Shock Waves manuscript No. (will be inserted by the editor)

Evolution of a laser-generated shock wave in iron and its interaction with martensitic transformation and twinning

I.V. Erofeev^{1}, V.V. Silberschmidt^{2}, A.A. Kalin^{1}, V.A. Moiseev^{1}, and I.V. Solomatin^{3}

1 Moscow Institute for Physical Engineering, 31 Kashirskoe Av., 115409 Moscow, Russia 2 Lehrstuhl A für Mechanik, TU München, Boltzmannstr. 15, D-85746 Garching b. München, Germany 3 Mining Institute, Ural Department of the Russian Academy of Sciences, 78A Karl Marx Street, 614007 Perm, Russia

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.

Key words: Nanosecond laser pulse, Martensitic transformation, Twinning, Localisation

1 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. (Averin et al. 1990, 1991). 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:

– significant level of energy and force threshold values and their dependence on pulse duration;

– marked instability;

– spatial non-uniformity and localisation in mesoscopic volumes (Koneva and Kozlov 1991).

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 (Pogrebnjak et al. 1989; Povalyaev et al. 1990); relativistic electrons; super-high-speed loading (Merzhievskii and Titov 1987; Naimark and Belyaev 1989) and laser radiation (Meyers et al. 1980; Clauer et al. 1981; Cottet and Romain 1982, 1984a, b, Burdonskii et al. 1984; Cottet et al. 1988; Cottet and Boustie 1989; Gromov et al. 1990; Eliezer et al. 1990).

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).

2 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} (Silberschmidt et al. 1989, 1990; Silberschmidt and Tanaka 1993; Belyaev et al. 1989) can be introduced as a microscopic parameter. A respective macroscopic parameter s_{ik} = ⟨m_{ik}⟩ is

Correspondence to: V.V. Silberschmidt

Fig. 1a,b. Dependence of free energy on parameter of martensitic transformation: a before and after transformation (curves 1 and 2, respectively) and parameter of twinning; b for various levels of stress

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} (Silberschmidt 1992) 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}.

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 (Silberschmidt et al. 1989, 1990; Silberschmidt and Tanaka 1993 and twinning (Silberschmidt 1992). A dependence of the free energy on macroscopic parameters, characterising the extent of martensitic transformation and of twinning are shown in Fig. 1. 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 plastic deformation and structural changes can be obtained in terms of the thermodynamics of irreversible processes. A dissipative function has the following form:

T P_{s} = σ_{ik}e_{ik}^{p} − \frac{∂F}{∂s_{ik}} \frac{∂s_{ik}}{∂t} − \frac{∂F}{∂d_{ik}} \frac{∂d_{ik}}{∂t} ≥ 0, (1)

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; σ_{ik} is a macroscopic stress tensor; e_{ik}^{p} = e_{ik} − e_{ik}^{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, \frac{∂F}{∂s_{ik}} , \frac{∂F}{∂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:

σ_{ik} = L_{iklm}^{11}e_{lm}^{p} − L_{iklm}^{12}ṡ_{lm} − L_{iklm}^{13}ḋ_{lm},
\frac{∂F}{∂s_{ik}} = L_{iklm}^{21}e_{lm}^{p} − L_{iklm}^{22}ṡ_{lm} − L_{iklm}^{23}ḋ_{lm},
\frac{∂F}{∂d_{ik}} = L_{iklm}^{31}e_{lm}^{p} − L_{iklm}^{32}ṡ_{lm} − L_{iklm}^{33}ḋ_{lm},  (2)

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. (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 (Silberschmidt et al. 1989, 1990; Silberschmidt 1992; Silberschmidt and Tanaka 1993) that the system (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 from transformations, the full free energy of the system can be written as

F = F_{0} + F_{1} + F_{2},  (3)

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 (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. 1, 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{aligned}F_{1} = & \frac{A_{1}}{2} s^{2} + \frac{B_{1}}{3} s^{3} + \frac{C_{1}}{4} s^{4} \\ & −(D_{1}σ − M(T − T_{p}))s, \end{aligned}  (4)

F_{2} = \frac{A_{2}}{2} d^{2} + \frac{B_{2}}{3} d^{3} + \frac{C_{2}}{4} d^{4} − D_{2}σd,  (5)

Fig. 2a–c. Evolution of shock wave profile: a, kinetics of martensitic transformation; b and of twinning; and c. Pulse duration 23 ns; time: 1, 20 ns, 2, 40 ns, 3, 60 ns

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.

3 Experimental study and formulation of the problem

Characteristic features of threshold phenomena localisation, as was shown by Averin et al. (1991), are mostly distinct for a pulse duration less than 30 ns. The monopulse 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 (Cottet and Boustie 1989).

Respective experimental methods are described in detail by Averin et al. (1991), 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 α = 30^{∘} 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 (Gromov et al. 1990; Cottet and Romain 1982) for the pressure estimation in SW. The pulse amplitude, P_{m}, can be approximated by the following relation:

P_{m} = kq^{n},  (6)

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 × 10^{12} W/cm^{2} for a pulse duration of 23 ns and 5 × 10^{12} W/cm^{2} for 3 ns. The data from Eliezer et al. (1990), with k = 1.3 × 10^{−5}, n = 0.4, suit such intensity values best. Then calculations based on (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 μm were used in the described experiments. The specimens were obtained by means of powder metallurgy with subsequent annealing at T = 900^{∘}C and air-cooling. The final size of grains in plates was approximately 100 μm. 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 (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 shockwave evolution and the kinetics of structural changes. A dimensionless form of these equations is

\frac{∂v^{∗}}{∂t^{∗}} = \frac{1}{ρ} \frac{∂σ^{∗}}{∂ξ} ,

\frac{∂σ^{∗}}{∂t^{∗}} = κ\frac{∂v^{∗}}{∂ξ} − \frac{1}{τ} σ^{∗} − γ_{1} \frac{∂s}{∂t^{∗}} − γ_{2} \frac{∂d}{∂t^{∗}} ,

\frac{∂s}{∂t^{∗}} = −\frac{1}{τ_{s}} \frac{∂F_{1}}{∂s} ,

\frac{∂d}{∂t^{∗}} = −\frac{1}{τ_{d}} \frac{∂F_{2}}{∂d} ,  (7)

where v^{∗} = \frac{t_{1}}{l} v_{z}; t^{∗} = \frac{t}{t_{i}}; σ^{∗} = \frac{σ_{zz}}{G}; G = \frac{E}{2(1+μ)}; ξ = \frac{z}{l};

κ = \frac{2}{3(2−m)}; m = \frac{3K}{3K+2G}; K = \frac{E}{1−2μ}; τ_{s} = \frac{L^{22}}{t_{i}}; τ_{d} = \frac{L^{33}}{t_{i}};

γ_{1} = \frac{L^{13}L^{22}G}{L^{11}L^{22}L^{33}−L^{22}(L^{13})^{2}−L^{33}(L^{12})^{2}} ;

γ_{2} = \frac{L^{12}L^{33}G}{L^{11}L^{22}L^{33}−L^{22}(L^{13})^{2}−L^{33}(L^{12})^{2}} . Here v_{z} is a rate vector component; ρ is the material density; z, ξ are normal and dimensionless co-ordinates; τ is the Maxwell relaxation time; E, Young’s modulus; μ,

Fig. 3. Evolution of pulse amplitude (pulse duration: 1, 3 ns, 2, 23 ns)

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 (Belyaev et al. 1989). 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≠j)→0.

Then, the boundary conditions are:

υ^{∗}(1, t) = 0,

σ^{∗} = \{ \begin{matrix}2σ̃_{a}t^{∗}, \\ 2σ̃_{a}^{∗}(1 − t^{∗}), \\ 0 \end{matrix}  \begin{aligned}t^{∗} & ≤ & 1/2, \\ 1/2 & < & t^{∗} ≤ 1, \\ t^{∗} & > & 1, \end{aligned}  (8)

where σ̃_{a} is the dimensionless pulse amplitude. Initial conditions have the following form:

s(ξ, 0) = 1, υ(ξ, 0) = σ^{∗}(ξ, 0) = d(ξ, 0) = 0. (9)

Boundary conditions (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 (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 σ_{a} = 80 and 160 GPa.

4 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. 2 for t_{i} = 23 ns. Exceeding the critical stress threshold results in the initiation of the α → ε transformation. It causes the step formation on the loading front. Such two-wave configuration is characteristic for I-type phase transitions. The reverse ε → α 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

Fig. 4. Effective time of twinning t_{t} (1) and twin length (2)

rear front of the shock wave (Fig. 2). The SW propagation is accompanied by a sharp decrease in its amplitude (Fig. 3 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. 2b). 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 α ↔ ε transformation. A sharp drop in the shock wave amplitude in the case of the 3-ns pulse makes the initiation of the α → ε 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. 2c). 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_{t}, the total time, when the wave amplitude in the given point is larger than the critical twinning stress, is given in Fig. 4. The similarity of the two curves proves a direct effect of the action time of overcritical load on twinning.

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.

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.

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