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\begin{center}
{\Large\bf            NANOSCOPICS OF DISLOCATIONS AND DISCLINATIONS    } \\
{\Large\bf                        IN GRADIENT ELASTICITY               } \\

                                                    \par\vspace{2cm}\noindent
                                     M.~Yu.~Gutkin                    
                                                    \par\vspace{2cm}\noindent
                    Institute of Problems of Mechanical Engineering      \\
                            Russian Academy of Sciences                  \\
                 Bolshoj 61, Vas.~Ostrov, St.~Petersburg 199178, RUSSIA  \\
                                                    \par\vspace{2cm}\noindent
                                       ABSTRACT
\end{center}
\par\noindent
The results of application of gradient theory of elasticity to a description 
of elastic properties of dislocations and disclinations are reviewed. The main
achievement made in this approach is the elimination of the classical 
singularities at defect lines and the possibility of describing short-range 
interactions between them on a nanoscale level. Non-singular solutions for 
elastic fields and energies of dislocations in an infinite isotropic medium
are represented in a closed analitycal form and discussed in detail. Similar
solutions for straight disclinations are also considered with application to
the specific case of disclination dipoles. A special attention is paid to the
nanoscopic behavior and stress fields of dislocations near interfaces.
Recent non-singular solutions for both the dislocation stresses and ``image''
forces on dislocations are demostrated in a general integral form and 
corresponding peculiarities in dislocation behavior near interfaces are 
discussed. 


                                                        \par\newpage\noindent
{\Large\bf Contents}
                                                \par\bigskip\bigskip\noindent
1. Introduction
                                                        \par\medskip\noindent
2. Governing equations of gradient elasticity
                                                        \par\smallskip
   2.1. Special gradient theory of elasticity           \par
   2.2. More general gradient theory of elasticity
                                                        \par\medskip\noindent
3. Nanoscale elastic properties of dislocations
                                                        \par\smallskip
   3.1. Straight dislocations in homogeneous media
                                                        \par\smallskip\ \ \ \ 
        3.1.1. A general solution                       \par\ \ \ \ 
        3.1.2. Screw dislocations                       \par\ \ \ \
        3.1.3. Edge dislocations                        
                                                        \par\smallskip
   3.2. Straight dislocations near interfaces
                                                        \par\smallskip\ \ \ \ 
        3.2.1. A general solution                       \par\ \ \ \
        3.2.2. Screw dislocations                       \par\ \ \ \
        3.2.3. Edge dislocations
                                                        \par\medskip\noindent
4. Nanoscale elastic properties of disclinations
                                                        \par\smallskip
   4.1. Individual disclinations --- a general solution
                                                        \par\smallskip
   4.2. Disclination dipoles
                                                        \par\smallskip\ \ \ \ 
        4.2.1. First-type twist disclinations           \par\ \ \ \
        4.2.2. Second-type twist disclinations          \par\ \ \ \
        4.2.3. Wedge disclinations
                                                        \par\medskip\noindent
5. Conclusions
                                                        \par\medskip\noindent
6. References


                                                        \par\newpage\noindent
1. INTRODUCTION
                                                \par\bigskip\bigskip\noindent
Traditional description of elastic fields produced by dislocations and 
disclinations is based on the classical theory of linear elasticity~[1--7]. 
In the isotropic case, the appropriate expressions for 
dislocation/disclination elastic fields are quite simple and broadly 
applicable to model the structure and mechanical behavior of various materials
and solid state systems [1,\ 7--12]. However, some components of these fields
are singular at the defect lines, a fact that limits the applicability of the
theory to describe situations where it is important to know the strained state
near defects. This concerns, for example, dislocation-disclination models for
grain boundaries, as well as dislocation-disclination models simulating 
metallic glass structures and nanocrystalline behavior where one deals with 
high-density ensembles of defects. Thus, this problem is similar to that in 
the case of cracks [13,\ 14] with the well-known singular expressions 
predicted by classical elasticity theory.
\par
The first attempts to modify the elastic fields of such defects within a 
continuum theory, were done by taking into account couple stresses [15--36]. 
Dislocations [15--22], disclinations [22--28] and cracks [29--36] have been 
considered in both Cosserat and multipolar media. The solutions found for the 
corresponding elastic strain and stress fields differ from the classical 
solutions but still possess singularities at the lines of dislocations and 
disclinations, as well as at the crack tips.
\par
Following the classification given by Kunin [37], the Cosserat and 
multipolar media may be considered as continua having weak non-locality in 
elastic properties represented through a higher-order gradient of the 
displacement field and an additional material constant with the dimension of 
square length. Consideration of defects in the continua having strong 
non-locality, when there is an integral relation between elastic strains and 
stresses, gives better results --- the singularities of stress fields 
disappear at dislocation [37--44] and disclination [44,\ 45] lines as well as 
at crack tips [41,\ 46,\ 47] but there are some hidden problems with the 
boundary conditions used and the convergence of the solution. It is worth 
noting that the most convincing solutions [37,\ 38] were found for 
dislocations in a model of quasicontinuum where the discrete structure of a 
solid body was taken into account. The dislocation stress fields were obtained
in a closed analytical form, they were equal to zero at the dislocation line 
reaching extreme values at a certain distance from the dislocation line and 
then diminishing with small decreasing oscillations around the classical 
solution. To solve the same problems within a non-local continuum model with 
an integral relation between stresses and strains, Eringen [39--41] introduced
{\it a~priori} a special form of the kernel in the aforementioned integral 
relation and found other solution formulae. In the case of screw dislocations
[39--41], the solutions were presented in analytical form which was reduced to
the classical elasticity solution when non-locality was assumed to vanish.
In the case of edge dislocations [41], the solution was given in an integral 
form which did not allow such a limiting transition. The oscillations which 
were characteristic for the quasicontinuum model [37,\ 38], did not seem to 
appear here for both of these cases. It is interesting to point out that the 
expressions for the displacement and strain fields remained the same as in the
classical theory of elasticity, with all typical singularities at the 
dislocation lines. Moreover, as mentioned earlier there are some problems with
the number of the boundary conditions used and the convergence of the solution
which, however, have not been explicitly discussed. The authors of [42--45] do
not report about these peculiarities of non-local solutions though they used 
Eringen's model of non-local continuum.
\par
Thus, the above non-local continuum models, among other things, can not avoid 
the sigularities in the displacement and strain fields. The quasicontinuum 
models [37,\ 38] avoid this difficulty, but they are hardly applicable to 
real materials and systems where it is necessary to account for interior and 
exterior boundaries, as this brings significant technical difficulties.
It follows that is important to search further for non-standard continuum 
models which would lead to results comparable with atomistic calculations and
related data of experimental observations. For example, it would be 
interesting to estimate the displacements and elastic strains near the defect
cores and compare them with real values obtained from TEM images and related 
computer simulations.
\par
We consider below another possible way to address this problem which is to use
gradient modifications of the classical linear theory of elasticity. Two 
different gradient theories (a special one and another more general one) 
proposed by Aifantis and co-workers [48--56] with applications to crack 
problems are shortly discussed in {\it Section~2}. New non-singular solutions
within the more general gradient theory of elasticity for dislocations are 
considered in {\it Section~3} while those for disclinations are represented 
in {\it Section~4}.
                                                \par\bigskip\bigskip\noindent
2. GOVERNING EQUATIONS OF GRADIENT ELASTICITY
                                                \par\bigskip\bigskip\noindent
{\large\bf  2.1. Special gradient theory of elasticity}   
                                                \par\bigskip\noindent
In 1965, Mindlin [57] proposed a linear theory describing deformation of 
elastic solids where the density of strain energy was the function of strain
as well as of its first and second gradients. Taking into account the second
gradient of strain, the author claimed the incorporation of both cohesive 
forces and surface tension into the linear elasticity. The corresponding
modification of Hooke's law reads [57]
\begin{equation}
     \mbox{\boldmath $\sigma $ }=\lambda
            \left(   \mbox{\boldmath $I$} - c_1\mbox{\boldmath $I$}\nabla^2
                   - c_2\nabla\nabla \right)
                 ({\rm tr}\,\mbox{\boldmath $\varepsilon $})
    + 2\mu\left( 1 - c_3\nabla^2 \right)
                          \mbox{\boldmath $\varepsilon $}\;,           % (1)
\end{equation}
where $\lambda $ and $\mu $ are the usual Lam\'{e} constants, 
$\mbox{\boldmath $\sigma $ }$ and $\mbox{\boldmath $\varepsilon $}$ denote
elastic stress and strain tensors, $\mbox{\boldmath $I$}$ is the unit tensor, 
$\nabla ^2$ is the Laplacian; $c_1$, $c_2$ and $c_3$ are three independent 
gradient coefficients. 
\par
Substitution of (1) into the usual equilibrium equation 
$\nabla\cdot\mbox{\boldmath $\sigma $}=0$ gives the following equation for the
vector of displacement $\mbox{\boldmath $u$}$ [58]
\begin{equation}
     \mu\left( 1 - c_3\nabla^2 \right)\nabla^2\mbox{\boldmath $u$}
     + \left\{ \lambda + \mu - [\lambda\,(c_1+c_2) + \mu c_3]\nabla^2\right\}
       \nabla\nabla\cdot\mbox{\boldmath $u$} = 0\;.                     % (2)
\end{equation}
\par
In the case, when the vector of displacement $\mbox{\boldmath $u$}$
have more than one non-vanishing components, (2) gives a system of coupled 
partial differencial equations which seems to be hardly solved. However, 
if there is the relation $c_1+c_2=c_3=c$, (2) results in 
\begin{equation}
    \left( 1 - c\nabla^2 \right)
    \left\{ \mu\nabla^2\mbox{\boldmath $u$}
     + (\lambda + \mu )\nabla\nabla\cdot\mbox{\boldmath $u$}\right\}
      = 0\;.                                                            % (3)
\end{equation}
\par
In the special case when $c_1=c_3=c$ and $c_2=0$, Eq.~1 transforms into 
\begin{equation}
\mbox{\boldmath $\sigma $}=\lambda ({\rm tr}\,\mbox{\boldmath $\varepsilon $})
                               \mbox{\boldmath $I$}+
                  2\mu \mbox{\boldmath $\varepsilon $}-c\nabla ^2
\left[\lambda ({\rm tr}\,\mbox{\boldmath $\varepsilon $})\mbox{\boldmath $I$}+
                 2\mu \mbox{\boldmath $\varepsilon $}\right].          %  (4)
\end{equation}
Namely this equation was initially proposed by Altan and Aifantis [48] to
eliminate the strain singularity at the mode-III crack tip. They also showed 
that for an atomic lattice, the gradient coefficient $c$ can be estimated [48]
as $\sqrt{c}\approx a/4$, where $a$ is the lattice constant. Substitution of 
(4) into the equilibrium equation $\nabla\cdot\mbox{\boldmath $\sigma $}=0$ 
leads again to (3). A physical derivation of (3) and (4) based on a 
mixture-like model for composite materials was provided by Aifantis [50] and 
later in more detail by Altan and Aifantis [54]. Ru and Aifantis [49] have 
found a simplified way to solve boundary-value problems in this special theory
of gradient elasticity described by (3) or (4), by reducing them to solving a
non-homogeneous Helmholtz equation with the ``source'' term given in terms of
well-known solutions for the same problems in classical elasticity. They have
also shown that the stress field in this theory of gradient elasticity remains
the same as in classical elasticity. Altan and Aifantis [54] have used a 
Fourier transform procedure to solve (3) in two dimensions including the 
mode-I and -II cracks problems.
\par
Application of this theory to crack problems has resulted [48--54] into the 
elimination of the classical singularities from the solutions for the elastic
displacement and strain fields at the crack tips. The stress components, 
however, remained as in the classical theory but this difficulty has been
considered as less severe than the strain singularity because the stress may 
not be rigorously defined at the atomic level near a discontinuity. 
\par 
Encouraged by these results, we applied the special gradient elasticity 
theory given by (4) to dislocations [59,\ 60] and disclinations [58]. As 
was the case with cracks, new gradient solutions for displacement [59,\ 60], 
strain fields [59,\ 60] and energies [58] of dislocations as well as for 
strain fields [58] of disclinations were non-singular at the defect lines. 
The corresponding stress fields were the same as in the classical 
theory of elasticity. 
                                                \par\bigskip\noindent
{\large\bf  2.2. More general gradient theory of elasticity}
                                                \par\bigskip\noindent
In unpublished work by Ru and Aifantis [55] (see also [56]) a simple extension
of the gradient elasticity model given by (4) was used to dispense with both 
strain and stress singularity at the dislocation core and at the crack tip. 
The constitutive equation of this theory reads
\begin{equation}
     \left( 1-c_1\nabla^2 \right)\mbox{\boldmath $\sigma $} =
     \left( 1-c_2\nabla^2 \right)
     \left[\lambda ({\rm tr}\,\mbox{\boldmath $\varepsilon $})
           \mbox{\boldmath $I$} +
      2\mu \mbox{\boldmath $\varepsilon $}\right],                     %   (5)
\end{equation} 
with two different gradient coefficients $c_1$ and $c_2$. In [55] a rather 
simple mathematical procedure analogous to the one contained in [49] has been 
outlined for the solution of (5) in terms of solutions of classical elasticity
for the same boundary-value problem. In fact, it is easily established (see
[49], also [58--60]) that the right hand side of (5) for the case of 
$c_1\equiv 0$, gives the classical solution for the
stress field which we denote here by $\mbox{\boldmath $\sigma $}^0$, while the
solution for the displacement is determined through the inhomogeneous 
Helmholtz equation given by
\begin{equation}
   \left( 1-c_2\nabla^2 \right)\mbox{\boldmath $u$}=\mbox{\boldmath $u$}^0\;,
                                                                       %   (6)
\end{equation}
where $\mbox{\boldmath $u$}^0$ denotes the solution of classical elasticity 
for the same traction boundary-value problem. Equation (6) implies a similar
equation for strain $\mbox{\boldmath $\varepsilon $}$ of the gradient theory
\begin{equation}
   \left( 1-c_2\nabla^2 \right)\mbox{\boldmath $\varepsilon $} =
                               \mbox{\boldmath $\varepsilon $}^0\;,    %   (7)
\end{equation} 
in terms of the strain $\mbox{\boldmath $\varepsilon $}^0$ of the
classical elasticity theory for the same traction boundary-value problem.
With the displacement or strain field thus determined (which is obviously 
independent of whether $c_1\equiv 0$ or $c_1\neq 0$), it follows that the 
stress field $\mbox{\boldmath $\sigma $}$ of (5) can be determined (for the 
case $c_1\neq 0$) from the equation
\begin{equation}
   \left( 1-c_1\nabla^2 \right)\mbox{\boldmath $\sigma $} = 
                               \mbox{\boldmath $\sigma $}^0\;,
                                                                       %   (8)
\end{equation} 
where $\mbox{\boldmath $\sigma $}^0$ denotes the solution obtained for the 
same boundary-value problem within the classical theory of elasticity.
\par
Thus, in order to solve equation (5), one can solve separately equations (7) 
and (8) by utilizing the classical solutions 
$\mbox{\boldmath $\varepsilon $}^0$ and $\mbox{\boldmath $\sigma $}^0$ 
provided that appropriate care is taken for the extra (due to the higher order
terms) boundary conditions or conditions at infinity. For dislocations and
disclinations in a homogeneous medium, this
problem solutions are accounted for by assuming that the strain and stress 
fields at infinity are the same for both the gradient and classical theory.
The approach has firstly been applied [55] to the cases of screw dislocations
and mode-III cracks where the asymptotic solutions at the dislocation line 
and crack tip have been found demonstrating the elimination of both strain and
stress singularities there. Recently, the gradient elasticity described by (5)
has been used to find nonsingular solutions for stress fields of dislocations
[61,\ 62] and disclinations [62,\ 63] in homogeneous solid. The boundary-value
problems of dislocations near interfaces within the gradient elasticity 
theory (5) have been solved in [64--66] where non-singular expressions have
been found for dislocation stress fields as well as for ``image'' forces on 
dislocations due to interfaces.

\newpage
                                                \par\bigskip\bigskip\noindent
3. NANOSCALE ELASTIC PROPERTIES OF DISLOCATIONS
                                                \par\bigskip\bigskip\noindent
{\large\bf 3.1. Straight dislocations in homogeneous media} 
                                                \par\bigskip\noindent
{\bf 3.1.1. A general solution }
                                                \par\bigskip\noindent
Consider a mixed dislocation whose line coincides with the $z$-axis of a 
Cartesian coordinate system. Let it's Burgers vector be
$\mbox{\boldmath $b$}=b_x\mbox{\boldmath $e$}_x+b_z\mbox{\boldmath $e$}_z$
thus determining the edge ($b_x$) and screw ($b_z$) components.
                                              \par\bigskip\noindent
{\underline{\it Classical solution}}
                                              \par\medskip\noindent
In the framework of classical elasticity theory, the total displacement field 
is described by
\begin{eqnarray}
     \mbox{\boldmath $u$}^0 &=&
           \frac{b_x\mbox{\boldmath $e$}_x+b_z\mbox{\boldmath $e$}_z}{2\pi }
           \left\{  \arctan\frac{y}{x}
                   +\frac{\pi}{2}{\rm sign}(y)[1-{\rm sign}(x)] \right\} 
                                                             \nonumber \\
      &+&  \frac{b_x}{4\pi (1-\nu )}
           \left\{  \mbox{\boldmath $e$}_x\,\frac{xy}{r^2}
                   -\mbox{\boldmath $e$}_y
                     \left[ (1-2\nu )\ln r + \frac{x^2}{r^2} \right]\right\},
                                                                       % (9)
\end{eqnarray}
where $\nu $ is the Poisson ratio, $r^2=x^2+y^2$. Here we use a single-valued 
discontinuous form suggested by de Wit [5]. The elastic strain field 
$\varepsilon^0_{ij}$ reads (in units of $1/[4\pi (1-\nu )]$) by [1,\ 5]
\begin{eqnarray}
   & &  \varepsilon_{xx}^0=-b_xy\,[(1-2\nu )r^2+2x^2]/r^4,\;\;\;\;
        \varepsilon_{yy}^0=-b_xy\,[(1-2\nu )r^2-2x^2]/r^4,         
                                                              \nonumber \\
   & &  \varepsilon_{xy}^0= b_xx(x^2-y^2)/r^4,\;\;\; 
        \varepsilon^0_{xz}=-b_z(1-\nu )y/r^2,\;\;\;
        \varepsilon^0_{yz}= b_z(1-\nu )x/r^2,                           % (10)
\end{eqnarray}
and the elastic stress field $\sigma^0_{ij}$ is (in units of 
$\mu /[2\pi (1-\nu )]$) [1,\ 5]
\begin{eqnarray}
   & & \sigma^0_{xx} = \varepsilon^0_{xx}(\nu =0),\;\;\;\;
       \sigma^0_{yy} = \varepsilon^0_{yy}(\nu =0),\;\;\;\;
       \sigma^0_{zz} = \nu (\sigma^0_{xx}+\sigma^0_{yy}), 
                                                              \nonumber   \\  
   & & \sigma^0_{xy} = \varepsilon^0_{xy},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,
       \sigma^0_{xz} = \varepsilon^0_{xz},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,
       \sigma^0_{yz} = \varepsilon^0_{yz},                             % (11)
\end{eqnarray}
Fields (9) ($y$-component), (10) and (11) are singular at the dislocation 
line. 
\par
The elastic energy $W^0$ of the dislocation per unit dislocation length is
[1]
\begin{equation}
     W^0 = \frac{\mu}{4\pi}\left( b^2_z+\frac{b^2_x}{1-\nu }\right)
                                   \ln\frac{R}{r_0}\;,                %  (12)
\end{equation}
where $R$ denotes the size of the solid and $r_0$ is a cut-off radius for the 
dislocation elastic field near the dislocation line. When $r_0\to 0$, $W^0$ 
becomes singular.
                                                        \par\bigskip\noindent
{\underline{\it Gradient solution}}
                                                        \par\medskip\noindent
Let us now consider the corresponding dislocation fields within the theory of 
gradient elasticity given by (5). As described in {\it Section~2.2},
one can obtain the solution of (5) by solving separately equations (6)--(8). 
They can be solved [58,\ 60--66] by using the Fourier transform method. 
Omitting intermediate calculations, we give here only the final results. For 
the total displacements, solution of (6) gives [59,\ 60--62]
\begin{eqnarray}
     \mbox{\boldmath $u$} &=& \mbox{\boldmath $u$}^0
          - \frac{b_x}{4\pi (1-\nu )}
            \left\{ [  \mbox{\boldmath $e$}_x\,2xy 
                     + \mbox{\boldmath $e$}_y\,(y^2-x^2) ]\,r^2\Phi_2
                     + \mbox{\boldmath $e$}_y\Phi_0 \right\}  
                                                           \nonumber \\
      & &\phantom{\mbox{\boldmath $u$}^0}    
          + \frac{b_x\mbox{\boldmath $e$}_x+b_z\mbox{\boldmath $e$}_z}{2\pi }
               \,{\rm sign}(y)
                 \int\limits_{0}^{+\infty}\frac{s\sin (sx)}{\frac{1}{c_2}+s^2}
                       e^{-|y|\sqrt{\frac{1}{c_2}+s^2}}ds\;,            % (13)
\end{eqnarray} 
where $\mbox{\boldmath $u$}^0$ is given by (9), 
$\Phi_0=(1-2\nu )\,K_0(r/\sqrt{c_2})$, 
$\Phi_2=[2c_2/r^2-K_2(r/\sqrt{c_2})]/r^4$, $K_n(r/\sqrt{c_2})$ is the 
modified Bessel function of the second kind and $n=0,1,\dots $ denotes the 
order of this function. 
\par
For the elastic strain, solution of (7) gives [59,\ 60--62] 
$\varepsilon_{ij}=\varepsilon_{ij}^0+\varepsilon _{ij}^{gr}$, where
$\varepsilon_{ij}^0$ are given by (10) and $\varepsilon_{ij}^{gr}$ (in units 
of $1/[2\pi (1-\nu )]$) by
\begin{eqnarray}
   & & \varepsilon_{xx}^{gr} =\phantom{-}
           b_xy\,[(y^2-\nu r^2)\Phi_1 + (3x^2-y^2)\Phi_2],\;\;\;\;
       \varepsilon_{xz}^{gr} =\phantom{-} b_z\,(1-\nu )yr^2\Phi_1/2\;,    
                                                          \nonumber \\
   & & \varepsilon_{yy}^{gr} =\phantom{-}
           b_xy\,[(x^2-\nu r^2)\Phi_1 - (3x^2-y^2)\Phi_2],\;\;\;\;
       \varepsilon_{yz}^{gr} =-\,b_z\,(1-\nu )xr^2\Phi_1/2\;,   
                                                          \nonumber \\
   & & \varepsilon_{xy}^{gr} =
          -b_xx\,[y^2\Phi_1 + (x^2-3y^2)\Phi_2],                      % (14)
\end{eqnarray}
where $\Phi_1=K_1(r/\sqrt{c_2})/(\sqrt{c_2}r^3)$.
For the stresses, the solution of (8) gives [61,\ 62]
$\sigma_{ij}=\sigma_{ij}^0+\sigma_{ij}^{gr}$, where $\sigma_{ij}^0$ are given 
by (11) and $\sigma_{ij}^{gr}$ (in units of $\mu /[\pi (1-\nu )]$) by
\begin{eqnarray}
  & & \sigma_{xx}^{gr} = 
               \varepsilon_{xx}^{gr}(\nu =0,c_2\leftrightarrow c_1),\;\;\;
   \sigma_{yy}^{gr} = \varepsilon_{yy}^{gr}(\nu =0,c_2\leftrightarrow c_1),
                                                                      \;\;\;
   \sigma_{zz}^{gr} = \nu (\sigma_{xx}^{gr}+\sigma_{yy}^{gr}),
                                                           \nonumber \\
  & &  \sigma_{xy}^{gr} = \varepsilon_{xy}^{gr}(c_2\leftrightarrow c_1),\;\;\;
       \sigma_{xz}^{gr} = \varepsilon_{xz}^{gr}(c_2\leftrightarrow c_1),\;\;\;
       \sigma_{yz}^{gr} = \varepsilon_{yz}^{gr}(c_2\leftrightarrow c_1). 
                                                                       % (15)
\end{eqnarray}
\par
The main feature of the solution given by (13)--(15) is the absence of any 
singularities in the displacement, strain and stress fields. In fact, when 
$r\to 0$, we have 
$K_0(r/\sqrt{c_k})|_{r\to 0}\to -\gamma + \ln\,(2\sqrt{c_k}/r)$, 
$K_1(r/{\sqrt{c_k}})\to\sqrt{c_k}/r$, 
$K_2(r/{\sqrt{c_k}})\to 2c_k/r^2-1/2$, ($k=1,2$) and, thus, $u_y$ is
finite, $\varepsilon_{ij}\to 0$, $\sigma_{ij}\to 0$. 
The fields of displacements (13) and strains (14) have been analysed in 
detail in [59,\ 60] within a special version of gradient elasticity theory 
($c_1\equiv 0$). The stress fields (15) have been examined in [61]. 
We discuss below the main features of (13)--(15) separately for screw
({\it Section~3.1.2}) and edge ({\it Section~3.1.3}) dislocations.
\par
Using (15), one can find the elastic energy of the dislocation within the 
gradient elasticity given by (5) as the work 
$W_s=-\frac{1}{2}\int\limits_V\sigma_{zy}\beta_{yz}^{*(cl)}\,dV$
(for screw dislocations) and 
$W_e=-\frac{1}{2}\int\limits_V\sigma_{xy}\beta_{yx}^{*(cl)}\,dV$
(for edge dislocations) done by the gradient-dependent dislocation stress
field (15) for producing the corresponding classical (for simplicity)
plastic distortion $\beta_{yi}^{*(cl)}=(b_i/2)\delta (y)[1-{\rm sign}\,(x)]$,
$i=x,z$ [5]. The final result reads [61]
\begin{equation}
     W = \frac{\mu }{4\pi (1-\nu ) }
    \left\{ \frac{b_x^2}{2} + \left[ b_x^2+(1-\nu )b_z^2 \right]
            \left( \gamma + \ln\frac{R}{2\sqrt{c_1}} \right)\right\},  % (16)
\end{equation}
where $\gamma =0.57721566\ldots $ is Euler's constant. Thus, we obtain a 
strain energy expression which is not singular at the dislocation line.
\par
It is worth noting [58] that the energy expression (16) contains only one 
gradient coefficient $c_1$ that looks strangely. A more general consideration 
would include also the corresponding gradient expressions for the plastic 
distortions (e.g.
$\beta_{yz}^{*(gr)}=(b_z/2)\delta (y)[1-{\rm sign}\,(x)]\,\left( 1 - 
e^{-|x|/\sqrt{c_2}}\right)$ given in [59] for screw dislocation). The 
resultant energy expressions 
$W_s=-\frac{1}{2}\int\limits_V\sigma_{zy}\beta_{yz}^{*(gr)}\,dV$ (screw) and 
$W_e=-\frac{1}{2}\int\limits_V\sigma_{xy}\beta_{yx}^{*(gr)}\,dV$ (edge) would
contain now both gradient coefficients $c_1$ and $c_2$.
                                                \par\bigskip\noindent
{\bf 3.1.2. Screw dislocations}
                                                \par\bigskip\noindent
Consider a screw dislocation whose line coincides with the $z$-axis of a 
Carthesian coordinate system (Fig.~1) and discuss briefly the main most 
interesting features [59,\ 61] of the gradient solution for dislocation fields
(13)--(15) which distinguish it from the well-known classical solution 
(9)--(11).
\par
First, let us compare the behavior of total displacements (9) and (13) on the
``cutting'' plane $y=0$. In the case of the screw dislocation (Fig.~1), the
displacement vector has the only non-vanishing component 
$w_z=b_z/(2\pi)\,w(x,y)$. When $y\rightarrow 0$, the integral in (13) may be 
evaluated in an explicit form [59] that gives 
\begin{equation}
     w(x,y\rightarrow 0)=\frac{\pi }{2}\,{\rm sign}\,(y)   
                         \left[1-{\rm sign}\,(x)
              \left(1-e^{-\frac{|x|}{\sqrt{c_2}}}\right)\right].       % (17)
\end{equation}
The additional term ``$-e^{-\frac{|x|}{\sqrt{c_2}}}$'' which appears in the 
gradient solution (17) leads to the smoothing of the total displacement 
profiles (curves $1'_{+}$ and $1'_{-}$ in Figs.~1 and 2), in contrast to the 
abrupt jumps occuring in these profiles in the classical solution 
(curves $1_{+}$ and $1_{-}$ in Figs.~1 and 2). It is interesting to note that 
the size of such a transition zone is approximately $10\sqrt{c_2}$ which gives
the value $2.5a$ for a crystalline lattice, i.e.~the usual size of the 
dislocation core. It follows that in gradient elasticity, the dislocation core
appears naturally as a result of a calculation defining the region of maximum 
strain around the dislocation line [59]. In classical elasticity, there is no 
definition for the dislocation core and the dislocation is treated as a linear
singularity of elastic stress and strain fields. 
\par
Second, it is interesting to compare the displacement field (17) with
the corresponding field derived within a model accounting for the 
atomic structure of a solid body [58]. In particular, we consider the 
Peierls-Nabarro dislocation model [1] which in our case gives the following 
expression for the total displacement $w_z^{(PN)}$ at $y\to 0$ 
\begin{equation}
     w_z^{(PN)}(x,y\to 0) 
               = \frac{b_z}{2\pi }\,{\rm sign}\,(y)
               \left(\frac{\pi }{2}-\arctan\frac{x}{a/2}\right),       % (18)
\end{equation}
where $a$ is the lattice constant. Using the estimate $\sqrt{c_2}\approx a/4$ 
[48,\ 50], we depict graphically (17) and (18) in Fig.~3. One can conclude 
that the gradient model gives much narrower dislocation core (about $2a$) 
than the Peierls-Nabarro model.
\par
Let us consider now a gradient solution for a dipole of screw dislocations 
[59]. As usually, such a solution may be found simply by a superposition of 
solutions for separate dislocations. The only reason for 
discussing this case here is due to a new interesting effect concerning the 
behavior of the total displacement of a dipole. Consider two parallel screw 
dislocations lying in the plane $y=0$ along the $z$-axis and crossing the 
$x$-axis at the points $x=-d$ and $x=0$. Let us assume the same Burgers 
vector $b_z$ and opposite tangent vectors $\pm l_z$. This means that they can
be treated as a limiting case of a rectangular gliding dislocation loop having
edge segments at $z=\pm\infty $. In the framework of classical dislocation 
theory, the field of total displacement for such a dislocation configuration 
is described by the vector $u_z=b_z/(2\pi )u(x,y)$, where
\begin{equation}
     u(x,y)=\arctan\frac{y}{x}-\arctan\frac{y}{x+d}+
     \frac{\pi}{2}\,{\rm sign}\,(y)\,[{\rm sign}\,(x+d)-{\rm sign}\,(x)]. 
                                                                        % (19)
\end{equation}
The gradient solution is described by the vector $w_z=b_z/(2\pi )w(x,y)$, 
where
\begin{equation}
     w(x,y)=u(x,y)-{\rm sign}\,(y)\,\frac{i}{2}
            \int\limits_{-\infty}^{+\infty}\left(1-e^{ids}\right)
            \frac{s}{\frac{1}{c_2}+s^2}\, 
            e^{-|y|\sqrt{\frac{1}{c_2}+s^2}}e^{isx}d\,s,               %  (20)
\end{equation}
and $u(x,y)$ is given by (19).
\par
When $y\rightarrow 0$,
\begin{equation}
     w(x,y\rightarrow 0)=\frac{\pi}{2}\,{\rm sign}\,(y)
          \left[{\rm sign}\,(x+d)\left(1-e^{-\frac{|x+d|}{\sqrt{c_2}}}\right)-
       {\rm sign}\,(x)\left(1-e^{-\frac{|x|}{\sqrt{c_2}}}\right)\right]. 
                                                                       %  (21)
\end{equation}
The graphs for $u(x,y\rightarrow 0)$ and $w(x,y\rightarrow 0)$ are presented 
in Fig.~2. Besides the evident difference in the form of these profiles, it is
important to note that the maximum value of the total displacement depend on 
the dipole arm $d$ for the gradient solution $w$, in contrast to the classical
solution $u$ which is independent of $d$. This dependence disappears when 
$d\ge d_0\approx 10\sqrt{c_2}$ where $d_0$ defines a new characteristic 
distance; namely, the radius of the ``strong short-range interaction'' between
dislocations. This gives $d_0\approx 2a$, a result consistent with intuition.
\par
As it has been pointed out in {\it Section~3.1.1}, the main feature of the 
gradient solution given by (14) and (15) is the absence of any singularities 
in both the strain and stress fields (previous models eliminated eather the 
strain or stress singularity but not both, see {\it Section~1}). It is 
interesting to note that the stress components $\sigma_{xz}$ and $\sigma_{yz}$
given by the superpositions of corresponding components in (11) and (15), 
are exactly the same as the ones obtained by Eringen [39--41] for the stress 
field of a screw dislocation within his version of non-local elastisity.
To illustrate the characteristic features of the gradient solution,  
the spatial distribution of elastic strains $\varepsilon_{iz}$ and stresses 
$\sigma_{iz}$\ \ ($i=x,y$) near the dislocation line [58,\ 61] is presented 
in Fig.~4. One can see that the gradient solutions for the elastic strains
and stresses attain their extreme values of approximately 
$\pm b_z/(10\pi\sqrt{c_2})$ and $\pm\mu b_z/(5\pi\sqrt{c_1})$ at a distance
$\approx\sqrt{c_2}$ and $\sqrt{c_1}$ from the dislocation line, respectively.
Using for the gradient coefficients $c_1$ and $c_2$ the estimate [48,\ 50] 
$\sqrt{c_1}\approx\sqrt{c_2}\approx a/4$ for a crystalline lattice, we find 
that ${\rm max}\left|\varepsilon_{iz}\right|\approx 12\%$ [59] and 
${\rm max}\left|\sigma_{iz}\right|\approx \mu /4$ [61] at a distance 
$\approx a/4$ from the dislocation line. It is also seen that the gradient 
solutions coincide with the classical ones far away from the dislocation core 
($r\ge r_0\approx 4\sqrt{c_k}$) [59]. 
\par
The elimination of singularity from the strain and stress fields permits us to
consider in detail the short-range nanoscale interaction between dislocations.
It has been shown above that such short-range interaction takes place when the
spacing $d$ between dislocations is smaller than 
$\approx 10\sqrt{c_2}\approx 2.5a$. Here we consider two simple dislocation 
configurations: a dipole and a pair of screw dislocations (Fig.~5) [58]. The 
elastic fields of these configurations are obtained as simple superpositions 
of the corresponding fields for individual dislocations. Fig.~6 shows the 
distribution of the strain and stress components, $\varepsilon_{yz}(x,0)$ and
$\sigma_{yz}(x,0)$, near the dislocation dipole (Fig.~6{\it a,b})
and dislocation pair (Fig.~6{\it c,d}). One can see that the strain and stress
are finite at the dislocation lines and tend to zero there when the magnitude 
of interdislocation spacing $d$ increases. Between dislocations or near them,
where the classical solutions (dashed curves) give unreasonably high strain 
and stress values, the gradient solutions (solid curves) give quite reasonable
values, $\varepsilon_{yz}\leq 25\%$ and $\sigma_{yz}\leq\mu /2$. In the case 
of dislocation dipoles, the strain and stress at the central point between
the dislocations do not exceed these levels and tend to zero when $d\to 0$ 
(Fig.~7). In the case of a dislocation pair, the levels of strain and stress
decrease between the dislocation lines as they glide to each other and remain
finite all the way up to the point where the two dislocations come together 
at the same line. We can conclude [58] that the short-range elastic 
interaction between dislocations is much smaller than that predicted from 
classical elasticity. This means that there is no significant energetic 
barriers for the processes of elementary nucleation of dislocation dipoles or
formation of superdislocations in high-density dislocation ensembles where 
the interdislocation spacing is of the order of a few nanometers.
                                                \par\bigskip\noindent
{\bf 3.1.3. Edge dislocations }
                                                \par\bigskip\noindent
Consider an edge dislocation whose line coinsides with the $z$-axis while the 
Burgers vector $\mbox{\boldmath $b$}=b_x\mbox{\boldmath $e$}_x$ is parallel 
to the $x$-axis of the Carthesian coordinate system. The most interesting 
question here is the behavior of the total displacement and elastic strain 
and stress near the dislocation line [60,\ 61]. 
\par
First, consider the components $u_x$ and $u_y$ of the total displacement 
given by (13), on the ``cutting plane'' $y=0$. When  $y\rightarrow 0$, we 
have [60]
\begin{equation}
     u_x(x,y\rightarrow 0)=\frac{b_x}{4}\,{\rm sign}\,(y)\,
                         \left[1-{\rm sign}\,(x)\,
               \left(1-e^{-\frac{|x|}{\sqrt{c_2}}}\right)\right],       % (22)
\end{equation}
\begin{equation}
    u_y(x,y\rightarrow 0)=\frac{b_x}{4\pi (1-\nu )}\left[-1-(1-2\nu )\ln |x| +
                           \frac{2c_2}{x^2}-
                         (1-2\nu )K_0\left(\frac{|x|}{\sqrt{c_2}}\right)-
               K_2\left(\frac{|x|}{\sqrt{c_2}}\right)\right].           % (23)
\end{equation}
Expression (22) coincides with equation (17) which 
describes the total displacement of a screw dislocation at $y=0$. The 
term ``$-e^{-\frac{|x|}{\sqrt{c_2}}}$'' which appears in the gradient 
solution (22) leads to the smoothing of the total displacement profile in 
contrast to the abrupt jump occuring in this profile in the classical 
solution (see Figs.~1 and 2). This means that in gradient elasticity, 
the dislocation core appears again naturally, directly from the calculations
as we have discussed this above for the screw dislocation. 
\par
Expression (23) contains the terms 
which are singular at the dislocation line. However, they compensate each 
other. To demonstrate this correctly, let us consider here the field of total
displacements created by a dipole of edge dislocations, thus avoiding the
logarithm of dimensional quantity. 
\par
Let two parallel edge dislocations lie in the plane $y=0$ along the $z$-axis 
and cross the $x$-axis at the points $x=-d$ and $x=0$. Let us assume the same
Burgers vector $b_x$ and opposite tangent vectors $\pm l_z$. This means that 
they can be treated as a limiting case of a rectangular gliding dislocation 
loop having screw segments at $z=\pm\infty $. Both classical and gradient 
solutions may be obtained by a simple superposition of the solutions for a 
separate dislocation. In particular, for $y\rightarrow 0$ we have [60]
\begin{eqnarray}
     u_x|_{y\rightarrow 0} &=& \frac{b_x}{2}\,{\rm sign}\,(y)\,
       \left[{\rm sign}\,(x+d)\,\left(1-e^{-\frac{|x+d|}{\sqrt{c_2}}}\right)-
     {\rm sign}\,(x)\,\left(1-e^{-\frac{|x|}{\sqrt{c_2}}}\right)\right], \\
                                                                        % (24)
     u_y|_{y\rightarrow 0} &=& \frac{b_x}{4\pi (1-\nu )}\;          
          \left[-(1-2\nu )\left\{\ln \frac{|x|}{|x+d|}+
           K_0\left(\frac{|x|}{\sqrt{c_2}}\right)-
           K_0\left(\frac{|x+d|}{\sqrt{c_2}}\right)\right\}\right.\nonumber \\
         & & \phantom{\frac{b_x}{4\pi (1-\nu )}\;}+\left.\frac{2c_2}{x^2}-
                                                   \frac{2c_2}{(x+d)^2}-
                K_2\left(\frac{|x|}{\sqrt{c_2}}\right)+
                K_2\left(\frac{|x+d|}{\sqrt{c_2}}\right)\right].        % (25)
\end{eqnarray}
The component $u_x(y\rightarrow 0)$ given by (24), coincides with the 
component $w_z(y\rightarrow 0)$ of a similar dipole of screw dislocations 
(see formula (21) and Figs.~1 and 2). From this analysis, we have concluded 
that in gradient elasticity a new characteristic distance appears, namely, the
radius of strong short-range interaction between dislocations. It follows
from the identity of expressions (21) and (24) that this conclusion is also 
valid for the case of edge dislocations [60].
\par
The component $u_y(y\rightarrow 0)$ given by (25) is regular everywhere 
including the dislocation lines. In fact, for $x\rightarrow 0$ we have
$K_0(|x|/\sqrt{c_2})|_{x\rightarrow 0}\rightarrow 
                               - \gamma - \ln\frac{|x|}{2\sqrt{c_2}}$ and 
$K_2(|x|/\sqrt{c_2})|_{x\rightarrow 0}\rightarrow 2c_2/x^2 - 1/2$,
where $\gamma =0.57721566\ldots $ is Euler's constant. Thus, 
\begin{equation}
     u_y(x=0,y=0)=\frac{b_x}{4\pi (1-\nu )}
          \left[(1-2\nu )\left\{\gamma + \ln\frac{d}{2\sqrt{c_2}} +
                K_0\left(\frac{d}{\sqrt{c_2}}\right)\right\} +
                    \frac{1}{2} - \frac{2c_2}{d^2} +
                K_2\left(\frac{d}{\sqrt{c_2}}\right)\right].            % (26)
\end{equation}  
It is seen from (26) that $u_y(0,0)$ increases with the dipole arm $d$ 
and equals to zero when $d=0$. The dependence of $u_y(x,y=0)$ on 
$d$ is presented in Fig.~8 which demonstrates the fact that the gradient 
solution is especially effective for describing the strong short-range 
interaction between dislocations. 
\par
The plots of Fig.~8{\it c} have been obtained numerically in [60] for a 
dislocation dipole with arm $d=100\sqrt{c_2}$ which may thus approximate the 
field of a separate dislocation. The gradient solution gives smaller 
displacements near the dislocation line 
($-0.3\sqrt{c_2}\leq x\leq 0.3\sqrt{c_2}$) and larger ones outside of this
region. With these plots, we have a possibility to compare the calculated 
value of the displacement $u_y(0,0)$ at the dislocation line with data from 
experimental observations and computer simulations. In considering $u_y$ for 
a dislocation dipole with arm $d=100\sqrt{c_2}$, we have the estimate 
$u_y(0,0)\approx 2.3\,b_x/[4\pi (1-\nu )]$ which gives $\approx 0.26\,a$ for
$b_x=a$ and $\nu =0.3$. This value is in a good agreement with the 
results of direct observation of edge dislocations near an Nb/Al$_2$O$_3$ 
interface [67], as well as with the results of computer simulations for the
core of an edge dislocation in $\alpha $-Fe [68]. It is worth noting that
this estimate can not be obtained within a Peierls-Nabarro dislocation model
[1] because the latter imposes that the $u_y$ displacement is identically
zero.
\par
Let us now consider the behavior of the elastic strain and stress components 
given by superpositions of formulae (10)-(11) with (14)-(15) near the 
dislocation line. The strain expessions (10) with (14) have been obtained in 
[60]. The stress expressions (11) with (15) reported in [61,\ 62], 
are represented in a closed form in contrast to the ones obtained in an 
integral form by Eringen [41] for the stress field of an edge dislocation 
within his version of non-local elastisity. The main feature
of the solution given by (10)-(11) with (14)-(15) is the absence of any 
singularities in the strain and stress fields (previous models eliminated 
eather the displacement and strain singularity or the stress singularity but 
not both). One can note some interesting details in the behavior of the
gradient solution [60]. 
\par
From Fig.~9, one can conclude that the component $\varepsilon _{xx}(x=0)$ is 
smaller than $\varepsilon _{xx}^0(x=0)$ for $0\leq |y|\leq 1.3\sqrt{c_2}$ and 
larger than $\varepsilon _{xx}^0(x=0)$ for $|y|\geq 1.3\sqrt{c_2}$. It 
achieves a maximum value $\approx 0.3$ (all strain values are given here in 
units of $b_x/[4\pi (1-\nu )\sqrt{c_2}]$) at $|y|\approx\sqrt{c_2}$ which 
gives $\approx $14\% for an atomic lattice. The component 
$\varepsilon _{yy}(x=0)$ is smaller than  $\varepsilon _{yy}^0(x=0)$ 
everywhere, it is equal to zero at $|y|\approx 0.6\sqrt{c}$ and achieves two 
extrema values of opposite signs (on the same side of the dislocation line) 
which are equal to $\approx 0.01$ ($\approx $0.5\%) at 
$\approx 0.2\sqrt{c_2}$, and $\approx 0.06$ ($\approx $3\%) at 
$\approx 4\sqrt{c_2}$. It is interesting to note that $\varepsilon _{yy}(x=0)$
is significantly smaller than $\varepsilon _{xx}(x=0)$, in contrast to the 
classical solution where 
$\varepsilon _{yy}^0(x=0)\equiv\varepsilon _{xx}^0(x=0)$. The behavior of the 
dilatation $\varepsilon =\varepsilon_{xx} + \varepsilon_{yy}$ is similar to 
the behavior of the strain component 
$\varepsilon _{xz}$ of a screw dislocation (see {\it Section~3.1.2}). The 
dilatation has maximum value of about $0.3$ ($\approx $14\%) at 
$|y|\approx\sqrt{c_2}$, it is equal to zero at the dislocation line and 
practically coincides with the classical solution outside of the dislocation 
core ($|y|>4\sqrt{c_2}$). The shear component $\varepsilon _{xy}$ is smaller 
than  $\varepsilon _{xy}^0$ everywhere, it approaches zero at the dislocation
line, achieves maximum values of about $0.25$ ($\approx $12\%) at 
$|x|\approx 1.5\sqrt{c_2}$ and practically coincides with  
$\varepsilon _{xy}^0$ far from the dislocation line ($|x|>10\sqrt{c_2}$).   
\par
A general (three-dimensional) view of the spatial distribution of elastic
strains near the dislocation line [58] is provided in Fig.~10
which demonstrates all aforementioned features of gradient solution for the
strain field.
\par
Similar spatial distribution of stress components $\sigma_{xx}$ and 
$\sigma_{yy}$ near the dislocation line [61] are represented in Fig.~11. 
The distribution of shear stress component $\sigma_{xy}$ is the same as that
of shear strain component $\varepsilon_{xy}$ (Fig.~10{\it c}) with appropriate
replacement of strain units of $b_x/[4\pi (1-\nu )\sqrt{c_2}]$ with stress 
units of $\mu b_x/[2\pi (1-\nu )\sqrt{c_1}]$ and coordinate units 
$(x,y)/\sqrt{c_2}$ with $(x,y)/\sqrt{c_1}$. Also, the distribution of 
hydrostatic stress component $\sigma =(\sigma_{xx}+\sigma_{yy}+\sigma_{zz})/3$
is the same as that of dilatation $\varepsilon $ (Fig.~10{\it d}) with 
similar replacement of units ($b_x/[4\pi (1-\nu )\sqrt{c_2}]$ with 
$\mu b_x(1+\nu )/[6\pi (1-\nu )(1-2\nu )\sqrt{c_1}]$, and $(x,y)/\sqrt{c_2}$ 
with $(x,y)/\sqrt{c_1}$). One can see that the gradient solutions for the 
stresses attain their extreme values ($|\sigma_{xx}|\approx 0.45\,\mu $ and  
$|\sigma_{yy}|\approx |\sigma_{xy}|\approx 0.27\,\mu $ for $b_x=a=4\sqrt{c_1}$
and $\nu =0.3$) at a distance $\approx a/4$ from the dislocation line.
It is also seen that the gradient solutions coincide with the classical ones 
far away from the dislocation core ($r\ge r_0\approx 4\sqrt{c_k}$) [60]. 
                                                \par\bigskip
Thus, in considering the straight dislocations in a homogeneous elastic 
isotropic solid within the gradient theory of elasticity described by (5), 
one can summarize that new gradient solution gives non-singular expressions
for dislocation displacement, strain and stress fields and elastic energies. 
It has been shown that for individual dislocations, 
the elastic strains and stresses are strictly equal to
zero at the dislocation lines and achieve their extreme values of 
$\approx (3\div 14)$\% and $\approx (\mu /4\div\mu /2)$, respectively, at a 
distance $\approx a/4$ from the dislocation line. Two characteristic 
distances appear naturally in this approach: $r_0\approx 4\sqrt{c_2}$ which 
may be viewed as the radius of dislocation core and $d_0\approx 10\sqrt{c_2}$
which may be viewed as the radius of strong short-range interaction between 
dislocations. 
                                                \par\bigskip\bigskip\noindent
{\large\bf 3.2. Straight dislocations near interfaces}
                                                \par\bigskip\noindent
A description of the elastic interaction of dislocations with interphase 
boundaries has been one of the key problems in the theory of defects, with 
applications to materials science and engineering and special
attention to polycrystalline, multilayered and thin-film solid systems
(e.g.~[1,\ 11,\ 12,\ 69--73]). This description is traditionally based on 
solutions of appropriate boundary-value problems in the classical linear 
theory of elasticity. The corresponding solutions provide the elastic fields 
of dislocations far from both the interface and the dislocation line, thus 
being satisfactory for the cases
when long-range elastic interactions are of interest. However, when 
short-range interactions are of interest, the classical solutions lead to
unreasonable results. These concern the elastic singularity at the dislocation
line, as well as the ``image'' force which acts on dislocations from the side
of an interface which also becomes singular when the dislocation approaches
the interface. Moreover, some components of the elastic stress field of a 
dislocation suffer jumps at the interface, a fact which may be acceptable from
a macroscopic point of view but physically unrealistic from a nano- or 
microscopic point of view. To avoid the aforementioned three difficulties, 
the boundary-value problem of a straight dislocation near a flat interface 
has been reconsidered [64--66] within the theory of gradient elasticity 
described by (5).
                                                \par\bigskip\noindent
{\bf 3.2.1. A general solution }
                                                \par\bigskip\noindent
Consider a flat interface which separates two elastic isotropic media
denoted by 1 ($x>0$) and 2 ($x<0$) with shear moduli $\mu_i$, Poisson ratios
$\nu_i$, and gradient coefficients $c_{1_i}$ and $c_{2_i}$, where $i=1,2$, 
respectively (Fig.~12). Let a straight dislocation having Burgers vector 
$\mbox{\boldmath $b$}=b_x\mbox{\boldmath $e$}_x + b_y\mbox{\boldmath $e$}_y
+ b_z\mbox{\boldmath $e$}_z$
goes through the point $(x=x',y=0)$ along the $z$-axis of a Cartesian 
coordinate system.
                                                \par\newpage\noindent
{\underline{\it Classical solution}}
                                                \par\medskip\noindent
In the framework of classical elasticity theory (where 
$c_{1_i}=c_{2_i}\equiv 0$), for $x'\ge 0$, the dislocation stress field is 
given [69] (in units of $\mu_1/[\pi (k_1+1)]$) in the medium~1 by
\begin{eqnarray}
   & & \sigma^{0(1)}_{xx} =
        b_x \left\{ - \,\frac{2y}{r^2_{-}} - \frac{4yx_{-}^2}{r^4_{-}} 
          + \frac{(A+B)y}{r^2_{+}} + \frac{4Ay(x'^2+x^2)}{r^4_{+}}  
          + \frac{32Ayxx'x_{+}^2}{r^6_{+}} \right\} \nonumber\\   
                                                                       % (A1b)
     & &\phantom{\sigma } +
        b_y \left\{ - \,\frac{2x_{-}}{r^2_{-}} + \frac{4x_{-}^3}{r^4_{-}} 
          + \frac{(A+B)x-(3A-B)x'}{r^2_{+}} 
          + \frac{4Ax_{+}(x'^2-6x'x-x^2)}{r^4_{+}} 
          + \frac{32Axx'x_{+}^3}{r^6_{+}} \right\},  \\               
                                                                  % (6)-> (27)
                                                                       % (A1c)
   & & \sigma^{0(1)}_{yy} =
        b_x \left\{ - \,\frac{2y}{r^2_{-}} + \frac{4yx_{-}^2}{r^4_{-}} 
          + \frac{(3A-B)y}{r^2_{+}}  
          + \frac{4Ay(3x'^2+4x'x-x^2)}{r^4_{+}}
          - \frac{32Ayxx'x_{+}^2}{r^6_{+}} \right\}  \nonumber \\
                                                                       % (A1d)
     & &\phantom{\sigma } +
        b_y \left\{ \frac{6x_{-}}{r^2_{-}} - \frac{4x_{-}^3}{r^4_{-}} 
          - \frac{(5A+B)x+(9A+B)x'}{r^2_{+}} 
          + \frac{4Ax_{+}(3x'^2+10x'x+x^2)}{r^4_{+}}  
          - \frac{32Axx'x_{+}^3}{r^6_{+}} \right\},  \\
                                                                  % (7)-> (28)
                                                                       % (A1e)
   & & \sigma^{0(1)}_{xy} =
        b_x \left\{ - \,\frac{2x_{-}}{r^2_{-}} + \frac{4x_{-}^3}{r^4_{-}} 
          + \frac{(3A-B)x-(A+B)x'}{r^2_{+}}
          + \frac{4Ax_{+}(x'^2+6x'x-x^2)}{r^4_{+}} 
          - \frac{32Axx'x_{+}^3}{r^6_{+}} \right\}  \nonumber\\   
                                                                       % (A1f)
     & &\phantom{\sigma^{0(1)}_{xy}}\, +
        b_y \left\{ - \,\frac{2y}{r^2_{-}} + \frac{4yx_{-}^2}{r^4_{-}} 
          + \frac{(A+B)y}{r^2_{+}} - \frac{4Ay(x'^2+4x'x+x^2)}{r^4_{+}}
          + \frac{32Ayxx'x_{+}^2}{r^6_{+}} \right\}, \\                 
                                                                  % (8)-> (29)
                                                                       % (A1g)
   & & \sigma^{0(1)}_{zz} = \nu_1\,(\sigma^{0(1)}_{xx} + \sigma^{0(1)}_{yy}),
                                                     \\           % (9)-> (30)
   & & \sigma^{0(1)}_{xz} = 
        b_z\,\frac{k_1+1}{2}\left\{ -\,\frac{y}{r^2_{-}}
          + \frac{\Gamma -1}{\Gamma +1}\frac{y}{r^2_{+}}\right\}, \\
                                                                  % (3)-> (31)
   & & \sigma^{0(1)}_{yz} =   
        b_z\,\frac{k_1+1}{2}\left\{\;\;\;\frac{x_{-}}{r^2_{-}}
          + \frac{\Gamma -1}{\Gamma +1}\frac{x_{+}}{r^2_{+}}\right\}, 
                                                                  % (3)-> (32)
\end{eqnarray}
and in the medium~2 by 
\begin{eqnarray}
   & & \sigma^{0(2)}_{xx} =
        b_x \left\{ \frac{(A+B-2)y}{r^2_{-}} 
               - \frac{4yx_{-}[(1-B)x-(1-A)x']}{r^4_{-}} \right\}\nonumber \\
                                                                       % (A1h)
   & & \phantom{\sigma^{0(2)}_{xx}}\, +
        b_y \left\{ \frac{(A+B-2)x+(B-3A+2)x'}{r^2_{-}} 
          + \frac{4x_{-}^2[(1-B)x-(1-A)x']}{r^4_{-}} \right\}, \\% (10)-> (33)
                                                                       % (A1i)
   & & \sigma^{0(2)}_{yy} =
        b_x \left\{ \frac{(3B-A-2)y}{r^2_{-}} 
               + \frac{4yx_{-}[(1-B)x-(1-A)x']}{r^4_{-}} \right\} \nonumber\\
                                                                       % (A1j)
   & & \phantom{\sigma^{0(2)}_{yy}}\, +
        b_y \left\{ \frac{(6-5B-A)x+3(B+A-2)x'}{r^2_{-}} 
          - \frac{4x_{-}^2[(1-B)x-(1-A)x']}{r^4_{-}} \right\}, \\% (11)-> (34)
                                                                       % (A1k)
   & & \sigma^{0(2)}_{xy} =
        b_x \left\{ \frac{(3B-A-2)x+(2-A-B)x'}{r^2_{-}}
          + \frac{4x_{-}^2[(1-B)x-(1-A)x']}{r^4_{-}} \right\}\nonumber \\
                                                                       % (A1l)
   & & \phantom{\sigma^{0(2)}_{xy}}\, +
        b_y \left\{ \frac{(A+B-2)y}{r^2_{-}} 
          + \frac{4yx_{-}[(1-B)x-(1-A)x']}{r^4_{-}} \right\}, \\ % (12)-> (35)
                                                                       % (A1m)
   & & \sigma^{0(2)}_{zz} = \nu_2\,(\sigma^{0(2)}_{xx} + \sigma^{0(2)}_{yy}),
                                                              \\ % (13)-> (36)
   & & \sigma^{0(2)}_{xz} = 
      - b_z\,\frac{\Gamma\,(k_1+1)}{\Gamma +1}\,\frac{y}{r^2_{-}}\;,
                                                              \\ % (4) -> (37)
   & & \sigma^{0(2)}_{yz} = \;\;\;   
        b_z\,\frac{\Gamma\,(k_1+1)}{\Gamma +1}\,\frac{x_{-}}{r^2_{-}}\;,
                                                                 % (4) -> (38)
\end{eqnarray}
where $x_{\pm }=x\pm x'$,\ \ 
$r^2_{\pm }=x_{\pm }^2+y^2$,\ \ $A=(1-\Gamma )/(1+k_1\Gamma )$,\ \ 
$B=(k_2-k_1\Gamma )/(k_2+\Gamma )$,\ \ $\Gamma = \mu_2/\mu_1$,\ \ 
$k_i=3-4\nu_i$,\ \ $i=1,2$.
\par
It is easily to see that the components $\sigma^{0}_{xx}$, $\sigma^{0}_{xy}$ 
and $\sigma^{0}_{xz}$ are continuous at the interface $(x=0)$ while the 
components $\sigma^{0}_{yy}$, $\sigma^{0}_{zz}$ and $\sigma^{0}_{yz}$
suffer jumps\ \  
$\left[\sigma^{0}_{kl}\right]=\sigma^{0(1)}_{kl}-\sigma^{0(2)}_{kl}$\ \  
there [64--66] which are (in units of $\mu_1/[\pi (1-\nu_1)]$)
\begin{eqnarray}
     \left[\sigma^{0}_{yy}\right]_{x=0} 
          &=& \frac{(A-B)\,b_xy-(3A+B)\,b_yx'}{x'^2+y^2}
           +  \frac{4A(b_xy+b_yx')\,x'^2}{(x'^2+y^2)^2}\;, 
                                                              \\ % (14)-> (39)
     \left[\sigma^{0}_{zz}\right]_{x=0} 
          &=& 
     \frac{\nu_1[(A-1)\,b_xy-(3A+1)\,b_yx']-\nu_2(B-1)(b_xy+b_yx')}{x'^2+y^2}
                                                               \nonumber \\
  &+& \frac{4\nu_1A(b_xy+b_yx')\,x'^2-2\nu_2(A-1)\,b_yx'^3}{(x'^2+y^2)^2}\;, 
                                                              \\ % (15)-> (40)
     \left[\sigma^{0}_{yz}\right]_{x=0} 
          &=& 
        b_z\,(1-\nu_1)\,\frac{\Gamma -1}{\Gamma +1}\,\frac{x'}{x'^2+y^2}\;. 
                                                                 % (5) -> (41)
\end{eqnarray}
Such jumps are expected from the macroscopic viewpoint of classical  
elasticity because these components do not give any contribution to the 
$x$-component of the elastic force which has to be in equilibrium at the 
interface. On the other hand, in considering the stressed state of an ideally
welded interface from a nanoscopic point of view, the nature of this jump is 
not quite clear. In fact, the atomic layers on both sides of the interface 
interact elastically not only with atoms of their own material but also
with atoms of the opposite material. Therefore, one has to assume the 
existence of a transitional zone of a few atomic layers where elastic 
interactions between atoms vary smoothly from stronger ones which are 
characteristic of one bulk material, to weaker ones which are characteristic 
of the other bulk material. It follows from this assumption, that stress
jumps like (39)--(41) is only a consequence of the approximation of classical 
continuum models which often become insufficient for describing nanoscale 
phenomena. To demonstrate this fact, we note that the stress jumps in 
(39)--(41) tend to infinity in the $xz$-plane when the dislocation approaches
the interface. It may thus be desirable for the interface stress jumps to be 
eliminated from the solution of this problem within any generalized theory of
elasticity aiming to consider nanoscale phenomena.
                                                \par\bigskip\noindent
{\underline{\it Gradient solution}}
                                                \par\medskip\noindent
Let us now consider the corresponding dislocation fields within the theory of 
gradient elasticity given by (5). As proposed in [55] and described also in 
[61--63] (see {\it Section~2.2}), one can obtain the solution of (5) by 
solving separately equations (7) and (8) for strain 
$\mbox{\boldmath $\varepsilon $}$ and stress $\mbox{\boldmath $\sigma $}$  
fields, respectively, in terms of the strain 
$\mbox{\boldmath $\varepsilon $}^0$ and stress $\mbox{\boldmath $\sigma$}^0$
fields of the classical elasticity theory for the same boundary-value problem.
Here we consider only the solution of equation (8) for stress field because 
it is especially important for applications.
\par
Equations (8) can be solved [61--66] by using the Fourier transform 
method. Rewrite the stress equation in the form
\begin{equation}
   \left( 1-c_{1_i}\nabla^2 \right)\mbox{\boldmath $\sigma $}^{(i)} = 
                      \mbox{\boldmath $\sigma $}^{0(i)}\;,       % (17)-> (42)
\end{equation}
where $\mbox{\boldmath $\sigma $}^{0(i)}$ are given by (27)--(38). Below we
will omit for simplicity the first index ``1'' at the gradient coefficient
$c_{1_i}$ in which case $c_1$ will refer to the material~1 and $c_2$ to 
the material~2. Basing on above notes to the classical solution as well as on 
conclusions of Ru and Aifantis [49,\ 55], we use [64--66] the classical 
boundary conditions
\begin{equation}
     \left[\sigma_{xl}\right]_{x=0} = 0\;,\;\;\;\;\; l=x,y,z\;,  % (18)-> (43)
\end{equation} 
and nine extra boundary conditions 
\begin{equation}
     \left[\sigma_{yy}\right]_{x=0} = 
     \left[\sigma_{zz}\right]_{x=0} = 
     \left[\sigma_{yz}\right]_{x=0} = 
     \left[\frac{\partial\sigma_{kl}}{\partial x}\right]_{x=0} = 0\;,
     \;\;\;\;\; k,l=x,y,z\;,                                     % (19)-> (44)
\end{equation} 
Last equation in (44) provide for smooth transitions of stress components 
through the interface.
\par
Omitting intermediate calculations, we give here only the final results. 
The gradient solution [64--66] reads \ \ 
$\sigma_{kl}^{(i)}=\sigma_{kl}^{0(i)}+\sigma_{kl}^{gr(i)}$,\ \ where
$\sigma_{kl}^{0(i)}$ are given by (27)--(38) and $\sigma_{kl}^{gr(i)}$ are
(in units of $\mu_1/[\pi (k_1+1)]$), for medium~1
\begin{eqnarray}
  \sigma_{xx}^{gr(1)} &=& 
     4b_x\left\{ y^3\Phi_1(r_{-}) + y(3x_{-}^2-y^2)\Phi_2(r_{-}) 
      -\frac{2Ac_1y}{r_{+}^6}
        \left[ 3x_{+}^2-y^2+24x'x_{+}\frac{x_{+}^2-y^2}{r_{+}^2}\right]
         \right.                                 \nonumber  \\           
     &+& \left.
        \int_0^{\infty }\frac{s^2\sin(sy)}{\lambda_1+\lambda_2}\,
        e^{-x\lambda_1}\left[ c_1s\frac{\lambda_2-\lambda_1}{\lambda_1}\,
        e^{-x'\lambda_1} 
       + \{Ac_1(\lambda_2+s)(1+2x's)-c'(\lambda_2-s)]\}\,e^{-x's} 
         \right]d\,s\right\}
                                                 \nonumber \\
     &+& 4b_y\left\{ - x_{-}y^2\Phi_1(r_{-}) 
                     - x_{-}(x_{-}^2-3y^2)\Phi_2(r_{-})+\frac{2Ac_1}{r_{+}^6}
         \left[ x_{+}(x_{+}^2-3y^2)
             - 6x'\,\frac{x_{+}^4-6x_{+}^2y^2+y^4}{r_{+}^2}\right]
         \right.                                 \nonumber \\
     &+& \int_0^{\infty }\frac{s^2\cos(sy)}{\lambda_1+\lambda_2}\,
         e^{-x\lambda_1}\left[ c_1(\lambda_2-\lambda_1)\,e^{-x'\lambda_1}
         \right.                                 \nonumber \\
& & \phantom{\int_0^{\infty }\frac{s^2\cos(sy)}{\lambda_1+\lambda_2}\,
         e^{-x\lambda_1}\left]\right.}   \biggl.\left.
       - \{Ac_1(\lambda_2+s)(1-2x's)+c'(\lambda_2-s)\}\,e^{-x's} 
         \right]d\,s\biggr\},                               
                                                              \\ % (20)-> (45)
   \sigma_{yy}^{gr(1)} &=& 
     4b_x\left\{ x_{-}^2y\Phi_1(r_{-}) - y(3x_{-}^2-y^2)\Phi_2(r_{-}) 
      +\frac{2Ac_1y}{r_{+}^6}
        \left[ 3x_{+}^2-y^2+24x'x_{+}\frac{x_{+}^2-y^2}{r_{+}^2}\right]
         \right.                                             \nonumber \\     
     &-& \left.
        \int_0^{\infty }\frac{\lambda_1^2\sin(sy)}{\lambda_1+\lambda_2}\,
        e^{-x\lambda_1}\left[ c_1s\frac{\lambda_2-\lambda_1}{\lambda_1}\,
        e^{-x'\lambda_1} 
       + \{Ac_1(\lambda_2+s)(1+2x's) - c''(\lambda_2-s)\}\,
        e^{-x's} \right]d\,s\right\}
                                                 \nonumber \\
    &+& 4b_y\left\{ - x_{-}^3\Phi_1(r_{-}) + x_{-}(x_{-}^2-3y^2)\Phi_2(r_{-}) 
         -\frac{2Ac_1}{r_{+}^6}\left[ x_{+}(x_{+}^2-3y^2)
             - 6x'\,\frac{x_{+}^4-6x_{+}^2y^2+y^4}{r_{+}^2}\right]
         \right.                                 \nonumber \\
     &-& \left.
        \int_0^{\infty }\frac{\lambda_1^2\cos(sy)}{\lambda_1+\lambda_2}\,
        e^{-x\lambda_1}\left[ c_1(\lambda_2-\lambda_1)\,e^{-x'\lambda_1} 
         \right.\right.                                 \nonumber \\
& & \phantom{\int_0^{\infty }\frac{\lambda_1^2\cos(sy)}{\lambda_1+\lambda_2}\,
              e^{-x\lambda_1}\left[\right.}\biggl.\left.
       - \{Ac_1(\lambda_2+s)(1-2x's) + (B+c's^2)(\lambda_2-s)/\lambda_1^2\}
         \,e^{-x's} \right]d\,s\biggr\},                     
                                                              \\ % (21)-> (46)
  \sigma_{xy}^{gr(1)} &=& 
     4b_x\left\{ -x_{-}y^2\Phi_1(r_{-}) - x_{-}(x_{-}^2-3y^2)\Phi_2(r_{-}) 
      +\frac{2Ac_1}{r_{+}^6}
        \left[ x_{+}(x_{+}^2-3y^2)
             + 6x'\,\frac{x_{+}^4-6x_{+}^2y^2+y^4}{r_{+}^2}\right]
         \right.                                             \nonumber \\     
     &+& \left.
        \int_0^{\infty }\frac{s\,\cos(sy)}{\lambda_1+\lambda_2}\,
        e^{-x\lambda_1}\left[ c_1s(\lambda_2-\lambda_1)\,e^{-x'\lambda_1} 
         \right.\right.                                      \nonumber \\     
 & & \phantom{\int_0^{\infty }\frac{s\,\cos(sy)}{\lambda_1+\lambda_2}\,
        e^{-x\lambda_1}\left[\right.}\biggl.\left.
       + \{B - Ac_1(\lambda_1^2+\lambda_2s)(1+2x's)
           - c's(\lambda_2-s)\}\,e^{-x's} \right]d\,s\biggr\}
                                                 \nonumber \\
     &+& 4b_y\left\{   x_{-}^2y\Phi_1(r_{-}) - y(3x_{-}^2-y^2)\Phi_2(r_{-}) 
        +\frac{2Ac_1y}{r_{+}^6}
         \left[ 3x_{+}^2-y^2-24x'x_{+}\frac{x_{+}^2-y^2}{r_{+}^2}\right]
         \right.                                 \nonumber \\
     &-& \left.
        \int_0^{\infty }\frac{s\,\sin(sy)}{\lambda_1+\lambda_2}\,
        e^{-x\lambda_1}\left[ c_1\lambda_1(\lambda_2-\lambda_1)\,
                                                             e^{-x'\lambda_1} 
        \right.\right.                                 \nonumber \\
 & & \phantom{\int_0^{\infty }\frac{s\,\sin(sy)}{\lambda_1+\lambda_2}\,
        e^{-x\lambda_1}\left[\right.}\biggl.\left. 
       + \{B + Ac_1(\lambda_1^2+\lambda_2s)(1-2x's)
             - c's(\lambda_2-s)\}\,e^{-x's} \right]d\,s\biggr\}, 
                                                          \\    % (22)-> (47)
  \sigma_{zz}^{gr(1)} &=& 
     4\nu_1\biggl\{(b_xy-b_yx_{-})\,r_{-}^2\Phi_1(r_{-})  \biggr.
                                                              \nonumber \\
     &-& b_x
        \int_0^{\infty }\frac{\sin(sy)}{\lambda_1+\lambda_2}\,
        e^{-x\lambda_1}\left[ s\,\frac{\lambda_2-\lambda_1}{\lambda_1}\,
        e^{-x'\lambda_1} \right.                              
                                                              \nonumber \\
     & &\phantom{b_x\int_0^{\infty }\frac{\sin(sy)}{\lambda_1+\lambda_2}\,
        e^{-x\lambda_1}\left[\right.}\biggl.
        + \{A(\lambda_2+s)(1+2x's)-[(B-1)\nu_2/\nu_1 +1](\lambda_2-s)\}\,
        e^{-x's} \biggr]d\,s
                                                 \nonumber \\
     &-& b_y
        \int_0^{\infty }\frac{\cos(sy)}{\lambda_1+\lambda_2}\,
        e^{-x\lambda_1}\left[ (\lambda_2-\lambda_1)\,e^{-x'\lambda_1} \right.
                                                 \nonumber \\
     & &\phantom{b_y\int_0^{\infty }\frac{\cos(sy)}{\lambda_1+\lambda_2}\,
         e^{-x\lambda_1}}  \biggl.\bigl.
        - \{A[s+\lambda_2(1-2x's)]+[(B-1)\nu_2/\nu_1 +1](\lambda_2-s)\}\,
        e^{-x's} \bigr]d\,s\biggr\},                   \\        
                                                                 % (23)-> (48)
  \sigma_{xz}^{gr(1)} &=& 
        b_z\frac{k_1+1}{2}\left\{\;\;\;yr^2_{-}\Phi_1(r_{-})
       + \int_0^{\infty}
             \frac{s\;\sin(sy)}{\lambda_1+\lambda_2}\;\;e^{-x\lambda_1}
             \left( \frac{\lambda_1-\lambda_2}{\lambda_1}\,e^{-x'\lambda_1}
                  + 2\,\frac{\Gamma -1}{\Gamma +1}\,e^{-x's} \right)d\,s
                                                             \right\},
                                                            \\   % (10)-> (49)
  \sigma_{yz}^{gr(1)} &=& 
        b_z\frac{k_1+1}{2}\left\{ - x_{-}r^2_{-}\Phi_1(r_{-})
      - \int_0^{\infty}
             \frac{\lambda_1\cos(sy)}{\lambda_1+\lambda_2}e^{-x\lambda_1}
             \left( \frac{\lambda_1-\lambda_2}{\lambda_1}e^{-x'\lambda_1}
                  + 2\frac{\Gamma -1}{\Gamma +1}e^{-x's} \right)d\,s
                                                             \right\},
                                                                 % (11)-> (50)
\end{eqnarray} 
and for medium~2
\begin{eqnarray}
  \sigma_{xx}^{gr(2)} &=& 
         8(B-1)c_2\,\{-\,b_xy(3x_{-}^2-y^2)+b_yx_{-}(x_{-}^2-3y^2)\}/r_{-}^6
                                                 \nonumber \\
     &-& 4b_x
        \int_0^{\infty }\frac{s^2\sin(sy)}{\lambda_1+\lambda_2}\,
        e^{x\lambda_2}\left[ 2c_1s\,e^{-x'\lambda_1} 
       + \{Ac_1(\lambda_1-s)(1+2x's)-c'(\lambda_1+s)\}\,e^{-x's} 
         \right]d\,s
                                                 \nonumber \\
     &-& 4b_y
        \int_0^{\infty }\frac{s^2\cos(sy)}{\lambda_1+\lambda_2}\,
        e^{x\lambda_2}\left[ 2c_1\lambda_1\,e^{-x'\lambda_1} 
       - \{Ac_1(\lambda_1-s)(1-2x's)+c'(\lambda_1+s)\}\,e^{-x's} 
         \right]d\,s\,,                          \nonumber \\        
     & &  \,                                                \\   % (24)-> (51)
  \sigma_{yy}^{gr(2)} &=& 
         8(B-1)c_2\,\{b_xy(3x_{-}^2-y^2)-b_yx_{-}(x_{-}^2-3y^2)\}/r_{-}^6
                                                             \nonumber \\
     &+& 4b_x
        \int_0^{\infty }\frac{\lambda_1^2\sin(sy)}{\lambda_1+\lambda_2}\,
        e^{x\lambda_2}\left[ 2c_1s\,e^{-x'\lambda_1} 
       + \{ Ac_1(\lambda_1-s)(1+2x's) - c''(\lambda_1+s)\}\,e^{-x's} 
         \right]d\,s
                                                             \nonumber \\
     &+& 4b_y
        \int_0^{\infty }\frac{\lambda_1^2\cos(sy)}{\lambda_1+\lambda_2}\,
        e^{x\lambda_2}\left[ 2c_1\lambda_1\,e^{-x'\lambda_1} 
        \right.                                              \nonumber \\
     & &\phantom{4b_y
        \int_0^{\infty }\frac{\lambda_1^2\cos(sy)}{\lambda_1+\lambda_2}}\left.
        - \{ Ac_1(\lambda_1-s)(1-2x's) 
          + (B+c's^2)(\lambda_1+s)/\lambda_1^2\}\,e^{-x's} \right]d\,s\,,
                                                              \\ % (25)-> (52)
  \sigma_{xy}^{gr(2)} &=& 
         8(B-1)c_2\,\{b_xx_{-}(x_{-}^2-3y^2)+b_yy(3x_{-}^2-y^2)\}/r_{-}^6
                                                             \nonumber \\
     &+& 4b_x
        \int_0^{\infty }\frac{s\,\cos(sy)}{\lambda_1+\lambda_2}\,
        e^{x\lambda_2}\left[ -2c_1\lambda_1s\,e^{-x'\lambda_1} 
        \right.                                              \nonumber \\
     & &\phantom{4b_x
        \int_0^{\infty }\frac{s\,\cos(sy)}{\lambda_1+\lambda_2}\,
        e^{x\lambda_2}\left[\right.}\left.
       + \{ B - Ac_1\lambda_1(\lambda_1-s)(1+2x's)
           + c's(\lambda_1+s)\}\,e^{-x's} \right]d\,s
                                                             \nonumber \\
     &+& 4b_y                                                        
        \int_0^{\infty }\frac{s\,\sin(sy)}{\lambda_1+\lambda_2}\,
        e^{x\lambda_2}\left[ 2c_1\lambda_1^2\,e^{-x'\lambda_1} 
        \right.                                              \nonumber \\
     & &\phantom{4b_y
        \int_0^{\infty }\frac{s\,\sin(sy)}{\lambda_1+\lambda_2}\,
        e^{x\lambda_2}\left[\right.}\left. 
       - \{ B + Ac_1\lambda_1(\lambda_1-s)(1-2x's)
           + c's(\lambda_1+s)\}\,e^{-x's} \right]d\,s\,,
                                                              \\ % (26)-> (53)
  \sigma_{zz}^{gr(2)} &=& 
     4b_x\nu_1\int_0^{\infty }\frac{\sin(sy)}{\lambda_1+\lambda_2}\,
        e^{x\lambda_2}\left[2s\,e^{-x'\lambda_1} 
        \right.                                              \nonumber \\
  & &\phantom{4b_x\nu_1\int_0^{\infty }\frac{\sin(sy)}{\lambda_1+\lambda_2}
        }\left.
        + \{A(\lambda_1-s)(1+2x's)-[(B-1)\nu_2/\nu_1+1](\lambda_1+s)\}\,
        e^{-x's} \right]d\,s
                                                 \nonumber\\
     &+& 
     4b_y\nu_1\int_0^{\infty }\frac{\cos(sy)}{\lambda_1+\lambda_2}\,
        e^{x\lambda_2}\left[2\lambda_1\,e^{-x'\lambda_1} 
        \right.                                              \nonumber \\
  & &\phantom{4b_y\nu_1\int_0^{\infty }\frac{\cos(sy)}{\lambda_1+\lambda_2}
        }\left.
        + \{A[s-\lambda_1(1-2x's)]-[(B-1)\nu_2/\nu_1+1](\lambda_1+s)\}\,
        e^{-x's} \right]d\,s\,,                        
                                                         \\      % (27)-> (54)
  \sigma_{xz}^{gr(2)} &=& 
        b_z\,(k_1+1)\int\limits_0^{+\infty}
             \frac{s\;\sin(sy)}{\lambda_1+\lambda_2}\;\;e^{x\lambda_2}
             \left( e^{-x'\lambda_1}
                  + \frac{\Gamma -1}{\Gamma +1}\,e^{-x's} \right)d\,s\;,
                                                         \\      % (12)-> (55)
  \sigma_{yz}^{gr(2)} &=& 
        b_z\,(k_1+1)\int\limits_0^{+\infty}
             \frac{\lambda_2\cos(sy)}{\lambda_1+\lambda_2}\,e^{x\lambda_2}
             \left( e^{-x'\lambda_1}
                  + \frac{\Gamma -1}{\Gamma +1}\,e^{-x's} \right)d\,s\;.
                                                                 % (13)-> (56)
\end{eqnarray}                                                               
where $\Phi_1=K_1(r/\sqrt{c_1})/(\sqrt{c_1}r^3)$,\ \ 
      $\Phi_2=[2c_1/r^2-K_2(r/\sqrt{c_1})]/r^4$, \ \
$c'=c_1+c_2(B-1)$,\ \ $c''=c_1+c_2(B-1)\lambda_2^2/\lambda_1^2$,\ \ and 
$\lambda_i=\sqrt{1/c_i+s^2}$, $i=1,2$.
\par
The gradient stress components $\sigma_{kl}^{(i)}$ given by the superposition 
of the classical ones (27)--(38) and gradient extra terms (45)--(56), are 
continuous at the interface ($x=0$). When $\mu_1=\mu_2=\mu $,\ \ 
$\nu_1=\nu_2=\nu $, and $c_1=c_2=c$ (the case of a homogeneous medium), they
are transformed into the superposition of (11) and (15). When $c_1=c_2\to 0$ 
(the limiting transition to the classical elasticity), the gradient extra 
terms (45)--(56) disappear. It is worth noting, that (45)--(56) contain 
specific terms caused only by a difference between the gradient coefficients 
$c_1$ and $c_2$ (see, for example, first subintegral terms).
\par
Below, we consider separately the cases of screw and edge disocations for the
following three types of interfaces: a ``purely elastic'' interface 
($\mu_1\neq\mu_2$,\ \ $\nu_1\neq\nu_2$,\ \ $c_1=c_2=c$), a ``purely gradient''
interface ($\mu_1=\mu_2$,\ \ $\nu_1=\nu_2$,\ \ $c_1\neq c_2$), as well as
a general ``mixed gradient elastic'' interface ($\mu_1\neq\mu_2$,\ \ 
$\nu_1\neq\nu_2$,\ \ $c_1\neq c_2$).
                                                \par\bigskip\noindent
{\bf 3.2.2. Screw dislocations}
                                                \par\bigskip\noindent
For screw dislocations, the general solution is given [64,\ 65] by the 
superpositon of the classical expressions (31), (32), (37) and (38), and 
gradient extra terms (49), (50), (55) and (56). 
                                                       \par\bigskip\noindent
\underline{{\it Purely elastic interface} ($\mu_1\neq\mu_2$,\ \ 
$\nu_1\neq\nu_2$,\ \ $c_1=c_2=c$)}
                                                       \par\bigskip\noindent
In this case, the gradient solution (in units of $\mu_1b_z/\,2\pi $) is
[64,\ 65]
\begin{eqnarray}
     \sigma_{xz}^{(1,2)} &=& \sigma_{xz}^{0(1,2)}
       + \frac{y}{\sqrt{c}\,r_{-}}\,K_1\left(\frac{r_{-}}{\sqrt{c}}\right)
       + \frac{\Gamma -1}{\Gamma +1}
         \int\limits_0^{+\infty}
             \frac{s}{\lambda }\,\sin(sy)\,e^{-|x|\lambda -x's}\,d\,s\;,
                                                            \\   % (14)-> (57)
    \sigma_{yz}^{(1,2)} &=& \sigma_{yz}^{0(1,2)}
    - \frac{x-x'}{\sqrt{c}\,r_{-}}\,K_1\left(\frac{r_{-}}{\sqrt{c}}\right)
      - \frac{\Gamma -1}{\Gamma +1}\,{\rm sign}(x)
          \int\limits_0^{+\infty}
             \cos(sy)\,e^{-|x|\lambda -x's}\,d\,s\;,
                                                                 % (15)-> (58)
\end{eqnarray} 
where $\sigma_{iz}^{0(1,2)}$ are determined by (31), (32), (37) and (38), and 
$\lambda =\sqrt{1/c+s^2}$. Both components are continuous at the interface,
in contrast to the classical solution where $\sigma_{yz}^0$ suffers
a jump given by (41). Figs.~13 and 14 illustrate this difference. It is seen 
that the magnitude of the jump increases as the dislocation approaches the 
interface. Also, the gradient solution is finite at the dislocation line, 
while the classical one is singular
there (Fig.~14). It is seen that the classical and gradient solutions coincide
far ($r>5\sqrt{c}$) from the interface or the dislocation line, while 
they are quite different at nanoscopic distances from them ($r<5\sqrt{c}$).
\par
When the dislocation lies directly at the interface ($x'=0$), the integrals in
(57)--(58) can be calculated in a closed form giving (in units of 
$\mu_1\mu_2b_z/[\pi (\mu_1+\mu_2)]$) 
\begin{equation}
     \sigma_{xz} =  -\frac{y}{r^2} +
          \frac{y}{\sqrt{c}\,r}\,K_1\left(\frac{r}{\sqrt{c}}\right),\;\;\;\;\;
     \sigma_{yz} =   \frac{x}{r^2} -
          \frac{x}{\sqrt{c}\,r}\,K_1\left(\frac{r}{\sqrt{c}}\right),  
                                                                 % (16)-> (59)
\end{equation}
where $r^2=x^2+y^2$. It is worth noting that the gradient solutions given by 
(59) for such an interface dislocation differ only by a factor 
$2\mu_2/(\mu_1+\mu_2)$ from those given by the corresponding components in
the superposition of (11) and (15) for a screw dislocation in a 
single-phase infinite medium as is the case in the classical theory of 
elasticity.
\par
Let us consider now the ``image'' force $F_x^{el}$ which acts on the 
dislocation unit length due to the interface (Fig.~12). The gradient solution 
(in units of $\mu_1b_z^2/\,2\pi $) reads [64,\ 65]
\begin{equation}
     F_x^{el}(x')\; =\; b_z\,\sigma_{yz}^{(1)}(x=x',\,y=0)\;
      = \;\frac{\Gamma -1}{\Gamma +1} \left\{ \frac{1}{2x'} 
     - \int\limits_0^{+\infty}e^{-x'(\lambda +s)}\,d\,s\right\}, % (17)-> (60)
\end{equation}
where the first term in the brackets is the classical singular solution and 
the second one is the extra gradient term. The numerical evaluation of (60) 
is presented in Fig.~15 where also a similar solution for $x'<0$ is plotted. 
It is seen that the classical singularity is eliminated from the gradient 
solutions which attain maximum values at a distance $\approx\sqrt{c}$ from 
the interface and tend to zero at the interface. 
\par
This result is especially instructive for the case of a free surface when 
$\Gamma =0$ (see the negative-valued curves in Fig.~15).
In fact, there is no image force when the dislocation lies at the free 
surface, the force emerges and increases when the dislocation begins to 
penetrate into the material (the estimated dislocation core radius is
$\approx 4\sqrt{c}$ [59]), achieves a maximum value and decreases when the 
dislocation moves inside the material. The last stage is also well described 
by the classical solution (Fig.~15) which, however, can not describe at all 
the abovementioned preceding stages. Within the gradient theory of (5), one 
can estimate a maximum shear stress $\tau_{max}=|F_x^{el}|_{max}/b_z$ which 
the screw dislocation has to overcome for penetrating into the material. From
Fig.~15, it is estimated that $\tau_{max}\approx\mu /\,2\pi$, i.e.~the value
of theoretical shear strength [1]. 
For the case of two bonded solids, the result of zero value at the interface 
(i.e., the appearance of an unstable equilibrium position there) is not as 
clear. 
                                                       \par\bigskip\noindent
\underline{{\it Purely gradient interface} ($\mu_1=\mu_2=\mu $,\ \ 
$\nu_1=\nu_2=\nu $,\ \ $c_1\neq c_2$)}
                                                       \par\bigskip\noindent
In this case, the corresponding gradient solution is given by the 
superposition of (31), (32), (37) and (38) with (49), (50), (55) and (56),
respectively, with $\Gamma =1$. Here we focus only on the appropriate 
``image'' force $F_x^{gr}$ which acts upon the dislocation due to the 
difference in the gradient moduli $c_1$ and $c_2$. This force is given 
(in units of $\mu_1b_z^2/\,2\pi $) by [64,\ 65]
\begin{equation}
     F_x^{gr}(x')\; = \; b_z\,\sigma_{yz}^{(1)}(x=x',\,y=0)\;
      =\; - \int\limits_0^{+\infty}
             \frac{\lambda_1-\lambda_2}{\lambda_1+\lambda_2}\,
             e^{-2x'\lambda_1}\,d\,s\;.                          % (18)-> (61)
\end{equation}
When $c_1>c_2$, i.e.~$\lambda_1<\lambda_2$, the integral in (61) is negative
and the force $F_x^{gr}$ is positive. This means that the dislocation is 
pushed away from the interface into the bulk of material~1 which has the 
larger gradient coefficient. This is in agreement with the gradient 
solution for the strain energy of a screw dislocation [58,\ 61]
$W=\mu b_z^2/(4\pi )\{\gamma + \ln(R/\,2\sqrt{c_1})\}$ (see also (16) with 
$b_x=0$); indeed, the larger $c_1$ is, the smaller $W$ is. Plots for  
$F_x^{gr}(x')$ are presented in Fig.~16 from which one can conclude that
this force has a short-range character and acts just near the interface. At 
the interface, it attains a maximum value which depends strongly on the ratio
$c_2/c_1$ (Fig.~16). 
                                                       \par\bigskip\noindent
\underline{{\it General mixed gradient elastic interface} 
($\mu_1\neq \mu_2$,\ \ $\nu_1\neq \nu_2$,\ \ $c_1\neq c_2$)}
                                                       \par\bigskip\noindent
In this case, the gradient solution is given by the superposition of (31), 
(32), (37) and (38) with (49), (50), (55) and (56), respectively, and the 
``image'' force $F_x$ (in units of $\mu_1b_z^2/\,2\pi $) is [64,\ 65]
\begin{equation}
     F_x(x') = 
       \frac{\Gamma -1}{\Gamma +1}\,\frac{1}{2x'} 
     - \int\limits_0^{+\infty}\frac{\lambda_1}{\lambda_1+\lambda_2}
       \left\{ \frac{\lambda_1-\lambda_2}{\lambda_1}\,e^{-2x'\lambda_1}
             + 2\,\frac{\Gamma -1}{\Gamma +1}\,e^{-x'(\lambda_1+s)} \right\}
                                    \,d\,s\;.                    % (19)-> (62)
\end{equation}
It is worth noting that $F_x$ is not a simple superposition of $F_x^{el}$
and $F_x^{gr}$ given by (60) and (61), respectively. However, it manifests 
their characteristic features (Fig.~17). Similar to $F_x^{el}$, the force in 
(62) is a nonsingular long-range force which coincides with the classical 
solution far ($|x'|>5\sqrt{c_1}$) from the interface. Similar to $F_x^{gr}$, 
it attains non-zero values at the interface which depend on both ratios 
$\mu_2/\mu_1$ and 
$c_2/c_1$. In fact, the sign and qualitative behavior of $F_x$ near the 
interface is entirely determined by $c_2/c_1$. For example, if $\mu_2>\mu_1$, 
there are three different types of behavior of $F_x$ (Fig.~17). When 
$c_2<c_1$, $F_x>0$ everywhere and attains maximum values near or at the 
interface. When $c_2=c_1$, $F_x\equiv F_x^{el}$ and becomes equal to 
zero at the interface (see above). When $c_2>c_1$, $F_x>0$ 
except at a small region around the interface. The size of this region which
depends on $c_2/c_1$ is about $\sqrt{c_1}$, and $F_x<0$ inside this region 
attaining its minimum value at the interface. When $c_2<c_1$, 
the dislocation is pushed from material~2 into material~1 and has no
equilibrium position. When $c_2=c_1$, it does the same but it has an unstable
equilibrium position at the interface. When $c_2>c_1$, a dislocation being
in material~2, is attracted to the interface and is locked at a stable 
equilibrium position $x'\approx -(0.2$--$0.8)\sqrt{c_1}$ near that interface,
while a dislocation located in material~1, has an unstable equilibrium
position $x'\approx (0.4$--$0.7)\sqrt{c_1}$ near the interface; being 
attracted to it within a small region $x'<(0.4$--$0.7)\sqrt{c_1}$ and pushed
away from it otherwise.
                                                \par\bigskip\noindent
{\bf 3.2.3. Edge dislocations}
                                                \par\bigskip\noindent
For edge dislocations, the general solution is given [66] by the 
superpositon of the classical expressions (27)--(30) and (33)--(36), and 
gradient extra terms (45)--(48) and (51)--(54). 
                                                       \par\newpage\noindent
\underline{{\it Purely elastic interface} ($\mu_1\neq\mu_2$,\ \ 
$\nu_1\neq\nu_2$,\ \ $c_1=c_2=c$)}
                                                       \par\bigskip\noindent
In this case, we consider the effects caused only by the 
difference in the elastic constants of the bonded media. Two main advantages 
of the gradient solution may be pointed out in this connection.
First, there are no singularities in the
stress components $\sigma_{kl}^{(i)}$ at the dislocation line. Second, there 
are no jumps like those given by (39)--(40) in $\sigma_{yy}^{(i)}$ and 
$\sigma_{zz}^{(i)}$ at the interface. This allows one to consider nanoscale 
short-range elastic interactions between dislocations and interfaces, 
in contrast to the classical singular solution (27)--(30) and (33)--(36),
where the components $\sigma_{yy}^{0(i)}$ and $\sigma_{zz}^{0(i)}$ suffer jump
discontinuities at the interface. This is
illustrated in Fig.~18 for the component $\sigma_{yy}^{(i)}(x,0)$ of a 
dislocation with Burgers vector $b_y$. It is seen that classical and 
gradient solutions coincide far ($r>10\sqrt{c}$) from the interface or the
dislocation line, while near them (within nanoscopic distances $r<10\sqrt{c}$)
they are quite different.
\par
Let us consider now the ``image'' force $F_x^{el}$ which acts upon the 
dislocation unit length by the interface (Fig.~12). For a dislocation with 
Burgers vector $b_x$, the gradient solution 
$F_x^{el}(x')=b_x\,\sigma_{xy}^{(1)}(x=x',0)$ reads (in units of 
$\mu_1b_x^2/[\pi (k_1+1)]$) [66]
\begin{equation}
     F_x^{el}(x')\; =
         -\,\frac{A+B}{2x'} + \frac{4Ac}{x'^3} 
         + 2c\int\limits_0^{+\infty}s\,[B(\lambda -s)-A(\lambda +s)(1+2x's)]\,
         e^{-x'(\lambda +s)}\,d\,s\,,                            % (28)-> (63)
\end{equation}
where $\lambda =\sqrt{1/c+s^2}$. The first term in this expression is the 
classical singular solution, while the remaining two are extra gradient terms.
The numerical 
evaluation of (63) is presented in Fig.~19 where also a similar solution for 
$x'<0$ is plotted. It is seen that the classical singularity is eliminated 
from the gradient solution which attains maximum values at a distance 
$\approx\sqrt{c}$ from the interface and has no jumps at the interface. 
\par
In the case of a free surface when $\mu_2=\nu_2=0$ (see the negative-valued 
curves of Fig.~19), the situation is just like with a screw dislocation (see
{\it Section~3.2.2}).
So, there is no image force when the dislocation lies at the 
free surface. The force appears and increases when the dislocation begins to 
penetrate into the material, achieves a maximum value and decreases
when the dislocation moves inside the material. Again, the last stage (for 
$x'>5\sqrt{c}$) is also well described by the classical solution (Fig.~19). 
Within the gradient theory (5), one can estimate a maximum shear stress 
$\tau_{max}=|F_x^{el}|_{max}/b_x$ which the edge dislocation has 
to overcome in order to penetrate inside the material. From Fig.~19, it is 
estimated that $\tau_{max}\approx\mu /\,2.8\pi$ (for $\nu =0.3$), i.e.~the 
value of theoretical shear strength [1]. 
                                                       \par\bigskip\noindent
\underline{{\it Purely gradient interface} ($\mu_1=\mu_2$,\ \ 
$\nu_1=\nu_2$,\ \ $c_1\neq c_2$)}
                                                       \par\bigskip\noindent
In this case, we consider the effects caused only by the difference in the 
gradient coefficients of the bonded media. Here we focus only on the ``image''
force $F_x^{gr}$ which acts upon the dislocation due to the difference
between the gradient coefficients $c_1$ and $c_2$. For a dislocation with
Burgers vector $b_x$, this force is given (in units of 
$\mu_1b_x^2/[\pi (k_1+1)]$) by [66]
\begin{equation}
     F_x^{gr}(x')\; = \; 4\int\limits_0^{+\infty}
                         \frac{s^2}{\lambda_1+\lambda_2}
         \left\{c_1(\lambda_2-\lambda_1)\,e^{-2x'\lambda_1}
                + (c_2-c_1)(\lambda_2-s)\,e^{-x'(\lambda_1+s)}\right\}d\,s\;.
                                                                 % (29)-> (64)
\end{equation}
A numerical evaluation of this integral shows that $F_x^{gr}$ is positive when
$c_2>c_1$ and negative when $c_2<c_1$ (Fig.~20). This means that an edge 
dislocation is pushed away from the interface into the bulk of the material
which has the smaller gradient coefficient, in contrast to the case of a screw
dislocation (see {\it Section~3.2.2}) which exhibits opposite behavior. The 
reasons for this difference is not clear as yet. From Fig.~20, one can 
conclude that $F_x^{gr}$ has a short-range character and acts just near the 
interface. At the interface, it attains a maximum value which depends 
strongly on the ratio $c_2/c_1$. 
                                                       \par\bigskip\noindent
\underline{{\it General mixed gradient elastic interface}
($\mu_1\neq\mu_2$,\ \ $\nu_1\neq\nu_2$,\ \ $c_1\neq c_2$)}       
                                                       \par\bigskip\noindent
In this case, the ``image'' force $F_x$ (in units of 
$\mu_1b_x^2/[\pi (k_1+1)]$) is [66]
\begin{eqnarray}
     F_x(x') &=& -\,\frac{A+B}{2x'} + \frac{4Ac_1}{x'^3} 
     + 4\int\limits_0^{+\infty}\frac{s}{\lambda_1+\lambda_2}
       \left[ c_1s\,(\lambda_2-\lambda_1)\,e^{-2x'\lambda_1} \right.
                                                          \nonumber \\
    & &\phantom{-\,\frac{A+B}{2x'} + \frac{4Ac_1}{x'^3} +}\left.
      + \;\{B-Ac_1(\lambda_1^2+\lambda_2s)(1+2x's) - c's(\lambda_2-s)\}\,
        e^{-x'(\lambda_1+s)} \right]d\,s\;.                      % (30)-> (65)
\end{eqnarray}
It is worth noting that $F_x$ is not a simple superposition of $F_x^{el}$
and $F_x^{gr}$ given by (63) and (64), respectively. This is illustrated in
Fig.~21. The force in (65) is a nonsingular long-range force which is 
continuous across the interface and coincides with the classical solution far 
($|x'|>5\sqrt{c_1}$) from the interface. Its values at the interface depend 
strongly on both ratios $\mu_2/\mu_1$ and $c_2/c_1$. In fact, the sign and 
qualitative behavior of $F_x$ near the interface may be determined by 
$c_2/c_1$. For example, for $\mu_2/\mu_1=3$, there are three different types 
of behavior for $F_x$ (Fig.~21). When $c_2>c_1$, 
$F_x>0$ everywhere and attains maximum values near or at the interface. 
When $c_2=c_1$, $F_x\equiv F_x^{el}$ (see above). When $c_2<c_1$,
$F_x>0$ except at a very small region around the interface. Its size depends 
on $c_2/c_1$ and is about $0.3\sqrt{c_1}$; $F_x<0$ inside this region and 
attains minimum values at the interface. Thus, when $c_2\geq c_1$, 
the dislocation is pushed from material~2 into material~1 and possess no
equilibrium position. When $c_2<c_1$ (e.g.~$c_2/c_1=0.3$), a dislocation being
in material~2 is attracted to the interface and may be locked at a stable 
equilibrium position $x'\approx -0.1\sqrt{c_1}$ near the interface,
while a dislocation located in material~1 possess an unstable equilibrium
position $x'\approx 0.2\sqrt{c_1}$ near the interface; being 
attracted to it within a small region $x'<0.2\sqrt{c_1}$ and pushed
away from the interface otherwise.
                                                       \par\bigskip
Thus, the gradient elasticity described by (5) has been employed to consider a
straight dislocation near a flat interface which separates two elastic 
media with different elastic constants and gradient coefficients. We have 
derived [64--66], in integral forms, solutions for the dislocation stress 
fields and for the ``image'' force which acts upon the dislocation by the 
interface. It has been shown that all stress components remain continuous 
across the interface, in contrast to the well-known classical solution [69] 
where three (one for a screw dislocation and two for an edge dislocation) 
stress components suffer jump discontinuities there. Far from the interface 
and the dislocation line (at distances $\gg 10\sqrt{c}$), gradient and 
classical solutions coincide. We have also dispensed with the classical 
singularity of the elastic ``image'' force acting upon the dislocation by the 
interface [69], and shown that it remains finite and continuous throughout. 
We have found an additional short-range elastic interaction between the 
dislocation and the interface due to the difference in the gradient 
coefficients of the media in contact. The additional ``image'' force acting 
upon the dislocation is finite and maximum at the interface. Under the action
of this force, the screw (edge) dislocation tends to penetrate into the 
medium with the larger (smaller) gradient coefficient. In the general case 
where both the elastic constants $\mu_i$ and the gradient coefficients $c_i$ 
are different for these media, the total ``image'' force exhibits quite 
different behavior near the interface depending on the ratios $\mu_2/\mu_1$ 
and $c_2/c_1$, while its long-range component remains as in the classical 
theory of elasticity.
                                                \par\bigskip\bigskip\noindent
4. NANOSCALE ELASTIC PROPERTIES OF DISCLINATIONS
                                                \par\bigskip\bigskip\noindent
In considering straight disclinations, it is reasonable to start from the 
solutions for the so-called screened disclination configurations [7,\ 12] 
because the classical elastic fields for individual disclinations in infinite
media contain the terms which are singular at infinity ($\sim r\ln r$ for 
displacements, $\sim\ln r$ for strains and stresses) and have no physical 
meaning (dimensional quantities under the logarithm signs). Among the various
screened disclination configurations, we have chosen that of disclination 
dipoles not only because they are described by simple expressions in classical
elasticity [5--7] but also because they are very convenient to determine
elastic interactions. Thus, the gradient solutions have been originally 
obtained for a disclination dipole within both the gradient theories described
by (4) [58] and (5) [62,\ 63]. For shorteness, we present below the general
solution for an individual disclination obtained as a limiting case of a
disclination dipole [58,\ 62,\ 63]. However, we also consider disclination 
dipoles in {\it Section~4.2} to discuss mostly important features of the 
gradient solution.
                                                \par\bigskip\noindent
{\large\bf 4.1. Individual disclinations --- a general solution}
                                                \par\bigskip\noindent
Consider a disclination of general type 
with Frank vector 
$\mbox{\boldmath $\omega $} = \omega_x\mbox{\boldmath $e$}_x
                            + \omega_y\mbox{\boldmath $e$}_y
                            + \omega_z\mbox{\boldmath $e$}_z$ in an infinite
elastic medium. The scalars $\omega_x$ and $\omega_y$ determine the twist 
components of the disclination while $\omega_z$ determines its wedge 
component. Let its line coincides with the $z$-axis of the Cartesian 
coordinate system and the Frank vector is applied at the origin of this 
coordinate system. For such an isolated disclination, both classical and 
gradient solutions themselves have no physical meaning because they are not 
screened but they may be used in modelling screened disclination 
configurations as basic elements [7,\ 12].
                                                \par\bigskip\noindent
{\underline{\it Classical solution}}
                                                \par\medskip\noindent
The classical solution for
elastic strain fields $\varepsilon^0_{ij}$ reads [5--7] (in units of 
$1/[4\pi (1-\nu )]$)
\begin{eqnarray}
   \varepsilon^0_{xx} &=&
        \omega_xzx\frac{2\nu r^2- x^2+y^2}{r^4}
      + \omega_yzy\frac{2\nu r^2-3x^2-y^2}{r^4}
      + \omega_z \left\{ (1-2\nu )\ln r + \frac{y^2}{r^2} \right\},
                                                               \nonumber \\
   \varepsilon^0_{yy} &=&
        \omega_xzx\frac{2\nu r^2-x^2-3y^2}{r^4}
      + \omega_yzy\frac{2\nu r^2+x^2- y^2}{r^4}
      + \omega_z \left\{ (1-2\nu )\ln r + \frac{x^2}{r^2} \right\},
                                                               \nonumber \\
   \varepsilon^0_{xy} &=&
        (\omega_yx-\omega_xy)z\frac{x^2-y^2}{r^4} 
      - \omega_z\frac{xy}{r^2},
                                                            \;\;\;
   \varepsilon^0_{xz}  =
                 \omega_y\frac{xy}{r^2}
               - \omega_x\left\{(1-2\nu )\ln r + \frac{y^2}{r^2} \right\},
                                                             \nonumber \\
   \varepsilon^0_{yz} &=& 
                 \omega_x\frac{xy}{r^2}
               - \omega_y\left\{(1-2\nu )\ln r + \frac{x^2}{r^2}\right\},
                                                                 % (14)-> (66)
\end{eqnarray}
and for the stress fields it may be written
(in units of $\mu /[2\pi (1-\nu )]$) as
\begin{eqnarray}
   & & \sigma^0_{xx} = \varepsilon^0_{xx}(\nu =0),\;\;\;
       \sigma^0_{yy} = \varepsilon^0_{yy}(\nu =0),\;\;\;
       \sigma^0_{xy} = \varepsilon^0_{xy},\;\;\;
       \sigma^0_{xz} = \varepsilon^0_{xz},\;\;\;
       \sigma^0_{yz} = \varepsilon^0_{yz},
                                                                \nonumber \\
   & & \sigma^0_{zz} = 
       -\omega_x\,z\,2\nu\frac{x}{r^2}
       -\omega_y\,z\,2\nu\frac{y}{r^2}
       +\omega_z\,   2\nu\ln r\;,                               % (15)-> (67)
\end{eqnarray}
where $r^2=x^2+y^2$.
Most of the components in (66) and (67) contain singular terms $\sim\ln r$.
                                                \par\bigskip\noindent
{\underline{\it Gradient solution}}
                                                \par\medskip\noindent
The gradient solutions have been originally obtained for a disclination 
dipole within both the gradient theories described by (4) [58] and (5) 
[62,\ 63]. Solving (7), we have finally for an individual disclination under 
consideration the strain field 
$\varepsilon_{ij}=\varepsilon^0_{ij}+\varepsilon_{ij}^{gr}$, where
$\varepsilon_{ij}^0$ are given by (66) and $\varepsilon_{ij}^{gr}$ 
(in units of $1/[4\pi (1-\nu )]$) by [58,\ 62,\ 63]
\begin{eqnarray}
    \varepsilon_{xx}^{gr} &=& 
         \omega_x\,2xz\left\{(y^2-\nu r^2)\Phi_1+(x^2-3y^2)\Phi_2\right\}
                                                                \nonumber \\
       &+& \omega_y\,2yz\left\{(y^2-\nu r^2)\Phi_1+(3x^2-y^2)\Phi_2\right\}
         + \omega_z\left\{\Phi_0+r^2(x^2-y^2)\Phi_2\right\},
                                                                \nonumber \\
    \varepsilon_{yy}^{gr} &=& 
         \omega_x\,2xz\left\{(x^2-\nu r^2)\Phi_1-(x^2-3y^2)\Phi_2\right\}
                                                                \nonumber \\
       &+& \omega_y\,2yz\left\{(x^2-\nu r^2)\Phi_1-(3x^2-y^2)\Phi_2\right\}
         + \omega_z\left\{\Phi_0-r^2(x^2-y^2)\Phi_2\right\},
                                                                \nonumber \\
    \varepsilon_{xy}^{gr} &=& 
         \omega_x\,2yz\left\{-x^2\Phi_1+(3x^2-y^2)\Phi_2\right\}
                                                                \nonumber \\
       &-& \omega_y\,2xz\left\{y^2\Phi_1+(x^2-3y^2)\Phi_2\right\}
         + \omega_z\,2xyr^2\Phi_2,
                                                                \nonumber \\
    \varepsilon_{xz}^{gr} &=& 
         \omega_x\left\{-\Phi_0-r^2(x^2-y^2)\Phi_2\right\}
       - \omega_y\,2xyr^2\Phi_2,
                                                                \nonumber \\
    \varepsilon_{yz}^{gr} &=& 
         \omega_y\left\{-\Phi_0+r^2(x^2-y^2)\Phi_2\right\}
       - \omega_x\,2xyr^2\Phi_2,                                 % (16)-> (68)
\end{eqnarray}
where $\Phi_i$ are the same as in {\it Section~3.1.1}. For the stress field, 
the solution of (8) gives $\sigma_{ij}=\sigma^0_{ij}+\sigma_{ij}^{gr}$, where
$\sigma_{ij}^0$ are given by (67) and $\sigma_{ij}^{gr}$ (in units of 
$\mu /[2\pi (1-\nu )]$) by [62,\ 63]
\begin{eqnarray}
  & & \sigma_{xx}^{gr} = 
               \varepsilon_{xx}^{gr}(\nu =0,c_2\leftrightarrow c_1),\;\;\;
   \sigma_{yy}^{gr} = \varepsilon_{yy}^{gr}(\nu =0,c_2\leftrightarrow c_1),
                                                           \nonumber \\
  & & \sigma_{zz}^{gr} = 2\nu\left\{
         (\omega_xx + \omega_yy)\,zr^2\Phi_1(c_2\leftrightarrow c_1)
                    + \omega_z\Phi_0(\nu =0,c_2\leftrightarrow c_1) \right\},
                                                           \nonumber \\
  & &  \sigma_{xy}^{gr} = \varepsilon_{xy}^{gr}(c_2\leftrightarrow c_1),\;\;\;
       \sigma_{xz}^{gr} = \varepsilon_{xz}^{gr}(c_2\leftrightarrow c_1),\;\;\;
       \sigma_{yz}^{gr} = \varepsilon_{yz}^{gr}(c_2\leftrightarrow c_1). 
                                                                 % (17)-> (69)
\end{eqnarray}
\par
Using the limiting transitions noted in {\it Section~3.1.1}, it is easily to 
show the total elimination of classical logarithmic singularity from elastic
fields (68) and (69). In the next sections we consider similar elastic fields
of disclination dipoles in detail and discuss their characteristic features 
separately for twist disclination of two types as well as for wedge 
disclinations.
                                                \par\bigskip\bigskip\noindent
{\large\bf 4.2. Disclination dipoles}
                                                \par\bigskip\noindent
We discuss below two interesting aspects, i.e.~the behavior of 
elastic strains near disclination lines and the short-range elastic
interaction between disclinations in a dipole. It is more convenient to 
discuss twist and wedge disclinations separately. 
\par
Consider a disclination dipole which consists of two parallel disclinations
with Frank vector $\pm\mbox{\boldmath $\omega $}$ 
($\mbox{\boldmath $\omega $} = \omega_x\mbox{\boldmath $e$}_x
                            + \omega_y\mbox{\boldmath $e$}_y
                            + \omega_z\mbox{\boldmath $e$}_z$). The scalars
$\omega_x$ and $\omega_y$ determine the twist components of the disclinations
while $\omega_z$ determines their wedge component (Fig.~22). Let the 
disclinations lie in the plane $y=0$ along the $z$-axis and cross the 
$x$-axis at the points $x=-d$ (negative disclination) and $x=0$ (positive 
disclination).  
                                                \par\bigskip\noindent
{\bf 4.2.1. First-type twist disclinations}
                                                \par\bigskip\noindent
We consider here a dipole of twist disclinations having the Frank vectors
$\pm\mbox{\boldmath $\omega $}=(\pm\omega_x,0,0)$. The elastic strain (stress)
components of such a dipole are given by the simple superpositions of the 
terms associated with $\omega_x$ in (66) and (68) ((67) and (69)), and similar
terms taken with the opposite sign [58,\ 63]. 
\par
Let us discuss first the behavior of the elastic strains and stresses 
near the line of the positive disclination ($x=0,y=0$). For $r\to 0$ we have  
for strains (in units of $\omega_x/[4\pi (1-\nu )]$) [58]
\begin{eqnarray}
     \varepsilon _{xx}\,|_{r\to 0} &=& 
         z\left\{ \frac{1-2\nu }{d} - \frac{4c_2}{d^3}  
             + \frac{2\nu }{\sqrt{c_2}}\,K_1\left(\frac{d}{\sqrt{c_2}}\right) 
             + \frac{2}{d}\,K_2\left(\frac{d}{\sqrt{c_2}}\right)\right\}\;,
                                                  \nonumber  \\   % (100)->
     \varepsilon _{yy}\,|_{r\to 0} &=& 
         z\left\{ \frac{1-2\nu }{d} + \frac{4c_2}{d^3}  
         - \frac{2(1-\nu )}{\sqrt{c_2}}\,K_1\left(\frac{d}{\sqrt{c_2}}\right) 
           - \frac{2}{d}\,K_2\left(\frac{d}{\sqrt{c_2}}\right)\right\}\;,
                                                   \nonumber \\   % (101)->
     \varepsilon _{xy}\,|_{r\to 0} &=& \varepsilon _{yz}|_{r_1\to 0} 
              \;\;=\;\; 0\;, 
                                                   \nonumber  \\   % (102)->
     \varepsilon _{xz}\,|_{r\to 0} &=& 
        - \frac{1}{2} + (1-2\nu )\left(\gamma +\ln\frac{d}{2\sqrt{c_2}}\right)
        + \frac{2c_2}{d^2}
        + (1-2\nu )\,K_0\left(\frac{d}{\sqrt{c_2}}\right)
        - K_2\left(\frac{d}{\sqrt{c_2}}\right),      
                                                   \nonumber  \\    % (103)
     \varepsilon\,|_{r\to 0} &=& 
          2(1-2\nu )\,z\left\{ \frac{1}{d} 
       - \frac{1}{\sqrt{c_2}}\,K_1\left(\frac{d}{\sqrt{c_2}}\right)\right\}\;,
                                                               % (104)-> (70)
\end{eqnarray}
and for stresses (in units of $\mu\omega_x/[2\pi (1-\nu )]$) [63]
\begin{eqnarray}
   & & \sigma_{xx}\,|_{r\to 0} = 
       \varepsilon_{xx}(\nu =0, c_2\leftrightarrow c_1)|_{r\to 0}\;,
                                                               \;\;\;\;\;\;\;
       \sigma_{yy}\,|_{r\to 0} = 
       \varepsilon_{yy}(\nu =0, c_2\leftrightarrow c_1)|_{r\to 0}\;,
                                                               \nonumber \\
   & & \sigma _{zz}\,|_{r\to 0} = 
         z2\nu\left\{ \frac{1}{d}  
        - \frac{1}{\sqrt{c_1}}\,K_1\left(\frac{d}{\sqrt{c_1}}\right)\right\}, 
                                                               \nonumber \\   
   & & \sigma_{xy}\,|_{r\to 0} = \sigma_{yz}\,|_{r\to 0} = 0\;, 
       \;\;\;\;\;\;\;
       \sigma_{xz}\,|_{r\to 0} = 
       \varepsilon_{xz}(c_2\leftrightarrow c_1)|_{r\to 0}\;,     % (13)-> (71)
\end{eqnarray}
\par
It is seen that the elastic strains and stresses are finite at the 
disclination line in contrast to the classical solutions (66) and (67), 
respectively, which are singular there. The values of the strain and stress 
components at the disclination line depend, in general, on the dipole arm~$d$.
\par
For $d\gg\sqrt{c_2}$ (long-range disclination interaction), the strain
components read (in units of $\omega_x/[4\pi (1-\nu )]$) [58]
\begin{eqnarray}
  & & \varepsilon _{xx}\,|_{r\to 0} = \varepsilon _{yy}\,|_{r\to 0} 
        = \frac{\varepsilon }{2}\,|_{r\to 0} =    
        (1-2\nu )\frac{z}{d}\;,      
                                                   \nonumber \\  % (105)->
  & & \varepsilon _{xy}\,|_{r\to 0} = \varepsilon _{yz}\,|_{r\to 0} 
        = 0\;,\;\;\;\;\;\;\;
      \varepsilon _{xz}\,|_{r\to 0} = 
     - \frac{1}{2} + (1-2\nu )\left(\gamma +\ln\frac{d}{2\sqrt{c_2}}\right)\;,
                                                                % (107)-> (72)
\end{eqnarray}
and the stress components are (in units of $\mu\omega_x/[2\pi (1-\nu )]$)
\begin{eqnarray}
   & & \sigma_{xx}\,|_{r\to 0} = \sigma_{yy}\,|_{r\to 0} = 
       \sigma_{zz}\,|_{r\to 0}/(2\nu) = \frac{z}{d}\;,    \nonumber \\
   & & \sigma_{xy}\,|_{r\to 0} = \sigma_{yz}\,|_{r\to 0} = 0\;, 
       \;\;\;\;\;\;\;
       \sigma_{xz}\,|_{r\to 0} = 
       \varepsilon_{xz}(c_2\leftrightarrow c_1)|_{r\to 0}\;.     % (13)-> (73)
\end{eqnarray}
\par
Fig.~23 provides the distribution of elastic strains and stresses in
the planes $y=0$ (Fig.~23{\it a--c,e}) and $x=0$ (Fig.~23{\it d}) near the 
line of the positive disclination located at the point $(0,0)$ of Fig.~22 when
$d=10^4\sqrt{c_k}$, where $k=1$ for stresses and $k=2$ for strains [58,\ 63].
The solid 
lines represent the gradient solution while the dashed lines represent the 
classical one. One can see that within the gradient elasticity, the elastic
strains and stresses are finite and much smaller near the disclination line, 
they achieve extreme values there and tend to the classical solution at 
distances far away from the disclination line ($r>5\sqrt{c_k}$).
\par
A general view on the distribution of elastic strains and stresses given by 
the classical and gradient elasticity, is provided in Fig.~24. The top 
pictures represent the classical solution, while the bottom pictures represent
the gradient solution. Fig.~24 clearly illustrates the elimination of 
classical singularities near the disclination line within the gradient theory.
\par
The absence of classical singularities permits to investigate short-range
elastic interactions between disclinations. For example, one can observe the 
strained states at two characteristic points of the disclination dipole,
e.g.~at the disclination line ($0,0$) and at the middle of the dipole 
($-d/2,0$). In the first case, one can use equations (70) and (71) which are
represented graphically in Fig.~25. It is seen that the elastic strains and
stresses vary non-monotonously with the variation of $d$ within this interval 
and they are strictly equal to zero at $d=0$ (annihilation of disclinations).
A similar conclusion may be drawn in the second case (Fig.~26). When 
$d\gg\sqrt{c_k}$, the strains and hydrostatic stress decrease monotonously 
with increasing $d$, i.e.~the characteristic behavior for the long-range 
disclination interaction for the gradient solution is the same as for the 
classical solution. When $d<10\sqrt{c_k}$, the strains and hydrostatic stress
vary non-monotonously with $d$ and becomes exactly zero at $d=0$, in contrast
to the classical solution which gives a monotonous singular dependence on $d$.
                                                \par\bigskip\noindent
{\bf 4.2.2. Second-type twist disclinations}
                                                \par\bigskip\noindent
Next, we consider a dipole of twist disclinations having the Frank vectors
$\pm\mbox{\boldmath $\omega $}=(0,\pm\omega_y,0)$. The elastic strain (stress)
components of such a dipole are given by the simple superpositions of the 
terms associated with $\omega_y$ in (66) and (68) ((67) and (69)), and similar
terms taken with the opposite sign [58,\ 63]. 
\par
Near the line of the positive disclination, at $r\to 0$, we have for strains
(in units of $\omega_y/[4\pi (1-\nu )]$) [58]
\begin{eqnarray}
   & &  \varepsilon _{xx}\,|_{r\to 0} = \varepsilon _{yy}\,|_{r\to 0}
        = \varepsilon\,|_{r\to 0}
        = \varepsilon _{xz}\,|_{r\to 0} = 0\;,\;\;\;\;\;\;\;\;
        \varepsilon _{xy}\,|_{r\to 0} = 
         z\left\{ - \frac{1}{d} + \frac{4c_2}{d^3}  
           - \frac{2}{d}\,K_2\left(\frac{d}{\sqrt{c_2}}\right)\right\}\;,
                                                     \nonumber \\ % (109)->
  & &  \varepsilon _{yz}\,|_{r_1\to 0} = 
          \frac{1}{2} + (1-2\nu )\left(\gamma +\ln\frac{d}{2\sqrt{c_2}}\right)
        - \frac{2c_2}{d^2}
        + (1-2\nu )\,K_0\left(\frac{d}{\sqrt{c_2}}\right)
        + K_2\left(\frac{d}{\sqrt{c_2}}\right),                % (110)-> (74)
\end{eqnarray}
and for stresses (in units of $\mu\omega_y/[2\pi (1-\nu )]$) [63]
\begin{eqnarray}
   & & \sigma_{xx}\,|_{r\to 0} = \sigma_{yy}\,|_{r\to 0} = 
       \sigma_{zz}\,|_{r\to 0} = \sigma_{xz}\,|_{r\to 0} = 0\;,
                                                               \nonumber \\
   & & \sigma_{xy}\,|_{r\to 0} = 
       \varepsilon_{xy}(c_2\leftrightarrow c_1)|_{r\to 0}\;,\;\;\;\;\;\;\;
       \sigma_{yz}\,|_{r\to 0} = 
       \varepsilon_{yz}(c_2\leftrightarrow c_1)|_{r\to 0}\;.    % (13)-> (75)
\end{eqnarray}
\par
We see again that the elastic strains and stresses are finite at the 
disclination line, in contrast to the classical solutions (66) and (67) which 
are singular there. The values of the strain (stress) components 
$\varepsilon _{xy}$ and $\varepsilon _{yz}$ ($\sigma_{xy}$ and $\sigma_{yz}$)
at the disclination line depend on the dipole arm $d$.
\par
For $d\gg\sqrt{c_k}$ (long-range disclination interaction), the strain
components read (in units of $\omega_y/[4\pi (1-\nu )]$) [58]
\begin{eqnarray}
    & &   \varepsilon _{xx}\,|_{r\to 0} = \varepsilon _{yy}\,|_{r\to 0} =
          \varepsilon\,|_{r\to 0} = \varepsilon _{xz}\,|_{r\to 0} = 0\;,
                                                         \nonumber \\
    & &   \varepsilon _{xy}\,|_{r\to 0} = - \frac{z}{d}\;,\;\;\;\;\;\;\;\;   
          \varepsilon _{yz}\,|_{r\to 0} = \frac{1}{2} 
            + (1-2\nu )\left(\gamma +\ln\frac{d}{2\sqrt{c_2}}\right),  
                                                                % (113)-> (76)
\end{eqnarray}
and the stress components are (in units of $\mu\omega_y/[2\pi (1-\nu )]$)
\begin{eqnarray}
   & & \sigma_{xx}\,|_{r\to 0} = \sigma_{yy}\,|_{r\to 0} = 
       \sigma_{zz}\,|_{r\to 0} = \sigma_{xz}\,|_{r\to 0} = 0\;,
                                                               \nonumber \\
   & & \sigma_{xy}\,|_{r\to 0} = - \frac{z}{d}\;, \;\;\;\;\;\;\;\;
       \sigma_{yz}\,|_{r\to 0} = 
       \varepsilon_{yz}(c_2\leftrightarrow c_1)|_{r\to 0}\;.            % (77)
\end{eqnarray}
\par
The distribution of elastic strains and stresses near the disclination line 
in this case is similar to the case considered in the previous subsection. In 
fact, since in the limiting case of large $d$, the strain (stress) components 
for the $\omega_x$-disclination transform into the strain (stress) components
for the $\omega_y$-disclination by the simple interchange of $x$ and $y$,
Fig.~23{\it a} may be viewed as representing the strain (stress) component
$\varepsilon_{yy}(0,y)$,
Fig.~23{\it b} --- $\varepsilon_{xx}(0,y)$,
Fig.~23{\it c} --- $\varepsilon     (0,y)$ ($\sigma     (0,y)$),
Fig.~23{\it d} --- $\varepsilon_{xy}(x,0)$ ($\sigma_{xy}(0,y)$), and
Fig.~23{\it e} --- $\varepsilon_{yz}(0,y)$ ($\sigma_{yz}(0,y)$), with the 
appropriate substitution of $\omega_x$ by $\omega_y$ in the measured units. 
In this case, the solid lines represent the gradient solution, while the 
dashed lines represent the classical one. 
\par
In a similar way, the general view on the distribution of elastic strains and 
stresses given by classical and gradient elasticity for 
$\omega_y$-disclinations, may be seen in Fig.~24 with the interchange of $x$- 
and $y$-axes. As a result, 
Fig.~24{\it a} may represent the strain component
$\varepsilon_{yy}$, 
Fig.~24{\it b} --- $\varepsilon_{xx}$ 
Fig.~24{\it c} --- $\varepsilon     $ ($\sigma     $),
Fig.~24{\it d} --- $\varepsilon_{xy}$ ($\sigma_{xy}$),
Fig.~24{\it e} --- $\varepsilon_{yz}$ ($\sigma_{yz}$), and
Fig.~24{\it f} --- $\varepsilon_{xz}$ ($\sigma_{xz}$), with the appropriate 
substitution of $\omega_x$ by $\omega_y$ in the measured units. The top 
pictures represent again the classical solution, while the bottom pictures 
represent the gradient solution. 
\par
In this case also, the absence of classical singularities permits to 
investigate short-range elastic interactions between disclinations. Again,
one can observe the 
strained states at two characteristic points of the disclination dipole,
i.e.~at the disclination line ($0,0$) and at the middle of the dipole 
($-d/2,0$). In the first case, one can use equations (74) and (75) which are
represented graphically in Fig.~27. Two non-vanishing strain (stress) 
components $\varepsilon_{xy}$ ($\sigma_{xy}$) --- ({\it 1}) and 
$\varepsilon_{yz}$ ($\sigma_{yz}$) --- ({\it 2}) 
vary non-monotonously with the variation of $d$ within the interval 
$d<10\sqrt{c_k}$ and are equal to zero at $d=0$ (annihilation of 
disclinations). A similar conclusion may be drawn in the second case 
(Fig.~28) for the only one non-vanishing strain (stress) component 
$\varepsilon_{xy}$ ($\sigma_{xy}$). When $d\gg\sqrt{c_k}$,
the strain (stress) values decrease monotonously with increasing $d$, 
i.e.~the characteristic behavior for the long-range disclination interaction 
for the gradient solution is the same as for the classical solution. When
$d<10\sqrt{c_k}$, the strain (stress) varies non-monotonously with $d$ and 
is equal to zero at $d=0$, in contrast to the classical solution which
gives a monotonous singular dependence on $d$.
                                                \par\bigskip\noindent
{\bf 4.2.3. Wedge disclinations}
                                                \par\bigskip\noindent
Consider a dipole of wedge disclinations having the Frank vectors
$\pm\mbox{\boldmath $\omega $}=(0,0,\pm\omega_z)$. The elastic strain (stress)
components of such a dipole are given by the simple superpositions of the 
terms associated with $\omega_z$ in (66) and (68) ((67) and (69)), and similar
terms taken with the opposite sign [58,\ 63]. 
\par
Near the line of the positive disclination, at $r\to 0$, the strains result in
(in units of $\omega_z/[4\pi (1-\nu )]$) [58]
\begin{eqnarray}
   & &  \varepsilon_{xx}\,|_{r\to 0} = \phantom{-}
          \frac{1}{2} - (1-2\nu )\left(\gamma +\ln\frac{d}{2\sqrt{c_2}}\right)
        - \frac{2c_2}{d^2}
        - (1-2\nu )\,K_0\left(\frac{d}{\sqrt{c_2}}\right)
        + K_2\left(\frac{d}{\sqrt{c_2}}\right),            
                                                   \nonumber \\ % (114)->
   & &  \varepsilon_{yy}\,|_{r\to 0} = 
        - \frac{1}{2} - (1-2\nu )\left(\gamma +\ln\frac{d}{2\sqrt{c_2}}\right)
        + \frac{2c_2}{d^2}
        - (1-2\nu )\,K_0\left(\frac{d}{\sqrt{c_2}}\right)
        - K_2\left(\frac{d}{\sqrt{c_2}}\right),            
                                                   \nonumber \\ % (115)->
   & & \varepsilon_{xy}\,|_{r\to 0} = \phantom{-}\,0\;,\;\;\;\;\;\;\;\;
       \varepsilon     \,|_{r\to 0} = 
        -\;2(1-2\nu )\left\{\gamma +\ln\frac{d}{2\sqrt{c_2}}
           + K_0\left(\frac{d}{\sqrt{c_2}}\right)\right\}\;,    % (117)-> (78)
\end{eqnarray}
and the stresses are given (in units of $\mu\omega_z/[2\pi (1-\nu )]$) by [63]
\begin{eqnarray}
   & & \sigma_{xx}\,|_{r\to 0} = 
           \varepsilon_{xx}(\nu =0,c_2\leftrightarrow c_1)|_{r\to 0}\;, 
                                                      \;\;\;\;\;\;\;\;
       \sigma_{yy}\,|_{r\to 0} = 
           \varepsilon_{xx}(\nu =0,c_2\leftrightarrow c_1)|_{r\to 0}\;, 
                                                               \nonumber \\
   & & \sigma_{xy}\,|_{r\to 0} = 
         \varepsilon_{xy}(c_2\leftrightarrow c_1)|_{r\to 0}\;,\;\;\;\;\;\;\;\;
       \sigma _{zz}\,|_{r\to 0} = 
       - 2\nu\left\{ \gamma + \ln\frac{d}{2\sqrt{c_1}}
                   + \,K_0\left(\frac{d}{\sqrt{c_1}}\right)\right\}\;. % (79)
\end{eqnarray}
\par
One can see that the elastic strains and stresses are finite at the wedge 
disclination line as is the case with twist disclinations, in contrast to the 
classical solutions (66) and (67) which are singular there. The values of the
strain and stress components at the disclination line depend again on the 
dipole arm $d$.
\par
For $d\gg\sqrt{c_k}$ (long-range disclination interaction), the strain
components transform (in units of $\omega_z/[4\pi (1-\nu )]$) into [58]
\begin{eqnarray}
    & & \varepsilon_{xx}\,|_{r\to 0} = \frac{1}{2} 
            - (1-2\nu )\left(\gamma +\ln\frac{d}{2\sqrt{c_2}}\right),
                                                         \;\;\;\;\;\;\;\;
        \varepsilon_{yy}\,|_{r\to 0} = - \frac{1}{2} 
            - (1-2\nu )\left(\gamma +\ln\frac{d}{2\sqrt{c_2}}\right),
                                                         \nonumber \\
    & & \varepsilon_{xy}\,|_{r\to 0} = 0\;,\;\;\;\;\;\;\;\;         
        \varepsilon     \,|_{r\to 0} =  
            -\;2(1-2\nu )\left(\gamma +\ln\frac{d}{2\sqrt{c_2}}\right),
                                                                % (121)-> (80)
\end{eqnarray}
and the stress components are (in units of $\mu\omega_z/[2\pi (1-\nu )]$)
\begin{eqnarray}
    & & \sigma_{xx}\,|_{r\to 0} = 
           \varepsilon_{xx}(\nu = 0,c_2\leftrightarrow c_1)|_{r\to 0}\;, 
                                                      \;\;\;\;\;\;\;\;
        \sigma_{yy}\,|_{r\to 0} = 
           \varepsilon_{xx}(\nu = 0,c_2\leftrightarrow c_1)|_{r\to 0}\;, 
                                                         \nonumber \\
    & & \sigma_{xy}\,|_{r\to 0} = 0\;,\;\;\;\;\;\;\;\;
        \sigma_{zz}\,|_{r\to 0} = 
        - 2\nu\left( \gamma + \ln\frac{d}{2\sqrt{c_1}}\right)\;.      % (81)
\end{eqnarray}
\par
Fig.~29 illustrates the distribution of elastic strains and hydrostatic stress
in the plane $y=0$ near the line of the positive disclination located at the 
point $(0,0)$ of Fig.~22 when $d=10^4\sqrt{c_k}$. The solid lines represent 
the gradient solution, while the dashed lines represent the classical one. One
can see that in the gradient elasticity, the elastic strains and stresses are 
finite at the disclination line, they achieve extreme values there and tend to
the classical solution far away from the disclination line.
\par
Fig.~30 provides a general view on the distribution of elastic strains and
stresses given by classical and gradient elasticity. The top pictures 
represent the classical solution, while the bottom pictures represent the 
gradient solution. Fig.~30 clearly demonstrates the elimination of classical 
singularities near the disclination line within the gradient theory.
\par
The short-range elastic interaction between wedge disclinations may be 
illustrated by observing the dependence of the strained state at 
the line of one of the dipole disclinations on the dipole arm $d$.
In doing so, one can use equations (78) and (79) which are represented
graphically in Fig.~31. It is seen that in the case of wedge disclinations,
the elastic strains and hydrostatic stress vary monotonously with $d$,
in contrast to the case of twist disclinations where they are non-monotonous
for $d<10\sqrt{c_k}$. Here, only the strain component $\varepsilon_{xx}$ 
achieves an extreme value at $d\approx\sqrt{c_2}$ but even this maximum has a
small value. However, the strain and stress components are also zero at $d=0$
(annihilation of disclinations), as is the case with twist disclinations.
\par
Another interesting feature when considering the short-range interaction
between wedge disclinations is the transformation of the elastic fields of a 
dipole of wedge disclinations into the elastic fields of an edge dislocation 
when the dipole arm $d$ becomes smaller than the scale unit $\sqrt{c_k}$.
Fig.~32 shows the distribution of the strain components 
$\varepsilon_{xx}$, $\varepsilon_{yy}$, and $\varepsilon $ 
(from top to bottom) for four subsequent positions of the negative wedge 
disclination 
$(-50\sqrt{c_2},0)$ --- ({\it a}), $(-5\sqrt{c_2},0)$ --- ({\it b}), 
$(-\sqrt{c_2},0)$ --- ({\it c}), and $(-0.01\sqrt{c_2},0)$ --- ({\it d}),
while the positive wedge disclination occupies the same position $(0,0)$.
Again, the solid lines represent the gradient solution, while the dashed lines
represent the classical one. We can see how the levels and profiles of the 
strain (stress) components are changing with decreasing $d$. The final 
pictures (Fig.~32{\it d}) give exactly the same strain distributions
as we have reported in [60] for an edge dislocation (see Fig.~9). In our 
co-ordinate system, this dislocation would have a Burgers vector 
$-b_y=-\omega_zd$. This means that in the gradient elasticity, edge 
dislocations may be modelled through dipoles of wedge disclinations, as is 
the case in classical elasticity [5--7].
                                                       \par\bigskip
Thus, within the gradient theory of elasticity described by (5), 
dipoles of straight disclinations of general type give zero or finite values 
for the elastic strains and stresses at the disclination lines. The finite 
values depend strongly on the dipole arm $d$ and show regular and monotonous 
(in the case of wedge disclinations) or non-monotonous (in the case of twist 
disclinations) behavior for short-range (when $d<10\sqrt{c_k}$) interactions 
between disclinations. When the disclinations annihilate ($d\to 0$), the 
elastic strains and stresses tend regularly to zero values. Far from the 
disclination lines ($r\gg 10\sqrt{c_k}$), gradient and classical solutions 
coincide. When the dipole arm $d$ is much smaller than the scale unit 
$\sqrt{c_k}$, the elastic fields of a dipole of wedge disclinations transform
into the elastic fields of an edge dislocation, as is the case in classical 
elasticity.
                                                \par\bigskip\bigskip\noindent
5. CONCLUSIONS
                                                \par\bigskip\bigskip\noindent
Thus consideration of dislocations and disclinations within the gradient 
theory of elasticity described by (5) results in a complete eliminations of 
singularities from the elastic fields and energies of dislocations as well as
from strains and stresses of disclinations at the defect lines. It has been 
shown that the elastic strains and stresses are strictly equal to zero at the
dislocation lines and achieve their extreme values of $\approx (3\div 14)$\% 
and $\approx (\mu /4\div\mu /2)$, respectively, at a distance $\approx a/4$ 
from the dislocation line. Two characteristic distances appear naturally in 
this approach: $r_0\approx 4\sqrt{c_2}$ which may be viewed as the radius of 
dislocation core and $d_0\approx 10\sqrt{c_2}$ which may be viewed as the 
radius of strong short-range interaction between dislocations. In considering
dislocations near interfaces, it has been shown that all stress components 
remain continuous across the interface, in contrast to the well-known 
classical solution [69] where three (one for a screw dislocation and two for 
an edge dislocation) stress components suffer jump discontinuities there. 
Also, the classical singularity of the elastic ``image'' force acting upon 
the dislocation by the interface [69], is eliminated from the gradient 
solution. The ``image'' force remains finite and continuous at the interface
and has an additional short-range component due to the difference in the 
gradient coefficients of the media in contact. Under the action of this 
component, the screw (edge) dislocation tends to penetrate into the 
medium with the larger (smaller) gradient coefficient. In the general case 
where both the elastic constants $\mu_i$ and the gradient coefficients $c_i$ 
are different for these media, the total ``image'' force exhibits quite 
different behavior near the interface depending on the ratios $\mu_2/\mu_1$ 
and $c_2/c_1$, while its long-range component remains as in the classical 
theory of elasticity. In considering disclination dipoles, one can conclude 
that non-vanishing at the disclination lines strains and stresses depend 
strongly on the dipole arm and tend regularly to zero values when the 
disclinations annihilate. In general, the gradient solutions permit one to 
calculate strains and stresses directly near a dislocation/disclination line 
and to analyze short-range interactions in dense ensembles of defects. The
results reviewed can be of advantage when constructing physical models of the
structure and mechanical behavior of metallic glasses and nanostructured 
materials as well as of conventional metals and alloys under large plastic
deformations.
                                                \par\bigskip\bigskip\noindent
{\bf Acknowledgements:} This work was supported by INTAS-93-3213/Ext and 
TMR/ERB FMRX CT 960062 and, in part, by the Russian Research Council ``Physics
of Solid-State Nanostructures" (Grant 97-3006). The author would like to 
thank Professor E.C.~Aifantis and Dr.~K.N.~Mikaelyan for permanent 
collaboration, valuable contributions, fruitful discussions and 
encouragements. 
                                                \par\bigskip\bigskip\noindent
6. REFERENCES
                                                \par\bigskip\bigskip\noindent
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\newpage
                                                       \par\noindent
FIGURE CAPTIONS
                                                \par\bigskip\noindent
Figure 1. Total displacements around a screw dislocation in an infinite body. 
The curves $1_{\pm }$ and $1'_{\pm }$ denote schematically respectively the 
profiles of total displacements $u_z(x,y=0\pm )$ for the classical solution 
and $w_z(x,y=0\pm )$ for the gradient solution along the plane $y=0$.
                                                \par\medskip\noindent
Figure 2. The profiles of total displacements for a separate screw dislocation
($1_{\pm }$ --- classical solution, and $1'_{\pm }$ ---  gradient 
solution) and for a dislocation dipole with arm $d$ ($2_{\pm }$--$6_{\pm }$  
--- classical solution, and $2'_{\pm }$--$6'_{\pm }$ --- gradient 
solution). The ratio $d/\sqrt{c_2}$ is taken as 
$\infty $~($1_{\pm }$,$1'_{\pm }$), 1~($2_{\pm }$,$2'_{\pm }$), 
5~($3_{\pm }$,$3'_{\pm }$), 10~($4_{\pm }$,$4'_{\pm }$), 
15~($5_{\pm }$,$5'_{\pm }$), and 20~($6_{\pm }$,$6'_{\pm }$).
                                                \par\medskip\noindent
Figure 3. Total displacements near the line of screw dislocation within
          the gradient model ({\it 1}) and the Peierls-Nabarro model
          ({\it 2}) when $y\to +0$.
                                                \par\medskip\noindent
Figure 4. The coordinate-depended parts of strain  
          $\varepsilon_{xz}^0$, $\varepsilon_{xz}$ --- ({\it a}),
          $\varepsilon_{yz}^0$, $\varepsilon_{yz}$ --- ({\it b}), or stress
          $\sigma_{xz}^0$, $\sigma_{xz}$ --- ({\it a}),
          $\sigma_{yz}^0$, $\sigma_{yz}$ --- ({\it b}) components near the 
          line of a screw dislocation which goes through the point ($0,0$). 
          The strain values are given in units of $b_z/(4\pi\sqrt{c_2})$ while
          the stress values in units of $\mu b_z/(2\pi\sqrt{c_1})$. The top 
          figures give the classical solutions 
          $\varepsilon_{iz}^0,\,\sigma_{iz}^0$ while the bottom ones give the
          gradient solution $\varepsilon_{iz},\,\sigma_{iz}$, ($i=x,y$).
                                                \par\medskip\noindent
Figure 5. The dipole ({\it a}) and pair ({\it b}) of screw dislocations.
                                                \par\medskip\noindent
Figure 6. The coordinate-depended parts of strain $\varepsilon_{yz}(x,0)$ or 
          stress $\sigma_{yz}(x,0)$ near the dipole ({\it a,b}) and pair 
          ({\it c,d}) of screw dislocations for different values of the 
          interdislocation spacing $d/\sqrt{c_k}=$ 
          2~({\it a}), 10~({\it b}), 4~({\it c}), and 1({\it d}), where $k=2$
          corresponds to strain distribution while $k=1$ to stress 
          distribution. The strain 
          values are given in units of $b_z/(4\pi\sqrt{c_2})$ while the stress
          values in units of $\mu b_z/(2\pi\sqrt{c_1})$. The dashed curves 
          represent the classical solution.
                                                \par\medskip\noindent
Figure 7. The strain and stress components, $\varepsilon_{yz}(d/2,0)$ and
          $\sigma_{yz}(d/2,0)/(2\mu )$, in the middle of the dipole of screw 
          dislocations via the dipole arm $d/\sqrt{c_k}$ for the case 
          $b_z=4\sqrt{c_k}$, where $k=2$ corresponds to strain distribution
          while $k=1$ to stress distribution. The dashed curve represents the
          classical solution.
                                                \par\medskip\noindent
Figure 8. The $y$-components of total displacement of a dipole of edge
dislocations along the plane $y=0$ when the dipole arm $d$ is equal to 
$1\sqrt{c_2}$ ({\it a}), $10\sqrt{c_2}$ ({\it b}) and $100\sqrt{c_2}$ 
({\it c}); 1---classical solution $u_y^0(x,y=0)$, 1$'$---gradient solution 
$u_y(x,y=0)$. The displacement values are given in units of 
$b_x/[4\pi (1-\nu )]$.
                                                \par\medskip\noindent
Figure 9. The strain components ({\it a})
$\varepsilon _{xx}^0(x=0,y)\equiv\varepsilon _{yy}^0(x=0,y)$---1,
\ \  $\varepsilon _{xx}(x=0,y)$---1$'$,\ \ 
$\varepsilon _{yy}(x=0,y)$---2$'$; \ \ ({\it b})  
$\varepsilon ^0 (x=0,y)$---1,\ \ $\varepsilon (x=0,y)$---1$'$; ({\it c})
$\varepsilon _{xy}^0(x,y=0)$---1,
\ \ $\varepsilon _{xy}(x,y=0)$---1$'$\ \ near the dislocation 
line. The strain values are given in units of $b_x/[4\pi (1-\nu )\sqrt{c_2}]$.
                                                \par\medskip\noindent
Figure 10. The strain components $\varepsilon_{xx}$ --- ({\it a}),
           $\varepsilon_{yy}$ --- ({\it b}), $\varepsilon_{xy}$ --- ({\it c}),
           and $\varepsilon $ --- ({\it d})  near the line of an edge
           dislocation which goes through the point ($0,0$). The strain 
           values are given in units of $b_x/(4\pi (1-\nu )\sqrt{c_2}$. The 
           top figures give the classical singular solutions while the
           bottom ones give the gradient regular solutions.
                                                \par\medskip\noindent
Figure 11. The stress components $\sigma_{xx}^0$, $\sigma_{xx}$ --- ({\it a})
           and $\sigma_{yy}^0$, $\sigma_{yy}$ --- ({\it b}) near the line of 
           an edge dislocation which goes through the point ($0,0$). The 
           stress values are given in units of 
           $\mu b_x/[2\pi (1-\nu )\sqrt{c_1}]$. The top figures give the 
           classical solutions $\sigma_{ij}^0$ while the bottom ones give the
           gradient solution $\sigma_{ij}$.
                                                \par\medskip\noindent
Figure 12. A straight dislocation near a flat interface.
                                                \par\medskip\noindent
Figure 13. Stress field ($\sigma_{yz}$) map for a screw 
           dislocation located at the point $(10,0)$ near a flat interface 
           (at $x=0$) separating two elastic media with 
           $\mu_2=2\mu_1$ and $c_1=c_2=c$. The stresses are given in units of 
           $\mu_1b_z/(2\pi\sqrt{c})$. The dashed contours represent the 
           classical solution $\sigma_{yz}^0$.
                                                \par\medskip\noindent
Figure 14. The stress component $\sigma_{yz}(x,y=0)$ near the line of a screw 
           dislocation located at a distance $x'/\sqrt{c}=10$~(a), 5~(b),
           2~(c) and 0~(d) from the interface (at $x=0$) of two elastic media 
           with $\mu_2=10\mu_1$ and $c_1=c_2=c$. The stress values are given 
           in units of $\mu_1b_z/(2\pi\sqrt{c})$. The dashed curves represent 
           the classical solution $\sigma_{yz}^0$.
                                                \par\medskip\noindent
Figure 15. The ``image'' force $F_x^{el}$ which acts upon the dislocation
           unit length due to the interface (at $x=0$) of two elastic media 
           with $c_1=c_2=c$ and $\mu_2/\mu_1=10$, $7$, $5$, $3$ and $0$ (from 
           top to bottom), as a function of 
           the dislocation position $x'/\sqrt{c}$. The force values are given 
           in units of $\mu_1b_z^2/(2\pi\sqrt{c})$. The dashed curves 
           represent the classical solution.
                                                \par\medskip\noindent
Figure 16. The ``image'' force $F_x^{gr}$ which acts upon the dislocation
           unit length due to the interface (at $x=0$) of two elastic media 
           with $\mu_2=\mu_1$ and $c_2/c_1=0.5$, $0.7$, $0.9$, $1.1$, $1.5$ 
           and $2$ (from top to bottom), as a function of the dislocation 
           position $x'/\sqrt{c_1}$ (a). The values of the image force 
           at the interface $F_x^{gr}(x'=0)$ as a function of the ratio 
           $c_2/c_1$~(b). The force values are given in units of 
           $\mu_1b_z^2/(2\pi\sqrt{c_1})$.
                                                \par\medskip\noindent
Figure 17. The general ``image'' force $F_x$ which acts upon the 
           dislocation unit length due to the interface (at $x=0$) of two 
           elastic media with 
           $\mu_2=3\mu_1$ and $c_2/c_1=0.3$, $0.5$, $0.7$, $0.9$, $1$, $2$, 
           $3$ and $5$ (from top to bottom), as a function of the dislocation 
           position $x'/\sqrt{c_1}$. The force values are given in units of 
           $\mu_1b_z^2/(2\pi\sqrt{c_1})$. The dashed curves represent the 
           classical solution.
                                                \par\medskip\noindent
Figure 18. The stress component $\sigma_{yy}(x,y=0)$ near the line of an
           edge dislocation with the Burgers vector $b_y$ located at distances
           $x'/\sqrt{c}=10$~(a), 5~(b), 2~(c) and 0~(d) from the interface 
           (at $x=0$) of two elastic media with $\mu_2=10\mu_1$, 
           $\nu_1=\nu_2=0.3$ and $c_1=c_2=c$. The stress values are given in 
           units of $\mu_1b_y/[\pi (k_1+1)\sqrt{c}]$. The dashed curves 
           represent the classical solution $\sigma_{yy}^0$.
                                                \par\medskip\noindent
Figure 19. The ``image'' force $F_x^{el}$ which acts upon the dislocation
           unit length by the interface (at $x=0$) of two bonded media with 
           $c_1=c_2=c$, $\nu_1=\nu_2=0.3$ and $\mu_2/\mu_1=10$, $7$, $5$, $3$ 
           and $0$ (from top to bottom) as a function of
           the dislocation position $x'/\sqrt{c}$. The force values are given 
           in units of $\mu_1b_x^2/[\pi (k_1+1)\sqrt{c}]$. The dashed curves 
           represent the classical solution.
                                                \par\medskip\noindent
Figure 20. The ``image'' force $F_x^{gr}$ which acts upon the dislocation
           unit length by the interface (at $x=0$) of two bonded media with 
           $\mu_2=\mu_1$, $\nu_1=\nu_2=0.3$ and $c_2/c_1=2$, $1.5$, $1.1$, 
           $0.9$, $0.7$ and $0.5$ (from top to bottom) as a function of the 
           dislocation position $x'/\sqrt{c_1}$. The force values are given in
           units of $\mu_1b_x^2/[\pi (k_1+1)\sqrt{c_1}]$.
                                                \par\medskip\noindent
Figure 21. The general ``image'' force $F_x$ which acts upon the dislocation 
           unit length by the interface (at $x=0$) of two bonded media with 
           $\mu_2=3\mu_1$, $\nu_1=\nu_2=0.3$ and 
           $c_2/c_1=5$, $3$, $2$, $1$, $0.9$, $0.7$, $0.5$ and $0.3$ (from top
           to bottom) as a function of the dislocation position 
           $x'/\sqrt{c_1}$. The force values are given in units of 
           $\mu_1b_x^2/[\pi (k_1+1)\sqrt{c_1}]$. The dashed curves represent 
           the classical solution.
                                                \par\medskip\noindent
Figure 22. A dipole of straight disclinations having the Frank vectors
           $\pm\mbox{\boldmath $\omega $}$. 
                                                \par\medskip\noindent
Figure 23. The components of strains
           $\varepsilon_{xx}(x,0)$ --- ({\it a});
           $\varepsilon_{yy}(x,0)$ --- ({\it b});
           $\varepsilon     (x,0)$ --- ({\it c});
           $\varepsilon_{xy}(0,y)$ --- ({\it d});
           $\varepsilon_{xz}(x,0)$ --- ({\it e}); and stresses
           $\sigma     (x,0)$ --- ({\it c});
           $\sigma_{xy}(0,y)$ --- ({\it d});
           $\sigma_{xz}(x,0)$ --- ({\it e}) near the line of the positive 
           first-type twist disclination when the dipole arm 
           $d=10^4\sqrt{c_k}$. The strain values are given in units of 
           $\omega_xz/[4\pi (1-\nu )\sqrt{c_2}]$ --- ({\it a--d}) and 
           $\omega_x/[4\pi (1-\nu )]$ --- ({\it e}), and the stress values,
           in units of 
           $\mu\omega_xz(1+\nu )/[6\pi (1-\nu )(1-2\nu )\sqrt{c_1}]$ --- 
           ({\it c}); $\mu\omega_xz/[2\pi (1-\nu )\sqrt{c_1}]$ --- ({\it d}),
           and $\mu\omega_x/[2\pi (1-\nu )]$ --- ({\it e}). The 
           dashed curves represent the classical solutions 
           $\varepsilon_{ij}^0$ and $\sigma_{ij}^0$.
                                                \par\medskip\noindent
Figure 24. The strain components 
           $\varepsilon_{xx}^0$, $\varepsilon_{xx}$ --- ({\it a});
           $\varepsilon_{yy}^0$, $\varepsilon_{yy}$ --- ({\it b});
           $\varepsilon     ^0$, $\varepsilon     $ --- ({\it c});
           $\varepsilon_{xy}^0$, $\varepsilon_{xy}$ --- ({\it d});
           $\varepsilon_{xz}^0$, $\varepsilon_{xz}$ --- ({\it e});
           $\varepsilon_{yz}^0$, $\varepsilon_{yz}$ --- ({\it f}); 
           and stresses
           $\sigma^0     $, $\sigma     $ --- ({\it c});
           $\sigma^0_{xy}$, $\sigma_{xy}$ --- ({\it d});
           $\sigma^0_{xz}$, $\sigma_{xz}$ --- ({\it e});
           $\sigma^0_{yz}$, $\sigma_{yz}$ --- ({\it f}),
           near the line of the positive first-type twist disclination when 
           the dipole arm $d=10^4\sqrt{c_k}$. The strain values are given in 
           units of 
           $\omega_xz/[4\pi (1-\nu )\sqrt{c_2}]$ --- ({\it a--d}) and 
           $\omega_x/[4\pi (1-\nu )]$ --- ({\it e,f}), and the stress values,
           in units of 
           $\mu\omega_xz(1+\nu )/[6\pi (1-\nu )(1-2\nu )\sqrt{c_1}]$ --- 
           ({\it c}); $\mu\omega_xz/[2\pi (1-\nu )\sqrt{c_1}]$ --- ({\it d}),
           and $\mu\omega_x/[2\pi (1-\nu )]$ --- ({\it e,f}).
           The top figures give the classical solutions $\varepsilon_{ij}^0$ 
           and $\sigma^0_{ij}$ while the bottom ones give the gradient 
           solutions $\varepsilon_{ij}$ and $\sigma_{ij}$.
                                                \par\medskip\noindent
Figure 25. The strain (stress) components 
           $\varepsilon_{xx}$ --- ({\it 1}),
           $\varepsilon_{yy}$ --- ({\it 2}),
           $\varepsilon     $ ($\sigma $) --- ({\it 3}),
           $\varepsilon_{xz}$ ($\sigma_{xz}$) --- ({\it 4}), and
           $\varepsilon_{xy}(\sigma_{xy})=\varepsilon_{yz}(\sigma_{yz})=0$ 
           --- ({\it 5}) at the line of the positive first-type twist 
           disclination via the dipole arm $d$. The dashed line~({\it 5}) 
           gives the zero level. The strain values are given in units of 
           $\omega_xz/[4\pi (1-\nu )\sqrt{c_2}]$ --- ({\it 1--3}) and 
           $\omega_x/[4\pi (1-\nu )]$ --- ({\it 4}), while the stress values,
           in units of
           $\mu\omega_xz(1+\nu )/[6\pi (1-\nu )(1-2\nu )\sqrt{c_1}]$ --- 
           ({\it 3}) and $\mu\omega_x/[2\pi (1-\nu )]$ --- ({\it 4}). 
                                                \par\medskip\noindent
Figure 26. The strain (hydrostatic stress) components 
           $\varepsilon_{xx}^0$, $\varepsilon_{xx}$ --- ({\it a}),
           $\varepsilon_{yy}^0$, $\varepsilon_{yy}$ --- ({\it b}), and
           $\varepsilon^0$, $\varepsilon $ ($\sigma^0$, $\sigma $) --- 
           ({\it c}) in the middle of the dipole of first-type twist 
           disclinations (the point ($-d/2,0$)) via the dipole arm $d$. The 
           dashed curves represent the classical solution 
           $\varepsilon_{ij}^0$ ($\sigma_{ij}^0$). The strain values are 
           given in units of $\omega_xz/[4\pi (1-\nu )\sqrt{c_2}]$, while the
           stress values, in units of
           $\mu\omega_xz(1+\nu )/[6\pi (1-\nu )(1-2\nu )\sqrt{c_1}]$.
                                                \par\medskip\noindent
Figure 27. The strain (stress) components 
           $\varepsilon_{xy}$ ($\sigma_{xy}$) --- ({\it 1}),
           $\varepsilon_{yz}$ ($\sigma_{yz}$) --- ({\it 2}), and
           $\varepsilon_{xx}(\sigma_{xx})=\varepsilon_{yy}(\sigma_{yy})=
          \varepsilon (\sigma )=\varepsilon_{xz}(\sigma_{xz})=0$ --- ({\it 3})
           at the line of the positive second-type twist disclination via the
           dipole arm $d$. The strain values are given in units of 
           $\omega_yz/[4\pi (1-\nu )\sqrt{c_2}]$ --- ({\it 1}) and 
           $\omega_y/[4\pi (1-\nu )]$ --- ({\it 2}), while the
           stress values, in units of 
           $\mu\omega_yz/[2\pi (1-\nu )\sqrt{c_1}]$ --- ({\it 1}) and 
           $\mu\omega_y/[2\pi (1-\nu )]$  --- ({\it 2}).
                                                \par\medskip\noindent
Figure 28. The only non-vanishing strain (stress) components 
           $\varepsilon_{xy}^0$ ($\sigma_{xy}^0$) (the dashed curve) and 
           $\varepsilon_{xy}$ ($\sigma_{xy}$) (the solid curve) in the middle
           of the dipole of second-type twist disclinations (the 
           point ($-d/2,0$)) via the dipole arm $d$. The strain values are
           given in units of $\omega_yz/[4\pi (1-\nu )\sqrt{c_2}]$, while the
          stress values, in units of $\mu\omega_yz/[2\pi (1-\nu )\sqrt{c_1}]$.
                                                \par\medskip\noindent
Figure 29. The strain (hydrostatic stress) components 
           $\varepsilon_{xx}(x,0)$ --- ({\it a}),
           $\varepsilon_{yy}(x,0)$ --- ({\it b}), and
           $\varepsilon     (x,0)$ ($\sigma (x,0)$)--- ({\it c}) near the 
           line of the positive wedge disclination when the dipole arm 
           $d=10^4\sqrt{c_k}$. The strain values are given in units of 
           $\omega_z/[4\pi (1-\nu )]$, while the stress values, in units of 
           $\mu\omega_z(1+\nu )/[6\pi (1-\nu )(1-2\nu )]$. The 
           dashed curves represent the classical solutions 
           $\varepsilon_{ij}^0$ ($\sigma^0$). 
                                                \par\medskip\noindent
Figure 30. The strain (stress) components 
           $\varepsilon_{xx}^0$, $\varepsilon_{xx}$ --- ({\it a}),
           $\varepsilon_{yy}^0$, $\varepsilon_{yy}$ --- ({\it b}),
           $\varepsilon^0$, $\varepsilon $ ($\sigma^0$, $\sigma $) --- 
           ({\it c}), and
           $\varepsilon_{xy}^0$, $\varepsilon_{xy}$ ($\sigma_{xy}^0$, 
           $\sigma_{xy}$) --- ({\it d}),
           near of the line of the positive wedge disclination when the 
           dipole arm $d=10^4\sqrt{c_k}$. The strain values are given in units
           of $\omega_z/[4\pi (1-\nu )]$, while the stress values, in units of
           $\mu\omega_z(1+\nu )/[6\pi (1-\nu )(1-2\nu )]$ --- 
           ({\it c}), and $\mu\omega_z/[2\pi (1-\nu )]$ --- ({\it d}). The top
           figures give the classical solution 
           $\varepsilon_{ij}^0$ ($\sigma_{ij}^0$) while the bottom ones give 
           the gradient solution $\varepsilon_{ij}$ ($\sigma_{ij}$).
                                                \par\medskip\noindent
Figure 31. The strain (stress) components 
           $\varepsilon_{xx}$ --- ({\it 1}),
           $\varepsilon_{yy}$ --- ({\it 2}), 
           $\varepsilon     $ ($\sigma $) --- ({\it 3}), and
           $\varepsilon_{xy}(\sigma_{xy})=0$ --- ({\it 4}) at the line of the
           positive wedge disclination via the dipole arm $d$. The strain 
           values are given in units of $\omega_z/[4\pi (1-\nu )]$, while the
           stress values, in units of
           $\mu\omega_z(1+\nu )/[6\pi (1-\nu )(1-2\nu )]$.
                                                \par\medskip\noindent
Figure 32. The transformation of the strain components $\varepsilon_{xx}$,
           $\varepsilon_{yy}$, and $\varepsilon $ (from top to bottom) of 
           a dipole of wedge disclinations into those of an edge 
           dislocation ({\it d}) having the Burgers vector 
           $-b_y=-\omega_zd$, when the dipole arm $d$ becomes smaller 
           than the scale unit $\sqrt{c_2}$. The dipole arm $d$ for four 
           subsequent positions of the negative wedge disclination is 
           equal to $50\sqrt{c_2}$ --- ({\it a}), $5\sqrt{c_2}$ --- ({\it b}),
           $\sqrt{c_2}$ --- ({\it c}), and $0.01\sqrt{c_2}$ --- ({\it d}).
           The strain values are given in units of 
           $\omega_z/[4\pi (1-\nu )]$. The dashed curves represent the 
           classical solution $\varepsilon_{ij}^0$.


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