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\begin{center}
{\large\bf CATASTROPHIC MICRODEFORMATIONS in CRYSTALLINE LATTICE.
STRUCTURE STABILITY and MODIFICATIONS.}
\vspace{1em}
{\large\bf E.L.Aero }
\vspace{1em}
Institute of Problems of Mechanical Engineering,
Russian Academy of Sciences, Saint-Petersburg, 199178.
e-mail: aero@microm.ipme.ru
\end{center}
\vspace{2em}
Essentially nonlinear theory of two dimensional lattice subjected to
\addtolength{\baselineskip}{2.mm}
intensive shear is presented. Two branches of deformations (acoustic and
pseudo optical) are considered. The deformation energy is shown to
consist of periodic and gradient terms. The equilibrium
equation in the sine-Helmholtz form is exactly solved. It demonstrates
some effects of bifurcations. he first, when homogeneous macrodeformation
is transformed to nonhomogeneous one
and some superstructure with great periods and new translation order
is formed. The second bifurcation divides two deformed states-
elastic and elastoplastic one when the nearest atomic order is altered
and new modification of crystalline lattice is formed. Some criteria
of local and global structure stability are established.
\vspace{2em}
1. GENERAL EQUATIONS.
One of the postulates in the continuum theory of deformations is that local
topology remains unchanged in the course of structural transformations.
The nearest environment is conserved without any rearrangement
of interatomic bonds. But sometimes so called martensite
transitions take place. Then lattice structure is modified and
polymorphic (bimorphic)
transitions are possible due to cooperative but not to diffusion atomic
rearrangements. Another example is twinning or double formation. Giving up
these and the other cases when classical model restrictions are not good it is
worthwhile to combine a continual approach with a discrete, lattice model.
Suppose that discrete degrees of freedom are presented by a specific lattice
potential - a periodic function of interatomic displacements $u$ rater than
their gradients. Interatomic or micro displacements are introduced in the
theory of
optical vibrations of ionic crystals too but only in infinitesimal approach. As for
the macro displacements $U$ of acoustic branch they correspond to a half
sum of atomic absolute displacements. A macroscopic field is obeyed to the
equations of continuum theory.
It is possible to present this model as
consisting of two mutually penetrating noncoherent sublattices with misfit
deformations. We have usual continuum when these sublattices are coherent
and in nonrecorded state they are divided by the constant value equal
to integer number of atomic periods, i.e. $u\,=\,n$. Here microdisplacement
$u$ is expressed in atomic units $b$. We are interesting in the case when
$u\,=\,u(x,y)$. If $u\,=\,1/2, \,U\,=\,0 $, then a lattice structure is in unstable state.
Transitions from continual state $u\,=\,0,\, U\,=\,U(x,y)$ to demorphic state
$U\,=\,U(x,y),\, u\,=\,u(x,y)$ may be realized due to bifurcation. If $u(x,y)
> 1/2$ then an effect of bond switching takes place. It may be accompanied
by local structure (morphology) transitions, defect formations and other
variations of lattice topology.
We shall develop a mesoscopic theory taking into account both degrees of freedom.
Some results are presented early in [1-3].
Let us introduce into a complete energy of deformation $\bar E$ both terms - atomic
periodic and continual gradient ones. Consider here the simplest example of
polyatomic layer with thickness $H$ subjected to shear along the $OY$ axis.
The structure of the layer is presented by a two-dimensional periodic lattice
with period $b$ and nonlinear interatomic interactions. Then the energy of a
layer portion of length $B$ is taking in the form
$$
\bar E = \int\nolimits_0^{H}\int\nolimits_0^B[(1/2)k_1(\partial u/\partial x-\varepsilon/b )^2 + (1/2)k_2(\partial u/\partial y)^2 + p(1 - \cos2\pi u)]\,dxdy.
\eqno (1.1)
$$
Here $u$ is the microdisplacement (in units of $b$) in the $Y$ direction.
Best of all $k_1\, and\, k_2$ are the shear micromodulus and the strength ones.
A positive value $p$ is the amplitude of the periodic inter particle potential
or cohesive force. As for the constant $\varepsilon$ it is macroscopic shear
deformation or x-gradient of macrodisplacement field $U$.
The third term in (1.1) represents the energy of a finite shear of rigid monolayers
(chains), and the first two terms account for their compliance (elasticity of
the 2D lattice in shear and tension) along the $Y$ axis. They represent continuum
forces, smooth on the mesoscopic scales defined below.
If we have rigid
monolayers then the simple condition when the field $u$ is constant means
nothing that the shear microdeformation is homogeneous and equal to
microdisplacement (in units of $b$) but not equal to macrodeformation
$\varepsilon$.
The equilibrium equation, corresponding to a minimum in functional (1.1) (when
$\varepsilon$ is constant) has the form
$$
k_1(\partial^2u/\partial x^2) + k_2(\partial^2u/\partial y^2) - p\sin2\pi u = 0
\eqno (1.2)
$$
This nonlinear equation, known as the sine-Helmholtz one, defines two
characteristic coherent lengths of the lattice
$$
l_1 = \sqrt {k_1/p} , \qquad l_2 = \sqrt {k_2/p}.
\eqno (1.3)
$$
They determine the mesoscopic length scales appearing in the present theory
in contrast to scaleless continuum theory. For $l_1\to 0$, we arrive at the
Frenkel- Kontorova one dimensional model - elastic chain of atoms on a periodic
rigid substrate. Our model is concerned with a system of parallel
chains interacting with each other but not with substrate which may be
absent at all. They can slip relative to one another and deform continuously.
We can see that in two dimension lattice single Frenkel-Kontorova s solitons
coalesce to special domain boundaries.
In fact we concern with high-gradient fields.
Macroscopic, low-gradient fields can be approximated (with very high accuracy),
to within one interatomic
distance, as an expression $ U\,=\,N b$, where $N$ is a grate integer.
It may be a weak discrete function of $x,y$. This approximation may be called as quantization
of macrofield. If it is assumed then macrofield is obeyed not only to continuum
equilibrium equation but the equation (1.2) too. Indeed, $sin(2\pi Nb) = 0$
and therefore (1.2) transits to the equation
of the continuum theory of elasticity which is satisfied. But microfield $u$ is
obeyed to the
equation (1.2) too. Consequently both are obeyed to (1.2). It means that there is
some bifurcation point of coexistence of these two modes of deformation.
Let us take into account a maximum
in the functional (1.1) also. Corresponding condition differs from (1.2) by a sign
before sine term, i.e.
$$
k_1(\partial^2u_+ /\partial x^2) + k_2(\partial^2u_+/\partial y^2) + p\sin2\pi u_+ = 0.
\eqno(1.4)
$$
$$
u_-(x,y)\,=\,u_+(x,y) + 1/2.
\eqno(1.5)
$$
Here $u_+\, and\, u_-$ are solutions of the equations (1.4) and (1.2) correspondently.
One has obvious relation (1.5) between them which means that these two fields
differ by a half of lattice period $b$. As a result, the lattice passes from a
stable to unstable configuration - some atoms shift from local wells to tops
of a potential atomic relief and the others, on the contrary, from tops to wells.
It is formally corresponds to the change in sign of interatomic forces. Clearly
that these processes are accompanied by bond rearrangement and changes in the
nearest neighbor environment. It is impossible in the continuum mechanics.
The second (exited) configuration may be stabilized indeed due to some boundary
conditions. Therefore both will be considered.
\vspace{1em}
2. ELASTIC CRITICAL SHEAR.
At the beginning we shall resolve these equations. Consider first of all a
critical elastic shear when plastic deformations
are absent. Let it reaches a maximum value on the boundaries such as
$$
u_-(0,y)=-1/2,\qquad u_-(nH,y)=(2n-1)/2,\qquad n=+1,2,...
\eqno(2.1)
$$
Clearly, we have interchain rigid shear (on the boundaries) by a half interatomic
distance. Inside of the layer of thickness H, interchain shear is less and
it becomes zero somewhere inside. We have elastic shear because $u\,<\,1/2$.
The solution, obeyed to this boundary conditions, is presented as (1.5) where
$u_+$ is a solution of the equation (1.4). The last may be resolved by separating
method after Lamb [3] that give us following doubly periodic function
$$
tang(\pi u_+/2)\,=\,+ tn(xK_1/H)\, dn(yK_2/B)/A cn(yK_2/B),\qquad A^2=\nu_2/\sqrt{1-\nu_1^2}.
\eqno(2.2)
$$
Two signs correspond to two opposite equivalent directions of microshear.
Here H and B are one-forth periods of the functions on the right-hand side, i.e.
the elliptic tangent (tn) and the elliptic cosine (cn). As for the delta-function of
Jacoby (dn), it has the period equals to 2B.
Best of all, $K_1\, and\, K_2$ are the complete elliptic integrals of the first kind,
dependent on the magnitudes of $\nu_1,\,\nu_2$ in a usual manner. Asymptotic
behavior of the elliptic functions is as follow [4]
$$
K\to \sqrt 16/(1-\nu^2) , \qquad tn\to \sinh,\qquad dn\to cn,\quad cn\to 1/cosh ,\qquad \nu\to 1,
$$
$$
K\to \pi/2,\qquad tn\to \tan,\quad dn^2\to (1-\nu^2\sin^2), \qquad \nu\to 0.
\eqno(2.3)
$$
Using the properties of the elliptic tangent function $tn0=0$ and $tnK\to\infty$,
one can see that solution (2.2), (3.2) satisfies to boundary conditions (2.1).
Obvious, this solution is satisfied by periodic boundary conditions (period 4B)
along the Y axis. Thus we have double periodic solution and double periodic
superstructure. It consists of rectilinear cells (domains) divided by domains
walls or boundaries.
\vspace{1em}
3. LOCAL STABILITY CONDITIONS.
Periods of the superstructure are not arbitrary. They depend not only
on $\varepsilon$, but on some properties of the lattice. Indeed, there are two algebraic
relations, so called
dispersion equations derived in [5]. They may by realized if substitute the
solution (2.2), (1.5) to the equation (1.2). Then one has
$$
S (k_1 K_1^2/H^2 A^2 + k_2 K_2^2/B^2) = - p ,\qquad S=(A^2-1)(1-\nu_2^2/A^2)/2,
\eqno(3.1)
$$
$$
k_1 K_1^2/H^2 = A^2k_2 K_2^2/B^2,\qquad p\ge0.
\eqno(3.2)
$$
They specify the existence conditions for the stable solution (2.2), (1.5). A local
stability is taken into account. Clearly, it takes place when $A<1$ because
$p>0$.
Analysis of these relations makes it possible to reveal all points of structural
transformations at arbitrarily large nonlinear displacements.
To this end, one should exclude the
constants of integration $\nu_1\,and\nu_2$. These can be expressed through
H and B using (3.1) and (3.2). As a result we obtain A as a function of H and B.
The relations between A,B and H can be conveniently visualized by plotting H
as a function of B at various values of the bifurcation parameter A, which
can be formally called the inverse amplitude of the displacement field -
according to its position in the expression (2.2).
To each A value corresponds some (H vs. B) curve which has the only asymptote, $B\,=\,B_i$,
and arrives at the terminal point $H_c,B_c$, approaching the asymptote in such a
way that
$$
H \ge H_c,\qquad B_c \le B < B_i.
\eqno(3.3)
$$
The limits are as following
$$
H_c = \pi l_1(1-A^2)/2A^2,
$$
$$
B_c = l_2(1-A^2)K_{22},\qquad B_i=\pi l_2\sqrt{1-A^2}.
\eqno(3.4)
$$
Here $K_{22}$ is the value of the function $K_2(\nu_2)$ in the point $\nu_2=A^2$.
Clearly, not all values of H and B are allowable - obvious limitations follow
these inequalities. The main result is that there are the upper limit of all
lengths of modulation B. It equals to $B_i$ when $A\,=\,0$, i.e. we have
simply $B\le1$. Consequently only modulated microshear is stable as soon as
inequalities (3.3) take place. As for a homogeneous elastic microshear
($B\to \infty$) it is impossible at all because does not satisfy to (3.3) or
to the equation (1.2). Note that a continuum theory of elasticity introduces no limits.
It is interesting to present the inequalities in (3.3) in such a manner
$$
{(\pi_2 k_1/4H^2)(1-A^2)^2/A^4} \le p,\qquad (k_2/B^2)(1-A^2)^2 K_{22}^2 \le p,
\eqno(3.5)
$$
$$
k_2(\pi/2B)^2(1-A^2) > p,
\eqno(3.6)
$$
The first two present conditions of local stability of microdeformations
but the last one is the criterion of their formation. In other
words, the energy of nonhomogeneous elongation along Y-direction must exceeds
potential barriers of a lattice due to
an effect of incompatibility of neighbor chains as in Frenkel-Kontorova s one-
dimensional model. If not so then microdeformations are impossible at all and
we have macrodeformations only. But
as it is clear from (3.6), deformed configuration becomes
stable when barriers are large enough to prevent destroying of a superstructure,
i.e. translation crystalline order.
Clearly, the wide spectre of mean space frequencies $1/B$ and $1/H$ is allowed,
i.e. a set of private resolutions take place. But in an external macrofield
only some of them are preferable. It will be demonstrated later.
\vspace{1em}
4. BIFURCATION and TEMPERATURE/DEFORMATION TRANSITIONS.
If inequalities (3.5) are not realized, a transition to the other lattice
structure (with the other topology) is possible. This process is going through
some bifurcation point where a continuum model is possible. If the crystalline potential $p$ tends to
zero (due to temperature, for example), then the set of equations (2.4), (2.5)
becomes a uniform homogeneous system
in two variables, $H^2/k_1K_1^2\, and\, B^2/k_2K_2^2$. If these variables
become zero (trivial solution) -which is not physically
meaningless - solution (2.2) will correspond to a uniform shear in a continuum.
A nontrivial solution to (2.4) and (2.5) can be found by equating the corresponding
determinant to zero, i.e.
$$
S\,=\,0
\eqno(4.1)
$$
Turning to the corresponding expression for S in (3.1), we find two roots of
equation (4.1):
$$
A\,=\,1,\qquad \nu_1^2+\nu_2^2\,=\,1;\qquad A\,=\,1,\quad \nu_1=0,\quad \nu_2=1
\eqno(4.2)
$$
Under these conditions, (2.2) degenerate into a series of partial solutions, some
coinciding with solutions in the continuum model (nonuniform shear) and some
not. Without going into details, we focus ourselves on a more essential aspect of the problem.
Indeed, at A = 1, there is bifurcation of solutions. At this point, $S\,=\,0$
as does the left-hand side of equation (2.4). The same occurs when the crystalline
potential $ p\,=\,0$. As mentioned above, this implies a transition from a
minimum to maximum of energy functional, i.e. from a stable to unstable lattice
configuration. It becomes stable at $A>1$ if one shifts the microdisplacement
by $1/2$ according to (1.5). Then a new
solution of elasto-plastic deformation may be constructed - see below.
Essentially, a similar bifurcation occurs when the crystalline potential does not
change, but some parameters reach some asymptotic values. Let us define effective
potentials as
$$
P_1\,=\, pH^2/k_1K_1^2\,=\,(H/K_1 l_1)^2 \qquad P_2\,=\,pB^2/k_2K_2^2\,=\,(B/K_2 l_2)^2
\quad(4.3)
$$
These potentials may also become zero at constant p when $H^2/k_1K_1^2$ or
$B^2/k_2K_2^2$ becomes zero, which corresponds to the transition to the
continuum field $u$ because $S\,=\,0$. Than we have two acoustic modes
($U$ , \,$u$). Inspite of the interaction between sublattices ( $p\ne0$),
they are deforming as two continua. With the new designations, the
dispersion relations (2.4) and (2.5) take the form
$$
S\,=\,P_2\,=\,A^2P_1\qquad or\qquad S (K_1/A)^2\,=\,(H/l_1)^2,\quad S K_2^2\,=\,(B/l_2)^2
\eqno(4.4)
$$
Clearly that the bifurcation point ($A\,=\,1$ or $S\,=\,0$) can be attained,
but only asymptotically,
when, at constant p, the effective potentials defined by (4.3) tend to zero.
If atomic potential $p$ characterizes stability of undeformed rigid lattice to
shear the values
$P_1\,andP_2$ can be thought of as potentials (or stability) of the collective
interaction between chain segments of length H and B belonging to neighboring
deformed chains or stability (strength) of deformed lattice. The
transition from individual to collective potentials is associated also with
deformation-induced changes of translation order in the system - atomic-
scale periodicity gives way to a mesoscopic-scale structure with larger
periods.
Let us to analyze the second conditions in (4.4). They demonstrate universal
scaling effects - the ratio of lengths ($H/l_1$ or $B/l_2$) is a function
of $S \sim(A^2-1)$, i.e. of a "distance" from bifurcation point $A\,=\,1$.
The other words a low of similitude takes place. The left-hand side parts of these
conditions are the scaling parameters. Taking into account that
$l\,=\,(\sqrt k/p)$ is a function of temperature, one can say that bifurcation
point (transition from region $A <1$ to region $A >1$) may be reached due to
temperature $p\,=\,0$) as well as due to deforming ($A\,=\,1$) according to (4.4). All processes in the
frame of our model obey this universal conditions.
The first two conditions in (4.4) demonstrate anisotropy of domain form. Each
of them (H- and B- domain) have different scale parameters so as
$Bl_1/Hl_2\,=\,A(K_2/K_1)$.
\vspace{1em}
5. ELASTO PLASTIC SHEAR with SLIDING.
If bifurcation point $A\,=\,1$ is overcome then the solution (2.2) becomes
unstable. But it may be modified for $A >1$ if one eliminates sublattice translation
1/2 of period from (1.5). Then one has
$$
tang(u_-/4)\,=\,+ tn(xK_1/H)\, dn(yK_2/B)/A cn(yK_2/B),\qquad A^2=\nu_2/\sqrt{1-\nu_1^2}.
\eqno(5.1)
$$
Essentially that now the other boundary conditions bay be satisfied. Using the
properties of the elliptic tangent function $tn0=0$ and $tnK\to\infty$,
one can see that
$$
u(0,y)=0,\qquad u(\,nH,y) = m ,\qquad m, n = +1,2,...
\eqno(5.2)
$$
Clearly, we are dealing with a rigid interchain shear (sliding) by one interatomic
distance $b$ on the boundaries, $x=+nH$ and $x=-n H$. But the interchain
shear becomes zero at $x=0$. If $u\le 1/2$ then plastic deformations
are neglect and one has elastic shear. It takes place in some zone near the
boundary $x\,=\,0$. The elasto-plastic frontier is defined by
a condition $tang(u_-/4)=1$. But it is not rectilinear. In two plastic regions
the bound rearrangement takes place but in an elastic one it is not.
It is interesting that sliding is irregular, because microfield $u(x,y)$
is not uniquely defined from (5.1) - m may by stochastic function of n!
If it is so one has so called shuffling mode of plastic deformation as in
theory of martensite transitions.
Obviously, the solution (5.1) is satisfied by periodic boundary conditions (period 4B)
along the Y axis. Thus we have double periodic solution and double periodic
superstructure. It consists of rectilinear cells (domains) divided by domains
walls or boundaries. They are consist of Frenkel-Kontorova s solitions or
misfit solitons. It is possible that due to them but not due to dislocations
may be realized above mentioned shuffling mode.
The periods of the superstructure depend on external field $\varepsilon$ or
$\sigma$ such as on
properties of the lattice as in elastic case. Indeed, there are two algebraic relations, so called
dispersion equations. They may by derived if substitute the
solution (5.1) to the equation (1.2). Then one has
$$
S (k_1 K_1^2/H^2 A^2 + k_2 K_2^2/B^2) = 2 p ,\qquad S=(A^2-1)(1-\nu_2^2/A^2),
\eqno(5.3)
$$
$$
k_1 K_1^2/H^2 = A^2k_2 K_2^2/B^2,\qquad p\ge0.
\eqno(5.4)
$$
The first condition is differ from (2.4) only by the sign of the right-hand side.
Therefore it is satisfied by the inequality $A>\,1$. Both expressions
specify the existence conditions for the stable solution (5.1). A local stability
is taken into account. The global stability will be considered later.
As for the conditions such as (4.3) and (4.4), they get opposite signs
before the value $S$ only. One can use total conditions introducing notation
$\parallel S \parallel$ for absolute value. Then one has for both cases
$$
\parallel S\parallel \,=\, A^2 P_1\,=\,P_2
\eqno(5.5)
$$
Obviously universal scale principle is valid in the case $A >1$ too.
Analysis of these relations makes it possible to reveal all points of structural
transformations at arbitrarily large nonlinear displacements.
To this end, one should exclude the
constants of integration $\nu_1$ and $\nu_2$. These can be expressed through
H and B using (5.3) and (5.4). As a result we obtain A as a function of H and B.
The relations between A, B and H can be conveniently visualized by plotting H
as a function of B at various values of the bifurcation parameter A.
To each A value some (H vs. B) curve corresponds, which resembles a hyperbola
with the asymptotes given by
$$
H \ge H_t, \qquad H_t = l_1(A^2-1)K_{11},
\eqno(5.6)
$$
$$
B \ge B_t, \qquad B_t = (\pi/2)l_2\sqrt{A^2-1}.
\eqno(5.7)
$$
Here $K_{11}$ is the value of the function $K_1(\nu_1)$ in the point
$\nu_1 = \sqrt(1-1/A^4)$.
Domain square HB is minimal when H and B are approximately equal to each
other and greater then $ H_t$ and $ B_t$. The density of energy $E/BH$ is maximal
when the square HB is minima. Obviously very anisotropic domains are preferable.
If $B\to \infty$ than $H \to H_t$. As for the last value it corresponds to
minimum of total energy $\bar E/BH$. Short periods B are impossible. It is the
essential difference from the elastic case $ A<1$. This rather simple mechanism
is realized when $A^2>2$. When $A^2<2$ coexistance region takes place where
short domains ($B < \pi/2$) are forming.
Energetic restrictions become visible if one presents (5.7), (5.6) in another
form such as
$$
k_1(K_{11}/H)^2 (1-1/A^2)^2\le p, \qquad (1-1/A^2)^2 \le P_1,
\eqno(5.8)
$$
$$
k_2(\pi/2B)^2(A^2-1)\le p,\qquad (A^2-1)\le P_2 .
\eqno(5.9)
$$
Clearly, elasto-plastic deformation becomes stable and long-scale translation
order exists, if atomic potential barriers
exceed a local deformation energy - right-hand side of this
inequality. The second inequalities in (5.9), (5.8) are presented via effective potentials
$P_1$ and $P_2$. They gives us the criteria of microstructure stability. If
it is not then
deformations destroy the translation order entirely. If $A\approx 1$ then
both effective barriers stabilizing the structure are low. When $A\gg 1$ one has
very high barrier $P_2$ only, i.e. very stable long domains.
\vspace{1em}
6. INTERACTION of MODES and STABILITY in EXTERNAL FIELD.
The equation (1.2) does not contain macrodeformations $\varepsilon$ and
macrodisplacements $U$. They depends on equilibrium conditions for
macroscopic body and generally speaking, play the role of " external field"
for the microdeformations. In [1,2] we considered the problem without
external fields.
Indeed the expression (1.1) may be presented after integration as
$$
\bar E/B H\,=\,\bar k_1(\varepsilon^2 - 2 \varepsilon\, \delta u\,b/H)/2 + E/B H
\eqno(6.1)
$$
$$
E\,=\, \int\nolimits_0^{H}\int\nolimits_0^B[(1/2)k_1(\partial u/\partial x )^2 + (1/2)k_2(\partial u/\partial y)^2 + p(1 - \cos2\pi u)]\,dxdy.
\eqno(6.2)
$$
Here $\bar k_1\,=\,k_1/b^2$ is a macroscopic shear modulus, $\delta u$ is the
difference of values of microdisplacements on neighbor
boundaries , par example $x\,=\,0,\,H$. In our cases considered below $\delta u\,=\,1$.
Then (1/H) is a mean microdeformation on a half period 2H.
But one can accept $\delta u\,=\,-1$ supposing the opposite sign of microshear,
i.e. -(b/H).
The second (crossed) term in (6.1) means that micro and macrodeformations
are dependent. Let us to define the macro stresses $\sigma$ as
$$
\sigma\,=\,\partial(\bar E/HB)/\partial \varepsilon\,=\,\bar k_1(\varepsilon - b\,/H).
\eqno(6.3)
$$
We took into account that $\delta u\,=\,1$.
Clearly that the second term in the brackets (b/H) is the average microdeformation
or simply - inner deformation weather the first is the external one. It equals to
$ \partial U/\partial x$ of cause. If microdeformations are zero then one
has macro fields $\sigma\, and\, \varepsilon$ only.
We suppose that interaction between macrofield and longitudinal modulations
is absent because the mean microdeformations are zero.
Therefore these modulations must be generated spontaneously according to
energy minimum conditions such as
$$
\partial(E/B H)/\partial B\,=\,0,\qquad \partial^2(E/B H)/\partial B^2\,>\,0.
\eqno(6.4)
$$
If it is so, one has an equilibrium value of modulation half period $2B_m$.
In elastic case ($A < 1$) $B_m\,=\,B_c$. But $B_m\to \infty$ when $A > 1$.
Let us consider the case $A <1$. Then $B_m$ ought be substituted to (6.1), (6.2). Then one has the energies $E_m\,,\,
\bar E_m$ as some functions of two variables, H\, and $\varepsilon$. Then
one can define mean microdeformation $b/H$ as a function of $\varepsilon$
or $\sigma$.
Let us to establish some relation between them. Suppose that external
microstresses are absent, i.e.
$$
\partial(\bar E_m/H B_m)/\partial(b/H)\,=\,-\bar k_1\varepsilon\,+\,\partial(E_m/H B_m)/\partial(b/H)\,=\,0.
\eqno(6.5)
$$
It is the necessary condition of an extremum of the complete energy. It may be presented
as the condition of equilibrium between macro and micro fields
$$
\varepsilon\,=\,(1/\bar k_1)\partial(E_m/B_m H)/\partial(b/H).
\eqno(6.6)
$$
Now one can define the second period H, but as a function of $\varepsilon$.
Let us to consider some conditions of superstructure stability in a macrofield.
as a function of two variables, $\varepsilon \, and \, (b/H)$. Then one must accept
$$
D\,=\,\bar k_1\partial^2(E_m/B_mH)/\partial(b/H)^2\,-\bar k_1^2\,>\,0.
\eqno(6.7)
$$
Here D is a determinant of matrix (2x2) of the second derivatives of the complete
energy $\bar E_m$. If
$$
\bar k_1\,>\,0,\qquad \partial^2(E_m/H B_m)^2\,>\,0,
\eqno(6.8)
$$
then one has a minimum.
If it is so then (6.6) is the condition of stable equilibrium between micro and
macrodeformations.
In other words, it defines mean microdeformations induced by
an external "macrofield" $\varepsilon$. Therefore the value of thickness $H$ is
not arbitrary. Equilibrium thickness depends on $\varepsilon$
and properties of a lattice. It means that we have
no real rigid boundaries but induced superstructure with inner "boundaries".
If one uses the condition (6.6) in (6.3) then macrostresses may be presented via
mean microdeformations only as
$$
\sigma\,=\,\partial(E_m/B_mH)/\partial(b/H)\,-k_1b/H.
\eqno(6.9)
$$
It is the condition of microshear generation in an external
field of shear stresses. If one expresses $(b/H)$ from (6.6) and substitutes it to
(6.9) then nonlinear function $\sigma via \varepsilon$ may be derived due to
microdeformations.
Some analysis of the function $E_m(H)$ shows us that
if $b/H\,=\,0$ then (6.6) gives a critical value of external "field" $\varepsilon_c$
. Otherwise, forced microdeformations appear when external " field" reaches a
threshold
$$
\varepsilon_t\,=\,[\partial(E_m/B_m H)/\partial(b/H)]_o
\eqno(6.10)
$$
Here the subscript at the right-hand side means that $b/H\,=\,0\, or\, H\to \infty$.
For spontaneous
microdeformations $\varepsilon_t\,=\,0$. Note that we have monotone increasing
function $(E_m/B_mH)$ of $(b/H)$. It means that microdeformations appear, if
$$
\varepsilon\,\ge\varepsilon_t.
\eqno(6.11)
$$
Thus in a frame of proposed model there are two deformation modes (macroscopic
and microscopic) when the threshold (6.11) are reached. Then a lattice is decomposed into
two sublattices and microdisplacements appear. As it is well known in ionic
crystals an electric field decomposes the lattice due to splitting of charges.
Obviously, the wide spectre of mean space frequencies $1/B\, and\, 1/H$ is allowed,
i.e. a set of private resolutions take place.
But only some of them are preferable. Indeed, the expression (6.4) or (6.6) defines
a single pair of periods $B_m\, and\, H$, if macrofield $\varepsilon\, or\,
\sigma$ is fixed ($A < 1$).
In the case $A > 1$ one has $ H = H_t,\, B\to \infty$.
\vspace{1em}
7. CONCLUSION.
Thus we analyzed giant mutual displacements of atomic chains, i.e. pseudo
optical deformation mode. It may be generated as a result of instability of
a lattice. Then some inner boundaries of shear sliding and space modulations
appear.
If $u\le 1/2$, one has elastic nonuniform shear. Then microdeformations
destroy translation symmetry of a lattice, but only in mesoscopic scales.
Some global symmetry appears as superstructure or domain structure with periods
$2H\,and\,4B$. Transversal boundaries consist of Frenkel-Kontorova s solitons,
divided by distance 2 B. Microdeformations are nonhomogeneous. They destroy
translating lattice order but only inside domains - in nanoscale. Domains
form superstructure with great periods. When external macrodeformations
$\varepsilon$ exist than interaction between acoustic and pseudo optical modes
take place. Than only one pair B, H realizes local and global stable
superstructure which is formed when some threshold $\varepsilon_t$ is reached.
It is the first bifurcation point.
The second one is realized when some parameter $A \to 1$. Than local stable
pseudo optical mode becomes possible if $u \ge 1$ on parallel boundaries
divided by distance 2 H.Then elastoplastic microshear takes place. The superstructure takes place too, but
the nearest atomic order is changed due to depining. In other words, catastrophic
reconstruction of a lattice takes place without it destroying - a double springs up.
The microdeformation (pseudo optical mode) generated under influence of macroscopic
shear deformation (acoustic mode) when some threshold is overcome. We analyzed
these problems basing on exact solutions of equilibrium equation of sine-Helmholtz
tape.
\vspace{2em}
REFERENCES
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Inorganic materials. Vol. 35, No. 8, 1999, pp. 860-862.
2.Aero E.L. Microscale deformations in 2-D lattice - structural transitions
and bifurcations under critical shear, Solid State Physics. Vol.42, No.6,
pp. 152-158, 2000.
3. Lamb, G.L.,Jr.,Elements of Soliton Theory, New York: Wiley, 1980. Translated
under the title Vvedenie v tepriyu solitonov, Moskow: Mir,1983.
4. Janke,E., Emde,F., and Losh, F., Tafel horerer Funktionen, Stuttgart: Tourner,
1960. Translated under the title Spetsial nye funktii, Moskow: Nauka, 1977.
5. Aero, E.l., Plane Boundary Problems for the Sine-Helmholtz Equation in the
Elasticity Theory of Liquid Crystals in Nonuniform Magnetic Fields,
Pricl. Mat. Mekh. Vol.60, No. 1, 1999, pp.79-87.
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