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\par\noindent
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\par\vspace{2cm}\noindent
\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
$c_20$ 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 $c_2c_1$, 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 (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 $c_20$ 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 (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
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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 $\varepsilon_{ij}^0$.
\end{document}