Materials Sciences and Applicatio ns, 2011, 2, 654-660
doi:10.4236/msa.2011.26090 Published Online June 2011 (http://www.SciRP.org/journal/msa)
The Analysis of Stiffness for Rubbery Metallic
Material Based on Mesoscopic Features
Hong Zuo1*, Hongbai Bai2, Yuhong Feng3
1MOE Key Laboratory for Strength and Vibration, School of Aerospace, Xi’an Jiaotong University, Xi’an, China; 2Department of
Gun Engineering, Ordnance Engineering College, Shijiazhuang, China; 3School of Material Science & Engineering, Xi’an Jiaotong
University, Xi’an, China.
Received November 23rd, 2010; revised December 15th, 2010; accepted May 17th, 2011.
ABSTRACT
In this article, deformation and mechanical response of a rubbery metallic material were investigated. First, the
mesoscopic structural properties of the material and its evolution during part producing were analyzed and described
in detail. Then the inherence relationship between the macroscopic mechanical properties and mesoscopic structural
characteristics were studied, in which the related mesoscopic structural characteristics were limited in the basic unit
(mm) scale such as the radius of metal wire and unit coil, etc. Furthermore, according to the mesoscopic properties of
the material, a curved beam unit based on the mesoscopic scale and shape factor was introduced to bridge the me-
chanical response and the mesoscopic parameters such as the beam orientation and spatial distribution. In the end, a
mesoscopic stiffness model was proposed, from which the macroscopic mechanical properties of material could be de-
duced from the mesoscopic characteristic size, shape and the mechanical properties of base metallic material.
Keywords: Rubbery Metallic Material, Stiffness Model, Mesoscopic Characteristic
1. Introduction
In recent years, a kind of rubbery metallic material
(RMM) has been applied as some kinds of seal material,
heat shield material, filters, gaskets and aircraft engine
mounts in a large number of industrial fields where crude
rubber material have been usually served, owing to its
excellent applicability of environment and longer life.
Among these applications, the most exciting lies in its
application in aerospace as a vibration absorber since the
space environment needs the absorber possesses a wide
range of temperature adaptability and more than several
years life, while crude rubber cannot competent for. Ex-
cept of this, this material also has lower density and ex-
cellent mechanical properties. Considering the mechani-
cal behavior such as the stiffness and deformation re-
sponding for the applied loads play an important role
during the damping material design and application, it
should be further investigated.
Although RMM has been applied as a damping part in
the aerospace for a longtime, there still need sufficient
investigation in several fields such as its application and
mechanical behavior. According to the studies from open
literature, we only find that fewer tests conducted by
Childs [1], which relate to this material for possible ap-
plication in the high-pressure fuel turbo pump on space
shuttle. The results of their bench test showed good
damping characteristics, but numerical quantities were
not reported. A computer rotor dynamic model, with
RMM damper installed in the supports was reported to
have much better performance than any other alternative
available. The use of RMM as vibration isolators was
described by Rivin [2] and Barnes [3], they suggested the
density of the RMM influences the stiffness to great ex-
tent when used RMM for aircraft engine mounts. They
also noted the damping characteristics of RMM are de-
pendent on the material selected. As a bearing damper in
a liquid hydrogen turbo pump, RMM was applied in the
LE-7 engine for the first time reported by Okayasu and
others [4]. The old turbo pump with no RMM damper
faced several vibration problems. Whereas, once RMM
“friction dampers” applied in the bearing supports of the
turbo pump, the machine can work in a high rate, larger
than its third critical speed. They reported that RMM
show a most effective source of damping compared with
other damping units. The experiment results from Wang
and Zhu [5] show RMM damper could control three
times imbalance than the squeeze film damper. Vance

The Analysis of Stiffness for Rubbery Metallic Material Based on Mesoscopic Features655
and co-workers [6-8] reported that RMM dampers in
parallel with a squirrel cage allow wider control of the
support stiffness. They showed RMM could work even at
cryogenic temperatures. They also conducted the endur-
ance tests on these dampers over six months and showed
little to no change in the damping characteristics. They
reports the dynamic characteristics of the material are
dependent on the radial strain amplitude and axial strain
(compression), and they also suggested the effect of axial
thickness of the mesh have to be studied.
It is well-known that material mechanical response is
usually influenced by many factors, e.g., the parameters
from forming procedures, mechanical properties and size
of the base mater. However, there are few investigations
in this field were obtained from the open literatures till
now. Especially, there has no study focused on the
mesoscopic characteristic and stiffness response was
proposed. Thus, the purpose of the present study is to
investigate the relationship between the mesoscopic fea-
tures of the material and macroscopic stiffness behavior
during the material subjected to a compressed loading
along the direction of its stamping axis during forming.
In this paper, through investigating the evolution rule
of several mesoscopic feature parameters such as shape,
size and spatial distribution of base mater coil before and
after stamping during part producing, a mesoscopic
stiffness model which could represent the mesoscopic
properties and macroscopic mechanical properties of the
material was proposed based on the mechanical analysis.
2. Mesoscopic Characteristics of RMM
Before discussing mesoscopic characteristics of RMM,
the detailed produced procedures should be expatiated
firstly as follows: first, the metal wire with its base mate-
rial as stainless steel，the refractory alloy or low tem-
perature alloy was wrapped to spring line. Then, the
spring line was knitted to body as roughcast of RMM
with different style, e.g., the three-dimension knitted
method or the wool clew circled method. Third, the
roughcast was molded to the designed part through cold
stamping. Whereas, it is the excellent environmental
adaptability of the base metal endow RMM with good
temperature adaptability and longer life same as base
material. On the other hand, this forming procedure for
roughcast could be also thought as three steps of
three-dimensional molding, i.e., the first step is to create
a “line” as the filamentous spring with equal pitch. The
next step is to knit a “plane” through a crossed or slanted
knitting technique by spring “line”, similarly to the
blanket knitting (Figure 1). The last step is to create
“body” through different techniques such as the “plane”
piling up.
Expect for the physical properties such as the me-
chanical properties, the friction coefficient, and the ex-
tent of temperature adaptability of the base material, the
mesoscopic characteristic parameters of RMM usually
contains the geometric size of the mesoscopic structure
and the characteristics of base material, for example, the
diameter of the metallic wire
s
d, the diameter of spirals
j
d, the nominal included angle of the definite metallic
wire with the plane of the compressed surface

, and
the relative density of RMM

.
According to the procedures of weave and cold
stamped extent, the relationship between the relative
density of RMM and mesoscopic structure feature pa-
rameter (the diameter of metallic wire, the diameter of
spirals and the compressed extent) could be derived ac-
cording to several proper assumptions: the homogeneous
assumption, the self-similar deformation assumption in
the mesoscopic view and the well-proportional deforma-
tion assumption during cold stamping stage. Examining
the mesoscopic structure features of the material before
cold stamping, it will be found that the typical meso-
scopic structure appears as an element cubic composed
with two crossed circles of metallic wire. This can be
thought as a self-similar elementary material unit for
RMM, as shown in Figure 2. According to the definition
of relative density

for porous material the ratio of the
volume of related base material with the porous material,
the relative density of a RMM is expressed as follows
according to the spatial arrangement of metallic wire
(Figure 3):

22
3
π1
2
j
s
js
dd
dd



 (1)
where
j
d denotes the pitch diameter of the spiral,
s
d
the diameter of the metallic wire, and

the included
Figure 1. The crossed weave or oblique weaves techniques
by the “line”.

The Analysis of Stiffness for Rubbery Metallic Material Based on Mesoscopic Features 656
Figure 2. The mesoscopic feature of metallic wire on the
side surface.
Figure 3. The spatial configuration of metallic wire.
angle of metallic wire. For the original roughcast before
cold stamping, μ is the compressed ratio during cold
stamping and can be related to the included angle of me-
tallic wire

in one circle wires, tan π

 . μ = 1
corresponds to the no cold stamping body while μ < 1
corresponds to the stage of cold stamping. For example,
for the case of no cold stamping,

= arctan(1/π)
(about 17.66˚) based on the assumption that two circle
wires crossed in a cubic body, and the relative density
0

can be expressed as follows:

22
03
π1
2
j
s
js
dd
dd



 (2)
During cold stamping, compression deformation along
the axis of cold stamping happened. And on the direc-
tions perpendicular to this axis, no deformation take
placed from the macroscopic view since the rigid con-
straint of material by the die. Therefore, the mesoscopic
deformation and shape change of metallic wire during
the macroscopic uniaxial compressed deformation can be
described as follows. First, the included angle between
the metallic wire and compressed plane

decreased
gradually with the macroscopic deformation increased.
Next, the metallic wire deformed in a compound of
twisting and bending, as shown in Figure 4. It is obvious
that the influence of this complex deformation on the
relative density of material is smaller than the decrease of
included angle. Thus, it is reasonable to evaluate the
change of relative density only through the change of
included angle.
Suppose
j
d and
s
d were given, the relative density
0

of material was determined by Equation (2). For
example, if
j
d is taken as 1.0 mm, and
s
d 0.1 mm
before cold stamping, then 0

is determined as 0.128.
In general, it is difficult to calculate the relative density
through Equation (1) since the compressed ratio is not
easily determined after cold stamping. However, the
relative density of the material before cold stamping can
be obtained by experimental measurement. Therefore, we
can determine the included angle corresponding to the
material after cold stamping through relative density.
Based on the definition of included angle of wire in the
material unit and the homogeneous assumption, the
self-similar assumption and the proportional deformation
assumption above (referred to Figure 5), the nominal
included angle of wire for material is defined by means
of the relation of included angle of wire with other pa-
rameters in Equation (1),

22
3
π1
arctan
2
j
s
js
dd
dd


 (3)
For example, with the same values of
j
d and
s
d, if

is increased to 0.2, then,

will decrease to 3.54˚.
In this case, the volume ratio μ of compressed model and
original model is nearly 0.19 from the geometrical rela-
tion of

and
j
d.
3. The Stiffness Model
Once the included angle of two crossed wires in a unit
was given, the spatial distribution of wires in a unit can
be determined. Given the mechanical properties of me-
tallic wire, the stiffness of this unit is derived based on
the deformation stiffness analysis (referred to Figure 5).
For the material RMM, it is reasonable to select two
crossed wire circles as a representative unit to represent
the material mesoscopic structure (referred to Figure 3).
In this material element, there exist eight same deformed
metallic wire segments distributed in the space of the
element and stacking each other. These segments have a

The Analysis of Stiffness for Rubbery Metallic Material Based on Mesoscopic Features657
X
Y
Z
α
(a)
X
Y
Z
(b)
F
f
α
(c)
dj/2
f
(d)
Figure 4. The small curve beam model.
length of quarter of the wire circle and can be thought as
a basic element comprising the representative unit of the
material according the space distribution (referred to
Figure 3). Thus, the stiffness of the unit is thought as the
basic element stacked in series by four layers and in each
layer there are two basic elements arranged parallel. Now
let’s check the deformation of this basic unit when the
compressed loading subjected, we can find that this basic
unit can be thought as part of a curve cantilever beam
illustrated in Figure 4. The constraint of curved cantile-
ver beam is as follows: the fixed constraint with six
freedoms at one end, and two forces stressed on the other
end of the cantilever, one is the pressure force along the
orientation of stamping direction, and the other is the
constraint force by the die around. In practice, this con-
straint force in the units witch don’t contact with the die
can be though as the frictional force owing to the relative
sliding in the surface of adjacent metallic wires (shown
in Figure 4). Thus, according to the basic deformation
analysis, the stiffness of the curve beam is derived as
follows:

1
42
3
cos322sin 2
24π8
16
s
e
jj s
df
kGE
dd d










(4)
Where G and E are the shear module and Young’s
module of the metallic wire respectively,

is the in-
cluded angle calculated by relative density of RMM, and
f is the friction coefficient of base material.
A representative unit in the material is composed by 8
basic elements (curved beams) with same deformation
manner. According to the contribution of this element to
the representative unit stiffness, the stiffness of one rep-
resentative unit is calculated here:

1
42
3
cos322sin 2
24π8
32
s
jj s
df
KGE
dd d










(5)
where

22
3
π1
arctan
2
j
s
js
dd
dd


. From the expression of
the stiffness of the unit, as soon as
j
d and
s
d were
given, for a metal wire with a definite E, G and f, the
stiffness of metallic rubber is then a function of

.
4. Discussion
According to the mesoscopic stiffness model proposed
above, the effect of each mesoscopic characteristic pa-
rameter such as
j
d,
s
d and

on the macroscopic
material stiffness could be further discussed.
4.1. The Influence of dj and ds
The pitch diameter
j
d of spiral of the wire is a main

The Analysis of Stiffness for Rubbery Metallic Material Based on Mesoscopic Features 658
X
Y
Z
Figure 5. The mesoscopic deformation process in RMM.
control parameter during material production. Although it
is well-known that it has a considerable effect on the total
stiffness of material, however this effect and its variation
were never exactly evaluated analytically in the past. In
general, the diameter of metallic wire
s
d is about tenth
of pitch diameter of spiral, thus the value of


j
s
dd
usually is estimated as
j
d approximately when evalu-
ating the influence of
j
d. Therefore, through the expres-
sion of Equation (5), the magnitude of stiffness of the
material is determined in inverse proportion to the fourth
power of
j
d approximately. The effect of the pitch di-
ameter of spiral
j
d on the stiffness according to the
relative experiments [9] was illustrated in Figure 6. In
the figure, the experimental result expressed as a solid
curve while the predicted result was denoted as a dashed
curve. From the figure, the variation of the stiffness vs.
increase of
j
d is consistent with the prediction by
Equation (5).
Figure 6. The influence of the diameter of the spiral of wire.
The diameter of metallic wire ds is another mesoscopic
material feature which influences the mechanical proper-
ties of the material and is determined during the material
design. From Equation (5), the stiffness of the material
RMM is in a direct proportion to the fourth power of
s
d,
and this can be compared with the experimental result
illustrated in Figure 7. As seem as in the Figure 6, the
experimental result of stiffness expressed as a solid curve
while the predicted result was denoted as a dashed curve.
From the figure, the increase trend of stiffness with
s
d
is measure also the direct proportion to the fourth power
of
s
d, the same as the predicted by the proposed model.
If we introduce another parameter r to denote the
ratio of the diameter of spiral
d
j
d and the diameter of
wire
s
d, then, Equation (5) is expressed as follows
Figure 7. The influence of the diameter of metallic wire.

3
32 1
rr
N
Kdd . (6)
1
2
cos322sin 2
24π8
f
NGE






 


. (7)
where

The Analysis of Stiffness for Rubbery Metallic Material Based on Mesoscopic Features659
le
he influence of mepic
Here, N expresses the mechanical properties of mate-
rial and the intensity of cold stamping, while r d ex-
presses the mesoscopic size properties of the original
roughcast material. It is obviously from Equation (6) that
a singparameter, the ratio of diameter of spiral and
wire r
d can represent tsosco
properties quantitively.
4.2. The Influence of

Remember that there exist inherent relation between
the macroscopic feature
an

and the mesoscopic pa-
rameter

, referenced to Equation (3) owing to the cold
stamped processing of the material. According to the
producti procedure of RMM, the magnitude of relative
density
on

of the material is controlled by the intensity
of compressed deformation along the direction of com-
pressed force, and has nothing to do with the deformation
along other directions. Now, let’s consider the other ex-
pression of Equation (5) as follows
1
2
132 1
24π44 cos
M
KGEE


fL




 





, (8)
where

3
32 1
rr
dd
1
M and considering tan 1



.
rm
ctice,
Here, L is a coefficient and can be deteined by
Equation (3). In pra the included angle

is very
small (< 5˚), thus cos

is equal to 1 approximately,
thus, the flexibility of RMM (1/K) is decreased inverse
proportionally with the relative density

increased.
However, on the other hand, if the included angle is very
small, the intervention between contacted wires is inten-
sively. Thus assumption of freedom deformation for
curve cantilever is no longer active, and this influen
4.3. The Influence of Friction Coefficient of
ith the frictio
ic wires f increased.
The main
co
l R
roportional to the fourth power of
ce
should be investigated further.
Metallic Wires f
During the material deformed under compressed loading,
its mesoscopic deformation experienced several stages.
After the early stage where no relative slide of contacted
wires, the influence of friction coefficient of metallic
wires f on the stiffness of the material emerged in the
deformation later. It is obviously from Equation (5) that
the friction coefficient of metallic wires will affect the
value of the stiffness of material. The relationship of the
friction coefficient of metallic wires f and the total stiff-
ness of material shows the larger friction coefficient will
induce the higher stiffness of the material. The flexibility
of RMM is decreased proportionally wn
coefficient of metall
5. Conclusions
From the mesoscopic stiffness model proposed in this
study, the influence of mesoscopic structure features and
material properties on the macroscopic stiffness of the
material have been evaluated quantitatively. According to
this model, it is possible to design the macroscopic prop-
erty of the material at early producing stage.
nclusions of this study are presented below:
1) The variation law of stiffness of the materiaMM
is in inverse p
j
d ap-
pr
al
po
oximately.
2) The stiffness of the materi RMM is in direct pro-
rtion to the fourth power of
s
d, although the value of
s
d is smaller than the value of
j
d.
3) The larger relative density of material and the larger
friction coefficient of metallic wires will strengthen the
aterial.
the Natural Science Foundation of Shanxi
(2005A19).
100,
macroscopic stiffness of the m
6. Acknowledgements
This work was supported by the National Defense Foun-
dation, the Natural Science Foundation of Xi’an Jiaotong
University and
REFERENCES
[1] D. W. Childs, “The Space Shuttle Main Engine High-
Pressure Fuel Turbopump Rotordynamic Instability Prob-
lem,” ASME Journal of Engineering for Power, Vol.
No. 1, 1978, pp. 48-57. doi:10.1115/1.3446326
[2] E. Rivin, “Vibration Isolation of Precision Machinery,”
SV, Sound and Vibration, Vol. 13, No. 8, 1979, pp. 18-23.
[3] A. B Barnes, “Improvement in Machinery for Woven
Wire Mesh,” Wire Industry, Vol. 51, No. 605, 1984, pp.
Propulsion Conference, Orlando, Florida,
Aerospace Power, Vol. 12, No. 2, 1997, pp.
t the ASME Turbo Expo, New Orleans, 4-7
actions of the ASME,
nal
403-404.
[4] A. Okayasu, T. Ohta, et al., “Vibration Problems in the
LE-7 Liquid Hydrogen Turbo Pump,” Proceedings of the
26th AIAA Joint
1990, pp. 1-5.
[5] X. Wang and Z. Zhu, “Ring-Like Metal Rubber Damper,”
Journal of
143-145.
[6] E. M. Al-Khateeb and J. M. Vance, “Experimental
Evaluation of a Metal Mesh Bearing Damper in Parallel
with a Structural Support,” ASME Paper 2001-GT-0247,
Presented a
June 2001.
[7] M. Zarzour and J. M. Vance, “Experimental Evaluation of
a Metal Mesh Bearing Damper,” Journal of Engineering
for Gas Turbines and Power, Trans
Vol. 122, No. 2, 2000, pp. 326-329.
[8] B. Ertas, E. M. Al-Khateeb and M. John, “Cryogenic
Temperature Effects on Metal Mesh Dampers and Liquid
Hydrogen Turbo Pump Rotor Dynamics,” AIAA Jour
& Proceedings, 2002, AIAA Paper AIAA-2002-4164.

The Analysis of Stiffness for Rubbery Metallic Material Based on Mesoscopic Features 660
International Journal of Mechanics and Materials in De-
[9] H. Zuo, Y. H. Chen, H. B. Bai and H. Sun, “The Com-
pression Deformation Mechanism of a Metallic Rubber,”
sign, Vol. 2, No. 3-4, 2005, pp. 269-277.
doi:10.1007/s10999-006-9007-x