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 METHOD SIMULATION OF NONLINEAR
VISCOELASTIC STRESS WAVE PROBLEMS
Wenqiang Wang, Zhiping Tang*, and Y. Horie+
* University of Science and Technology of China, Hefei,Anhui 230027 China
+ Los Alamos National Laboratory, Los Alamos, New Mexico 87545 USA
Abstract. ADEM(Discrete Element Method) simulation of nonlinear viscoelastic stress wave problems is
carried out. The interaction forces among elements are described using a model in which neighbor
elements are linked by a nonlinear spring and a certain number of Maxwell components in parallel. By
making use of exponential relaxation moduli, it is shown that numerical computation of the convolution
integral does not require storing and repeatedly calculating strain history, so that the computational cost is
dramatically reduced. To validate the viscoelastic DM2 code[l], stress wave propagation in a Maxwell rod
with one end subjected to a constant stress loading is simulated. Results excellently fit those from the
characteristics calculation. The code is then used to investigate the problem of meso-scale damage in a
plastic-bonded explosive under shock loading. Results not only show "compression damage", but also
reveal a complex damage evolution. They demonstrate a unique capability of DEM in modeling
heterogeneous materials.
INTRODUCTION
A PRACTICAL NONLINEAR
VISCOELASTIC CONSTITUTIVE
EQUATION
In continuum mechanics, the solution of most
viscoelastic stress wave problems must resort to
numerical techniques such as the characteristics,
finite difference, and finite element methods. Only
when the initial and boundary conditions are very
special, could analytical results be obtained [2].
A new approach, totally different from the
continuum mechanics based methods, to the solution
of viscoelastic stress wave problems, is presented in
this paper. The approach is based on the so-called
discrete element method(DEM) [1]. At present, the
DEM is the most popular technique for solving
granular material problems. However, its potential in
solving continuum problems, especially its unique
capacity of tackling heterogeneous media has not yet
been fully explored.
In the following, first a viscoelastic DEM
model is presented, then a sample calculation is
given to demonstrate the efficiency of the approach.
A variety of nonlinear viscoelastic constitutive
equations have been established for years. However,
most of them are difficult to use in practice.
To overcome this difficulty, a simple yet
reasonable phenomenological model is used in this
study. The model, shown in figure 1, comprises a
nonlinear spring and a certain number of Maxwell
components being connected in parallel.
FIGURE 1. A phenomenological model for nonlinear
viscoelasticity
371
From figure 2, it can be seen that, at time step
«-l, (3) becomes:
The total stress of the model can be written as:
N
(1)
n-l
/=!
(4)
where <JE is the stress of the nonlinear spring, (Jl
is the stress of the l-th Maxwell component. TV is the
number of the Maxwell components.
Be
crEE = ——— -
k=\
at time step n, (3) becomes,
n-l
(2)
k=\
where S is strain, B and ^ are material constants.
*
^_ g
, =El [exp( ——— }s
*
*/
+(n)
A^.
n-l
An
= exp(—)£/£exp(—
(3)
*l
where / is the current time. El and Tt are the spring
modulus and the relaxation time of the l-th
Maxwell component, respectively.
Equation (3) is a kind of convolution integral
[2].
'
t(k)
~* \ •(
k=\
(5)
Comparison of (4) and (5) yields
cr\n} =exp(--
(6)
n
NUMERICAL COMPUTATION OF
CONVOLUTION INTEGRAL
This is a recursive relation between 07 and
<J\ , from which it is seen that, the value of (Jl
at any time step can always be obtained from (Jl at
the previous time step. Thus, by making use of
exponential relaxation modulus, we prove that,
numerical computation of the convolution integral
does not require storing and repeatedly calculating
strain history, therefore the computational cost can
be saved dramatically.
Numerical computation of convolution
integral in general form usually requires storing and
repeatedly calculating strain history at each time step.
This is a very heavy burden for the simulation of a
large system with a great number of elements and
time steps. Simulation may even become
impracticable under such circumstances. However, it
is found in this study that such difficulty does not
exist in numerical calculation of equation (3).
INTERACTION FORCES BETWEEN
NEIGHBORING ELEMENTS
In DEM simulations, a real system being
studied is divided into separate elements. The
governing equations of the discrete system are based
on Newton's second law. Various results including
mechanical, physical, and chemical properties can be
obtained by solving these equations.
In general, forces acting on an element
include those from outside of the system and those
from its neighboring elements in the system. It is
crucial to correctly determine the neighborhood
relationships among elements and calculate the
interaction forces between neighboring elements.
Interaction models are obviously dependent of
material behaviors, that is, different behaviors such
as elasticity, plasticity, viscoelasticity, and
viscoplasticity, etc., require different models.
1 integral function
at time step n-1
2integral function
at time step n
FIGURE 2. The integration scheme for equation (3)
372
In this study, the phenomenological model
illustrated in figure 1 is taken as the interaction
model between two neighboring elements in a
nonlinear viscoelastic system. So, the system can be
regarded as a network in which each pair of
neighboring elements are linked by a nonlinear
spring and a certain number of Maxwell components
in parallel. Hence, interaction force per unit area
between two neighbors can be obtained from
equation (1). This value, multiplied by the
interaction area between the two neighbors, then
gives the interaction force. The details are given in
[3].
A SAMPLE CALCULATION
To validate the viscoelastic DM2 code, stress
wave in a semi-infinite Maxwell rod with one end
subjected to a constant stress loading is simulated,
and the result is compared with the characteristics
calculation.
The parameters of the model material being
simulated are,
pQ =l.27g/cm\El = 4.S67xlOl°dyne/cm\
r1=2xlO~65.
The applied stress is,
cr* =l.6x\Q9 dyne/cm2.
The radius of the element used in the DEM
simulation is 0.001 cm.
Figure 3 shows the stress and particle velocity
profiles at different locations of the rod. The four
locations are from the impact end of the rod 0, 0.2,
0.4 and 0.6cm, respectively.
It is seen that the DEM results excellently fit
those from the characteristics calculation. The code
is then used to investigate the problem of meso-scale
damage in a plastic-bonded explosive under shock
loading. Results not only show "compression
damage", but also reveal a complex damage
evolution. This part of our work was given in [3] and
will be published elsewhere.
1.5E+09 -
CONCLUSIONS
A new approach, which is based on the
discrete element method, to the solution of
viscoelastic stress wave problems, is presented in
this paper. Several simulations performed using the
modified DM2 code have shown that this is a
promising technique for solving viscoelastic stress
wave problems.
REFERENCES
1.
2.
3.
FIGURE 3. Stress wave profiles in a semi-infinite
Maxwell rod
373
Tang,Z.R, Horie, Y., and Psakhie, S. G, in High
Pressure Shock Compression of Solids IV, edited
by L Davidson, Y. Horie, and M. Shahinpoor
(Springer,New York, 1997), p.143.
Christensen, R. M., Theory of Viscoelasticity, An
introduction, 2nd ed., Academic Press, 1982.
Wang Wenqiang, Discrete Element Method and Its
Use in Analysis of Response of Materials and
Structures. Ph.D dissertation, University of
Science and Technology of China, October 2000.