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\newcommand{\beps}{\mbox{\boldmath $\varepsilon$}}
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\def \eps{\varepsilon}      \def \tr{{\rm tr}}      \def \arsh{{\rm sinh}}
\def \phi{\varphi}          \def \Cof{{\rm Cof}}    \def \arch{{\rm cosh}}
\def \area{{\rm area}}      \def \det{{\rm det}}    \def \sign{{\rm sign}}
\def \vol{{\rm vol}}        \def \cond{{\rm cond}}  \def \Zint{{\rm int}}
\def \dom{{\rm dom}}        \def \rank{{\rm rank}}  \def \diam{{\rm diam}}
\def \length{{\rm length}}  \def \arg{{\rm arg}}
\def \supp{{\rm supp}}      \def \Diag{{\rm Diag}}  \def \deff{{\rm def}}
\def \sh{{\rm sh}}          \def \Arch{{\rm Arch}}  \def \min{{\rm min}}
\def \ch{{\rm ch}}          \def \tg{{\rm tg}}
\def \arctg{{\rm arctg}}    \def \arcsin{{\rm arcsin}}
\def \const{{\rm const}}    \def \arccos{{\rm arccos}}
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\begin{document}
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\begin{center}
{\large\bf
                          ENERGY ESTIMATIONS
              OF PHASE TRANSFORMATIONS  UNDER THE ACTION
             OF A SPHERICALLY CONVERGING COMPRESSION WAVE \\
}
\vskip5mm

{\large I.A. Brigadnov, A. B. Freidin
\\ \vskip1mm D.A. Indeitzev, N.F. Morozov, Yu.V. Petrov}

\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip5mm


\begin{quote}
A model  for  description  of  deformation  processes  initiated  by phase
transformations in a  ball  subjected  to  the  action  of  a  spherically
converging compression wave of high power is proposed. Explanation for the
effect of origination of a cavity in the center of a ball is given.
\end{quote}

\large

\vskip4mm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf 1. Formulation of the problem  and  basic hypotheses.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
A continuous  homogeneous ball of radius $R$ made of isotropic material is
subjected to the spherically symmetric radial-compressing  action  applied
to the surface [1].

The equilibrium  equation  of continuous medium is of the form
\mbox{ $\rho \ddot \mathbf{u} - \nabla \cdot\bsig = 0$}, where
$\nabla$ is the  differential Hamilton's  operator, $\bsig$  is
the Cauchy (true) stress tensor; $\rho$ is the density of
material, and $\mathbf{u}(t,\mathbf{r})$ is the displacement
vector, $\mathbf{r}$ is the point of a ball, the upper dots denote
the partial derivative with respect to time $t$.

At the initial moment the ball is at rest,  i.e.
$\mathbf{u}(0,\mathbf{r}) = \dot \mathbf{u}(0,\mathbf{r}) =%\equiv
0$. At $t>0$ the ball surface is subjected to the normal
spherically symmetric action ${\sigma_r}_{|_{r=R}}= F(t)$, where
$r$ is the radial coordinate, and $F(t)$ is the law of
transformation of impact pulse in time. The displacement in the
ball center is left bounded. Under the impact action of high power
and with allowance for phase transitions, the material is
described by the unknown tensor constitutive relation, $\bsig={\bf
S}\{\nabla \mathbf{u}\}$, where braces point to  the operational
dependence which has to include the possibility of phase
transformations. We do not concretize the dependence and further
develope an approximate energetic approach. The approach is based
on the approximation of the true wave process by basic functions,
which are defined from the solution of the preliminary simplified
initial boundary-value problem by the following hypothesis:

1) All the values are solely dependent on the radius:  i.e. the
displacement is radial  and  up  to the initiation of the
discontinuity the Cauchy stress tensor is a spherical one:
$\sigma=-p \,\mathbf{I}$,  where $p$ is  the  hydrostatic pressure
and $\mathbf{I}$ is unit tensor.

2) By  virtue  of  a weak compressibility of iron,  the
deformation of the ball is described by the linear strain tensor
$\beps $.

3) Material  is  described  by  the  Hooke  linear model,
\mbox{$p = -k_0 \tr(\beps)$}, where $k_0$ is the bulk modulus.

As a result,  the basic functions are found from solution of the
classical problem  on  the  acoustical  wave  propagation  in  a
ball.  Depending on retention of a variety  of  terms  in the
expansion  of  wave  solution, different  models of real physical
processes can be constituted,  such as: 1) retention of the first
term (the compression  wave)  alone  permits  to elucidate  the
occurrence of a cavity in the ball center due to formation of a
gaseous nucleus surrounded  with  a  liquid  layer and a solid
crust thereupon, 2) retention just the second term (the
rarefaction wave) allows to explain the origination of the cavity
due to cavitation in  the  center of  a  liquid  nucleus formed
during melting under the action of the first compression wave,  3)
retention of the next following terms is worth-white for
sufficiently  continuous  pulse  loading.  As  for  the  adequacy
of different models, this problem can be solved by experimental
verifications.


\vskip2mm\noindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf 2. Estimation of sizes of the liquid layer and sublimation
nucleus.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Within the framework of the first  model,  the  processes  arisen
in the course  of  the  first  running  of  the compression wave
are considered. In this case
 $$
 p(t,r)= {R \over r} \, F \left( t-{{R-r} \over c} \right),
         \quad t \in \left( 0, {R \over c} \right) ,
 $$
where $c$ is the rate of front  motion [2]. It is assumed that at
the moment within the central domain of the ball,  a transition of
the potential strain energy into  the  internal  energy  takes
place and as a result, melting and sublimation can arise.

For melting and sublimation of mass $m$ the energies
\mbox{$Q^{(i)} = m \, c_p^{(i)}$  $(i=1,2)$}  are needed,  where
$c_p^{(1)}$ and $c_p^{(2)}$ are the specific energies of melting
and sublimation, respectively. Therefore, within  the  framework
of accepted hypotheses,  the radius of the sublimation nucleus,
$r_a=aR$, and outer radius of the liquid layer, $r_b=bR$ are found
by solution of the following nonlinear equations
%=========================================================================
$$
 \int\limits_0^a F^2(Rx/c) \, dx = \frac{_2}{^3} \rho k_0 c_p^{(2)}
 a^3,\quad
 \int\limits_a^b F^2(Rx/c) \, dx =
 \frac{_2}{^3} \rho k_0 c_p^{(1)} (b^3-a^3). \eqno{(1)}
$$
%=========================================================================

According to [1],  function  $F(t)$  is  a fast decreasing
function, therefore,  it is easily shown that the system of
equations (1) has always  a real pair of roots $0 \leq a \leq b$.
The following cases are realized in accordance with location of
the roots,  namely: 1) $a=b=0$ --- the whole ball remains solid,
2) $0=a<b<1$ --- a liquid nucleus surrounded with a solid crust is
formed,  3) $0<a<b<1$ --- a  gaseous  nucleus surrounded with a
liquid layer and solid crust thereupon is formed,  4) $0<a<1\leq
b$ --- a gaseous nucleus surrounded with a liquid layer is formed,
5) $a \geq 1$ --- the  ball  totally transforms  into  a gaseous
state, i.e. it is evaporated.  Note that the system (1) has always
trivial solution $a=b=0$, and  the two  last variants lead to the
complete destruction of the ball.


\vskip2mm\noindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf 3. Process in the solid crust after phase transitions.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
It is assumed that the sublimate pressure $p_a$ is equalized along
the radius practically  momentary. The pressure can be estimated
by the condition of equality for the sublimation energy,
$Q^{(2)}=\rho V_a c_p^{(2)}$ and the potential energy  of
compressed  gas  $p_a V_a$,  where  $V_a$  is the volume of a
sublimate nucleus. The pressure at the inner surface of a solid
crust is given by  the law of hydraulics. As a result, the
following estimations are true
%=========================================================================
$$
 p_a= \rho c_p^{(2)},
\hskip10mm
 p_b= \eta^2 p_a
\hskip10mm
(\eta=a/b).
                                                                \eqno{(2)}
$$
%=========================================================================

The solid crust begins to blow up under the action of the applied
pressure. In this case the stress tensor is not spherical, that
can lead  to an irreversible  shear deformation. Simultaneously,
within  the  ball the processes of sublimate condensation on the
inner  surface  of  the  liquid layer  and growing the solid crust
due to recrystallization of the liquid layer are carried out.  As
a result,  the pressure under the growing crust drops and the
deformation is stopped.  After cooling,  the outer radius of the
ball grows up to $R_* = R + u_r(R)$.

Irreversible deformation of the solid crust is estimated by the
solution of  the  problem  on  blowing up of the spherical layer
under the internal pressure for the material  relation  of  the
rigid-plastic  type [3]
%=========================================================================
$$
 \beps^D= {{\sigma_s} \over {2\mu\alpha}} \,
         H \left( {{|\bsig^D|}\over{\sigma_s}} -1 \right)
           \left( {{|\bsig^D|}\over{\sigma_s}} -1 \right)
         {{\bsig^D}\over{|\bsig^D|}} ,
                                                                \eqno{(3)}
$$
%=========================================================================
where $\beps^D$ and $\bsig^D$ are the deviatoric parts of the
strain and stress tensors respectively, $\mu$ is the shear
modulus; $\alpha> 0$ and $\sigma_s>0$ are the dynamic parameter of
strain hardening and yield  stress  respectively, $H(x)$ is the
Heaviside function.

>From the  plastic  incompressibility  condition
$\tr(\beps)=u_r'+2u_r/r=0$, the radial displacement
$u_r(r)=C/r^2$, is derived,  where  the  constant $C$ is
determined from the boundary condition on the internal surface of
the solid crust $\sigma_{rr}(r_b)=-p_b$.

So, for a relative increase of the outer solid crust radius the
following estimation holds true
%=========================================================================
$$
 \delta_R= {{u_r(R)} \over R} =
           {{b^3 \sigma_s}\over{2 \sqrt{6} \mu \alpha}} H(s-1)(s-1) ,
                                                                \eqno{(4)}
$$
%=========================================================================
where
$$
 s=  \max_{r}  \left\{ |\bsig^D|/\sigma_s \right\}
  = (3/2)^{1/2} \eta^2 (p_a/\sigma_s )
 $$
is the relative intensity of tangential stresses  on  the inner
surface of the solid crust. For $s \leq 1$ the outer ball radius
is not grown, since $\delta_R =0$.  It is easily to verify that
within the framework of the model considered a cavity can not be
formed at $\eta \leq \eta_*=(2/3)^{1/4} (\sigma_s/p_a)^{1/2}$. In
this case, examination of the second model is needed.

The relative radius  of  a  spherical  cavity  is  found  by  the
law of conservation of mass
%=========================================================================
$$
 \delta_* = {{r_*} \over R} =
              \left[ \left( 1+\delta_R \right)^3 - 1 -
                     \left( {\rho \over {\rho_*}} -1 \right) b^3
              \right]^{1/3},
                                                                \eqno{(5)}
$$
%=========================================================================
where \mbox{$\rho_*$}  is  the  density  of  the recrystallization
layer.


\vskip2mm\noindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf 4. Numerical results.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
An  example of analysis of the derived relations was performed for
the iron ball of radius $R=0{.}05$ {\it m} with the following
parameters [4]: $k_0 \approx 108$,  $\mu \approx 84$ and $\sigma_s
\approx 1$ {\it GPa}; $\alpha  \approx  0{.}1$ {\it relative
units}, $\rho \approx 7{.}6 \cdot 10^3$ {\it kg/m$^3$}, $c_p^{(1)}
\approx 1{.}3 \cdot 10^4$ and $c_p^{(2)} \approx 3{.}5 \cdot 10^5$
{\it J/mol}, molar weight of iron $m_0 \approx 6 \cdot 10^{-2}$
{\it kg/mol}.

The simplest   approximation   of   the   impact   effect   $F(t)
=  F_0 e^{-t/\lambda}$ with the parameter \mbox{$\lambda=0{.}15$}
$\mu s$ was considered [1].

Relations of the relative values of sublimation nucleus radius $a$
(Curve 1),  outer  liquid  layer radius $b$ (Curve 2) and
spherical cavity radius $\delta_*$ under the assumption of
$\rho_*=\rho$ (Curve 3)  for  different amplitudes $F_0$ are
presented in Figure 1.

For the typical amplitude $F_0=40$  {\it GPa} [1] the relative
values of the sublimation nucleus and inner liquid layer radius $a
\approx 0{.}14$   and   $b  \approx 0{.}53$ correspond to the
third from the above-mentioned states of the ball. In this case
the sublimate pressure  $p_a \approx 44{.}3$ {\it GPa} and the
relative increase in the ball outer radius \mbox{$\delta_R \approx
0{.}01$}, whence, on  the assumption that $\rho_*=\rho$ the
estimation for the relative radius of the spherical cavity,
$\delta_* \approx 0{.}32$,  is followed, that is in a good
agreement with experimental data [1].

Recrystallization of the liquid phase proceeds at the pressure
$p_a$ and temperature $T_a$ which can be estimated by the law for
ideal gas
%=========================================================================
$$ T_a=p_a\frac{3m_0}{2\rho R_0}=\frac{3c_p^{(2)}m_0}{2R_0}\approx
    3.75\cdot 10^3 \,{\rm K},
$$
%=========================================================================
where $R_0=8.31$ {\it J/degree$\cdot$mol} is the universal gas constant.

The presented model can be refined due to:  1) a more accurate
estimation of the potential compression energy at the moment of
focusing by using non-linear elasticity models, 2) taking into
account the temperature variation during adiabatic compression and
also the temperature and  pressure influences upon the specific
melting and sublimation heats, 3) a more accurate estimation of
the sublimate pressure, 4) taking into account the sublimate
condensation and melt recrystallization in the course of blowing
up  of  the solid crust  that  is resolved into solution of the
connected Stephan's problem and visco-plastic flow;  5)
consideration  of the phase composition of the recrystallization
layer.

{\bf Acknowledgement}.  This  work  was  supported  by  Russian
Foundation for Basic Research (grants   N 98-01-01054,
00-15-99273).


\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}

\bibitem{1}
Metals and minerals research in spherical shock-wave recovery experiments.
(Ed. B.I.Litvinov). ONTI RFNC-VNIITF, Snezhinsk, Russia, 1996. 71 p.

\bibitem{2}
{\it L.I.    Slepyan}.    Nonstationary    elastic    waves.
Leningrad: Suodostrojenije, 1972. 374 p. (in Russian)

\bibitem{3}
{\it V.D.Klyushnikov}.  Mathematical theoty  of  plasticity.  Moscow:  MSU
press, 1979. 208 p. (in Russian)

\bibitem{4}
Physical magnitudes. Handbook (Eds. I.S.Grigoriev, E.Z.Meilikhov).
Moscow: Energoatomizdat, 1991. 1232 p. (in Russian)

\end{thebibliography}
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\vskip10mm
{\it Institute for Problems in Mechanical Engineering

of Russian Academy of Sciences.

Bolshoy pr. V.O. 61, St.~Petersburg, 199178, Russia}


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\end{document}
