INTRODUCTION
Metal rubber’s wires are molded
and pressed into required shape. It has the elasticity and damping characteristics
like rubber material and at the same time keeps the broad adaptability of the
metal material to the environment. It can be used as sealing component in severe
environment with extremely high and low temperature, high pressure, high vacuum
and erosive medium as shown in Fig. 1.
Up to now metal rubber is mainly used in national defense industry. This situation
can be explained by its complex characteristics. The inner structure of metal
rubber is similar with macromolecule and the constitutive equation of metal
rubber is nonlinear (Zhongying, 2000). And theoretical
model of metalrubber was studied (Guo et al., 2004),
the paper presents the constitutive relation of metal rubber and application
research of metal rubber was studied (Zhao et al.,
2006). Therefore its elasticity and damping characteristics are very complicated.
Previous attempts on prediction of metal rubber performance are mostly based
on mathematical model of constitutive equation. Although much progress has been
made, it is hard to describe its characteristics with sufficient accuracy. In
our research of metal rubber, it nearly impossible to find an appropriate mathematical
model as shown in Fig. 2.
Without the help of mathematic model, a lot of experiments are necessary for
the design of metal rubber. Yet this leads to great cost. To make a new type
of metal rubber for sealing, for example, a set of new molds has to be made
for laying wires, stamping semifinished product and wrapping with polytetrafluoroethylene.
And then experiments have to be done to test the performance of samples made
by the new molds.

Fig. 1: 
A sample of metal rubber 

Fig. 2: 
A set of molds for metal rubber 
The new product cannot be massively produced until the samples satisfy the
requirement. Usually five or six sets of molds have to be made in this process.
This leads to great waste of time and resource and outlays are considerable.
That is why it is mostly used in national defense industry which cares less
about the cost. However the metal wire, the material of metal rubber, is cheap.
If the cost of design process can be reduced, undoubtedly metal rubber will
have more applications (Bai, 2002).
In this work, artificial neural network is introduced for the design of metal
rubber. ANNs are used in a wide range of engineering and nonengineering applications,
such as, pattern recognition, as well as behavior prediction and function approximation.
The characteristic feature of ANNs is that they are not programmed; they are
trained to learn by experience. It can be seen that ANNs can be used as an alternative
to mathematical models of metal rubber. In this paper coefficients of constitutive
equation of metal rubber are predicted by ANNs. If the coefficients cannot fulfill
the requirements. The design can not be modified in advance. Therefore the success
rate of design can be improved (Irie and Miyake, 1988).
COEFFICIENTS OF CONSTITUTIVE EQUATION
In the design process, the most important work is to find the coefficients
of constitutive equation of metal rubber. Once the coefficients are gotten,
mechanical, sealing and other performance can be deduced. Usually the constitutive
equation of nonlinear material can be expressed as follows:
where, x is the compressive displacement, F (x) is the force exerted on metal
rubber and a_{0}, a_{i}, b_{0}, b_{i} are coefficients
respectively. In Eq. 1 the first and second terms describe
the fourth terms which describe the dynamic characteristics.
Up to now metal rubber is mainly used as sealing component and the dynamic
characteristic is mostly of little importance. Omitting the third and fourth
term, then:
For static characteristic, it is found that cubic nonlinearity of displacement
is enough to describe the characteristic of material. So Eq. 2
becomes to:
Due to the memory characteristic, the forces of loading and unloading stage
are different from each other. Equation (3) was displaced
with:
and
where, the index notation, l represents the loading stage and t represents
the unloading stage, respectively.
In our research, the leastsquares method was used for the parameter identification
of experiment data. It finds the coefficients of polynomial F (x) that fits
the experiment data in a least squares sense. For each set of N forcedisplacement
data (x_{1}, y_{1}), (x_{2}, y_{2}),... and
(x_{n}, y_{n}) obtained in the static experiments, the approximated
by a polynomial of degree 3: The coefficients a_{0}, a_{1},...a_{3}
were obtained by minimizing the function:
NEURAL NETWORK PREDICTOR MODEL
The coefficients of constitutive equation had to be obtained from experiment
data. In this paper ANNs are introduced to predict those coefficients before
a new set of molds are made. If the coefficients cannot fulfill the requirements,
the design can not been modified. Therefore the success rate of design can be
improved (Li et al., 2011a).
An artificial neural network can be regarded as a black box which is able to
produce certain output data as a response to a specific combination of input
data. It is an information processing paradigm that is inspired by the way biological
nervous systems process information. By receiving the data for an existing system,
ANN can be trained to learn the internal relationships that govern that system
and predict its behavior without any physical equations (Li
et al., 2005). The major advantage of ANNs, compared to traditional
polynomial mapping, is that they are able to perform nonlinear mapping of multidimensional
functions, i.e. relationships from many inputs to many outputs. It can be seen
that design of metal rubber just belongs to this problem (Zhang
et al., 2007).
One ANN which has received most attention is the Backpropagation Network (BPN)
(Li et al., 2011b), as is shown in Fig.
3. BPNs have hierarchical feed forward network architecture. In the classical
structure of a BPN, output of each layer is sent directly to each neuron in
the layer above. While there can be many layers, the processing can be done
with a minimum of three layers: one layer receives and distributes the input
pattern, one middle or hidden layer captures the nonlinearities of the input/output
relationship and one layer produces the output pattern (Wang
et al., 2011). BPNs are trained by repeatedly presenting a series
of input/output pattern sets to the network. The network gradually learns the
input/output relationship of interest by adjusting the weights to minimize the
error between the actual and predicted output patterns of the training set.
The trained network is usually examined through a test set of data to monitor
its performance and validity. When the mean squared error of the test set reaches
a minimum, network training is considered completely and the weights are fixed.
The back propagation neural network with three layers, inputhidden i
output, is used. It has 5 inputs, 15 intermediate nodes and 6 outputs. The input
variables are: average density of metal rubber ρ_{a}, diameter
of wired d_{w}, external radius of metal rubber r_{e}, internal
radius of metal rubber r_{i}, height of metal rubber r_{h}.
The outputs are coefficients of constitutive equation according to the formula:
where,
are the input vectors,
are the connection weights between layers and θ_{j }are bias weights.
And activation function s is a Sigmodial function as follows:

Fig. 3: 
The structure of BPN with a hidden layer 
By using the algorithmgeneralized gradient descent search technique, BPA adjusts
the eights of the network and the threshold of each neuron recurrently according
to the criterion that the cost function is minimized. The cost function is meansquared
error between the actual outputs:
And the target outputs:
Metal rubber constitutive relation coefficient material density, shape, the
relation of the factor, it is difficult to describe accurate mathematical model.
This paper is ready to BP neural network based on the constitutive relation
coefficient estimates, resulting in the constitutive relationship of metallic
rubber following BP neural network estimated metal rubber material constitutive
Method.
EXPERIMENTS AND RESULTS
The experiments were carried out on a MTS810 universal testing machine. The
article has been applied about material density changes of the estimate of the
constitutive relation of metal rubber material. To the metal rubber material
do static experiment, specimen for without heat, r_{i} = 4 mm, r_{e
}= 12.5 mm (Li, 2006; Liu, 1997).The
metal wire is made of 1Cr18Ni9Ti and its elastic modulus is 198,000 Mpa. A total
of 19 groups of experiment data are used. Fifteen groups of data are used for
the training of the network and the others are used for validation. Static equipment
adopted in the experiment for WDW1002 type electronic universal testing machine,
precision grade of 0. 5, maximum load of 100 kN, load measuring accuracy can
reach 0. 1 n, displacement of measuring accuracy can reach 0. 001 mm.
As a rule of thumb, optimization of the neural network condition was performed
by trialanderror by adjusting various parameters. These include the learning
epoch size, the learning rate and momentum constants. The controlled error is
0.005. The ANN achieved a stable state after 1298 cycles of training. Finally,
the trained network was tested through four groups of experiment data. The results
are shown in Fig. 47.

Fig. 4: 
Forcedisplacement relationship for a metal rubber product
with ρ_{a} = 2.11g cm^{3}, d_{w} = 0.10 mm,
r_{e} = 5.0 cm, r_{i} = 4.6 cm, r_{h} = 0.4 cm:
1pridiction, 2experiment 

Fig. 5: 
Forcediaplacement relationship for a metal rubber product
with ρ_{a} = 2.11 g cm^{3}, d_{w} = 0.10 mm,
r_{e} = 6.0 cm, r_{i} = 4.6 cm, r_{h} = 0.4 cm:
1pridiction, 2experiment. 
It can be seen that the prediction from the ANN shows agreement with experiment
data. The accuracy can satisfy the requirement of the design of metal rubber.
Now ANNs have been used in our design process and play an important role. Before
a new set of molds is made, coefficients of metal rubber made by the molds will
be predicted. If the result does not satisfy the requirement, the design will
be modified. In most cases one set of molds is enough for the production of
new type of metal rubber with the help of ANN.

Fig. 6: 
Forcediaplacement relationship for a metal rubber product
with ρ_{a} = 2.516 g cm^{3}, d_{w} = 0.12
mm, r_{e} = 8.0 cm, r_{i} = 7.2 cm, r_{h} = 0.8
cm: 1pridiction, 2experiment 

Fig. 7: 
Forcediaplacement relationship for a metal rubber product
with ρ_{a} = 2.977 g cm^{3}, d_{w} = 0.15
mm, r_{e} = 10.0 cm, r_{i} = 9.0 cm, r_{h} = 1.0
cm:1pridiction,2experiment 
Only in a few cases two sets are needed. ANN reduces the cost of molds and
the need for extensive experiments.
CONCLUSION
In the past, the main method of designing metal rubber was the mathematical
model. However, they cannot describe the performance of metal rubber with sufficient
accuracy. In this paper, artificial neural network was used to predict the coefficients
of constitutive equation and showed a good congruence with the experiment data.
ANN method is suitable for the design of metal rubber. Besides the ANN model
is much faster and easier to use. It has been used in our design process and
is helpful to reduce the cost.
For future work in the direction of this study the main concern is integration
of more parameters as inputs and enhancing the accuracy of the model.