## Abstract

In this paper, we consider dislocations in the framework of first as well as second gradient theory of elasticity. Using the Fourier transform, rigorous analytical solutions of the two-dimensional bi-Helmholtz and Helmholtz equations are derived in closed form for the displacement, elastic distortion, plastic distortion and dislocation density of screw and edge dislocations. In our framework, it was not necessary to use boundary conditions to fix constants of the solutions. The discontinuous parts of the displacement and plastic distortion are expressed in terms of two-dimensional as well as one-dimensional Fourier-type integrals. All other fields can be written in terms of modified Bessel functions.

## 1. Introduction

In the theory of classical elasticity, it is well known that different displacement fields of a dislocation give the same elastic distortions but different plastic distortions. Thus, the displacement of a dislocation cannot be a physical state quantity. The reason why is that the displacement field of a dislocation is a multi-valued function. For example, the multi-valued displacement of a screw dislocation may be given in terms of(1.1)In classical elasticity, the displacement is fixed up to a constant displacement. One may reduce such a multi-valued function to a single-valued one by a branch cut which is of mathematical convenience, only (e.g. DeWit 1973; Teodosiu 1982). The single-valued function is then discontinuous because of the jump at the branch cut. In general, the branch cut may be arbitrary. The surface of the branch cut can be identified with the cut plane of the Volterra process. Thus, this branch cut cannot be detected experimentally and the elastic strains and stresses do not depend on it (e.g. Kröner 1993). This fact is nothing but the main message of the Volterra theorem. Due to the discontinuity, the total distortion consists of a plastic part which depends on the cut plane.

Modified solutions for displacements of dislocations were given in the framework of gauge theory of dislocations by Edelen (1996) and Lazar (2002*a*,*b*, 2003*a*). In this theory, one does not have a unique governing equation for the displacement and plastic distortion fields because they are not physical state quantities. Only, the elastic distortion and the dislocation density tensors are state quantities, which are gauge invariant. To determine and to modify the displacement, one may use local translations as gauge transformation and a gauge fixing. Thus, in gauge theory of dislocations the displacement and the plastic distortion are not unique due to physical reasons.

A slightly different theory of extended elasticity is the so-called gradient elasticity. Strain gradient theories were introduced by Mindlin (1964, 1965) and Mindlin & Eshel 1968). Gradient elasticity is a generalization of linear elasticity which includes higher-order terms to account for microstructural effects. In a strain gradient theory, the strain energy depends on the elastic strain and gradients of the elastic strain. Due to the gradient terms, such a theory contains additional coefficients with the dimension of a length, which are called gradient coefficients. Ru & Aifantis (1993) postulated a governing equation for the displacement vector in the framework of a special gradient elasticity. Later, Lazar & Maugin (2005) showed how and under which conditions one can derive such an equation in gradient elasticity from a variational principle. In such a gradient elasticity, which is called first gradient elasticity of Helmholtz type, the field quantities must satisfy the following inhomogeneous Helmholtz equations:(1.2)(1.3)(1.4)(1.5)where *ϵ* is a positive gradient parameter. , , and are the displacement vector, plastic distortion, elastic distortion and dislocation density tensor, respectively, calculated in the theory of classical elasticity. On the other hand, , , and are the displacement vector, plastic distortion, elastic distortion and dislocation density tensor, respectively, calculated in the theory of gradient elasticity. They fulfil(1.6)(1.7)where is called the total distortion. Corresponding equations are valid for the fields in classical elasticity. Gutkin & Aifantis (1996, 1997, 1999) found solutions of equations (1.2)–(1.4) for screw and edge dislocations. They used the one-dimensional Fourier transform with respect to *x* together with special ‘physically motivated’ boundary conditions. Unfortunately, the solutions for the displacement and the plastic distortion do not satisfy equations (1.2) and (1.3) at *y*=0 and the plastic distortion is still singular due to Dirac delta terms. But, of course, a solution should be a solution everywhere. Only, their elastic distortions fulfil (1.4) everywhere and agree with the results given by Lazar (2003*b*) in the gauge theory of dislocation and in the theory of elastoplasticity. This is one motivation to re-investigate that matter.

In the second gradient elasticity, one can show that the field quantities fulfil the following inhomogeneous partial differential equations of fourth order (Lazar *et al*. 2006):(1.8)(1.9)(1.10)(1.11)which can be factorized into the inhomogeneous bi-Helmholtz equations(1.12)(1.13)(1.14)(1.15)with(1.16)and(1.17)where *γ* is a second gradient parameter of higher order. Only rigorous solutions of equations (1.14) and (1.15) have been given quite recently for dislocations by Lazar *et al*. (2006). But no solutions of (1.12) and (1.13) are given in the literature.

In this paper, we will give exact solutions for screw and edge dislocations which satisfy equations (1.2)–(1.5) and (1.12)–(1.15) everywhere. We are using the two-dimensional Fourier transform to solve these equations. We are able to give closed form solutions for the elastic distortion tensor and the dislocation density tensor and one-dimensional integral representations for the displacement vector and the plastic distortion tensor. In our framework, it is not necessary to use boundary condition to fix the gradient term of the solution.

## 2. Screw dislocation

### (a) Classical solution

We consider a screw dislocation whose Burgers vector and dislocation line coincide with the *z*-axis. We use a discontinuous displacement field which has the symmetry(2.1)Then it is given by (e.g. Leibfried & Dietze 1949)(2.2)where has the range . Here, is a single-valued function with a discontinuity represented by a branch cut (see figure 1). Thus, the branch cut is given for *y*<0,(2.3)In the degenerate case, when *y*=0(2.4)So, we have at *y*=0: and .

The total distortion is just the gradient of the displacement (2.2)(2.5)(2.6)where *δ* denotes the Dirac delta function and is the Heaviside step function defined by(2.7)Because the second part of (2.5) is discontinuous and singular at the branch cut, it may be identified with the plastic distortion. The plastic distortion gives rise to a dislocation density of the single screw dislocation according to(2.8)It means that the dislocation is concentrated at .

### (b) Gradient solution

The governing equations (1.12)–(1.15) will be solved by using the two-dimensional Fourier transform. The two-dimensional Fourier transform (Sneddon 1951) is defined by(2.9)and the inverse Fourier transform by(2.10)where . Taking the representation of (2.2) as a two-dimensional Fourier integral (Mura 1982)(2.11)and substituting it into equation (1.12), we get an algebraic equation for . With the help of the inverse Fourier transform, we obtain the expression for . The solution for the displacement reads(2.12)The total distortion is calculated as(2.13)With the help of the Fourier integral representations (Wladimirow 1971)(2.14)(2.15)where , denotes the modified Bessel function of order *n* and introducing the function(2.16)it can be decomposed into the elastic distortion(2.17)and the plastic distortion(2.18)In the limits and , equation (2.18) converts into the classical plastic distortion given in the Fourier integral representation(2.19)For the sake of convergence of the -integral in equation (2.19), it is customary to give an infinitesimally negative imaginary part. In addition, the plastic distortion (2.18) is a rigorous solution of (1.13) written in the double Fourier integral form. The other non-vanishing component of the total distortion reads(2.20)which has the decomposition(2.21)and . Like in classical elasticity, the screw dislocation has only one non-vanishing component of the plastic distortion tensor. The non-vanishing component of the plastic distortion gives rise to the following dislocation density(2.22)In the limits and , (2.22) goes to the classical expression (2.8). Additionally, (2.22) is a rigorous solution of (1.15). Equations (2.17) and (2.21) are proper solutions of the inhomogeneous bi-Helmholtz equation (1.14). We note that equations (2.17), (2.21) and (2.22) agree with the expressions recently given by Lazar *et al*. (2006) by means of the stress function method.

In order to simplify the expressions for the displacement and the plastic distortion, we integrate out the variable in equations (2.12) and (2.18). So the double Fourier-type integral can be reduced to a single Fourier integral. Because of the symmetry of the single Fourier integrals the displacement may be written as Fourier-sine integral and the plastic distortion as Fourier-cosine integral. In addition, we re-label for convenience. In this way, the displacement is given as(2.23)where denotes the signum function defined by(2.24)or in terms of the Heaviside function(2.25)The plastic distortion reads(2.26)It is non-singular unlike the expressions for the plastic distortion calculated by Gutkin & Aifantis (1996). Of course, the cosine representation (2.26) of the plastic distortion gives the correct form of the dislocation density (2.22). The Fourier-cosine integral representation for the dislocation density has the form(2.27)Using the Fourier-cosine integral(2.28)the closed-form expression (2.22) is recovered in a consistent way. For further applications, the single Fourier-type integrals (2.23) and (2.26) may be evaluated numerically, e.g. by fast Fourier transform algorithms.

When , the integrals (2.23) and (2.26) may be evaluated in an explicit form. Using the integral relations(2.29)(2.30)the displacement has the form(2.31)The gradient terms which appear in (2.31) lead to a smoothing of the displacement profile unlike the jump occurring in the classical solution (see figure 2). Of course, the smoothing depends on the two gradient coefficients and . The asymptotic limits are(2.32)With this smoothing of the displacement profile the width of the dislocation (dislocation core radius) may be defined as a function of the gradient parameters and .

By using the integral(2.33)the plastic distortion reads(2.34)In contrast to the classical plastic distortion which is singular at *x*=0 due to in equation (2.19), equation (2.34) is smooth there (see figure 3).

For completeness, the expressions for the first gradient elasticity are listed. They are obtained from the second gradient results in the limit and . Of course, they satisfy equations (1.2)–(1.5). They read(2.35)(2.36)(2.37)(2.38)(2.39)Let us compare these results with the formulae given by Gutkin & Aifantis (1996). Equations (2.36) and (2.37) have the same form as their result. On the other hand, equations (2.35) and (2.38) look similar as their expression and, anyway, they have an important difference. Due to the choice instead of for an additional term, appears in equations (2.35) and (2.38). This Heaviside term guarantees the satisfying of the inhomogeneous Helmholtz equations and it cancels the classical -terms which would appear due to the differentiation of the -function. In addition, it can be seen in figures 2 and 3 that the second order expressions are smoother than the first order results.

The elastic strain, stress and the higher order stresses of a screw dislocation in the second gradient elasticity have been given by Lazar *et al*. (2006).

## 3. Edge dislocation

### (a) Classical solution

Consider now an edge dislocation whose Burgers vector is parallel to the *x*-axis and the dislocation line is along the *z*-direction. We are using the classical displacement with the branch cut at *x*=0 and for *y*<0 given by Leibfried & Lücke (1949) (e.g. Seeger 1955)(3.1)(3.2)where is given by equation (2.2) and *ν* is the Poisson's ratio. The displacement (3.1) was also given by Nabarro (1967). But his differs by a constant, because he used instead of . Of course, this constant value is not significant in elasticity theory. We note that only is discontinuous due to the jump. All other parts of the displacements (3.1) and (3.2) are continuous. In addition, the first part of (3.2) has a logarithmic singularity. The expression (3.2) was also given by Mura (1969) and DeWit (1973). The non-vanishing components of the elastic distortion are (DeWit 1973)(3.3)(3.4)(3.5)(3.6)which are singular at *r*=0. The plastic distortion reads(3.7)which is caused by the jump of . The dislocation density tensor has the following non-vanishing component (e.g. DeWit 1973)(3.8)

### (b) Gradient solution

Using the same technique to solve the inhomogeneous bi-Helmholtz equations (1.12)–(1.15) for the edge dislocation as in §2 for the screw dislocation, we obtain rigorous solutions for all field quantities. Here, we just give the results in order to avoid the same technical details. The solution for the displacement reads(3.9)(3.10)where is given by (2.12) and (2.23). The displacement (3.10) is plotted in figure 4. It is interesting to note that if we substitute instead of in the second part of equation (3.10), the expression that depends on the gradient coefficients remains the same. The elastic distortion is calculated as(3.11)(3.12)(3.13)(3.14)The plastic distortion is given by(3.15)It can be seen that it has the same form as the plastic distortion (2.18) and (2.26) for the screw dislocation. The dislocation density of a single edge dislocation reads(3.16)which has the same form as the expression for a screw dislocation (2.22).

Finally, we give the expressions for the first gradient elasticity obtained from the second gradient results in the limit and . They fulfil equations (1.2)–(1.5). The displacements have the following form:(3.17)(3.18)where is given by (2.31). Equation (3.17) has a similar form as the expression given by Gutkin & Aifantis (1997). Only, the expression for the discontinuous function is different because we used . Equation (3.18) is in full agreement with the formula given by Gutkin & Aifantis (1997). The elastic distortion is obtained as(3.19)(3.20)(3.21)(3.22)which agree with the formulae given by Lazar (2003*a*,*b*). The plastic distortion and the dislocation density read(3.23)

(3.24)

The elastic strain, stress and the higher order stresses of an edge dislocation in the second gradient elasticity have been recently calculated by Lazar *et al*. (2006).

## 4. Conclusions

In this paper, we used special theories of first and second strain gradient elasticity. The second strain gradient theory is a generalization of the first strain gradient elasticity with only one gradient parameter. This gradient theory of second order has two gradient coefficients, only. Using the Fourier transform technique, rigorous solutions for the displacement, plastic distortion, elastic distortion, dislocation density of screw and edge dislocations have been derived in the theory of gradient elasticity. The gradient solutions for the elastic distortion and the dislocation density are given in closed form in terms of modified Bessel functions and they agree with results obtained with a slightly different mathematical technique. The formulae for the displacement and plastic distortion have been obtained as double Fourier as well as single Fourier integrals. An advantage of our solutions for the displacement and plastic distortion is that they are solutions of the governing equations everywhere unlike the solutions obtained earlier by Gutkin & Aifantis (1996, 1997) which do not satisfy the corresponding inhomogeneous Helmholtz equations at *y*=0. Our solutions for the plastic distortion of screw and edge dislocations are non-singular and they do not contain Dirac delta terms. In addition, we have found our solutions without using boundary conditions unlike Gutkin & Aifantis (1996, 1997).

## Acknowledgements

M.L. acknowledges the support by the Laboratoire de Modélisation en Mécanique and the Université Pierre et Marie Curie in Paris. G.A.M. benefits from a Max-Planck-Award for international cooperation (2002–2006).

## Footnotes

- Received September 2, 2005.
- Accepted February 21, 2006.

- © 2006 The Royal Society