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THE INVERSE OF MATERIAL PROPERTIES OF FUNCTIONALLY GRADED PLATES BASED ON THE GUIDED WAVES

Jian-gong YU1, Wei-qiang MA2 1 School of Mechanical and Power Engineering, Henan Polytechnic University, Jiaozuo 454003, P.R. China

E-mail: yu@emails.bjut.edu.cn

2Department of Information Engineering Hennan Vocational College of Chemical Technology, Zhengzhou 450042, P.R. China

E-mail: hnhxxm@163.com

Using guided wave characteristics of dispersion and multi-mode, an inverse method based on neural network (NN) is presented to determine the material properties of Functionally Graded Materials (FGM) plates. The group velocities of several lowest modes at several lower frequencies are used as the inputs of the NN model; the outputs of the NN are the distribution function of the volume fraction of the FGM plate. The Legendre polynomials method is used to calculate the dispersion curves for the FGM plate. Through numerical simulation, the inversion work is carried out very easily and a satisfying result is obtained

Keywords: Material properties, plate, functionally graded materials, guided waves, neural network, dispersion

1. Introduction

Because of the gradual change of material property, functionally Graded Materials (FGM) are used to different applications and working environments, such as defense, aerospace and automobile industries etc. It is important to have a fast and cheap technique for evaluating the FGM after fabrication and in service to verify that the actual material properties match those of design.

Some literatures have dealt with this problem. Nakamura et al. [1] used the Kalman filter technique along with instrumented micro-indentation to estimate FGM film. BUTCHER et al. [2, 3] used ultrasonic pulse echo to measure the elastic properties of FGM. Chakraborty et al. [4] used spectral formulation for material property estimation by nonlinear optimization. But the application of finite spectral element has many limits so far. Han et al. [5] used a Genetic Algorithm (GA) to estimate material properties of FGM. Liu and Han [6, 7] used a neural network (NN) along with layer element to estimate material properties of FGM. They also combined GA and nonlinear least square method [8] to evaluate the material properties of FGM. The methods of Han and Liu [4-8] divided the continuous FGM into some inhomogeneous layers, and the results they obtained are discontinuous.

In this paper, a NN procedure is applied to characterize the material property of FGM plates from

guided wave group velocity. The input data used for this inverse procedure are group velocities of three lowest modes at six lower frequencies. The Legendre polynomials method [9] is employed as the forward solver to calculate the dispersion curves. The output of the NN is a polynomial, which is the fitting function of the distribution function of volume fraction.

2. Statement of the Problem

The FGMs are usually microscopically heterogeneous and are typically made from two components, such as metals and ceramics. Its essence is that there is a gradient field (a function of a parameter in space) in the material. The property of the combined materials can be obtained using a micro-mechanics model, such as the rule-of-mixture,

1

( ) ( )n

i ii

P x PV x=

=∑ , ( )( ) 1iV x =∑ (1)

where )(xVi and iP denote respectively the volume fraction of the ith material and the corresponding property of the ith material. So, the properties of the graded material composed of two components can be expressed as

( )2 1 2 1( ) ( )P x P P P V x= + − . (2)

Therefore, the characterization of the material property of FGM is actually equivalent to the characterization of

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volume fractions. The gradient field function ( )(xV ) of FGMs can be lest-square fitted to a mth order polynomial,

∑=

≈m

i

ii xaxV

0)( (3)

mode 1

mode 2

mode 3

Figure 1. Effect of the coefficient e on the group velocity dispersion curves

Normalizing Eq. (3), we get

∑=

≈m

i

ii xaxV

1)( 10 ≤≤ x , 1)(0 ≤≤ xV . (4)

Then, the distribution function of volume fraction is determined by the m coefficients.

Guided wave has the characteristics of dispersion and multi-mode, which is determined by the material properties. We can build the nonlinear map between the material properties and dispersion curves by NN. So, the material properties can be inversed by imputing the values of several points in group velocity (group velocity can be measured directly from experiment) dispersion curves.

The effects of varying the distribution function of volume fraction on the group velocity dispersion curves are studied in detail. Take a 5th order polynomial ax+bx2+cx3+dx4+ex5 as the distribution function. One of the five coefficients is varied from -0.3 to 0.5 keeping the other coefficients 0.1. Figs. 1 gives the dispersion curves of first three modes for FGM plates composed of silicon nitride and stainless steel (under surface is pure silicon nitride and the thickness is 1cm; their material properties can be found in Ref. [5]) when coefficient e vary. It shows that the coefficient have appreciable influence on the group velocity dispersion curves. The other coefficients have similar influence. The figures are not shown here. From Figure 1, it can be seen that significant change occurs in amplitude and the change is very regular when the coefficients are varied. The effect of varying the coefficient on the group velocities thus is obviously reflected. Besides, the regularity of the changes makes that the inversion processing can be easily carried out.

stainless steel

silicon nitride

Figure 2. Group velocity dispersion curves for homogenous plates

3. Inversion Strategy

3.1. Input of NN model

Input of the NN model should be easily and accurately measured and should be sensitive to the change of the volume fractions of the FGM plate. Group velocities of three lowest modes at several low frequencies of an FGM plate are selected as the inputs in this paper. Its sensitivity has been verified. Group velocities can be

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easily measured by separately inducing n-cycle sine waves modulated by a Hanning window at low frequencies. The frequencies should be low because there are too many modes at high frequencies and not every mode can be easily measured at high frequencies. Moreover, the distance between two frequencies can not be very short. The group velocities at two adjacent frequencies are close, which is adverse to the inversion process. Besides, at the chosen frequencies there should be as possible as the same mode numbers in FGM plates of different distribution function.

Take the FGM plate composed of silicon nitride and stainless steel as an example to illustrate the choice of frequencies. Because the characteristics of waves in FGMs are intervenient between its components, dispersion curves for homogenous silicon nitride and stainless steel plate should be consulted as shown in Figure 2 (inner radius and thickness of the plates is 9cm and 1cm). According to Figure 2, when frequency is below 0.26MHz, there are two modes at least and three modes at most in the FGM plates composed of silicon nitride and stainless steel. So, the frequencies can be chosen at 0.05, 0.1, 0.15, 0.2 and 0.25MHz so that the mode number of these FGM plates is 2 or 3. In fact, for these FGM plates to be measured, the cut-off frequencies of the forth mode must be higher than 0.26MHz. The highest frequency can be chosen at 0.3MHz or higher. At those frequencies that there is no mode 3, the group velocities of mode 3 is defined 0.

Instead of carrying out an actual experiment, the measured group velocities are simulated using the Legendre polynomials method from the actual volume fraction distribution.

3.2. Training sample

The training samples of the NN model consist of a number of sets of inputs and outputs. These training samples should cover all possible sets of distribution function of volume fractions. Obviously, it is impossible to generate every set of volume fractions distribution function. However, the volume fraction is usually controlled in the process of fabricating FGM, therefore a rough distribution of the volume fraction is known when the FGM is fabricated. Usually, we can know the approximate volume fractions of the upper and under surface and the distribution function is monotone or other. In this way, the number of sets of inputs and outputs is cut down greatly.

3.3. Neural Network Architecture

Any mapping from an input to output can be simulated by a multilayered feed-forward neural network with back propagation algorithm with a gradient-descent based learning algorithm. So, it has been widely used in many fields. Here, we use the three layered feed-forward neural network combined with a recurrent neural network.

The number of input nodes equals the number of input data values. The output layer produces the result. For the selection of neuron numbers of the hidden layers, there is not a best theorem so far. Here, it is selected as 2n+1 (Kolmogorov theorem, n is the number of input data values).

4. Numerical Simulations

The overall inversion method is evaluated through FGM plates composed of stainless steel and silicon nitride which thickness is1cm. The volume fractions of upper and under surface are assumed to be pure stainless steel and silicon nitride and the volume fractions of silicon nitride is monotonously degressive and there is no inflexion in the distribution function.

A total of 106 samples are calculated, in which 100 samples are used for training and the remaining 6 samples are used for testing. The training error is controlled at 1E-8. All the values of inputs and outputs in the training samples are normalized according to Eq. (4). The distribution functions of 100 training samples are divided to 4 groups:

)1/()1( −− nnx ee : n=1, 1.4, 1.8, 2.2, 2.7, 3.3, 3.8, 4.4, 5.0, 5.7, 6.4, 7.1, 8.0, 9.0, together 14 training samples.

11

2

2

+−+−

−

−

nn x

: n=0.05, 0.08, 0.12, 0.17, 0.23, 0.30, 0.38,

0.48, 0.58, 0.68, 0.79, 0.92, 1.6, 1.9, 2.3, 2.8, 3.5, 4.4, 5.8, 7, 8.5, 10, 12, 15, 20, 25, together 26 training samples.

nx : n=0.3, 0.33, 0.37, 0.4, 0.44, 0.48, 0.52, 0.57, 0.62, 0.67, 0.73, 0.78, 0.84, 0.89, 1.1, 1.2, 1.33, 1.46, 1.59, 1.71, 1.84, 1.98, 2.13, 2.3, 2.45, 2.6, 2.8, 3.1, 3.5, 4, together 30 training samples.

1)1( ++− nx : n=0.3, 0.33, 0.37, 0.4, 0.44, 0.48, 0.52, 0.57, 0.62, 0.67, 0.73, 0.78, 0.84, 0.89, 1.1, 1.2, 1.33, 1.46, 1.59, 1.71, 1.84, 1.98, 2.13, 2.3, 2.45, 2.6, 2.8, 3.1, 3.3, 3.5, together 30 training samples.

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The distribution functions of 20 testing samples are divided to 2 groups. The first group has 4 samples:

)1/()1( −− nnx ee : n= 6.2; 11

2

2

+−+−

−

−

nn x

: n=8;

nx : n=0.45; 1)1( ++− nx : n=3;

Obviously, although the first group testing samples are not training samples, they belong to the four training sample groups. The other two testing samples (the second group) are

]2

[ xSin π; 13.0

13.03

3

+−+−

−

− x

.

They do not belong to the four training sample groups. As is mentioned above, the frequencies of group

velocities inputted are 0.05, 0.1, 0.15, 0.2, 0.25 and 0.3MHz. So, there are 18 input data in a sample. Actually, in all training samples, the group velocities of the third mode at 0.05, 0.1 and 0.2MHz are all zero. They are unnecessary data. So, the group velocities of the third mode only selected at 0.2, 0.25 and 0.3MHz. Therefore, the number of input nodes of the NN architecture is 15 and the neuron numbers of the hidden layer is 31. Distribution functions of volume fraction are fitted to five-order polynomials ax+bx2+cx3+dx4+ex5 so that neuron numbers of the output layer is 5.

The inverse error is defined as

)()()( xPxPxP ife −= (5)

Figure 3 is the sine function and its inverted result. Figure 4 is the error distributions of the inversed results. It can be seen that the inversed results are accurate. From them we can see that the inversed errors of the second group testing samples are not larger than those of the first group. In all the testing samples, the maximum error point occurs on the sample 45.0x . It belongs to the first group and is below 8%.

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

]2/[ xSin πInversed polynomial

Figure 3. Group velocity dispersion curves for homogenous plates

0.2 0.4 0.6 0.8 1

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0.2 0.4 0.6 0.8 1-0.00025

0.00025

0.0005

0.00075

0.001

0.00125

0.2 0.4 0.6 0.8 1

-0.04

-0.02

0.02

0.04

0.06

0.08

0.2 0.4 0.6 0.8 1

0.005

0.01

0.015

0.02

0.025

0.2 0.4 0.6 0.8 1

0.01

0.02

0.03

0.04

0.05)(xPe

0.2 0.4 0.6 0.8 1

0.0025

0.005

0.0075

0.01

0.0125

0.015

0.0175

Figure 4. Group velocity dispersion curves for homogenous plates

5. Conclusions

An inversion method has been used to determine the material properties of FGM plates from their guided circumferential wave group velocities at low frequencies. The method combines a three layered feed-forward neural network with back propagation algorithm and a recurrent neural network. Because of the sensitivity of the group velocities to material properties and the regularity of the changes of group velocities correspond to the varying of material properties, the inversion work are carried out easily. Through numerical examples we validate that for the graded plates with the monotonous graded distribution function without inflexion point, the inversed results are satisfying whatever the testing samples belong to the training sample groups.

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Acknowledgments

The work was supported by the National Natural Science Foundation of China (No. 10802027).

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