LLNL-PROC-491549
Propagation of rarefaction pulses in
particulate materials with strain-softening
behavior
E. B. Herbold, V. F. Nesterenko
July 29, 2011
Shock Compression of Condensed Matter - 2011
Chicago, IL, United States
June 26, 2011 through July 1, 2011
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1
PROPAGATION OF RAREFACTION PULSES IN PARTICULATE
MATERIALS WITH STRAIN-SOFTENING BEHAVIOR
E.B. Herbold
1
, V.F. Nesterenko
2,3
1
Lawrence Livermore National Laboratory, L-236, P.O. Box 808, Livermore, CA, 94550, USA
2
Department of Mechanical Engineering, UC, San Diego, La Jolla, CA 92093-0411, USA
3
Materials Science & Engineering Program, UC, San Diego, La Jolla, CA 92093-0412, USA
Abstract. We investigate rarefaction waves in nonlinear periodic systems with a ‘softening’ power-law
relationship between force and displacement to understand the dynamic behavior of this class of
materials. A closed form expression describing the shape of the strongly nonlinear rarefaction wave is
exact for n = 1/2 and agrees well with the shape and width of the pulses resulting from discrete
simulations. A chain of particles under impact was shown to propagate a rarefaction pulse as the
leading pulse in initially compressive impulsive loading in the absence of dissipation. Compression
pulses generated by impact quickly disintegrated into a leading rarefaction solitary wave followed by an
oscillatory train. Such behavior is favorable for metamaterials design of shock absorption layers as well
as tunable information transmission lines for scrambling of acoustic information.
Keywords: strongly nonlinear wave, softening behavior, rarefaction wave
PACS: 05.45.Yv, 46.40.Cd, 43.25.+y, 45.70-n
INTRODUCTION
It is well known that certain materials exhibit an
elastic or viscoelastic softening behavior under load.
Discrete periodic materials with a “normal” power-
law relationship between force and displacement,
F
n
for n > 1 (elastic strain hardening), have
been shown to support compression solitary waves
in periodic granular assemblies [1-11]. When a
dynamic force is significantly larger than the initial
force applied to grains, these materials are
considered strongly nonlinear and recent
investigations have considered their use as
information carriers and waveguides [3,4,8]. The
long wave approximation of the equations of motion
for a discrete chain supports a stationary wave
solutions traveling in one-dimensional chains or
ordered two and three-dimensional arrays of
particles in the absence of dissipation [1,2]. It was
proven in [2] that stationary solitary and shock
waves should form in discrete materials with a
general force-displacement relation that stiffens with
displacement. Conversely, a discrete chain with an
interaction law exhibiting general softening
behavior (first considered in [12]) supports
rarefaction solitary shock-like waves [2]. In the
case of discrete softening materials without tensile
strength, the propagation of fracture waves follow
directly behind rarefaction pulses.
We investigate the behavior of stationary
rarefaction/release waves in discrete periodic
materials composed of point masses and an
elastically softening interaction law. An exact
solution of the long wave approximation with n =
1/2 is presented for stationary rarefaction solitary
waves and is compared to numerical simulations.
Stationary rarefaction waves have also been
investigated in magnetized Hall plasmas and are
thought to explain observed anomalous behavior
due to a changing electric field [13].
A softening behavior (decreasing of elastic
modulus with strain) is observed under certain
conditions of loading in a wide range of materials
from polymer foams [14] and rubber [15] to actin
networks in biological tissues [16]. In general, the
response of materials with a “softening” behavior
share several common responses under compressive
loading: a viscoelastic softening behavior
2
characteristic of configuration changes in polymer
chains [15] or the collapse of cell-wall structures in
polymer foams [14] followed by a stiffening
behavior attributed to the bulk resistance to further
deformation. Here, we consider a non dissipative
interaction law between particles in a one-
dimensional lattice. In experiments this may be
realized for foams where the softening behavior is
due to reversible elastic collapse of cell walls.
Particle motion may be described by a function of
relative particle displacements
(u
i
-u
i+1
) , where u
i
is
the displacement. The system of discrete equations
for a chain of identical particles with a power law
potential is
,
1,1,,1,1
iiiiiiiii
AAu
(1)
where A is an effective stiffness constant and
i1,i
u
i1
u
i
n
for 0 < n < 1. Initial displacements
caused by an external force may also be included in
displacement u
i
. The conditions for propagating
stationary rarefaction waves in generalized discrete
“softening” materials (and with a specific power-
law interaction) are presented in [2].
The long−wave approximation for Eq. (1) is
introduced in a way similar to the case of elastic
hardening materials (e.g. particle contact interaction
with a Hertzian potential) by assuming that the
particle diameter, a, is significantly less than the
propagating wavelength L. The result from
applying the long wave approximation to Eq. (1) is,
u
tt
c
n
2
u
x
n
na
2
6 n 1
u
x
(n 1)/2
u
x
(n1)/ 2
xx
x
.
(2)
A derivation of Eq. (2) can be found in [2]. The
long wave sound speed c
0
in the chain of particles is
found through the linearization of Eq. (2),
c
0
c
n
n
0
(n 1)/2
. Stationary solitary rarefaction
waves may propagate in a discrete chain of particles
if the chain is initially subject to a constant
compression force, f
0
, creating an initial strain ξ
0
.
The reduction of Eq. (2) assumes a stationary wave
propagating with speed V,
y
y y
n3
/ n1
y
n1
/ n1
C
2
0,
(3)
for arbitrary values of n. In Eq. (3), y is a reduced
form of the strain (ξ = -u
x
),
y c
n
/V
(n
1)/(n
1)
(n1)/2
,
where c
n
2
= Aa
n+1
is a parameter with units of speed
and η is the normalized coordinate traveling with the
speed of the solitary wave V,
x /a 6(n 1)/n.
Eq. (3) can be rewritten for a nonlinear
oscillator moving in an effective “potential field”,
d
2
y /d
2
dW /dy,
where W(y) is defined as
W y
1
2
y
2
n
1
4
y
4 / n1
C
3
y
2/ n 1
.
(4)
An analogy can be made that a “particle” in the
“potential field” W(y) moves from its initial position
(y
1
, corresponding to ξ
0
) to the position in the wave
corresponding to y
min
(related to ξ
min
) and back to y
1
.
In general, y
1
corresponding to the case when a
minimum strain is equal zero may be expressed only
as a function of the power-law exponent
y
1
2n/n 1
(1
n)/2(1
n)
[2,3].
For an anomalous softening interaction between
particles, 0 < n < 1, rarefaction solitary waves exist
when C
3
= 2C
2
/(n+1) is bound by [2, 12],
n
2
1
2
n
n /(1n)
C
3
n 1
2
2n
n 1
n / 1n
.
(5)
The value of C
3
defines system behavior between
weakly and strongly nonlinear regimes by the lower
and upper bound of C
3
, respectively. The width of
the wave increases in the weakly nonlinear case
compared to the strongly nonlinear case.
The exact solution for the long wave
approximation can be found for the case where the
minimum strain is equal to zero and C
3
is given by
the maximum value in Eq. (5). In the system of
reference moving with the wave and centered at the
minimum value of strain, an exact solution can be
obtained for n = 1/2 and C
3
= -1/6 by solving Eq.
(3),
y 2/3
3/ 2
tanh
3
2
/6
.
(6)
The corresponding equation for the strain is,
0
tanh
4
x /a
.
(7)
The exact solution Eq. (7) predicts a symmetric
3
pulse that starts from the initial strain value ξ
0
,
decreases to zero and then returns. The
characteristic pulse length is equal to 7a (for a cut-
off of ξ/ξ
0
=0.98) and does not depend on the
amplitude of the solitary wave similar to the case for
compressive solitary waves in a “sonic vacuum”
where n > 1 [2].
Fig. 1 shows that the width of the solitary
rarefaction wave increases with 0 < n < 1. A closed
form expression for solitary may be constructed for
rarefaction waves for general powers of n. The
amplitude is equal to y
1
and the width of the wave
increase for values of n in the interval 0 < n < 1,
y
2n
n 1
1n
/2 1n
tanh
1n /n
3 n 1 / n 1 n
.
(8)
The two remaining comparisons in Fig. 1 are made
between Eq. (8) and the numerical solution of Eq.
(3) for n = 1/5 and 4/5. The amplitudes of each
pulse from Eq. (8) are the same, but the widths of
the pulses are slightly underestimated. However,
Eq. (8) is a simple closed form approximation of the
general form for the long wave approximation.
Figure 1. (color online) Equation (6) is compared
with the numerical solution of Eq. (3) for three
different values of n. The amplitudes of the solitary
rarefaction waves are exact, but the widths of the
pulses increase for 0 < n < 1.
It is interesting to find the relationships between
the phase speed V and the strain ξ. The phase speed
can be found using the properties of the potential
function; W(y=y
min
) = W(y=y
1
) and
W /
y
yy
1
0
.
The speed of the rarefaction wave is [2],
V
r
c
n
0
min
2 n
0
n 1
min
n 1
n 1
0
n
min
n 1
1/ 2
.
(9)
The rarefaction wave with a minimum strain
(ξ
min
) equal to zero is a special case where the long
wave sound speed represented by c
0
at the point of
zero strain is infinite for n < 1, but the solitary
rarefaction wave speed is finite. We consider the
validity of the rarefaction solitary wave solution in
long wave approximation with this singularity point
by comparison with results for discrete chain. It
should be mentioned that a similar situation exists
for the case with strongly nonlinear compressive
solitary waves in “normal” material (n>1).
Figure 2. Solitary rarefaction strain wave in a chain
of 800 particles compressed with a static force of 1
N and n = 3/4 impacted by one particle at 6 m/s.
The initial disturbance disintegrates into wave
packets, periodic waves and the leading rarefaction
solitary wave after travelling ~600 particles. The
strain is offset by 1.5 for each time for visual clarity.
Numerous numerical investigations using n>1
agree well with the results obtained using the long-
wave approximation [2, 5-9]. For example, the
classical Hertzian interaction between perfectly
elastic spherical particles is a special case of Eq. (3)
where n = 3/2. Additionally, the ratio of the solitary
wave speed, V
s,r
to the sound speed c
0
is infinite.
This ratio is positive V
s,r
/c
0
> 1 for 0 < n < 1,
meaning the solitary rarefaction wave speed is
supersonic [2].
In Fig. 2 impact velocity of 6 m/s was specified
for the first particle to which a constant force is