the material far from the elliptic boundary is completely different, as shown in figure 7, pertaining to an isotropic incompressible material at null prestress.

Because the dislocation activity is triggered by a rise in the shear stress, and this occurs for highly prestressed materials along the preferred directions shown in figure 6, along these the dislocation activity tends to be strongly promoted. Therefore, this activation will again generate an increment in shear stress along the same directions, thus producing a sort of ‘cascade effect’, which will cluster dislocation formation along shear bands. This effect may explain the fact that the amount of slip taking place on active shear bands may be up to three orders of magnitude greater than that produced by a single dislocation [13].

## 6. Conclusions

Prestress has been shown to be an important factor in the mechanics of dislocation clustering, in ductile metals deformed near the shear band formation. A new solution for an edge dislocation, valid for incremental nonlinear elasticity, with the current state taken as homogeneous, shows emergence of highly localized deformation patterns, when the material is deformed near the boundary of ellipticity loss, which may trigger ‘cascade’ dislocation activation along shear band directions. Although this conclusion is limited by the assumption of homogeneity of the prestress (which is the only way to arrive at analytical solutions), it may correctly model the situation when a dislocation dipole is emitted.

## Acknowledgements

Partial support from ICMS (Edinburgh, 2010) is acknowledged. D.B. and L.A. gratefully acknowledge partial support from Italian Prin 2009 (prot. 2009XWLFKW-002); G.M. acknowledges support from the FP7 research project PIAP-GA-2011-286110.

## Appendix A. Notes on the photoelastic experiment reported in figure 1

Photoelastic experiments have been performed with a circular (with quarterwave retarders for 560 nm) polariscope (dark field arrangement and equipped with a white and sodium vapour lightbox at *λ*=589.3 nm, purchased from Tiedemann & Betz), designed by us and manufactured at the University of Trento (see http://www.ing.unitn.it/dims/ssmg/). Photographs have been taken with a Nikon D200 digital camera equipped with an AF-S micro Nikkor (70–180 mm, 1:4.55.6D) lens. The photoelastic material is a 5 mm thick platelet obtained from a commercial two-part epoxy resin (Crystal Resins by Gedeo, 305 Avenue du pic de Bretagne, 13420 Gemenos, France). The orthotropic material has been obtained by cutting (with a circular saw, blade HSS-DMo5 63×0.3×16 Z128 A) 0.3 mm thick and 2 mm deep parallel grooves (at a distance of 2.5 mm) in the resin sample, a technique previously used by O'Regan [19] on photoelastic coatings. The dislocation has been created with two 0.5 mm thick steel platelets in contact with each other at one side and attached to the resin on the other side. The platelets (placed horizontally and aligned parallel to the dashed line in the figure) have been forced to slide against each other to generate the stress field near an edge dislocation.

- Received December 28, 2012.
- Accepted February 28, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.