CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
NUMERICAL SIMULATION OF ELASTIC-VISCOUS-PLASTIC
PROPERTIES, POLYMORPHOUS TRANSFORMATIONS AND SPALL
FRACTURE IN IRON
A.V.Petrovtsev,V.A.Bychenkov,G.V.Kovalenko
Russian Federal Nuclear Centre - Institute of Technical Physics, P.O. Box 245, Snezhinsk,
Chelyabinsk region 456770, Russia
Abstract. The presentation is devoted to a model that describes the dynamic response of iron to stress waves
of various intensities considering elastic-viscous-plastic properties, polymorphous a-s transformation and
spall fracture. Model parameters were selected through the numerical simulation of a great number of
experiments. They ensure the real structure of stress waves in this material, including wave amplitudes in a
multi-wave (elastic precursor, phase precursor, principal plastic wave) configuration and strain rates at the
wave fronts in loading and unloading. The presentation discusses some results obtained in the simulation of a
number of explosive and shock-wave experiments and data on the influence of polymorphous transitions on
spall fractures in iron.
codes and determined their parameters through the
numerical simulation of many experiments.
INTRODUCTION
The wide use of iron and steel in engineering and
the fact that geophysics considers iron as one of the
basic elements in the Earth core arise the great
interest to iron research. Iron manifests a wide range
of behavioral features and properties in dynamic
processes that allows its use as an important model
material in studies into the response of solids to
shock waves. All these stimulated detailed studies
that have recently provided scientists with a lot of
experimental and evaluated data.
The goal of this work was to develop a model that
could accurately describe the real structure of stress
waves in the material, including amplitudes in a
multi-wave (elastic precursor, phase precursor, main
plastic wave) configuration and strain rates at the
wave fronts in compression and release for loads of
different intensity. It is in need to calculate the state
variations of material particles in the fracture area
properly. We implemented the kinetic models of
elastic-viscous-plastic properties, polymorphous
transitions and spall fractures in our hydrodynamic
KEY ELEMENTS OF THE MODEL
The thermodynamic formulation of iron
properties is based on a thermodynamically
complete multi-phase equation of state [1] implemented in a tabular form. For states in the region of
phase mixture, we assume thermodynamic equilibrium as an equality of phase temperatures and
pressures and use traditional mixture relations [2].
Changes in phase concentrations in Lagrangian
particles are determined using the relaxation
equations of transition kinetics based on the limiting
equilibrium [2] or metastable [3] surfaces of phase
concentrations. We generalized these equations to a
case of several transformations proceeding
simultaneously. Fig. 1 shows the non- equilibrium
Hugoniot constructed using the equations of state for
a- and s-phases of iron, our tabulated functions for
metastable concentrations for the direct a-s
transition and yield strength.
591
1.4
1.6
1.8
Time (mcs)
FIGURE 2. Velocity profiles for the sample-window interface in
Experiment 15 [4] (sapphire impactor of thickness hp3.185mm
at velocity W=0.4825 km/s, iron sample of thickness
h(F6.304mm, sapphire window in the VISAR measurements).
Shock strength a"13"* 10 GPa. Dots show experimental points and
solid and dashed lines do calculations considering Bauschinger
effect and the elastic-perfect-plastic model.
FIGURE 1. Iron Hugoniot. Experiment: O - [4], II - [5], • - [6].
Calculation: the solid line shows our data, the dashed line does
those in [3].
The plastic flow region is described by PrandlReiss equation associated with the flow law by
Mises. Yield strength and shear modulus depend on
material phase composition and porosity caused by
the formation and growth of damages [7]. Yield
strength for solid material is dependent on the strain,
strain rate and thermodynamic parameters of state
[8,9]. Exponential form of the strain rate
dependency is taken similar to that in [10]. Inelastic
character of unloading is treated with an
interpolation of the effective shear modulus [8,11].
The model allows for a complex strain history
according to [11].
Spall fractures are described using constitutive
equations derived from the viscous law for the
plastic flow near pores [12,13].
They appeared close to those obtained in [3] but
we had to use shorter transition times because of
accounting for the viscous component of shear
strength. We managed to describe profiles for all
waves in the three-wave configuration properly (see
Fig. 3). Results obtained in the numerical modeling
of these and also other experiments where wave
profiles were measured in samples less than 5-6 mm
thick [6,17,18], prove the inference made in [3] that
relaxation is not linear at the initial stage of stress
wave propagation.
Data from different shock-wave (impacting with
slab) and explosive (detonating HE layers)
experiments on spall fracture in iron and steels were
used to determine parameters of the fracture model,
to estimate spall strength and its dependence on
strain rate and polymorphous ot-s transition. As an
initial step, we made calculations using the tensile
threshold model. Our estimates for experiments
where the loading proceeded in the a-phase region
show values of critical tensile stresses a*«-2.5 GPa
for a max *6GPa [15], a*«-(3.3...3.4) GPa for
a max «7GPa [14] and amax« 8 and 12 GPa [18].
Apparently the difference is caused by the fact that
data [15] refer to the stage of incipient damage,
though other properties of studied iron could be also
of influence. Figure 4 shows an example of this
information.
NUMERICAL MODELING OF EXPERIMENTS
Parameters of the elastic-plastic part of the model
were chosen using results obtained in the numerical
modeling of experiments undertaken to mea-sure
stress profiles in iron loaded in its a-phase [4, 14,
15]. Data from the deep analysis [16] of shock
structure for Experiments 15 and 16 [4] were used to
calibrate the model for shock strength amax«1013 GPa. Information for Experiment 15 [4] (Fig.2)
was taken from very smart data on the Bauschinger
effect.
Like in [3], we determined kinetic parameters of the
polymorphous ot-8 transition in iron by comparing
calculated data with experimental free surface
velocity profiles [4] for amax« 17.. .30 GPa.
592
very narrow region where the fracture proceeded so called smooth spall [20]. The comparison of
measured and calculated results showed that critical
tensile stresses in the presented experiments proved
to be a*«~(2.8.. .4.3) GPa depending on the HE layer
thickness. For particles in the spall plane, the shock
strength was amax«28-32 GPa. Our data on spall
strength of iron loaded into the region of a-s
transition agree with those obtained in [18] for the
same material through an analysis of wave profiles
a*«-3.5 GPa for amax« 16.5 GPa. But our estimates
for a* for loading in the a-phase region are much
greater than those presented in [18]. By our data the
increase of spall strength in iron due to the
polymorphous transition is less significant.
Spall strength of iron loaded into the region of the
reversible a-s transition was estimated through the
numerical modeling of explosive experiments [19]
where samples made of iron similar to that used in
[18] were loaded by the normal detonation of HE
layers 30-90 mm thick. In the experiments [19]
profiles of stress waves generated in the barriers
damping spall layers were measured. These
1.4
1.6
1.8
2
Time (mcs)
FIGURE 3. VISAR free surface velocity profile in Experiment 5
[4] (iron impactor hp6.314mm, W=1.292km/s, iron sample
h0=6.314mm. Shock strength (^^23.6 GPa. Dots show
experimental points, the solid line - calculation for ti2~T2i-25
ns, the dashed line - calculations [3] for Ti2=T2F30 ns.
0.4
24
26
27
FIGURE 5. Longitudinal stress profiles in iron sample particles
h=0,1,2,3,4 and 5 mm in the simulation of explosive experiments
[19] (plane wave generator, HE layer 60 mm thick, Cu shield
10 mm thick, iron sample 6 mm thick). Fracture is not allowed.
0.3
0.2
The polymorphous transition influences stress
wave profiles in the sample. Thus it can significantly
influence fracturing if even the latter proceeds in the
region where the load was insufficient to cause
phase transitions. In the experiments where wedge
samples were loaded by the sliding detonation of
thin PE layers [6] (Fig.6) the polymorphous
transition proceeds only in a thin layer adjacent to
the load surface. It changes the rate of the increase in
tensile stress amplitudes in the wave reflected from
the free surface, the strain rate of material particles,
and characteristics of the fracture zone.
When the amplitude of stress wave exceeds
amax«60 GPa its loading part is one-wave and unloading part have two peculiarities corresponding to
e-y (region of almost continuous flow) and y-ot
f 0.15
i0.1
0.05
0.05
25
Time (mcs)
0.35
0.15
0.2
0.25
0.35
Time (mcs)
FIGURE 4. VISAR free surface velocity profile in Experiment
[17] with iron Armco-80 (iron impactor hi=3 mm,
W=0.36 km/sec, iron sample ho=6 mm), cr"***? GPa. Dots show
experimental points, the solid line - calculations for <j»=3.3 GPa
and the dashed one - calculations with the kinetic spall model.
data reveal information about characteristics of the
first spall layer. Fig. 5 shows that tensile stresses in
the samples were produced by the rarefaction shock
wave that led to high rates of material tension and a
593
ACKNOWLEDGMENTS
This work was stimulated by research under
Contract 1000530009-35 RFNC-VNIITF with
LANL.
REFERENCES
1
2
3
Time (mcs)
4
1.Dremov, V.V., Kutepov, A.L., Petrovtsev, A.V., and
Sapozhnikov, A.T., this Conference.
2.Andrews, D. J., J. Comp. Phys. 7, 310-326 (1971).
3.Boettger, J. C, and Wallace, D. C, Phys. Rev. B 55,
2840-2849(1997).
4.Barker, L. M., and Hollenbach, R. E., J. Appl. Phys.
45,4872-4887(1974).
5.Bancroft, D., Peterson, E.L., and Minshall, S.,
J. Appl. Phys. 27, 291-298 (1956).
5
FIGURE 6. Longitudinal stress profiles in iron sample particles
h^ 1,3,5,7,9,11 and 13 mm for experiments with wedge samples
loaded by the sliding detonation of thin HE layers [6]. Effective
sample thickness is h(Fl4mm. Solid and dashed lines show
calculations, respectively, with and without allowance for
polymorphous a-s transition in iron. Fracture is not allowed.
6.Kozlov, E.A., Shock Adiabat Features, High
Pressure Research 10, 541-582 (1992).
7.Johnson, J. N., J. Appl. Phys., 52, 2812-2824 (1981).
8.Steinberg, D.J, Cohran, S.G., and Guinan, M.W., J.
Appl. Phys. 51, 1498-1504 (1980).
9.Steinberg, D.J., and Lund, C.M., J. Appl. Phys. 65,
1528-1533(1989).
10.Tonks, D.L., J. Appl. Phys. 66, 1951-1960 (1989).
11.Moss, W.C., and Glenn, L.A., in Shock Waves in
Condensed Matter - 1983, edited by J.R. Asay et al.,
American Institute of Physics, 1984, pp. 133-136.
12.Belov, N.N., Korneyev, A.I., and Nikolayev A.P., Rus.
J. of Applied Mechanics and Technical Physics
(PMTF) 3, 132-136(1985)
B.Tonks, D.L., Vorthman, J.E., Hixson, R.S., Kelly, A.,
and Zurek, A.K., in Shock Compression of Condensed
Matter-1999, edited by M.D. Furnish et al., American
Institute of Physics, 2000, pp.329-332.
14. Arnold, W., in High Pressure Science and Technology1993, edited by S.C. Schmidt et al., American Institute
of Physics, 1993, pp. 1035-1038.
15.Kanel, G.I., and Shcherban, V.V., Rus. J. Physics of
Combustion and Explosion (FGV) 4, 93-103 (1980).
16.Tonks, D.L., Los Alamos National Laboratory Report
LA-12068-MS, 1991.
17.Barker, L. M., and Hollenbach, R. E., J. Appl. Phys.
43,4669-4675(1972).
18.Veeser, L.R., Gray, G.T., III, Vorthman, J.E.,
Rodriguez, P.J., Hixson, R.S., and Hayes, D.B., in
Shock Compression of Condensed Matter-1999, edited
by M.D. Furnish et al., American Institute of Physics,
2000, pp.73-76.
19.Kozlov, E.A., Zhugin, Yu.N. et al., to be published.
20. Ivanov, A.G., Novikov, S.A., Rus. J. Experimental
and Theoretical Physics (ZhETF) 40, 1880-1882
(1961).
(rarefaction shock wave) transitions (see Fig.7)
because unloading path in the P-T plane lies, by
temperatures, above the triple point of a-, s- and yphase equilibrium. It is of great importance to obtain
experimental data characterizing the above
peculiarities of wave profiles in order to draw
corresponding phase equilibrium curves (especially
for the line of s-y transition) because data from static
measurements are very uncertain.
100
90
80
70
60
50
40
10
0.02
0.04
0.06
0.08
Time (mcs)
0.1
0.12
FIGURE 7. Stress wave profile in the iron sample for
a^^SOGPa (iron impactor hpLSmm, W=3.2 km/s, iron
sample h(F5 mm). Particle coordinate h=l mm. i-l ns for all
transition.
594