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\begin{document}
%\pagestyle{empty}
\title{Localized phase transition wave in a bistable lattice}
%\title{Bridged crack in a bistable lattice}
\author{Andrej V. Cherkaev$^a$, Leonid I. Slepyan$^b$, ...}
\date{$^a${\em Department of Mathematics, The University of Utah \\
Salt Lake City, UT 84112, USA}\\
$^b${\em Department of Solid Mechanics, Materials and Systems\\
Tel Aviv University,
P.O. Box 39040, Ramat Aviv 69978 Tel Aviv, Israel}}


\maketitle

\vspace{5mm}\noindent {\bf Abstract} \\
\\Discrete square-cell and triangular-cell lattices with bistable
bonds are considered. Initially, the bond follows the Hooke's law
as the first branch of the force---elongation diagram, but when
the elongation reaches the critical value, the tensile force drops
and the transition to the other branch occurs. The latter branch
can be placed in parallel with the former (mathematically this
case is simpler) or have any inclination. In particular, it may
correspond to a crack with free faces. In such an initially
stresses structure, a spontaneous transition wave can propagate
between neighboring lines of particles. Steady-state transition
waves in infinite irreversible elastic and viscoelastic lattices
are studied analytically, while numerical simulations are used for
transient waves in finite reversible and irreversible lattices.
Besides, the related continuous model is examined. The goal is to
determine the conditions which allow such a wave to exist, the
transition wave speed and the energy dissipation in such processes
--- in dependence of the structure parameters and the load type
and level.


\section{Introduction}
Lattices consisting of particles connected by massless bistable
bonds are considered. The bonds are assumed to obey a two-phase
stress-strain relation, such that the phase transition is
accompanied by a jump discontinuity in the tensile force. In
addition, the phases may differ by the tangent modulus.

In such an initially stressed lattice, as a result of a
disturbance, the phase transition can arise as a {\em localized
wave} propagating between two neighboring lines of the lattice
particles. This wave results in the unloading of the outer bonds
which thus remain to be in the initial phase. So, we face a
one-dimensional phase transition wave propagating in a
two-dimensional lattice. It looks like a propagating crack bridged
by fibres as in fracture of fibre-reinforced composites [Budiansky
et al. (1986), Willis (1993), Meda and Steif (1994a,b), Movchan
and Willis (1996, 1997a,b, 1998), Huang et al. (1999), Nemat-Nasser
and Luqun (2001), Biner (2002)].

The present formulation integrate those used in the works on
one-dimensional phase transition waves in bistable chains [Slepyan
and Troyankina (1984, 1988), Puglisi and Truskinovsky (2000),
Slepyan (2000, 2001b), Balk et al. (2001a,b), Charlotte and
Truskinovsky (2002), Ngan and Truskinovsky (2002)] and fracture of
two-dimensional lattices [Slepyan (1981, 2000, 2001a,c),
Kulakhmetova et al. (1984), Fineberg et al. (1991, 1992), Marder
and Liu (1993), Marder and Gross (1995), Kessler and Levine (1998,
2001),  Fineberg and Marder (1999), Slepyan et al. (1999), Kessler
(2000), Gerde and Marder (2001), Pechenik at al. (2002), Slepyan
and Ayzenberg-Stepanenko (2002)]. The goal of this study is to
determine conditions of the existence of such a transition wave,
the dependence of its speed on the initial stress and the energy
dissipated during the dynamic transition.

In a special case, where the phases differ only by a jump in the
tensile force, while the tangent modulus is the same for both
phases, the bonds are assumed to obey a standard-material
viscoelastic law. In a general case, elastic bistable lattices are
considered.

Square- and triangular-cell lattices are examined. Note that the
square-cell lattice is intended to model anti-plane
shear; however, a hypothetic plane deformation with no
displacements in the $x$-direction --- with the same formulation
and results --- can also be assumed. We use the latter viewpoint
when it is more convenient; for example, when
displacements and forces are to be shown on a plane figure.


\section{Formulation}
Consider a lattice which massless bonds obey the following
force-elongation relation (Figure 1):
\beq
T = \Gm \,Q ~~~(t<t_*),
~~~T=\Gm \,Q -P_*-(1-\Gg)\Gm \,(Q-Q_c)=\Gg \Gm\, Q - P^*~~~(t>t_*),\n
P^*=P_*-(1-\Gg)\Gm\, Q_c,~~~ Q<Q_c~~~(t<t_*),~~~Q=Q_c~~~(t=t_*).
\eeq{03a1}

In these relations, $T$ is the tensile force in a bond, $Q$ is the
bond elongation and $Q_c$ is the critical elongation. For $t<t_*$
the bond is in the initial phase characterized by stiffness $\Gm$.
The phase transition occurs at $t=t_*$ when the elongation first
reaches the critical value. At this moment a jump in the tensile
force arises, $[T]=T(t_*+0)-T(t_*-0)=-P_*<0$, and the bond comes
into the other phase where the tangent `modulus' $dT/dQ=\Gg\Gm,
\Gg>0$. Note that the phase transition moment, $t-t_*$, is
different for different bonds. A one-dimensional elastic chain
with such an irreversible behavior of its bond was considered by
Slepyan and Troyankina (1984, 1988). A reversible two-phase chain
was considered by Balk et al. (2001a,b). The above relation and
the below considerations concern to a lattice under extension.
Clearly, the same formulation and results are applicable for the
lattice under compression or shear.

Initially, the lattice (presented in Figure 2) is uniformly
stressed in such a way that the elongation of the $y$-directed or
inclined bonds (as in a square-cell or a triangular-cell lattice,
respectively), $Q_0$, is below the critical value, $Q_c$,
while the $x$-directed bonds are not stressed. Note that in the
case of mode II state of a triangular-cell lattice, lengthening and
shortening alternate. In this case, the critical value is assumed
to be $\pm Q_c$ for these deformations, \res.

It is reasonable to assume that the phase transition wave arisen
by a disturbance is localized as a bridged crack between the
neighboring lines of particles (denoted as $n=0$ and $n=-1$).
Indeed, the negative jump in the tensile force acts to unload the
$y$-directed or inclined bonds outside this damaged layer, while
the initially unstressed $x$-directed bonds have a great safety
factor. It is also assumed that the wave propagates with a
constant speed $v$. This means that the time-interval between the
transformation of the neighboring bonds is $a/v$, where $a$ is the
lattice cell size. The steady-state regime is considered with the
transformed layer at $\Gn \equiv (x-vt)/a <0$, where the particle
coordinates are $x=am, m=0,\pm 1, ...\,$.

Thus, we face a steady-state problem for a lattice which bonds are
in the initial phase except those placed between lines $n=0$ and
$n=-1$ at $\Gn<0$. A solution is required which satisfies the
dynamic equations for such a `damaged' lattice and the following
inhomogeneous conditions at infinity: the strain field coincides
with a given initial one at $y=\pm\infty$ and $\eta= +\infty$.
The solution presents the elongation of a bond at the moment of the
above-mentioned transformation, $\Gn=0$, as a function of
the wave speed, $v$, and the initial elongation. The equality
\beq
Q(\Gn) = Q_c ~~~(\Gn=0)
\eeq{016}
serves for the determination of the speed as a function of the initial
elongation.

For our goal it is convenient to reformulate this problem as
follows. Consider a uniform lattice all the bonds of which are in
the initial phase. Introduce self-equilibrated couples of external
forces applied to the particles connected by the crack line bonds.
Each couple of forces is directed along the corresponding bond
outwards as shown in Figure 3. Referring to \eq{03a1} choose the
forces as
\beq
P(\Gn)= P_*+(1-\Gg)\Gm\,
([Q(\Gn)-Q_c]~~~(\Gn<0),~~~P(\Gn)=0~~~ (\Gn\geq 0).
\eeq{03a2}
The crack line intact bonds together with these external forces
act on the `crack-face' particles, $n=0$ and $n=-1$, the same as
if the bonds are in the second phase. Hence, such a reformulation
does not concern the lattice dynamic behavior.

Note that the case
\beq
P_*=\Gm\, Q_c,~~~Q_0=\Gg=0
\eeq{03a3}
corresponds to a free-face crack considered in Slepyan (2001a,c).
Another remarkable case, $\Gg=1$, will be considered below.

\section{Governing equation}
To proceed introduce a steady-state fundamental solution, $\CQ(\Gn)$,
corresponding to the above-mentioned self-equilibrated load. In these
terms, the additional elongation of a crack-line bond caused by the
crack-face external forces is
\beq
\GT(\Gn)=Q(\Gn)-Q_0=\CQ(\Gn)*P(\Gn),
\eeq{03a4}
where $*$ is the convolution symbol. In terms of the Fourier transform,
this equality becomes
\beq
\GT^F(k)=\CQ^F(k)P^F(k).
\eeq{03a5}

Denote
\beq
\GT^F=\GT_++\GT_-,~~~\GT_\pm=\int_{-\infty}^\infty \GT(\Gn)H(\pm\Gn)e^{ik\Gn}d\Gn,
\eeq{03a6}
where $H$ is the unit step function. The same subscript, $+$ ($-$),
will be used for the right-side (left-side) Fourier transform of any
function. Referring to \eq{03a2}
\beq
P^F=P_-=P_*(0+ik)^{-1}+(1-\Gg)\Gm\,\gl[\GT_--\GT_c(0+ik)^{-1}\gr],
\eeq{03a7}
where $\GT_c=Q_c-Q_0$. Now equation \eq{03a5} can be presented in the form
\beq
\GT_++\gl[1-(1-\Gg)\Gm\, \CQ^F\gr]\GT_-=\gl[P_*-(1-\Gg)\Gm\,\GT_c\gr]\f{\CQ^F}{0+ik}.
\eeq{03a8}

The fundamental solution, $\CQ^F(k)$, appearing in this equation can be
expressed in terms of the known lattice-with-a-crack Green functions
denoted by $L(k)$ in Slepyan (2001a,c).
With this in mind we rearrange relation \eq{03a8} to obtain
the corresponding equation for the lattice with a crack at $\Gn<0$.
Represent
\beq
P^F(k) = \Gm\, Q_-(k) + p^F(k),
\eeq{033}
where $p\,(\Gn)$ is an arbitrary function. The first term
in the right-hand side of \eq{033} compensates the tensile forces in the
considered bonds at $\Gn<0$ and hence, as far as the external forces
are defined by Eq.\,\eq{033}, this
corresponds to a lattice with a crack at $\Gn<0$ loaded at $n=0$ and
$n=-1$ by forces $p\,(\Gn)$.

In the considered case $\GT=Q$, since $Q_0=0$ \eq{03a3}, and
Eq.\,\eq{03a5} leads to
\beq
L(k)Q_+(k)+Q_-(k)=[L(k)-1]\f{p^F}{\Gm}
\eeq{035}
with
\beq
L(k)=\gl[1-\Gm\, \CQ^F\gr]^{-1}.
\eeq{036}
In terms of the elongations, Eq.\,\eq{035} coincides with the governing
equations for crack dynamics [see Eq.\,(36) in Slepyan (2001a) and
Eq.\,(11) in Slepyan (2001c)]. Thus, the expressions for $L(k)$
presented in these papers [Eq.\,(37) and Eq.\,(53), \res] can be
used to find $\CQ^F$ from \eq{036}. It is
\beq
\CQ^F=\f{1}{\Gm}\gl[1-\f{1}{L(k)}\gr],
\eeq{03a8a1}
where for mode
III unbounded square-cell lattice
\beq
L(k)=\f{r(k)}{h(k)}
\eeq{037}
with
\beq
h(k)=\sqrt{2(1-\cos k)
+Y},~~~r(k)=\sqrt{h^2(k)+4}
\eeq{038}
and
\beq
Y=(0+ikV)^2,~~~0+ikV =\lim_{s\to +0}(s+ikV), ~~~V=\f{v}{c}\,,~~~
c=\sqrt{\f{a^2\Gm}{M}}.
\eeq{039}
Here and below $a$ is the bond
length and $M$ is the lattice particle mass. Function $L(k)$ for a
square-cell lattice strip can be found in Slepyan et al. (1999).

For a triangular-cell lattice
\beq
L(k)=\f{h_\GD(k)}{r_\GD(k)}\,,
\eeq{0310}
where
\beq
h_\Delta = \f{F(n_2)\sqrt{n_1^2-1}- F(n_1)\sqrt{n_2^2-1}}{n_2-n_1}\,,\n
r_\Delta = 3\sqrt{n_1^2-1} \sqrt{n_2^2-1},\n
{}F(n_{1,2}) = 3(\cos k/2 -n_{1,2})^2 + 6\sin^2 k/2 (1\pm \cos k/2)
(1\mp n_{1,2})\n
+Y[(1\pm\cos k/2)(1\mp n_{1,2}) + 1-n_{1,2}\cos k/2].
\eeq{0311}
In the expression for $F(n_{1,2})$, upper signs correspond to mode I, while
lower signs correspond to mode II. Besides,
\beq
n_{1,2}=\gl(1+ 2\sin^2 k/2  + \frac{2}{3}Y\gr)\cos k/2
\mp \gl[\frac{1}{9}Y^2 - 4\sin^2\frac{k}{2}
\left(\sin^2\frac{k}{2} + \frac{1}{3}Y\right)^2\gr]^{1/2}.
\eeq{0312}
The rule of the determination of signs of the radicals in the above relations
[the argument of a radical is continuous in a pre-limiting case, $s>0$, see \eq{039}],
as well as the asymptotes for $k\to 0$ and $k\to \pm\infty$, can be obtained
from the corresponding papers. In particular,
\beq
\f{h(k)}{r(k)}\to\f{h_\GD(k)}{r_\GD(k)}\to 1~~~(k\to\pm\infty).
\eeq{0313}

Eq.\,\eq{03a8} can now be rewritten as
\beq
\CL(k) \GT_++\GT_-=\f{P^*[\CL(k)-1]}{(1-\Gg)\Gm\, (0+ik)}
\eeq{03a13}
with
\beq
P^*=P_*-(1-\Gg)\Gm\, \GT_c
\eeq{03a14}
and
\beq
\CL(k)=\f{L(k)}{\Gg L(k)+1-\Gg}.
\eeq{03a15}

Note that the initial strain does not appear in the governing equation
\eq{03a13}. The problem may thus be considered as a homogeneous one,
that is in terms of the additional elongations vanishing at $x\to +\infty$
and $y\to \pm\infty$,
as if the force-elongation diagram has its origin at $Q=Q_0, T=\Gm\, Q_0$
as shown in Figure 1 by dotted axes. In these terms, the force $P$
in \eq{03a2} is a function of $\GT$ as the difference between the
initial and the other phases, while $P^*$ \eq{03a14} is that at $\GT=0$
(Figure 4). An {\em effective initial elongation} can be introduced
for $\Gg<1$ as
\beq
Q^0=\f{P^*}{(1-\Gg)\Gm}
\eeq{0315-1}
which is also shown in Figure 4.

\section{Solution for {\bf $0 < \BGg \bf < 1$}}
Eq.\,\eq{03a13} can be solved using the Wiener-Hopf technique. In
this connection first note some features of the function $L(k)$
\eq{03a15}. Under the causality principle this function should be
considered as the limit ($s\to +0$) of $L(s+ikV,k)$ [see Slepyan
(2001a)]. It follows from the {\em Theorem of the fundamental
solution} [Slepyan (2001c)] that if $s>0$, as $k$ runs over the
real axis from $-\infty$ to $\infty$, the function $L(k)$ forms a
closed contour on the complex plane, $\Re L(k)+i\Im L(k)$, leaving
the negative half-axis, $\Re L(k) \leq 0$, in the outer domain.
This evidences that \beq \mbox{Ind} L(k)=\f{1}{2\pi}\gl[\mbox{Arg}
L(\infty)-\mbox{Arg} L(-\infty) \gr]=0. \eeq{03a16} It follows
that $\Gg L(k) +1-\Gg$ possesses the same features at least for
$0\leq \Gg \leq 1$. Hence, in this range of $\Gg$ \beq \mbox{Ind}
\CL(k)=0. \eeq{03a17} Further, $L(\pm\infty)=1$ and it can be seen
in \eq{03a15} that this equality is still valid for $\CL(k)$. The
function $\CL(k)$ can thus be factorized using the Cauchy-type
integral, namely for $0< \Gg<1$ \beq \CL(k)=\CL_+\CL_-,\n
\CL_\pm=\exp \gl[\pm\f{1}{2\pi i}\inti \f{\ln
\CL(\xi)}{\xi-k}d\xi\gr], \eeq{03a18} where $\Im k >0$ for $\CL_+$
and vice versa. In particular, it follows that \beq \CL_\pm \to
1~~~(k\to \pm i\infty). \eeq{03a19}

Noting that $\CL(0)=1/\Gg~[L(0)=\infty]$ we now can rearrange
Eq.\,\eq{03a13} in the form
\beq
\CL_+(k)\GT_++\f{\GT_-}{\CL_-(k)}=\f{Q^0[\CL_+(k)-1/\CL_-(k)]}
{0+ik}=C_+(k)+ C_-(k),\n
C_+(k)= \f{Q^0[\CL_+(k)-\CL_+(0)]}{0+ik}\,,~~~
C_-(k)= \f{Q^0[\CL_+(0)-1/\CL_-(k)]}{0+ik}\,,\n
\CL_\pm (0)=\f{1}{\sqrt{\Gg}}\CR^{\pm 1},~~~\CR=\exp\gl[\f{1}{\pi}
\int_0^\infty\f{\mbox{Arg}\,\CL(\xi)}{\xi}d\xi\gr].
\eeq{0321}
The functions $C_+(k)$ and $C_-(k)$ have no singular points in the upper
and lower half-planes, \res. Now, following the Wiener-Hopf
technique, we can separate the corresponding
functions in Eq.\,\eq{0321} and find a solution as follows:
\beq
\GT_+=\f{C_+(k)}{\CL_+(k)}\,,~~~\GT_-= C_-(k)\CL_-(k)
\eeq{0322}
which for $\Gn=0$ yields
\beq
\GT(0)=\lim_{p\to\infty}p\,\GT_+(ip)= \lim_{p\to\infty}p\,\CT_-(-ip)=
Q^0\gl[\CL_+(0)-1\gr]\n
= Q^0\gl(\f{1}{\sqrt{\Gg}}\CR-1\gr).
\eeq{0323}
In the considered wave, if it exists, $\GT(0)=\GT_c$. This presents
an equation
\beq
Q^0\gl(\f{1}{\sqrt{\Gg}}\CR-1\gr)=\GT_c
\eeq{0323a}
which defines the wave speed. If no wave speed satisfies this equation
the wave does not exist.
This equation has a limiting expression for $\Gg\to 1$. In this case
\beq
\mbox{Arg}\,\CL = \arctan \f{(1-\Gg)\Im L}{\Gg|L|^2+(1-\Gg)\Re L}\sim
(1-\Gg)\f{\Im L}{|L|^2}=-(1-\Gg)\Im \gl(\f{1}{L}\gr),\n
\CR\sim 1-\f{1-\Gg}{\pi}\int_0^\infty \Im \gl[\f{1}{L(k)}\gr]\f{dk}{k}
\eeq{0324}
and in the limit Eq.\,\eq{0323a} becomes
\beq
\f{P^*}{\Gm}\gl\{\f{1}{2}-\f{1}{\pi}\int_0^\infty\Im
\gl[\f{1}{L(k)}\gr]\f{dk}{k}\gr\}=\GT_c.
\eeq{0324a}
Below this result will be obtained directly for $\Gg=1$.

The limits for $\Gn\to \pm\infty$ follows from \eq{0322} [also see \eq{03a14}
and \eq{0321}] as
\beq
\GT(+\infty)=\lim_{p\to 0}p\,\GT_+(ip)=0,\n
\GT(-\infty)=\lim_{p\to 0}p\,\GT_-(-ip)= \f{P^*}{\Gg\Gm}
\eeq{0324b}
as it should be, namely, it is the static elongation corresponding
to the first phase at $\Gn>0$ and the second phase at $\Gn<0$.

\section{The case {\bf $\BGg  = 1$}}
Such a bistable model is remarkable for the fact that the
transition from the one branch to the other does not change the
equations. It results only in self-equilibrated couples of fixed
external forces applied to the particles connected by the
corresponding bond, Figure 3. These forces are directed outwards
to decrease the tensile force in accordance with the gap between
the branches. Thus, an intact uniform lattice can be considered,
and only the right-hand part of the dynamic equation (which is
independent of the lattice displacements) reflects the transition.
This simplifies greatly the problem considered. Namely, it now is
not a mixed boundary value problem. Besides, in this case the
superposition is valid allowing multiple line transitions to be
considered. The elastic version of such a relation was introduced
in Balk et al (2001a,b) with the use for the dynamics of a chain.
Truskinovsky ???

In this case, Eq.\,\eq{03a8} defines the elongation in an explicit
form
\beq
\GT^F(k)=\f{P_*\CQ^F}{0+ik}=\f{P_*}{\Gm\,(0+ik)}\gl[1-\f{1}{L(k)}\gr].
\eeq{03a8a}
Since
\beq
\Re L(-k)=\Re L(k), ~~~\Im L(-k)=-\Im L(k), ~~~1/L(0)=0,~~~ L(\pm\infty)=1,
\eeq{042}
the inverse transform
\beq
\GT(\Gn)=\f{1}{2\pi}\inti \GT^F(k)e^{-ik\Gn}dk
\eeq{041}
leads to
\beq
\GT(0) = \f{P_*}{\Gm}\gl[\f{1}{2} - \f{1}{\pi}\int_0^\infty
\Im \gl(\f{1}{L(k)}\gr)\f{dk}{k}\gr],\n
\GT(\infty) = 0, ~~~\GT(-\infty)=\f{P_*}{\Gm}
\eeq{043}
which coincides with the limiting results, $\Gg\to 1$
[see Eqs. \eq{0324a} and \eq{0324b}].

\subsection{Viscoelastic lattice}
For this special type of the force-strain dependence, if the
branches differ only by the `elastic part' of the tensile force
(as discussed below), the elastic-viscoelastic correspondence
principle is valid.

In an initial range of the extension, the bonds are assumed now
to satisfy the standard material viscoelastic relation:
\beq
T+\Gb\f{d T}{dt}=\Gm\,\gl(Q+ \Ga\f{d Q}{dt}\gr),
\eeq{011}
where $\Ga$ and $\Gb$ are the creep and relaxation times, \res.
This relation corresponds to a unit presented in Figure 5, where
the left spring
stiffness is $\Gk_1$, the right spring stiffness is $k_2$ and the
viscous resistance of the element successively connected with the
right spring is $c$. In these terms
\beq
\Gm=k_1,~~~\Ga=\f{c}{\Gk_1}+\f{c}{\Gk_2}\,,~~~\Gb=\f{c}{\Gk_2}.
\eeq{012}

Note that relation \eq{011} can also be represented in the form
\beq
T=\f{\Gm}{\Gb}e^{-t/\Gb}\int_{-\infty}^t\gl[Q(\Gj)+
\Ga\f{dQ(\Gj)}{d\Gj} \gr]e^{\Gj/\Gb}d\Gj.
\eeq{011a}

When the elongation reaches the critical
value, $Q_c$, the left spring is assumed to slip by a fixed value, such
that its
elongation becomes $Q-Q_*$, where $Q$ remains to be the elongation of the
unit and
\beq
\Gk_1Q_*=\Gm\, Q_*=P_*=\,\mbox{const}.
\eeq{013}
It is assumed that
\beq
P_*\leq \Gm\, Q_c,
\eeq{014}
that is the tensile force in the left spring remains to be nonnegative.
Thus, after this {\em transformation} the stress-strain relation
\eq{011a} becomes
\beq
T=\f{\Gm}{\Gb}e^{-t/\Gb}\int_{-\infty}^t\gl[Q(\Gj)+ \Ga\f{dQ(\Gj)}{d\Gj}
\gr]e^{\Gj/\Gb}d\Gj -P_*.
\eeq{015}

In accordance with the elastic-viscoelastic principle [or
directly from \eq{011}], the viscoelastic
solution can be obtained from the elastic one \eq{03a8a} if
$\Gm$ is changed to the complex modulus $\Gm\,\hat{E}$ with
\beq
\hat{E}=\f{1+ikV_\Ga}{1+ikV_\Gb}\,, ~~~V_\Ga=\f{v\Ga}{a}\,,~~~ V_\Gb=\f{v\Gb}{a}.
\eeq{051}
Accordingly, the function $Y$ \eq{039} must be replaced by
\beq
Y=\f{(0+ikV)^2}{\hat{E}}.
\eeq{052}
Formula \eq{043} for $\GT(0)$ becomes
\beq
\GT(0) =\f{P_*}{\Gm}\gl\{\f{\Gb}{2\Ga} - \f{1}{\pi}\int_0^\infty
\Im \gl[\f{1}{\hat{E}L(k)}\gr]\,\f{dk}{k}\gr\}.
\eeq{053}

\subsection{Massless viscoelastic lattice}
This model relates to a very low speed. If the inertia term is neglected,
$L(k)$ becomes a real periodic function and the equality $L(\pm\infty)=1$ in
\eq{042} does not hold. We should return to Eq.\,\eq{03a8a} which for
the considered massless viscoelastic lattice becomes
\beq
\GT^F(k)=\f{P_*}{\Gm\,(0+ik)\hat{E}}\gl[1-\f{1}{L(k)}\gr]
\eeq{03a8c}
with $Y=0$. In this case, if $\Gb> 0$ the elongation is discontinuous at
$\Gn=0$ as follows directly from the behavior of the viscoelastic unit
under discontinuous loading. The inverse transform leads to
\beq
\GT(\pm 0) =\f{P_*}{\Gm}\gl\{\f{\Gb}{2\Ga}\gl[1 \mp \gl(1-<\f{1}{L}>\gr)\gr]\gr.\n
+ \gl.\gl(1-\f{\Gb}{\Ga}\gr)\f{1}{\pi}\int_0^\infty
\f{dk}{(1+k^2)L(k/V_\Ga)}\gr\},
\eeq{054}
where $<1/L>$ is the averaged value of this periodic function. For the
square-cell lattice it is
\beq
<\f{1}{L}>=\f{1}{\pi}\int_0^\pi \f{\sin k/2}{\sqrt{1+\sin^2 k/2}}\,dk=\f{1}{2}\,,
\eeq{061}
while for modes I and II triangular-cell lattice it is approximately equal to 1/3.

The integral in \eq{054} can be asymptotically represented as
\beq
\f{1}{\pi}\int_0^\infty\f{dk}{(1+k^2)L(k/V_\Ga)}\to
\left\{ \begin{array}{ll}
1/2 <1/L>& (V_\Ga\to 0) \\
~~~~~~~0 & (V_\Ga\to\infty).
\end{array}\right.
\eeq{062}
Regarding to the asymptote $V_\Ga\to\infty$ it is assumed
that $v$ remains well below the elastic wave speed, $c$, not to contradict the
quasi-static formulation. This can be possible if $C_\Ga =\Ga c/a >>> 1$.

Note that the relation
\beq
\GT(+0)=\GT_c
\eeq{063}
must be used now. This relation has the following limiting expressions:
\beq
\GT(+0)=\f{P_*}{2\Gm}<\f{1}{L}>=\GT_c~~~(V_\Ga=0),\n
\GT(+0)=\f{\Gb}{\Ga}\f{P_*}{2\Gm}<\f{1}{L}>=\GT_c~~~(V_\Ga=\infty).
\eeq{064}
It can be seen that the resistance to the considered localized wave
propagation increases with its speed in an initial range of the ratio $v/c$.
This increase is most pronounced for small ratios $\Gb/\Ga$. For $\Gb=0, \Ga>0$
the resistance tends to infinity when $V_\Ga\to\infty$.
Such a boundary layer type of the solution is similar to that obtained for
the quasi-static crack growth in a viscoelastic lattice [Slepyan et al. (1999)].

This solution evidences that a stable region, where $Q(0)$ decreases as
the speed increases, does exist if $\Gb/\Ga<1$. This conclusion, however,
is based on the quasi-static formulation, which is valid if the viscosity
times fall into the heart of the static-amplitude-response domain; that is,
if $\Ga c/a$ is large enough, while $\Gb/Ga$ is small [see Slepyan (2000)].

In the elastic case, it follows that
\beq
\GT(0)=\f{P_*}{2\Gm}<\f{1}{L}>
\eeq{0214}
and hence the quasi-static regime corresponds to the initial elongation
\beq
Q_0=Q_c-\f{P_*}{2\Gm}<\f{1}{L}>.
\eeq{0215}

\subsection{Energy release}
The phase transformation path on the `macrolevel' corresponds to the
horizontal segment presented in Figure 6. The above analysis of the lattice
shows explicitly how the energy barrier (triangle ABC) is overcome. Triangle
CDE represents the released energy. The resulting energy released in this
process per bond, represented by the difference of the areas of the latter
and former triangles. For the quasi-static growth it is
\beq
G^0\approx \f{1}{6\Gm}P_*^2~~~ (\mbox{modes I and II}),\n
G^0=\f{1}{4\Gm}P_*^2~~~ (\mbox{mode  III}).
\eeq{0216}
It should be noted that low-speed regime for an elastic lattice
can be impossible. With regard to the crack growth this phenomenon
is discussed in Marder and Gross (1995) and Slepyan (2000). However,
under the influence of viscosity a stable growth within an initial
region of the speed may become possible [Slepyan et al. (1999),
Slepyan (2000)].

In a general case,
\beq
G^0=\f{\Gm}{2}\gl[\f{1}{\Gg}\gl(\f{P_*}{\Gm}-\GT_c\gr)^2-\GT_c^2\gr].
\eeq{0216a}

\subsection{Continuous elastic model}
The related steady-state problem for a continuous elastic medium can
be formulated as that for the growing crack which faces interact with
each other by a traction proportional to the crack opening and
loaded by distributed external forces, $p$, --- similar to those for
lattices. We may base on the equation
\beq
L(k)[\Gs_+(k)+ \Gs_-(k)]+ u_-(k)=0~~~ [u_+(k)=0]
\eeq{081}
with
\beq
\Gs_-=\Gk u_--\f{p}{0+ik}
\eeq{082}
and (for a subcritical crack speed)
\beq
L(k)=\f{f(V)}{\Gm\sqrt{k^2+0}}\,,
\eeq{083}
where $\Gs(\Gn)$ is the traction acting on the upper elastic half-plane and
$f(V)$ is a function of the nondimensional speed, $V$, different
for different fracture modes (see, for example, Freund, 1990, or Slepyan, 1990).
This leads to the equation
\beq
\CL(k)\Gs_++ u_-=\f{\CL(k)p}{0+ik}
\eeq{084}
with
\beq
\CL(k)= \f{L(k)}{\Gk L(k)+1}.
\eeq{085}
Using a standard way of considerations the Wiener-Hopf equation \eq{085} can
by resolved with the result as
\beq
\Gs_+(k)= \f{1-S_+(0)/S_+(k)}{0+ik}\,,~~~
u_-(k)= \f{S_+(0)S_-(k)}{0+ik}\,,\n
S_+(k)=[\Gk-i\Gm k/f(V)]^{-1/2}\exp\gl[\f{1}{2\pi i}\inti \f{\ln S_0(\xi)}{\xi-k}\,d\xi\gr]
~~~(\Im k>0),\n
S_-(k)=[\Gk+i\Gm k/f(V)]^{-1/2}\exp\gl[-\f{1}{2\pi i}\inti \f{\ln S_0(\xi)}{\xi-k}\,d\xi\gr]
~~~(\Im k<0),\n
S_0(k)=\f{\sqrt{\Gk^2+\Gm^2k^2/f^2(V)}}{\Gk +\Gm\sqrt{k^2+0}/f(V)}\,,\n
S_\pm(0)=\f{1}{\sqrt{\Gk}}\,,~~~S_\pm(k)\sim \sqrt{\f{f(V)}{\Gm(0\mp k)}}~~~(k\to\pm i\infty).
\eeq{086}
It follows that
\beq
\Gs_+(k)\sim \sqrt{\f{\Gm}{\Gk f(V)}}\f{p}{\sqrt{0-ik}} ~~~ (k\to i\infty),\n
u_-(k)\sim \sqrt{\f{f(V)}{\Gk\Gm}}\f{p}{(0+ik)^{3/2}} ~~~ (k\to -i\infty),\n
u_-(k)\sim \f{p}{\Gm (0+ik)}~~~(k\to 0)
\eeq{087}
with the asymptotes of the original functions as
\beq
\Gs(\Gn)\sim p\sqrt{\f{\Gm}{\pi \Gk f(V)\Gn}}~~~(\Gn\to +0),\n
u(\Gn)\sim 2p\sqrt{\f{-f(V)\Gn}{\pi\Gk\Gm}}~~~(\Gn\to -0),\n
u(\Gn)\to \f{p}{\Gk}~~~(\Gn\to -\infty).
\eeq{088}

The crack tip asymptotes in \eq{088} result in the speed-independent
energy release
\beq
G=\f{p^2}{\Gk}.
\eeq{089}
This value is a half of the total work of the external forces; the other
half is spent as the work against the resistance to the crack opening.
This equality is valid here since, in contrast to the lattice model,
no dissipative waves exist in the continuous model under a subcritical
crack speed.
\\
\\
{\bf Acknowledgment}
\\
\\
This research was supported by The Israel Science Foundation, grant
No. 28/00-1, and ARO, Grant No: 41363-M.

\vskip 18pt
\begin{center}
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\newpage\noindent
CAPTIONS
\\
\\
{}Figure 1. The force---strain diagram. The dotted axes origin corresponds
to the initial state of the lattice.
\\
\\
{}Figure 2. The lattices: (\Ba) Square-cell lattice; (\Bb) Triangular-cell
lattice. The wave path bonds are shown by dotted lines.
\\
\\
{}Figure 3. External forces applied to the intact lattice to model the
jump in the bond tensile force:  (\Ba) Square-cell lattice; (\Bb) Triangular-cell
lattice.
\\
\\
{}Figure 4. The force---strain diagram and the effective initial elongation.
\\
\\
{}Figure 5. The standard material unit.
\\
\\
{}Figure 6. The macrolevel phase transition path and the energy release.

\end{document}
