DSpace at VNU: An approach for obtaining approximate formulas for the Rayleigh wave velocity

DSpace at VNU: An approach for obtaining approximate formulas for the Rayleigh wave velocity tài liệu, giáo án, bài giản...

Trang 1

An approach for obtaining approximate formulas

for the Rayleigh wave velocity

a

Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam

b

Institute for Geosciences, Friedrich-Schiller University Jena, Burgweg 11, 07749 Jena, Germany Received 25 July 2006; received in revised form 1 February 2007; accepted 6 February 2007

Available online 17 February 2007

Abstract

In this paper, we introduce an approach for ﬁnding analytical approximate formulas for the Rayleigh wave velocity for isotropic elastic solids and anisotropic elastic media as well The approach is based on the least-square principle To dem-onstrate its application, we applied it in order to obtain an explanation for Bergmann’s approximation, the earliest known approximation of the Rayleigh wave velocity for isotropic elastic solids, and used it to establish a new approximation By employing this approach, the best approximate polynomials of the second order of the cubic power and the quartic power

in the interval [0, 1] were found By using the best approximate polynomial of the second order of the cubic power, we derived an approximate formula for the Rayleigh wave speed in isotropic elastic solids which is slightly better than the one given recently by Rahman and Michelitsch by employing Lanczos’s approximation Also by using this second order polynomial, analytical approximate expressions for orthotropic, incompressible and compressible elastic solids were found For incompressible case, it is shown that the approximation is comparable with Rahman and Michelitsch’s approximation, while for the compressible case, it is shown that our approximate formulas are more accurate than Mozhaev’s ones Remarkably, by using the best approximate polynomials of the second order of the cubic power and the quartic power

in the interval [0, 1], we derived an approximate formula of the Rayleigh wave velocity in incompressible monoclinic mate-rials, where the explicit exact formulas of the Rayleigh wave velocity so far are not available

Keywords: Rayleigh waves; Rayleigh wave velocity; Rayleigh wave speed; Approach of least squares; The best approximation; Approximate formula; Approximate expression

1 Introduction

have been intensively studied and exploited, due to wide applications in seismology, acoustics, geophysics, materials science, nondestructive testing, telecommunication industry and so on

doi:10.1016/j.wavemoti.2007.02.001

* Corresponding author Tel.: +84 4 5532164; fax:+84 4 8588817.

E-mail addresses: pcvinh@vnu.edu.vn (P.C Vinh), p.mali@uni-jena.de (P.G Malischewsky).

www.elsevier.com/locate/wavemoti

Trang 2

For the Rayleigh wave, its velocity is a fundamental quantity which interests researchers in seismology, geo-physics, and in other ﬁelds of physics and the material sciences It is discussed in almost every survey and monograph on the subject of surface acoustic waves in solids

Due to the signiﬁcance of the Rayleigh wave velocity in practical applications, researchers have attempted

to ﬁnd its analytical approximate expressions which are of simple forms and accurate enough for practical purposes That is why a lot of approximations of the Rayleigh wave speed appeared in the literature (see, e.g [3,4,6]) However, as indicated by Mozhaev [4], most of them were reported without any indication of the derivation procedure So it is interesting to have a method which can provide the explanations for these approximations

analytical approximate expressions for the Rayleigh wave speed This approach can give the derivation of pre-viously proposed approximate formulas and establish new approximate formulas as well As a ﬁrst application

of this procedure let us present an explanation of Bergmann’s approximation, the earliest approximate expres-sion of the Rayleigh wave speed in isotropic elastic solids

In order to create new approximations we can start either from the explicit exact formulas for the Rayleigh wave speed, or from the secular equations of the Rayleigh waves For the ﬁrst possibility, as an example, by using our approach we derived an approximate formula for the form of the third order polynomial, of the Ray-leigh wave speed in isotropic elastic solids for the range [1, 0.5] of Poisson’s ratio m It is shown that this result

best approximate second order polynomial in the interval [0, 1] established by our approach, we have obtained: (i) An approximate formula of the Rayleigh wave speed in isotropic elastic solids and it is shown that this

(ii) An approximate expression of the Rayleigh wave speed in incompressible orthotropic elastic solids

(iii) Approximate formulas of the Rayleigh wave velocity in compressible orthotropic elastic media and they

(iv) Remarkably, by using the best approximate (in the sense of least squares) second order polynomials of

materials with more complicated symmetry, namely the incompressible monoclinic materials with the

are not available

It is noted, that only recently explicit exact formulas of the Rayleigh wave speed have been published for

2 Least-square approach

As mentioned, there is a need for obtaining analytical approximate expressions of the Rayleigh-wave speed for the practical work in the laboratory or elsewhere These should be more simple than the exact one and suﬃciently accurate This is, mathematically, related to the approximation problem of a given function which can be formulated as follows:

that:

best approximation of f with respect to V If V is a ﬁnite dimensional linear subspace or a compact subset of X,

550 P.C Vinh, P.G Malischewsky / Wave Motion 44 (2007) 549–562

Trang 3

The most applicable cases are the cases in which X = L2[a, b] or X = C[a, b], where L2[a, b] consists of all functions measurable in (a, b), whose squared value is integrable on [a, b] in the sense of Lebesgue, and

whose norms are deﬁned, respectively, as follows:

kuk ¼

a

and

kuk ¼ max

m2½a;bjuðmÞj; u2 C½a; b:

a

h2V

a

p where:

a

represents the deviation of the function h from the function f on the interval [a, b] or the distance between h

p

is called the average error of the approximate solution of the

chosen such that g(m) has a simple form Since polynomials are considered as the simplest functions, V is

[a, b] and

i¼1

func-tional I(h) then becomes a diﬀerentiable function of y in the closed interval [a, b], so it attains its minimum

The prime denotes here the ﬁrst derivative

3 Application of the least-square approach

3.1 Mathematical basis of Bergmann’s approximation

Trang 4

3 8x2þ 8ð3 2cÞx 16ð1 cÞ ¼ 0; ð8Þ which satisﬁes:

where:

precisely one real solution

3 p

where:

p

p started appearing in the literature long ago As we know, the earliest and

form:

whose elements have the form:

m

the form:

where:

0

ð17Þ

ðmÞ

p

Trang 5

It is easy to verify that the system(17)with coeﬃcients deﬁned by(18)has precisely one solution, namely:

So, the best approximation of x(m) with respect to V (in the sense of least squares) in the interval [0, 0.5] is:

that Bergmann’s approximation is the best approximation of x(m) (in the sense of least squares), in the interval

Remark 1 The approximation of Bergmann is very good for positive values of Poisson’s ratio, but completely fails for negative values Materials with negative values of Poisson’s ratio, so-called auxetic materials, really

ðmÞ

p

ratio, which was found by trial and error, he has found the following approximation:

An explanation for his result, which originates from the least-square approach, was given by Pham Chi Vinh

approximation of x(m) (in the sense of least squares), in the interval [1, 0.5], with respect to the class of Taylor expansions of x(m) up to the third power at the values belong to the interval [1, 0.5] In this case, the set V contains functions:

ð1ÞðyÞ

ð2ÞðyÞ

ð3ÞðyÞ

with respect to y

3.2 A new approximation for the Rayleigh wave speed

To show the eﬀectiveness of the least-square approach, in this subsection we give a new approximation of x(m) on the interval [1, 0.5] in the form of a polynomial of the third order For this purpose, naturally we choose V as the set of all polynomials of order not bigger than 3:

condition:

we obtain the following system of linear equations:

8

>

<

>

:

ð25Þ

Trang 6

a¼ 0:0439059; b¼ 0:0350168; c¼ 0:192422; d¼ 0:87384: ð26Þ Thus, the best approximation of x(m) in this case is:

It is evidence because the class of Taylor expansions of x(m) up to the third power at the values which belong

to the interval [1, 0.5] is a subset of the set of all polynomials of order not bigger than 3

Fig 1shows plots of x(m) and its approximation g3(m) deﬁned by(27)in the interval [1, 0.5] It is very dif-ﬁcult to distinguish one from the other

about (0, 0.4) which is very important for geophysical applications

In this subsection, ﬁrst we want to ﬁnd a second order polynomial which is the best approximation of the

follows:

where a, b, c are constants In order to ﬁnd constants a, b, c corresponding to the best approximation, we

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Poisson’s Ratio ν

Fig 1 Plots of x(m) (solid line) and its approximation g (m) (dotted line) in the interval [1, 0.5].

Trang 7

Iða; b; cÞ ¼a

2

From the condition:

it follows:

ð2=5Þa þ ð1=2Þb þ ð2=3Þc ¼ 1=3;

ð1=2Þa þ ð2=3Þb þ c ¼ 2=5;

ð2=3Þa þ b þ 2c ¼ 1=2:

8

>

Hence, the desired second order polynomial is:

in the interval [0, 1], in the sense of least squares For example, the best approximation second order

232

[14,17]) Now we show that the unique solution of (1*) is:

First, we observe that: among all polynomials q(z) of the nth degree whose leading coefﬁcient is unity:

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Poisson’s Ratio ν

Fig 2 Percentage errors of g 3 (m) (solid line), x m (m) (dashed line) and Rahman–Michelitsch’s approximation (dash-dot line) in the interval [1, 0.5] Percentage error = |1 g(m)/x(m)| · 100%, g(m) is an approximation of x(m).

Trang 8

qðzÞ ¼ znþ an1zn1þ þ a1zþ a0; ð36Þ

max

z2½1;1jq0ðzÞj < max

z2½1;1

leads to the conclusion: among all polynomials of the nth order whose leading coefﬁcient is unity, the shifted

3.4 An approximation for the Rayleigh wave speed in compressible isotropic elastic solids

approximate formula of the Rayleigh wave speed in compressible isotropic elastic media

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q

ðmÞ

p

in term of Poisson’s ratio:

p

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p 13ð1 mÞ

s

Fig 3shows that approximate expression(41)is slightly more accurate than that obtained recently by

By using the interval [0.474572, 0.912622], instead of the one [0, 1], analogously as above, we obtain the fol-lowing approximation:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p 23:677ð1 mÞ

s

Remark 4

(i) By using the interval [0.763932, 0.912622], instead of the one [0, 1], quite analogously, we obtain the

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p 21:94067ð1 mÞ

s

Trang 9

(ii) Independently, Li[21]has found the approximation(41), also using (33).

3.5 An approximation for the Rayleigh wave speed in incompressible orthotropic elastic solids

For incompressible orthotropic elastic solids, the secular equation of the Rayleigh waves is of the form (see,

where:

p

whose solution belonging to (0, 1) is:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q

p

we obtain:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q r

0 0.5 1 1.5 2 2.5 3 3.5

4x 10

—3

Poisson Ratio ν

Fig 3 Absolute errors of the approximation x*(solid line), x vm2 (dash-dot line) and Rahman–Michelitsch’s approximation given by (7) in

[3] (dashed line) Absolute error = |x(m) g(m)|, g(m) is an approximation of x(m).

Trang 10

where x*i() is the approximation of x() which is deﬁned by the explicit formula (38) in[10].

FromFig 4, it is shown that the approximation (46)is comparable with the one given by Rahman and

3.6 An approximation for the Rayleigh wave speed in compressible orthotropic elastic solids

For compressible orthotropic elastic solids, the secular equation of the Rayleigh waves is of the form (see,

where:

2d2

ardðrd þ 2Þ

c is the Rayleigh wave speed, q is the mass density The dimensionless material parameters are deﬁned by:

whose solution satisfying 0 < z < min{1, r} is:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q

ffiffiffiffiffiffiffiffiffiffiffi

p , where:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q

v u

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Epsilon

Fig 4 Absolute errors of the approximation x*ideﬁned by (46) (solid line) and Rahman–Michelitsch’s approximation given by (9) in [3]

(dashed line).

Trang 11

Fig 5 shows that the approximation(52)is better than Mozhaev’s ones given by (16) and (20) in[4]in the indicated range of the parameters

Now we start from another secular equation of the Rayleigh waves in compressible orthotropic elastic

where:

ffiffiffiffiffiffiffiffiffiffiffiffiffi

r

a

p

a

p

and

a

p

a

p

Case 1: 0 < t < 1(0 < r < 1):

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a

p

a

p ð1 dÞ

p

ffiffiffiffiffiffiffiffiffiffiffi

p , here:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Þ

s

Case 2: t > 1(r > 1):

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

δ

Fig 5 Plots of the approximation x1*(1, 2, d) (solid line), Mozhaev’s given by (16) (dotted line with points), (20) (dash-dot line) in [4] at

a = 1,r = 2,d 2 [0.4, 0.95] and their exact values deﬁned by (3.28) in [12] (dashed line).

Trang 12

a0u3 u2þ a2uþ 1 ¼ 0; 0 < u < 1; ð59Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q

p , in which:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1Þ

rÞ

s

3.7 An approximation for the Rayleigh wave speed in incompressible monoclinic elastic materials

these materials, the dispersion equation of the Rayleigh waves in the explicit form was found recently by

0 11

11s0

66 ðs0

16Þ2; b¼

66

11

0 16

11

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

δ

Fig 6 Plots of the approximation x2*(1, 0.5, d) (solid line), Mozhaev’s given by (16) (dotted line with points), (20) (dash-dot line) in [4] at

a = 1,r = 0.5,d 2 [0.4, 0.95] and their exact values deﬁned by (3.28) in [12] (dashed line).

Trang 13

ðb2þ 1:5b3þ 1:714Þz2þ ðb1 0:6b3 0:914Þz þ b0þ 0:05b3þ 0:086 ¼ 0: ð67Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q

a=q

p

Fig 7 shows the plots of the approximation x4* at c = 0.1, b2 [0, 0.5] and its exact values which were

acceptable

4 Conclusions

In this paper, the approach of least squares is recommended for obtaining analytical approximate expressions of the Rayleigh wave speed By employing this method we can give the mathematical basis

to the previously proposed approximations and establish new approximate formulas as well As examples,

we used it in order to explain Bergmann’s approximation, the oldest known approximation of the Rayleigh wave speed in isotropic elastic materials, and create a new approximate formula for these materials in the interval [1, 0.5], and it is shown that it is a good approximation It is noted that, starting from the expli-cit exact formulas for the Rayleigh wave speed, we can construct approximate expressions in diﬀerent forms corresponding to chosen diﬀerent sets V By this method we have found the best approximate

equations by these second order polynomials, we have obtained new approximate formulas of the Rayleigh wave speed for isotropic elastic solids, incompressible and compressible orthotropic elastic materials,

exact formulas for the Rayleigh wave speed so far are not available It is shown that all these approximate formulas are more accurate than those proposed previously, except the incompressible orthotropic case

[0, 1], in the space C[0, 1] (which Rahman and Michelitsch referred to Lanczos’ approximations) to get analogous approximations

0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97

β

γ = 0.1

Fig 7 Plots of the approximation x4*at c = 0.1, b 2 [0, 0.5] (dashed line) and its exact values (solid line).

Định dạng
Số trang	14
Dung lượng	248,5 KB