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
FACTORS INFLUENCING THE SHAPE OF THE FRACTURE WAVE
INDUCED BY THE ROD IMPACT OF A BRITTLE MATERIAL
A.D. Resnyansky1 and N.K. Bourne2
1
Weapons Systems Division, Aeronautical and Maritime Research Laboratory,
DSTO, PO Box 1500, Salisbury SA 5108, Australia
2
Royal Military College of Science, Cranfield University, Shrivenham, Swindon, SN6 8LA, UK
Abstract. A fracture wave in a brittle material is a continuous fracture zone which may be
associated with the damage accumulation process during the propagation of shock waves. In
multidimensional structures the fracture wave may behave in an unusual way. The high-speed
photography of penetration of a borosilicate (pyrex) glass block by a hemispherical-nosed rod (1)
shows a visible flat wave forming as the fracture front. The role of the complex stress state and
kinetic description of the damage accumulation are analysed to describe the process of the
impact. The DYNA2D hydrocode and a kinetic strain-rate sensitive model (2) are employed.
the impact superimposed by the kinetic
accumulation of damage during the fracture wave
formation.
The role of the complex stress state is analysed
with a kinetic approach to the damage accumulation
in order to describe the fracture wave formation.
The damage kinetics is verified with available onedimensional tests (5). The choice of an equivalent
stress for the damage kinetics treating the complex
stress state is discussed in detail. The present paper
analyses the test results observed in (1). The work
employs a damage accumulation model of the phase
transition type developed earlier (2). The model was
incorporated within the DYNA2D-hydrocode.
Results of the calculations demonstrate that good
agreement with the experiment may be taken into
account with the treatment of complex stress state
within the constitutive approach in the manner of a
cup model.
INTRODUCTION
Studies of the fracture waves in brittle materials
become increasingly popular due to the application
of ceramics, high strength glass, concrete etc.
In considering the failure phenomenon, special
attention is paid to the formation and behaviour of a
continuous fracture zone within or behind shock
wave which Kanel named a fracture wave (3).
Feature of the process is the material failure in a
compression zone, because fracture wave follows
the shock wave.
The fracture wave phenomenon is associated
with accumulation of damage that requires a kinetic
constitutive description. In the multi-dimensional
case of ballistic impact, fracture waves may behave
in an unusual manner. Investigations have used
high-speed imaging of impacted borosilicate glass
(pyrex) in various papers (1, 4). In (4) a periodic
structure of fracture waves in colliding glass rods
has been observed. In (1) the ballistic impact of a
pyrex glass block by a rod with a hemi-spherical
nose resulted in development of a fracture wave that
appeared to be headed by a flat region.
This behaviour is apparently related to the
specific role of a complex stress state in the area of
MODEL
The model (2) uses the representation of a brittle
material as a two-phase 'mixture' of two
constituents. Each of the components is managed by
its own strain-rate sensitive model of the relaxation
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type (6). A damage parameter c is responsible for
the non-equilibrium transition of one phase to
another. The 'undamaged' phase corresponds to the
parameter value c=0. c=l associates the material
with a state of continuous 'totally damaged'
material, which is a hypothetical material saturated
with 'damage' that means that no further
degradation can occur in the material. The
'damaged' state differs from the original one only in
its mechanical properties. For example, in the given
case the second phase of the pyrex glass has the
same density and bulk modulus, but a significantly
reduced shear modulus. Yield limits of the second
constituent should be also essentially reduced for
the whole range of strain rates. The combined
material is governed by a model with transition
kinetics for the damage parameter which allow
smooth connection of the two phases during the
fracture process.
here I/A is a kinetics dependence fit to experimental
stress-strain curves, a for stress, and T for
temperature.
The equations of state for the damaged material
are a combination of equations of state (EoS) for the
phases according to a phenomenological approach
(a substitute of the mixture rule).
The model is employed by an in-house onedimensional code and is incorporated within LSDYNA2D (9).
ONE-DIMENSIONAL ANALYSIS
The chosen damage kinetics has been verified
with available experiments (the stress measurements
by manganin gauges) of the plane impact of a block
of pyrex glass with a copper flyer plate (5).
Calculation of the stress states are shown in Fig. 2
for two velocities of impact. The calculations clearly
demonstrate appearance of the transition related to
damage and which can be associated with the
fracture wave. The calculation results (curves Cl
and C2) are in a good agreement with the test data
(curves El and E2 from (5)).
1.6
l
oS
0.8
0
f
tr
<o
i
i
530
2 4 6
e,%
2 4 6
s,%
FIGURE 1. Calculated stress-strain curves (left) and associated
concentration parameter (right) used for the modelling of the
pyrex glass at the strain rates 103 s"1 (7) and 10~2 s"1 (2).
s,3
The model is an example of a two-phase
phenomenological approach which can consider
many situations where two phases described by their
own properties can be connected through the
transition/concentration parameter (7, 8).
The model for each of the phases is a strain-rate
sensitive model described in (6). The model
includes:
(i)
conservation laws;
(ii)
constitutive equation for relaxation of shear
stress;
(iii) the two phase model is completed with a
kinetics for the damage parameter c:
250
El
-1
Time
FIGURE 2. Stress profiles in plates of pyrex glass for impact by
copper flyer plate with velocities 250 m s"' (curves Cl and El)
and 530 m s'1 curves C2 and E2).
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stress states at which the material fail is between the
curves 1 and 2 in Fig. 3. A simple loading
confirmation of this can be found in Fig. 1 where
two stress limits correspond to a high strain rate and
a quasi-static case.
Now we analyse the case of impact of a block of
pyrex glass by a steel rod with the impact velocity
of 536 m/s (1). The rod is of 10 mm diameter and
has a hemi-spherical nose. The failure wave
photographs taken in (1) are shown in Fig. 4. The
5 mm cross-mark is used for location of the fracture
wave position.
ANALYSIS OF BALLISTIC IMPACT
Treating stresses in a multi-dimensional
geometry is not simple. Traditional fracture criteria
for complex stress states elaborate an equivalent
stress as a criterion of failure. For brittle materials
two stress modes, which are the shear stress a and
pressure p, are considered essential. Therefore for
engineering materials like concrete, soil, etc. a
closed curve (a cap fracture model (10)) is usually
drawn in the shear stress-pressure space which is a
limiting surface for stresses the material can sustain
at quasi-static loadings. In this case the equivalent
stress (JeqvCos p) is a functional combination of the
shear stress and pressure which restricts the set of
accessible states within the zone creqv< (Jcrit. Here
creqv=orcrit can be considered as a fracture criterion.
For the case of shock wave loading there is not a
single level curve creqv=crcrit. Conventionally for the
processes of different duration a separate criterion
0"eqv=0'Crit can be drawn in the (a, p)-space. We
consider a cap model presentation suitable for our
current purpose. In this case a chosen quasi-static
criterion could be drawn as curve 1 in Fig. 3. Then a
theoretical strength curve would be a curve 2 in Fig.
3. However for impact processes at a finite duration
corresponding curve would be between the curves 1
and 2.
FIGURE 4. The high-speed photographs (1) of the high-velocity
impact of a pyrex glass block by a semi-spherical-nose projectile.
2 jusec
shear stress
pressure
FIGURE 3. Schematic of contours of equivalent stress CFeqV=
const for the quasi-static and theoretical strength
FIGURE 5. Development of the damage contours in a block of
Pyrex glass after impact by a semi-spherical-nose rod with the
velocity of 536 m/s.
Application of these intermediate criteria to a real
problem is almost impossible but this idea may be
easily realised within the kinetic approach we
employ. We consider that the functional form of
equivalent stress in the above form can be used
instead of a simple state stress a in the constitutive
equations of the model. With this approach the
Result of the corresponding calculation with
DYNA2D-hydrocode employing the model
implemented is shown in Fig. 5. The failure wave
front is visible as three successive contours of the
damage concentration levels (c=3/6; c=4/6, and
c=5/6). The contours are very close to each other
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because of the specific kinetics chosen for this case;
sharp stress drop and concentration rise in Fig. 1 for
the case of uniform deformation illustrates this. The
fracture wave in the calculation is being developed
with approximately the same velocity of 1.8 km/s as
observed in the test (1).
In order to understand the nature of the fracture
wave a calculation was conducted for similar
problem with a flat-nose projectile. Results of this
calculation are shown in Fig. 6.
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compression zone. Taking both mechanisms with
the kinetic approach can be done with suitable
choice of equivalent stress in the constitutive
equations.
REFERENCES
1. Bourne, N. K., Forde, L., and Field, J. E., "High-speed
photography and stress-gauge studies of the impact and
penetration of plates by rods," in Proc. 2Td International
Congress on High-Speed Photography and Photonics1996. edited by D. L. Paisley, Proc. SPIE Vol. 2869,
1997. pp. 626-635.
2. Imomnazarov, Kh. Kh., Resnyansky, A.D., and
Romensky, E.I., "Dynamic Damage Model for
Viscoelastic Material", in ACAM99, the Second
Australasian Congress on Applied Mechanics, 10-12
February, 1999, Canberra, Australia, U-078, pp. 1-6.
3. Kanel, G.I., Molodets, A.M. and Dremin, A.N.,
Combust Explos. Shock Waves 13, 772-777 (1977).
4. Murray, N.H., Bourne, N.K., Field, J.E., and
Rosenberg, Z. "Symmetrical Taylor Impact of Glass
Bars", in Shock Compression in Condensed Matter-1997.,
edited by S.C. Schmidt et al, AIP Conference
Proceedings 429, New York, 1998, pp. 533-536.
5. Bourne, N. K., Rosenberg, Z., Mebar, Y., Obara, T.,
and Field, J. E., J de Physique IV Colloq. C8, 4, 635-640
(1994).
6. Godunov, S.K. and Romensky, E.I., Elements of
Continuum Mechanics and Conservation Laws (in
Russian), Novosibirsk, Nauchnaya Kniga Publ., 1998.
7. Resnyansky, A.D., Milton, B.E., and Romensky, E.I.,
"A Two-Phase Shock-Wave Model of Hypervelocity
Liquid Jet Injection into Air", in Proc. JSME Centennial
Grand Congress (Int. Conf. on Fluid Engnrg), JSME
ICFE-97-228, Tokyo, Japan, 13-16 July, 1997, pp 943947.
8. Resnyansky, A.D. and Romensky, E.I., "Using a
homogenization procedure for prediction of material
properties and the impact response of unidirectional
composite", in Proceedings llth Int. Conf. on Composite
Materials, Vol. II: Fatigue, Fracture and Ceramic Matrix
Composites, (Ed. Murray L. Scott), Gold Coast,
Queensland, Australia, 14th-18th July, 1997, Woodhead
Publishing Ltd, 1997, pp. 552-561.
9. Hallquist, J.O., User's manual for DYNA2D An explicit
two-dimensional hydrodynamic finite-element code with
interactive rezoning, Lawrence Livermore National
Laboratory, UCID-18756, Rev. 2, 1984.
10. Chen, W.F. and Baladi, G.Y. Soil Plasticity,
Developments in Geotechnical Engineering, 38, Elsevier
Sci. Publ., NY, 1985.
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5-FIGURE 6. Development of the damage contours in a block of
Pyrex glass after impact by a flat-nose rod with the velocity of
536 m/s.
The calculation demonstrates that two actual
fracture zones are being formed during the impact.
The first one attached to the free surface of the glass
target is associated with tensile stress (negative
pressure) and is typical for conventional materials.
The second one, visible as bigger inner fracture
zones, is in the tail of propagation of significant
shear stresses and is not typical for conventional
material because it is in the compression zone.
However, this fracture zone in the 45° direction to
the impact line is typical for ceramics and other
brittle materials studied in detail. Apparently
superposition of these effects at the impact by the
hemi-spherical-nose projectile gives this unusual
effect.
CONCLUSION
A phase-transition model developed for
description of damaging brittle material can be used
for calculation of fracture waves in glasses.
Formation of the flat fracture front in pyrex glass
at the high-velocity impact by a hemi-spherically
nosed rod can be a result of superposition of the
fracture waves determined by two mechanisms. The
first one is conventional and associated with tensile
and shear deformation in the tensile zone. The
second one is associated with shear in the
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