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+% example.tex
+% An example input file demonstrating the sw option of the SVJour
+% document class for the journal: Shock Waves
+% (c) Springer-Verlag HD
+%-----------------------------------------------------------------------
+%
+\documentclass[sw,bibself]{svjour}
+\usepackage{graphics}
+
+\newcommand{\fakefignlabel}[2]{\refstepcounter{figure}\label{#2}%
+\addtocounter{figure}{-1}\def\thefigure{#1}}
+
+\journalname{Shock Waves}
+
+\begin{document}
+
+\title{Evolution of a laser-generated shock wave in iron\\
+and its interaction with martensitic transformation\\
+and twinning}
+\titlerunning{Evolution of a laser-generated shock wave in iron}
+
+\author{I.V. Erofeev\inst{1} \and V.V. Silberschmidt\inst{2} \and
+A.A. Kalin\inst{1} \and V.A. Moiseev\inst1 \and I.V. Solomatin\inst3}
+
+\mail{V.V. Silberschmidt}
+
+\institute{Moscow Institute for Physical Engineering,
+31 Kashirskoe Av., 115409 Moscow, Russia\and
+Lehrstuhl A f\"ur Mechanik, TU M\"unchen, Boltzmannstr. 15,
+D-85746 Garching b. M\"unchen, Germany\and
+Mining Institute, Ural Department of the Russian Academy of Sciences,
+78A Karl Marx Street, 614007 Perm, Russia}
+
+\date{Received 9 August 1994 / Accepted 30 June 1997}
+
+
+\abstract{
+Effects of shock waves
+(generated by a nanosecond laser pulse in plates of
+Armco-iron) on structural changes are analysed. Localisation of processes of
+martensitic transformation and twinning -- for various values of
+laser pulse duration -- is studied both experimentally and numerically.
+A proposed model accounts for interaction of shock wave propagation and
+structure changes. Realisation of martensitic transformation and
+twin formation influences wave front modification. A stress amplitude
+decrease with increasing distance from a microcrater determines,
+together
+with the pulse duration, a character of spatial localisation of
+structural changes. Numerical results are compared with experimental data
+and serve as a basis for additional interpretation of phenomena.
+\keywords{Nanosecond laser pulse,
+Martensitic transformation, Twinning, Localisation}}
+
+\maketitle
+
+
+\section{Introduction}
+Generation of a shock wave (SW) with an amplitude of 50--200 GPa in
+metals
+by a nanosecond laser pulse is accompanied by a series of phenomena:
+plasma generation, plastic deformation, phase transitions, spall damage,
+etc. (\cite[1991]{ref1}). Characteristic
+features of an initiation
+and evolution of these processes are determined by parameters of
+external action and material properties. Despite the differences,
+all these phenomena also have some common features:
+\begin{itemize}
+
+\item
+significant level of energy and force threshold values and
+their dependence on pulse duration;
+
+\item
+marked instability;
+
+\item
+spatial non-uniformity and localisation in mesoscopic volumes
+(\cite{ref13}).
+\end{itemize}
+
+\noindent
+These similarities allow to examine these divers processes in terms
+of a general approach.
+Investigations of threshold phenomena under the SW generation
+in metals were carried out for various types of the concentrated
+energy flux (CEF): high-energy ion beams (\cite{ref17,ref18});
+relativistic electrons; super-high-speed loading (\cite{ref14,ref16})
+and laser radiation (\cite{ref15,ref5}; \cite[1984a,\,b]{ref8},
+\cite{ref4,ref7,ref6,ref12,ref11}).
+
+A parameter $Q = q/E$
+($q$ - intensity of CEF, $E$ - Young modulus of
+target material) was proposed by Averin et al. (1990; 1991)
+as a characteristic of the CEF -- metal interaction.
+Another important parameter is a pulse duration $t_{i}$,
+which sufficiently influences generation,
+development, and localisation of threshold phenomena
+for a level of $Q$ = $10^{5} - 10^{6}$ m/s
+(this level corresponds to experimental conditions under study).
+
+
+
+\section{Model}
+Propagation of a shock wave generated by a nanosecond laser pulse in
+Armco-iron is accompanied by martensitic transformation (MT) and twinning.
+These processes are the main deformation mechanisms because the plastic
+relaxation time (and the characteristic time of thermoconductivity) is
+sufficiently large compared to the pulse duration.
+An adequate account of these deformation processes presupposes
+an introduction of additional internal variables for characterisation
+of structural changes. Evidently, such variables must be of the same tensor
+order as traditional deformation parameters.
+
+A deformation response to martensitic transformation is linked with
+the reconstruction of the crystalline lattice. A symmetric second-order
+tensor $m_{ik}$ (\cite[1990]{ref21}; \cite{ref23,ref3})
+can be introduced as a microscopic parameter.
+A respective macroscopic parameter $s_{ik} = \big< m_{ik}\big>$
+is an independent thermodynamic variable
+and is obtained by the averaging of $m_{ik}$ over all possible states
+in a representative unit cell.
+In contrast to martensitic transformation, twinning is a shear process
+and thus the traceless second-order tensor $g_{kl}$
+(\cite{ref20}) is introduced as a microscopic
+parameter. A macroscopic parameter $d_{kl}$
+which describes the deformation reaction of a medium to twinning can be
+obtained by means of averaging $g_{kl}$.
+
+\begin{figure}
+\fakefignlabel{1a,b}{fig1}
+\centering\leavevmode\includegraphics{sw059f1.ps}
+\caption{Dependence of free energy on parameter of
+martensitic transformation: ${\bf a}$ before and after transformation
+(curves 1 and 2, respectively) and parameter of
+twinning; ${\bf b}$ for various levels of stress}
+\end{figure}
+
+An analysis of the system's thermodynamic potential is a prerequisite
+for the elaboration of macroscopic constitutive equations. Statistical
+thermodynamics was used for the study of characteristic features of the
+free energy change under martensitic transformation (\cite[1990]{ref21};
+\cite{ref23} and twinning (\cite{ref20}). A dependence of the free
+energy on macroscopic parameters, characterising the extent of
+martensitic transformation and of twinning are shown in
+Fig.\,\ref{fig1}. A number, position, and depth of energy minima
+corresponding to stable states of the system are determined by levels of
+temperature and acting force.
+
+Macroscopic equations describing the interaction of\break
+plastic deformation and
+structural changes can be obtained in terms of the thermodynamics of
+irreversible processes. A dissipative function has the following form:
+%
+\begin{equation}
+TP_{s} = \sigma_{ik} e_{ik}^{\mathrm{p}}
+- {{\partial F} \over {\partial s_{ik}}}
+{{\partial s_{ik}} \over {\partial t}}
+- {{\partial F} \over {\partial d_{ik}}}
+{{\partial d_{ik}} \over {\partial t}} \ge 0,
+\label{eq:1}
+\end{equation}
+%
+where $T$ is the temperature; $P_{s}$ is an entropy production,
+which is positive for
+irreversible processes and equal to zero for stable states according to
+the second law of thermodynamics; $\sigma_{ik}$
+is a macroscopic stress tensor;
+$e_{ik}^{\mathrm{p}} = e_{ik} - e_{ik}^{\mathrm{e}}$
+is an irreversible part of a strain rate tensor (indices e and p are
+used for elastic and plastic parts, respectively);
+$F$ is the free energy,
+${ {\partial F} \over {\partial s_{ik} } },
+{ {\partial F} \over {\partial d_{ik} } },$
+are thermodynamic forces, acting on the system, when the values of
+respective parameters differ from equilibrium values.
+Macroscopic constitutive equations can be derived from the dissipative
+function in approximations of local equilibrium and direct proportionality
+of the thermodynamic forces and fluxes:
+\begin{eqnarray}
+\sigma_{ik}&=&L_{iklm}^{11} e_{lm}^{\mathrm{p}}
+- L_{iklm}^{12} \dot{s}_{lm}
+- L_{iklm}^{13} \dot{d}_{lm} ,\nonumber\\
+%
+{ {\partial F} \over {\partial s_{ik} } }&=&
+L_{iklm}^{21} e_{lm}^{\mathrm{p}}
+- L_{iklm}^{22} \dot{s}_{lm}
+- L_{iklm}^{23} \dot{d}_{lm} ,\nonumber\\
+%
+{ {\partial F} \over {\partial d_{ik} } }&=&
+L_{iklm}^{31} e_{lm}^{\mathrm{p}}
+- L_{iklm}^{32} \dot{s}_{lm}
+- L_{iklm}^{33} \dot{d}_{lm} ,
+\label{eq:2}
+\end{eqnarray}
+%
+where $L_{iklm}^{rq}$ are kinetic coefficients, the
+matrix of which (in respect to $r$ and $q$)
+is symmetric and positively determined due to the Onsager reciprocal
+relations. The dot over parameters means time differentiating.
+
+
+The system of Eqs.~(\ref{eq:2}) contains a relaxation equation for
+stresses
+and two kinetic equations for parameters of martensitic transformation and
+twinning. Thus, the changes in the stress state and spatio-temporal
+evolution of structural changes can be studied for arbitrary loading
+conditions in terms of boundary value problems. The analysis of metal
+behaviour under quasi-static loading has shown
+(\cite[1990]{ref21}; \cite{ref20,ref23}) that the system (\ref{eq:2})
+describes the main properties of structural transformations:
+deformation hysteresis of MT and its shift along the temperature axis under
+the load change, polar character of twinning realisation, etc.
+
+Considering an additive character of contributions\break from
+transformations, the full free energy of the system can be written as
+%
+\begin{equation}
+F = F_{0} + F_{1} + F_{2},
+\end{equation}
+%
+where $F_{0}$ is a part of the free energy that does not depend on the
+structural variables and, thus, it does not influence the form of the state
+laws in (\ref{eq:1}); $F_{1}$ and $F_{2}$ are free energy parts
+correlated, respectively, to martensitic transformation and twinning.
+
+A dependence of the free energy on the structural parameters, obtained in
+statistical thermodynamics analysis shown in Fig.\,\ref{fig1}, can be
+approximated
+by polynomials of respective parameters. For the uniaxial case studied
+in this paper, these approximations can be written in the following form:
+\begin{equation}
+\begin{array}{rl}
+F_{1}={}&\displaystyle \frac{A_{1}}{2} s^{2}
++ \frac{B_{1}}{3} s^{3}
++ \frac{C_{1}}{4} s^{4}\\[2mm]
+&- \Big (D_{1} \sigma - M \big (T - T_{p} \big) \Big) s,
+\end{array}
+\end{equation}
+\begin{equation}
+F_{2} = {{A_{2}} \over 2} d^{2} + {{B_{2}} \over 3} d^{3}
++ {{C_{2}} \over 4} d^{4} - D_{2} \sigma d,
+\end{equation}
+where $A_{i}, B_{i}, C_{i}, D_{i}, M > 0$,
+$(i = 1,2)$ are material parameters, which are functions of
+temperature and mechanical structural parameters in a common case;
+$T_{p}$ is the transformation temperature.
+
+\section{Experimental study and formulation\\%ill\break
+of the problem}
+Characteristic features of threshold phenomena localisation,
+as was shown by \cite{ref2}, are mostly distinct for a pulse
+duration less than 30 ns. The mono-pulse laser action can be considered
+as a surface one and the radiation--material interaction is then
+accounted
+for by a simulation of the pressure change on the frontal surface of
+the target (\cite{ref6}).
+
+\begin{figure}
+\fakefignlabel{2a--c}{fig2}
+\centering\leavevmode\includegraphics{sw059f2.ps}
+\caption{Evolution of shock wave profile:
+${\bf a}$, kinetics of martensitic transformation;
+${\bf b}$ and of twinning; and ${\bf c}$.
+Pulse duration 23 ns; time: 1, 20 ns, 2, 40 ns, 3, 60 ns}
+\end{figure}
+
+Respective experimental methods are described in detail by
+\cite{ref2}, Merzhievskii and Titov
+(1987), and Burdonskii et al. (1984).
+Two lasers were used for shock wave generation in metals.
+Their characteristics are: pulse energy $E_{i}$ up
+to 60 J and up to 100 J with a pulse duration $t_{i}$
+of 23 ns and 3 ns, respectively.
+The pulses had a triangular form with a uniform energy distribution over
+a focus point. These experiments were carried out in a vacuum camera
+at a
+residual pressure of 1 Pa. The specimens were radiated under an angle
+$\alpha = 30^{\circ}$ between a laser pulse direction
+and a perpendicular to the target surface
+in order to exclude a re-refraction from the target upon the optical
+system.
+
+High pressure, a short pulse duration, and a small square of a loaded
+region complicate a direct measurement of the shock wave amplitude in
+the studied case. Thus, an experimental data treatment is used together
+with the numerical modelling (\cite{ref12,ref8}) for the pressure
+estimation in SW. The pulse amplitude, $P_{m}$, can be approximated by
+the following relation:
+\begin{equation}
+P_{m} = kq^{n},
+\label{eq:6}
+\end{equation}
+where $n = 0.4$--0.8, and $k$ is an empirical coefficient.
+The values of parameters $n$ and $k$
+depend upon the radiation intensity.
+In the case under study the intensity equals
+2 $\times$ 10$^{12}$ W/cm$^{2}$ for a pulse duration
+of 23 ns and 5 $\times$ 10$^{12}$ W/cm$^{2}$ for 3 ns.
+The data from Eliezer et al. (1990), with
+$k = 1.3 \times 10^{-5}$, $n = 0.4$,
+suit such intensity values best.
+Then calculations based on (\ref{eq:6}) give $P_{m}$ = 80 GPa for 23 ns
+and $P_{m}$ = 160 GPa
+for 3 ns.
+
+Plates of Armco-iron of width $l = 500$~$\mu$m
+were used in the described experiments.
+The specimens were obtained by means of powder metallurgy
+with subsequent annealing at $T = 900 ^{\circ}$C
+and air-cooling.
+The final size of grains in plates was approximately $100~\mu$m.
+After radiation, the specimens were separated along the direction of
+the shock wave propagation and a metallographic analysis was
+carried out.
+Structural changes were also studied using microhardness tests at different
+points of target section.
+
+For numerical simulations of threshold phenomena in shock waves
+generated by the laser pulse radiation of the Armco-iron plate the
+system (\ref{eq:2}) was used together with the pulse conservation law.
+With the plate width being sufficiently less than the two other
+dimensions, a uniaxial analysis (along the width of the plate) can be
+utilised for the investigation of characteristic features of the
+shock-wave evolution and the kinetics of structural changes. A
+dimensionless form of these equations is
+\[
+{ {\partial v^{*}} \over {\partial t^{*}} } = {1 \over {\rho}}
+{ {\partial \sigma^{*}} \over {\partial \xi} },\]
+\[
+{ {\partial \sigma^{*}} \over {\partial t}^{*} } =
+\kappa { {\partial v ^{*}} \over {\partial \xi} } -
+{1 \over {\tau}} {\sigma}^{*} -
+{\gamma_{1}} { {\partial s} \over {\partial t}^{*} } -
+{\gamma _{2}} { {\partial d} \over {\partial t}^{*} },\]
+\[
+{ {\partial s} \over {\partial t}^{*} } =
+- {1 \over {\tau}_{s}} { {\partial F _{1}} \over {\partial s}
+},\]
+\begin{equation}
+{ {\partial d} \over {\partial t}^{*} } =
+- {1 \over {\tau}_{d}} { {\partial F _{2}} \over {\partial d} },
+\end{equation}
+
+\noindent
+where $v^{*} = {{t_{1}} \over l} v_{z}$;
+$t^{*} = {t \over t_{i}}$;
+$\sigma^{*} = {{\sigma_{zz}} \over G}$;
+$G = {E \over {2(1+{\mu})}}$;
+${\xi} = {z \over l}$;
+${\kappa} = { {2} \over {3(2-m)}}$;
+$m = { {3K} \over {3K+2G}}$;
+$K = {E \over {1-2{\mu}}}$;
+${\tau}_{s} = {{L^{22}} \over {t_{i}}}$;
+${\tau}_{d} = {{L^{33}} \over {t_{i}}}$;
+
+\noindent
+${\gamma}_{1} = {{L^{13} L^{22} G} \over
+{L^{11} L^{22} L^{33} - L^{22} {(L^{13})^{2}} - L^{33} {(L^{12})^{2}}}}$;
+
+\noindent
+${\gamma}_{2} = {{L^{12} L^{33} G} \over
+{L^{11} L^{22} L^{33} - L^{22} {(L^{13})^{2}} - L^{33} {(L^{12})^{2}} }}$.
+
+\noindent
+Here $v_{z}$ is a rate vector component; ${\rho}$ is the material
+density;
+$z, {\xi}$ are normal and dimensionless co-ordinates;
+${\tau}$ is the Maxwell relaxation time; $E$, Young's modulus;
+${\mu}$, Poisson coefficient; $L^{ij}$ ( $i,j$ = 1,2,3)
+are scalar parameters -- the first terms of expansion of kinetic
+coefficients $L_{klmn}^{ij}$
+with respect to structural parameters (\cite{ref3}).
+Considering the independence of MT and twinning processes
+and accounting for the difference in characteristic times of relaxation
+and structural transformation, one can assume $L^{ij}$
+$(i >1, i {\not=} j) {\rightarrow} 0$.
+
+Then, the boundary conditions are:
+\[
+{\upsilon}^{*} (1, t) = 0,
+\]
+\begin{equation}
+{\sigma}^* = \left\{\begin{array}{l}
+2 \tilde {\sigma}_{\mathrm{a}}t^{*},\\[2pt]
+2 \tilde {\sigma}_{\mathrm{a}}^{*}{(1-t^{*})},\\[2pt]
+0
+\end{array}\right.\qquad
+\begin{array}{rcl}
+t^{*}&\leq&1/2,\\[2pt]
+1/2&<&t^{*} \leq 1,\\[2pt]
+t^{*}&>&1,
+\end{array}
+\label{eq:8}
+\end{equation}
+%
+where $\tilde\sigma_{\mathrm{a}}$ is the dimensionless pulse
+amplitude. Initial conditions have the following form:
+\begin{equation}
+s({\xi},0) = 1, \upsilon({\xi},0) = \sigma^{*}({\xi},0) = d({\xi},0) = 0.
+\label{eq:9}
+\end{equation}
+
+Boundary conditions (\ref{eq:8}) correspond to the case of the triangle
+stress
+pulse action on the frontal target surface and the wave reflection
+from the free (rear) surface. The conditions in (\ref{eq:9}) characterise
+a material's initial state, the martensite phase with the absence of
+twins. The numerical simulation was carried out for the interval of
+a pulse duration from 3--30 ns and $\sigma_{\mathrm{a}}$ = 80 and
+160 GPa.
+
+\begin{figure}
+\centering\leavevmode\includegraphics{sw059f3.ps}
+\caption{Evolution of pulse amplitude (pulse duration:
+1, 3 ns, 2, 23 ns)}\label{fig3}
+\end{figure}
+
+\section{Discussion}
+Numerical analysis allows us to analyse the peculiarities of the
+shock wave
+propagation and the change of its configuration. The interaction of
+processes of structural changes and wave propagation results
+in the specificity of the threshold phenomena localisation for
+a given interval of pulse duration and values of SW amplitude.
+
+The calculated SW configuration and structural transformation kinetics
+are shown in Fig.\,\ref{fig2} for $t_{i}$ = 23 ns.
+Exceeding the critical stress
+threshold results in the initiation of
+the $\alpha \rightarrow \varepsilon$ transformation.
+It causes the step formation on the loading front. Such two-wave configuration
+is characteristic for I-type phase transitions. The reverse
+$\varepsilon \rightarrow \alpha$ transformation
+in the unloading begins under the lower stress level because of
+the hysteresis in martensitic transformation. Thus, the shorter step
+is being formed on the rear front of the shock wave (Fig.\,\ref{fig2}).
+The SW propagation is accompanied by a sharp decrease in its amplitude
+(Fig.\,\ref{fig3} presents corresponding results of numerical
+simulation)
+and an increase in the distance between the loading and unloading fronts
+because of the relaxation. These two processes determine specific features
+of the initiation of structural transformations and their localisation.
+The shock wave propagation is accompanied by the shift of the zone of
+the reversible MT (Fig.\,\ref{fig2}b). The width of this zone grows with
+the increase in the distance from the microcrater in connection with
+the change of the SW configuration. Results of numerical simulation
+correlate to experimental data obtained by the microhardness measurements
+in different points of the specimen. A decrease in the wave amplitude to
+a stress value less than the critical one results in the formation of a
+localised finite zone of reversible
+$\alpha \leftrightarrow \varepsilon$
+transformation. A sharp drop in the shock wave
+amplitude in the case of the 3-ns pulse makes the initiation of the
+$\alpha \rightarrow \varepsilon$
+transformation impossible and is the reason for the absence of the MT
+zone in the specimens loaded with such a short pulse.
+
+A twin formation process is characterised by the sufficiently lower level
+of threshold stress. Thus, it occurs in all intervals of pulse duration
+and the localisation zone of twinning is wider, compared with
+MT (Fig.\,\ref{fig2}c). An insufficient decrease in the twinning
+parameter with
+the wave propagation is due to the so-called elastic twinning --
+a partial reversibility of the twinning process. The width of the zone
+with twins is determined at the moment when the decreasing wave
+amplitude becomes less than the critical stress necessary for twin formation.
+The observed absence of twins near the bottom of the microcrater
+for the case $t_{i}$ = 3 ns is linked with the small width of the SW
+at the initial stage of its propagation and, consequently,
+with an insufficiet action time of the stress,
+which is higher than the critical one, in this region. The
+irreversibility
+of the twinning allows the direct measurement of the twin length.
+A comparison between the twin-length change, obtained from experimental
+observations, and the calculated value of $t_{\mathrm{t}}$, the total
+time,
+when the wave amplitude in the given point is larger than the critical
+twinning stress, is given in Fig.\,\ref{fig4}. The similarity of the two
+curves
+proves
+a direct effect of the action time of overcritical load on twinning.
+
+\begin{figure}
+\centering\leavevmode\includegraphics{sw059f4.ps}
+\caption{Effective time of twinning $t_{\mathrm{t}}$ (1) and twin length (2)}
+\label{fig4}
+\end{figure}
+
+Thus, the proposed model allows the evolution of the shock wave
+(generated by the nanosecond laser pulse) and its interaction with
+structural transformations (MT and twinning) to be analysed.
+The initiation of the structural changes is caused by the overcoming of
+the threshold stress value in the loading front, and their localisation
+is linked with a rapid decay of the wave amplitude during its propagation.
+The structural transformations, in turn, change the shock wave
+configuration and result in the division of the loading and
+unloading fronts into sections, the height of the dividing point being
+correlated to the critical values of stress.
+
+\begin{acknowledgement}
+Two of the authors (VVS and IVS) gratefully
+acknowledge Prof. O.B. Naimark, Dr. V.V. Belyaev and L.V.Filimonova
+for fruitful discussions.
+\end{acknowledgement}
+
+\begin{thebibliography}{88.}
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+Moscow Institute for Physical Engineering, Moscow (in Russian)
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+\end{document}