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diff --git a/macros/latex/contrib/springer/svjour/sw/example/example.tex b/macros/latex/contrib/springer/svjour/sw/example/example.tex new file mode 100644 index 0000000000..15a810c943 --- /dev/null +++ b/macros/latex/contrib/springer/svjour/sw/example/example.tex @@ -0,0 +1,601 @@ +% 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.} +\bibitem[Averin et al. 1990]{ref1} +Averin VI, Gromov VI, Erofeev MV, Kalin AA, Kuznetsov MS, Moiseev VA, +Ostafitchuk VP, Pitchurin EP (1990) Threshold Phenomena and Modification +of Structure of Al Under Laser Impulse Action. +Moscow Institute for Physical Engineering, Moscow (in Russian) +\bibitem[Averin et al. (1991)]{ref2} +Averin VI, Gromov VI, Erofeev MV, Kalin AA, Kuznetsov MS, Moiseev VA, +Ostafitchuk VP, Pitchurin EP (1991) Threshold phenomena and modification +of metal structure and properties under the action of nanosecond +laser pulses. Bulletin of the Academy of Sciences of the U. S. S. 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