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
COMPUTER SIMULATION OF THE PROPAGATION OF SHORT
SHOCK PULSES IN CERAMIC MATERIALS
Vladimir A. Skripnyak, Evgeniya G. Skripnyak, and Tat'yana V. Zhukova
Department of Mechanics of Solids, Tomsk State University, 36, Lenin Ave., Tomsk 634050, Russia
Abstract. The propagation of shock pulses of submicrosecond duration in A12O3, ZrO2, and SiC ceramics is investigated by the numerical simulation method. Contributions of various mechanisms of structure evolution to stress relaxation in brittle ceramics are discussed. A theoretical formalism is suggested
for the prediction of dynamic strength of polycrystalline and nanocrystalline ceramics with different porosity and grain size.
INTRODUCTION
METHOD OF COMPUTER SIMULATION
Use of ceramic nanopowders allow materials to
be synthesized with the required grain size, porosity, and structural homogeneity. It is well known
that at static loads the fracture toughness, compressive strength, tensile strength, and flexural strength
of modern nanocrystalline ceramic materials come
close to those of metal alloys and steels. However,
experimentally the mechanical behavior of nanocrystalline ceramics at dynamic loads has not yet
been determined. Are there simultaneous increase
of the dynamic strength and the dynamic fracture
toughness of nanocrystal ceramics? Some researchers suggest that when the grain size is smaller than
100 nm, the dislocations can not support the inelastic deformation of structural elements in polycrystalline ceramics. At the dynamic loads, microcracks are nucleated in ceramics and cause the
strength decrease. Therefore, a study of the influence of the structural factors on mechanical behavior of ceramic materials under high strain rates
is of practical interest.
The micromechanical approach was used for
computer simulation of mechanical behavior of
polycrystalline ceramics under shock loading. Ceramics are considered as a continuous medium at
the macroscopic level, and as a structural medium
at the mesoscopic level. A mechanical state at the
macroscopic level is described by a system of conservation equations in continuum mechanics. The
stress and strain fields are significantly nonuniform
at the mesoscopic level. The stress and strain parameters at the macroscopic level are different from
those at the mesoscopic level. Ceramic materials
are considered as brittle when the temperature
during deformation does not exceed 20% of the
melting temperature. The model [1] relates the kinetics of inelastic deformation of ceramics to
structural changes. The numerical method [2] is
used for calculation in the present work.
MODEL
The main structural elements of polycrystalline
ceramics are grains and crystalline blocks. Voids
751
are the model parameters.
It is assumed that the shear inelastic strain
caused by microcracks nucleation and growth is
given by the expression [1]
and cracks are considered as structural defects at
the mesoscopic level. These structural defects significantly influence the mechanical behavior of ceramics, because they cause stress concentrations at
the mesoscopic level. Two different stresses are
used in the model. The average stress in the representative volume of the material is the stress at the
macrolevel. The stress in the solid phase of ceramics is considered as stress at the mesoscopic level. It
is denoted by the superscript m. The connection
between the stresses at the macroscopic and the
mesoscopic levels is given by the formula
^ =cr™ exp(-oc/oc*),
e" ( m ) =(2/3)R 0
where RO is the nucleated cracks size, and N is the
rate of cracks nucleation.
Therefore, the kinetics of inelastic strain in
brittle ceramics is defined by the kinetics of cracks
nucleation and growth. The nucleation of microcracks is described by the formula
(1)
N=[(S u -S**)/ri 2 ]H(N) ,
where a is the specific volume of pores and plane
cracks and a * is an empirical constant.
For A12O3 and ZrO2, a* = 0.1 describes well the
dependence of the static strength of ceramics with
porosity. The pressure Pm is calculated by the MieGruneisen equation of state [3], and the deviator
stress tensor is calculated by the relaxation equation:
DS i j /Dt=2>i(e i j -e i j n ),
(7)
where S U = [ ( 2 / 3 ) S ™ S™] 1 / 2 , S** is the critical
shear stress of cracks nucleation, N is the density of
cracks, and r|2 is the model parameter. For a plane
shock, the components of the deviator for the stress
tensor are 82-83 — 1/28!, therefore, S U =(4/3)T.
In the model, the dynamic strength is related to
the critical shear stress by the expression
S**=(3/2)Y0exp(-o/0.1).
It is assumed, that the parameter Y0 is connected with the grain size (d) and the Burgers vector (b) by the relation Y0= 2xiQOT (d/b)"0'1. Theoretical shear strength of the crystal lattice Tteor is estimated in the context of Frenkel's theory [4].
The growth of the average microcracks size is
described by the equation
(2)
where D/Dtis Jaumann's derivative, ji is the effective shear modulus of the damaged medium, e^
is the deviator for the effective strain rate tensor,
andejjis the deviator for the effective inelastic
strain rate tensor.
The damage parameter a during inelastic deformation can be written in the form
d/(l-ot)=ek,
(3)
— (P(m)-P*)H(P(m)- P*), (8)
where s£k is the inelastic volumetric strain rate.
The evolution of the inelastic shear strain at the
mesoscopic level under compression causes the reduction of the initial volume of pores upon shock.
The volumetric inelastic strain rate caused by the
crack healing under compression (4) and opening
of cracks under tension (5) is given by expression
where r|3, rj4, P*, and S* are the model parameters.
We assume that the relaxation of shear stress is
caused by two mechanisms, namely, by crack
growth (curve 1 in Fig. 1) and by nucleation of new
cracks (curve 2), which have different relaxation
times. Critical stresses 8* and S** of activation of
this mechanisms are also assumed different. The
critical shear stress S * is given by the relation
ejk -{S^m)e;](m)/[P(m)(l-a/a*)]}H(a)H(P(m)), (4)
8™ = STiR^NRHOP^ -P*),
(6)
(5)
S*=S*+rj 5 P (m) H(P (m) ),
where H (•) is the Heaviside function, R is the average crack size, N is the density of cracks, R is the
velocity of the cracks size growth, and T| } and P*
where r|5 and S0*are parameters of the model.
752
(9)
0.15, and 0.2 p,s after the impact. When the amplitude of the shock (13 GPa) was less than the Hugoniot elastic limit (HEL), the crack density (curve C)
and damage parameter a (curve D) changed only
slightly. A decrease in the initial crack density resulted in the increase of the calculated spall
strength. The density of cracks (curve A) rose
sharply and the damage parameter a (curve B) decreased when the amplitude of the shock (18 GPa)
was higher than the HEL. A decrease in specific
volume of voids a caused the growth of the shear
strength in shock. A decrease in grain size and increase in void and crack size lowers the stress concentration at the mesoscopic level. In this case, the
HEL (CTHEL) and the dynamic strength
Y=-aHEL(l-2v)/(l-v) increased. Figure 3 shows
the calculated dynamic strengths for ceramics with
different initial porosity. The experimental data in
Fig. 3 were borrowed from [5].
The values of r|5 and S0* are estimated from experimental data on static strengths at tension and
compression. The initial microcrack density N0, R,
andoc0 characterize the initial damage of polycrystalline ceramics. The parameters r|i, r|2, r|3, and
T|4 control the kinetics of inelastic deformation.
U. IZ ~
0.08-
1
2
=1.
^5
0.04-
V
If
O.OO-
/
J
>s*
- 6 - 4 - 2
FIGURE 1. Calculated dependence of shear stress on the strain
rate in ZrO2+3mol% Y2O3 ceramics with 1 -|um grain size.
Their values are determined by numerical simulation of the propagation of shock pulses. The criterion oc > 0.3 is used to detect macroscopic failure.
NUMERICAL RESULTS AND DISCUSSION
The model described by Eqs. (l)-(9) is used for
computer modeling of the dynamic phenomena in
ceramics with different initial porosity and grain
size.
o.o
20-
-1.2
-0.8 $
4-
-0.4
0.1
0.2
Equation (10) can be used to estimate the dynamic
strength for polycrystalline and nanocrystalline
A12O3 and ZrO2 ceramics
o
8-
0.0
0.4
FIGURE 3. Solid and dashed curves are the results of calculation.
1.6
16-
0.2
Y(a)=Yteor(d/b)-01exp(-oc/0.1).
(10)
The Burgers vector b equal to 4.76-10"8 and 5.14
• 10"8 cm was used for A12O3 and ZrO2 respectively.
The grain size d is in the same units as b. The estimate Yteor = 2 Tteor gives Yteor = 20.67 GPa for
A12O3 and Yteor =10.96 GPa for ZrO2. The maximum theoretical estimates of the HEL are 20.6 and
29.9 GPa for ZrO2 and A12O3 ceramics, respectively. True values of the HEL will be much lower.
0.0
x. cm
FIGURE 2. Calculated damage parameters in SiC ceramics
under shock loading.
Figure 2 shows the calculated profiles of stress
waves with amplitudes of 18 and 13 GPa in SiC ceramics. Ceramics had the initial damage a0 =
0.015.
Profiles (1-4) correspond to times of 0.05, 0.1,
753
In single crystal sapphire and ruby and nanocrystalline ceramics based on A12O3 and ZrO2, the fast
relaxation time of shear stress is about 10-20 ns.
The shear stresses approach the theoretical shear
strength of the crystal lattice when the duration of
the strong shock is less than 50 ns. Under deformation in brittle ceramics, the dissipation of mechanical energy results in the work on inelastic
strain and increase in the surface energy of cracks.
Inelastic deformation of some ceramic materials
may be caused by the martensitic phase transition.
The model predicts high values of the HEL when
crystalline blocks are considered as structural elements of the single crystals. The calculated HELs
are in good agreement with experimental data for
polycrystalline alumna and sapphire (12-21 GPa
[5]). Figure 4 shows predicted QHEL for ceramics
with different grain size. The experimental data
were borrowed from [5].
40
30 CO
Q_
CONCLUSIONS
10 -
-2
1) The computer simulation predicts that the decrease of the grain size in polycrystalline alumna
from 100 mm to 10 nm causes the 4-fold increase
of the HEL.
2) When the amplitude of shock does not exceed
the HEL, the density of cracks in ceramics slowly
decreases in shock front. In this case, the shear
strength and the spall strength of polycrystalline
ceramics may increase.
3) Inelastic deformation of ceramic materials can
be caused by the nucleation of microcracks at shock
compression above the HEL.
0
log (dX d
FIGURE 4. Calculated dependence of the Hugoniot elastic
limit of ceramics on the grain size.
Simulations suggest that inelastic deformation
is negligible in polycrystalline ceramics at high
shock amplitude when the pulse duration is comparable to the relaxation time. Under these conditions, the actual spall strength of polycrystalline ceramics is comparable to the theoretical strength of
the crystalline lattice at tension.
REFERENCES
1 2 345
1. Skripnyak, V. A. and Skripnyak, E. G., "Computer
Modeling of Mechanical Behavior of Constructional
Ceramics under Shock Loading," in New Models and
Numerical Codes for Shock Waves Processes in Condensed Matter-1997, edited by I.G. Cameron, AWE
Hunting - BRAE, Oxford, UK, 1997, pp. 26-36.
2. Zhukova, T. V., Makarov, P. V., Platova, T. M., and
Skripnyak, V. A., Fiz. Goreniya i vzryva 23, N.I, 2934, (1987). [in Russian].
3. Ahrens, T. J., "Equation of State," in High-Pressure
Shock Compression of Solids, ed. J.R. Asay, M. Shahinpoor, Springer-Verlag, New-York, 1993, pp.75-113.
4. Macmillan, N. H., "The Ideal Strength of Solids," in
Atomistic of Fracture, edited by R. Latanision , J. R.
Pickens, Plenum Press, New.-York., 1983, pp. 95-164.
5. Kanel, G. I, Razorenov, S. V., Utkin, A. V., and Fortov, V. E., Shock-Waves Phenomena in Condensed
Medium, [in Russian], Yanus, Moskow, 1996.
0.201-
111II
'^
0.00
4
logos'1)
8
12
FIGURE 5. Calculated dependence of the shear stress on the
strain rate in AJ^Os ceramics with grain sizes 100 (1), 10 (2), and
1 urn (3), 100 (4), and 10 nm (5).
Figure 5 shows the calculated shear stress for
ceramics with different grain size as a function of
strain rate. The calculated results predict not only
the growth of shear strength but also the decrease
of the relaxation time in nanocrystalline ceramics.
754