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
DISCRETE ELEMENT MODELING FOR SHOCK PROCESSES OF
HETEROGENEOUS MATERIALS
Z. P. Tang and W. W. Wang
University of Science and Technology of China, Hefei, Auhui, 230026, R R. China
Discrete element method (DEM) is one of the new-concept numerical simulation methods under
developing. In this paper, the theoretical advances of DEM achieved by this group have been presented.
They include the dispersion and dissipation effects, which related to the element size, the nonlinear
viscoelastic model and the gas phase element (GPE) model. Some typical simulation and experimental
results at grain scale under shock loading are reported. Since DEM is not a well-developed method,
particularly it does not have a systematic and strict mathematical proof for its theoretical basis, it can not
guarantee the correction and accuracy of the simulation results. The limitation of DEM and the needs for
further study are analyzed and discussed in this presentation.
INTRODUCTION
paper, some theoretical advances and applications
achieved in this group more recently have been
reported, such as the dispersion and dissipation
effects related to the element size, the nonlinear
viscoelastic modeling, and the gas phase element
modeling, etc.
However, DEM is far from perfection and is still
under developing, its basic shortages will be
discussed in the summary section.
Discrete Element Method (DEM) is one of the
new-concept numerical methods and has deep
potential for further developing and application.
Since DEM does not explicitly obey continuum
equations, this method can be applied to investigate
the complex processes and mechanisms, such as
dynamic behaviors of discontinuous materials and
multi-phase mixtures, local heterogeneity, large
deformation and structural failure etc.
The idea of using DEM for simulation of discrete
media was proposed as early as the end of 1960s.
Recently, Tang el a/lul have developed a 2-D
meso-scale DEM theory called "DM2" (Discrete
Meso-element Dynamic Method) and applied it to
simulate the shock synthesis of powder mixture with
chemical reaction. The simulated results reveal
clearly the meso-scale processes and mechanisms of
the mass mixing and hot spot forming during pore
collapse, which initiate the chemical reaction. In that
development, a criterion of time step was also
derived to assure the stability of calculation. In this
DISPERSION AND DISSIPATION OF DEM
CALCULATION
The discretization in space and time will cause
the effect of dispersion and dissipation as DEM is
used to simulate continuous media. Hence, there
exits discrepancy between the calculated and
analytical results. To find a general rule for better
understanding and control of this effect,
systematical investigation is necessary.
Dispersion Effect for Dynamic Linear
Elasticity
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For an arbitrary element o in a 2D DEM system
with triangular close packing crystalline, its
movement obeys Newton's law as
where w is the circular frequency, E is the Young's
modulus, k is the wave number, r is the radius of
element, and a is the angle of vector k respect to jc
axis. The dispersion relations with the element
radius r and the direction of k, based on Eqs. (2) and
(3), are listed in Table 1. The dispersion is nonlinear.
The cut-off frequency and different phase speed will
induce the oscillation and dispersion of wave
profiles. Reducing r can reduce dispersion, but will
increase the element number in the system and the
accumulation error. A suggested criterion for
choosing element size will be:
mUn =
(1)
VJ
win;
=!/,'
where m is the mass of the element o, f \ is the
interaction force of the element / applied to the
element o, u is the displacement, x and jy are the axes.
For linear elastic material and the assumption of
harmonic wave, one can obtain the dispersion
solution of Eq. (1) as follows^
3 - (cos 20\ + 2 cos 0, cos Ql) ±
r = xCQ/wM
where C0 is the sound speed, WM is the estimated
maximum frequency of the problem, and the
(2)
•^(cos 20, - cos 6>, cos 02 ) 2 + 3(sin 0, sin 0, ) 2
*,=
(4)
parameter K e (0,1).
(3)
TABLE 1. Dispersion along x and y Direction for Linear Elastic Triangle Close Packing Crystalline (Q=
k J.X
k//x
Transverse wave
w2
HW
(Cut-off
frequency)
CP
(Phase
velocity)
. kr
sin 2 —
Rr 0 ^ 2
V3C
2
Longitudinal wave
r 2 (9-8sin
Q
s
2
r
4V2^/3r
kr
kr
2
r1
c0
VI
Dispersion and Dissipation of DEM for Linear
Viscoelastic Problem
• "T
^t ',
v/3C
2
2
V3C0
r
\sm — kr\
S1
kr
^kr
2
^/Jv-0
Co
V3r
Q ^-8sin^isinfi
V3
.
oirj 2
*"
cm 2
Co •""
V3
gr
yo
o
Longitudinal wave
. Jlkr
* r )sin 2 * r
2
2
V3C0
. . kr.
|sm-|
</3C 0 —— ^-
Transverse wave
4pi/^
V3C 0
- •2
kr
2(C2+D2)
Dynamic viscoelastic problem has the physical
dispersion and dissipation. However, the effect of
DEM may distort the real physical nature if the size
of element is not properly chosen. After a series of
derivation, it yields the following relations for a
general Maxwell material
2(C2+D2)
(7)
4C
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(6)
where k* and a * are the equivalent wave number
and attenuation factor, E\ and r/ are the constants
of the Ith Maxwell body in parallel, k* and a * trend
to k and a of linear viscoelasticity as r (also j)
approaches 0. By setting/=/ as the upper limit of/,
it yields the upper boundary for element size as
r
1
"
— >
(C,r,) 2
v
/ //
max " " " / ; „ ,
= £, f exp(*
It can be proved that there exists an iteration
formula between the time steps n-1 and n for the
hereditary integration of Eq. (12) as follows:
<r]n} = exp(-— )
/o\0
x ?
(12)
(13)
V ;
Eq. (13) shows that it does not need to record and
recalculate the whole history of time steps for a
general Maxwell model. This can reduce lots of
memory and CPU time.
where C; = Ef / pQ . In practical calculation,
r = *rmax, /ce(04).
Simulation of ID Strain Waves in PBX
DEM SIMULATION FOR NONLINEAR
VISCOELASTIC MATERIALS
PBX is cohered with HMX. Fig. 1 is the
calculated result for the symmetric plate impact of
232m/s with the DEM. Eqs. (10)-(13) of the
nonlinear viscoelastic model are adopted in the
calculation. 3 Maxwell bodies are used in the model
with the different relaxation times. The constitutive
parameters used in the simulation are B = 9GPa,
Constitutive Equations of Nonlinear
Viscoelasticity and Algorithm of Hereditary
Integral
proposed a constitutive equation of
nonlinear viscoelasticity called ZWT model as
t-S^ .
{
fexp(——r
o
i
= 2.4
(9)
£/=0.067Gpa,
E2 fexp(
^
where the first three terms on the right side represent
the nonlinear elastic response and the last two
integrals are the Maxwell bodies with the different
relaxation times. This model has been successfully
applied to predict the dynamic responses for various
viscoelastic materials such as polymers, cements
and composites. Based on the idea of ZWT model,
we suggested a new nonlinear viscoelastic model
including a term of nonlinear spring and a term of
the general Maxwell body as
a=
T3 = QAjus. Fig. 1 shows that the calculated wave
profiles fit the experimental result well.
(10)
FIGURE 1. Simulated ID strain wave profiles in PBX
(dash line: calculated, solid line: experimental)
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Meso-Scale Simulation of PBX Damage under
Shock Compression
Fig. 2 is the micro observation of PBX, in which
the larger grains are the brittle HMX crystals.
Experimental investigation reveals that PBX will be
comminuted during shock compression, which
could be called "compression damage." Fig. 3(a) is
the initial configuration for the DEM simulation.
The loading stress is 750 MPa applied upward at the
bottom boundary. A brittle elastic model and a
viscoelastic model with three Maxwell bodies in
parallel were applied for the HMX elements (black)
and the binder matrix elements (gray), respectively.
Fig. 3 (b) and (c) show the evolution of cracks
(small triangles in the figure) before the wave front
reaches the free surface. The calculated stress fields
demonstrate that there is no any tension state in the
sample at those moments. Hence, the damage occurs
definitely under the compression condition and is
attributed to the shear effect.
FIGURE 2. Micro-observation of PBX
(a) Initial configuration for the DEM simulation
GAS PHASE ELEMENT MODELING
Basic Model for Gas Phase Element (GPE)
The interaction between the GPEs can be
described with the following equations of state
p = Apr
(polytropic gas)
(b)t=15ns
(14)
(JWL)
i(l——)exp(-* 2 v)R2v
(c) t=27ns
FIGURE 3. DEM simulation of the compression damage of PBX
05)
where p is the pressure, p is the density, y is the
adiabatic index, v is the relative volume, E0 is the
specific internal energy, the other symbols are the
material parameters.
The big problem is how to determine the
neighborhood relationship between GPEs, since
they always have repulsive force each other
whatever how far away they are. We made an
assumption that two GPEs keep their neighborhood
till the interaction force decreases to 1% of their
initial value. Thus we can define this region as the
"influence zone (IZ)". For polytropic gas with y
=3 this distance is about 4.4 r for 2D case, where r
is the radius of the GPE. Given a GPE, only those in
its IZ can be its interaction neighbors. However, if
there is an element between them, they will not be
the link neighbors. Fig. 4 explains this relationship,
where the element I inserted between C and D. Then
C and D are not the interaction neighbors. We call
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this as "screening effect (SE)".
FIGURE 5. Experimental observation of the damage and
surface ripples for an optical film under laser irradiation
FIGURE 4. Gas phase element model (hollow circle), Solid
circles are the elements of condensed phase.
0.0002
Simulation of the Optical Film Damage under
Intensive Laser Irradiation
0
-0.0002
-0.0004
-0.01
The thickness and inclusions will greatly
influence the property of an optical film. Fan et al[sl
reported that an intensive laser pulse caused the
explosion of inclusions and created craters and
surface ripples for a 1.06 urn reflecting film (Fig. 5).
The diameters of the laser spot and inclusions are
about 300 and 201* m, respectively. The laser power
density is 2.83xl0 10 W/cm2 and the duration time is
5 ns. The configuration for DEM simulation is
shown in Fig. 6. The hollow circles in the middle are
the GPEs. The inclusion turns into the plasma phase
immediately after absorbing laser heat and the initial
pressure calculated is about 5.11Gpa.[6]
Figs. 6(c) and (d) show the forming of the crater
and surface ripples. The diameter and depth of crater
fit the experimental data perfectly. The calculated
amplitude and wave length for ripples are about 10
nm and 18.6^m, respectively, assuming the film is
melting or close to melting under laser irradiation.
The calculated amplitude (IQnm) fits the
-0.005
0
0.005
0.01
X(cm)
(a) Initial configuration
(b) Central part, the hollow circles in the middle are the GPEs
0.002
0
-0.01
-0.005
X(cm)
(c) Calculated result at 27ns
experimental datum (\0.\nm) quite well The
calculated wave length is about the half of the
experimental one (30.9//w). The possible reasons
are that the simulation we did is 2D, the real case
should be 3D; and the simulation configuration has
the boundary effect. It is interested that the
re-checking the experimental condition shows that
the film temperature reached as high as 2000K due
to the error of the thickness of the film. This
temperature is beyond the melting point of the film
material and confirms our assumption.
(d) Calculated surface ripples at 40ns
FIGURE 6. DEM simulated results of optical film
SUMMARY AND DISCUSSION
In this paper, some theoretical advances of DEM
achieved by this group have been reported, such as
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the dispersion and dissipation effects, which are
related to the element size, the nonlinear viscoelastic
modeling, and the gas phase element modeling.
Some typical simulations are also presented and
compared with the experiments. Other advances
such as the quasi-static DEM and the development
of 3D code will be published elsewhere due to the
limited space here.[ 7'8]
However, the DEM is a new method under
developing and has lots of shortages. For example,
the selection of the element size r seems arbitrary
and artificial without a physical criterion. It
corresponds to artificially introduce a characteristic
length in the problem. What is the effect of this
length on the accuracy of the calculation? No one
answers. The interaction models between elements
are arbitrary and artificial too and there is no
experimental and/or theoretical proof on them. It is
mainly depending on the experience at present. The
orderly arrangement in DEM has the an-isotropic
feature in nature and can not model the mechanical
behavior for isotropic materials well, etc. Our
research gives the element size dependent criterion
of the time step determination theoretically,111 which
guarantees the stability of numerical calculation. We
have derived the limitation for element size from the
dispersion relationship of wave propagation through
a crystalline sites.[3] We also improved the method
of checking the neighboring elements and reduced
the CPU time and RAM greatly (from geometrical
to almost linear relation).[9] However, comparing to
the conventional numerical methods, particularly to
FEM, the DEM method is far from maturity. The
main problem is that there is no systematically and
strictly mathematical proof for DEM theory, hence
there is no guarantee to inssure the accuracy and
reliability for the simulated results. This is the key
for future research and development of DEM.
REFERENCES
1.
2.
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9.
Tang, Z. P., Horie , Y., and Psakhie, S. G, "Discrete
meso-element modeling in powders," in High
Pressure Compression of Solids IV, Response of
Highly Porous Solids to Shock Loading, edited by L.
Davison et al, Springer-Verlag, New York, 1997, pp.
143-176.
Tang, Z. P., Horie, Y, and Psakhie, S. G, "Discrete
meso-element dynamic simulation of shock response
of reactive, porous solids," in Shock Compression of
Condensed Matter-1995, edited by S.C. Schmidt et al,
AIP, New York, 1996, pp. 657-660
Wang, W. W., Discrete element method and its use in
analysis of response of materials and structures, Ph.
D dissertation, University of Science and technology
of China, 2000.10.
Tang, Z. P., Study of the mechanical property of
epoxy resin at high strain rates, M. S. Thesis,
University of Science and Technology of China,
1981.9.
Fan, Z. X., and Hu, H. Y, private communication,
2000.
Phipps, Jr. C.R. et al, J. Appl Phys., 64, 1083(1988).
Tang, Z. P., and Wang, W. Q., "Discrete element
modeling for linear viscoelastic materials at
quasi-static condition," CADMAT'2001, Tomsk,
Russia, 2001.
Tang, Z. P., 3-dimensional DEM theory and its
application to impact mechanics, to be appeared in
Science in China (E), 44(4), (2001).
Chen, L.B., Hu, X. J., and Tang, Z. P., An improved
algorithm for determining neighborhood for DEM
simulation, J. of Computational Mech,, 17(2000),
497
ACKNOWLEDGEMENTS
Dr. Horie is sincerely thanked for his helpful
advice and discussion. This work was supported by
the National Natural Science Foundation (19772050)
and the High Tech Project of China.
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