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\begin{document}
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\date{}
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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}
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\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
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{\bf 1. Formulation of the problem and basic hypotheses.}
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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
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{\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) $0=a** 0$ 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
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\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}
**