Toroidal wave in multilayered spherical curved plates

Journal of Sound and Vibration 332 (2013) 2816–2830

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Toroidal wave in multilayered spherical curved plates Jiangong Yu a,b,n, J.E. Lefebvre c,d,e, L. Elmaimouni f,g a

School of Mechanical and Power Engineering, Henan Polytechnic University, Jiaozuo 454003, PR China Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany c Univ Lille Nord de France, F-59000 Lille, France d UVHC, IEMN-DOAE, F-59313 Valenciennes Cedex 9, France e CNRS, UMR 8520, F-59650 Villeneuve d’Ascq, France f ´ Polydisciplinaire d’Ouarzazate, Univ. Ibn Zohr, 45000 Ouarzazate, Morocco ERSITA, Faculte g LMTI, Faculte´ des Sciences d’Agadir, Universite´ Ibn Zohr, BP 28/S Agadir, Morocco b

a r t i c l e in f o

abstract

Article history: Received 25 May 2012 Received in revised form 17 December 2012 Accepted 27 December 2012 Handling Editor: L. Huang Available online 31 January 2013

For the purpose of ultrasonic nondestructive inspection of layered spherical curved plates, characteristics of guided waves in multilayered spherical curved plates are studied using an improved orthogonal polynomial series method. Conventional polynomial series method has been used to solve wave motion in the multilayered flat plate for about four decades. However, it can deal with the layered structure only when the material properties of two adjacent layers do not change significantly. It also can not obtain correct and continuous normal stress profiles. Numerical comparisons verify that the improved polynomial method overcomes these drawbacks. Then, the influences of radius to thickness ratio, volume fraction and stacking sequence on dispersion curves and stress distributions are discussed. It is found that the radius to thickness ratio, the wave speed of component material and the position of the component material determine the distribution of mechanical energy at higher frequencies. & 2013 Elsevier Ltd. All rights reserved.

1. Introduction The spherical curved plates are widely used in the fields of pressure vessels and spherical domes of power plants. For the purpose of guided ultrasonic nondestructive testing and evaluation (NDT&E), characteristics of guided waves in spherical curved plates must be understood. As early as in 1969, Shah et al. [1] used shell-theory to analyze elastic waves in three-dimensional hollow spheres. Gaunaurd and Werby [2,3] derived dispersion curves for fluid loaded spherical shells. Kargl and Marston [4] also worked on the Lamb-like wave in isotropic spherical shells. Wang et al. [5] studied the dynamic responses in orthotropic laminated spherical shells subjected to arbitrary radial dynamic load by means of finite Hankel transforms and Laplace transforms. Towfighi and Kundu [6] and Yu et al. [7] studied wave propagation in anisotropic spherical curved plates. Using conventional orthogonal polynomial method, Yu et al. [8] studied the guided waves in functionally graded spherical curved plates. To meet various working environments, various laminated structures are developed. For the purposes of design, optimization and NDT&E, appropriate theoretical models and efficient solution methods are desired to investigate the behaviors of wave propagation in layered structures. There have been many methods to solve this problem, such as the

n

Corresponding author at: School of Mechanical and Power Engineering, Henan Polytechnic University, Jiaozuo 454003, PR China. E-mail address: [email protected] (J. Yu).

0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2012.12.032

J. Yu et al. / Journal of Sound and Vibration 332 (2013) 2816–2830

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finite element method [9,10], the transfer matrix method [11,12], the global matrix method [13], the reverberation-ray matrix method [14], the layer element method [15,16], the scattering-matrix method [17], the orthogonal polynomial series method [18,19], and so on. Furthermore, the wave propagating loss of laminated structures also received attentions. Ghineta and Atalla [20] developed the statistical energy analysis technique to study the the wave transmission loss of curved laminates and sandwich composite panels. Manconi and Mace [21] studied the wave loss of viscoelastic laminated panels using a wave finite element method. The orthogonal polynomial series approach has been proposed to solve wave problem for more than forty years. It has two specificities. Firstly, it directly incorporates the boundary conditions into the equations of motion by the device of assuming position-dependent material physical constants [22]. Secondly, the motion equations are converted into a matrix eigenvalue problem thanks to an expansion of the independent variables in an appropriate series of orthonormal polynomials: this makes it possible the semi-variational determination of the frequencies of modes and associated profiles. Moreover, because of the orthogonality of the polynomial series, there exist plenty of zeros in the eigen-matrix, which makes the solving easy. So far, the orthogonal polynomial method has been used to solve wave propagation in many kind of structures, from Laguerre polynomial for half infinite media [23,24] to Legendre polynomial for finite thickness structures [25,26], from homogeneous structures [27] to multilayered structures [18,28] and to functionally graded structures [29,30], from pure elastic structures [31] to various multi-field coupled structures [32,33], from flat plate [19] to various curved structures [34–36]. It can be said that orthogonal polynomial series method is very versatile in calculating guided waves in various structures. However, it has not been used for the multilayered curved structures. Importantly, it can deal with the layered plates only when the material properties of two adjacent layers do not change significantly and can not obtain correct and continuous normal stress profiles. This follows from the fact that the conventional orthogonal polynomial method uses a single polynomial expansion which is continuous in level and in slope over the entire structure even at the frontier between two adjacent layers resulting in continuous mechanical displacement distributions in level and in slope and therefore discontinuous normal stress distributions because of different elastic constants of two adjacent layers. For such a real structure, it is well known that the true or physical mechanical displacement is continuous at the interface between two adjacent layers, but its derivatives are not. These discontinuous derivatives with different elastic constants on each side of the interface allow the normal stress components to be continuous. This paper proposes an improved orthogonal polynomial series method to investigate the guided wave propagation in multilayered spherical curved plates. The improved method overcomes the limitations of conventional polynomial method. The validity of the improved orthogonal polynomial method is verified through numerical comparisons. The influences of the radius to thickness ratio, volume fraction and stacking sequence on dispersion curves and stress distributions are discussed. 2. Mathematics and formulation of the problem Consider a traction free N-layered spherical plate with a total thickness h, as shown in Fig. 1. In the spherical coordinate system (r,y,j), a and b are the inner and outer radii. The radius to thickness ratio is defined as Z ¼ b/h. We assume the crystal axes are coincident with the coordinate axes. For the wave propagation considered in this paper, the body forces are assumed to be zero. Thus, the dynamic equation for the layered spherical plate is governed by @T rr 1 @T ry 1 @T rj 2T rr þT ry cot yT yy T jj @2 ur þ ¼ rðr Þ 2 þ þ r @y rsin y @j @r r @t   @T ry 1 @T yy 1 @T yj 3T ry þ cot y T yy T jj @2 u ¼ rðr Þ 2y þ þ þ r @y rsin y @j r @r @t @T rj 1 @T yj @ 2 uj 1 @T jj 3T rj þ 2T yj cot y þ ¼ rðr Þ 2 þ þ r @y rsin y @j @r r @t

(1)

where Tij and ui are the stress and elastic displacements, respectively; r is the density of the material. The relationship between the general strain and general displacement can be expressed as @ur 1 @uy ur 1 @uj ur cot y , eyy ¼ þ uy , þ , ejj ¼ þ r @y rsin y @j @r r r r     1 1 @uy @uj 1 1 @ur 1 @uj , eyj ¼ þ uj cot y , erj ¼ uj þ 2r sin y @j 2r sin y @j 2 @r @y   1 1 @ur @uy uy  ery ¼ þ 2 r @y @r r

err ¼

where eij are the strains.

(2)

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J. Yu et al. / Journal of Sound and Vibration 332 (2013) 2816–2830

N

,

N bd

c ac 2

ca

cb d,

1

ca

2



cb d,

1



ϕ

a

0

h1 h2

hN b

θ

r

Fig. 1. Schematic diagram of a multilayered spherical plate showing the coordinate system.

The boundary and continuity conditions for a traction-free multilayered structure require that: (1) the mechanical displacement and the normal component of stress should be continuous at the interfaces; (2) the normal component of the stress should be zero at the inner and outer surfaces. By introducing the rectangular window function pa,hN ðr Þ ( 1, a r r r hN pa,hN ðr Þ ¼ , 0, elsewhere the stress-free boundary conditions (Trr ¼Try ¼Trj ¼ 0 at r ¼h0 ¼a constitutive relations of the plate (Datta and Hunsinger [23]): 9 2 8 C 11 ðr Þ C 12 ðr Þ C 13 ðr Þ C 14 ðr Þ T rr > > > > > > > 6 > > C 23 ðr Þ C 24 ðr Þ C 22 ðr Þ T yy > > > 6 > > > 6 > > = 6 < T jj > C 33 ðr Þ C 34 ðr Þ 6 ¼6 T yj > C 44 ðr Þ > 6 > > > > 6 > > > 6 > > T rj > symmetry > > 4 > > > ; :T >

and r ¼hN ¼ b) are automatically incorporated in the C 15 ðr Þ C 25 ðr Þ C 35 ðr Þ C 45 ðr Þ C 55 ðr Þ

ry

9 38 C 16 ðr Þ > > err pa,hN ðr Þ > > > > 7> > > eyy C 26 ðr Þ 7> > > > > > 7> > > = < 7 e C 36 ðr Þ 7 jj 7 2 e C 46 ðr Þ 7> > yj > > > 7> > > > 2erjpa,h ðrÞ > > C 56 ðr Þ 7 > > 5> N > > > > : 2ery pa,hN ðr Þ ; C 66 ðr Þ

(3)

where Cij are the elastic coefficients. The elastic coefficients of the multilayered spherical plate are expressed as C ij ðr Þ ¼

N X

C nij phn1 ,hn ðr Þ,

(4a)

n¼1

where N is the number of the layers and C nij are the elastic constants of the Nth material. Similarly, the mass density can be expressed as

rðr Þ ¼

N X

rn phn1 ,hn ðrÞ:

(4b)

n¼1

According to Kargl and Marston [4] and Towfighi and Kundu [6], the wave front on the surface of a spherical shell is assumed to be toroidal. In addition, to study wave propagation in a spherical plate segment from point A to point B, the two points A and B can always be aligned along the equator of a sphere by adjusting the positions of the north and south poles. Therefore, to study the wave propagation between two points in a spherical plate segment, it is sufficient to solve

J. Yu et al. / Journal of Sound and Vibration 332 (2013) 2816–2830

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the governing equations for y ¼ p/2 only. Thus the propagating wave is independent of y. Then the displacement components of this toroidal wave can be written as     (5a) ur r, y, j,t ¼ exp ikbjiot U ðr Þ     uy r, y, j,t ¼ exp ikbjiot V ðr Þ

(5b)

    uj r, y, j,t ¼ exp ikbjiot W ðr Þ

(5c)

U(r), V(r) and W(r) represent the amplitude of vibration in the r, y, j directions respectively. k is the magnitude of the wave vector in the propagation direction, and o is the angular frequency. Substituting Eqs. (2), (3), (4), (5) into Eq. (1), the governing differential equations in terms of displacement components can be obtained. When the wave propagating direction is coincident with crystal axis of the orthotropic material, the governing differential equations are as follows: ! N N N X X  2 00 X n n n 0 r U þ 2rU C 11 phn1 ,hn ðr Þ þ C 12 phn1 ,hn ðr Þ þ C 13 phn1 ,hn ðr Þ U n¼1 N X



C n22 phn1 ,hn ðr Þ þ

n¼1

n¼1

N X

C n33 phn1 ,hn ðr Þ þ 2

n¼1 2

þr U

0

N X

C n11

n¼1

!

N X n¼1

!0

phn1 ,hn ðrÞ þ rU

n¼1

N X

C n12

N X

2

C n23 phn1 ,hn ðr Þ U þ ðkbÞ U

C n55 phn1 ,hn ðr Þ

n¼1

!0

N X

phn1 ,hn ðrÞ þ rU

n¼1

C n13

!0

phn1 ,hn ðrÞ

n¼1

! N N N X X  X n n n 0 þ ikb rW W C 55 phn1 ,hn ðr Þikb C 22 phn1 ,hn ðr Þ þ C 33 phn1 ,hn ðr Þ W n¼1

n¼1

N  X C n13 phn1 ,hn ðr Þ þikbrW þ ikb rW 0 þ W n¼1



r 2 V 00 þ 2rV 0

N X

N X

n¼1

!0

C n13 phn1 ,hn ðr Þ

¼ o2 r 2 U

n¼1

" C n66 phn1 ,hn ðr ÞV 2

n¼1

N X

þ r 2 V 0 þ rV

N  X

N X

C n23 phn1 ,hn ðr Þ þ ðkbÞ

n¼1

!0

C n66

¼ o2 r 2 V

phn1 ,hn ðrÞ

rn phn1 ,hn ðrÞ

(6a)

n¼1

C n66 phn1 ,hn ðr Þ þ

n¼1



N X

n¼1

2

N X

# C n44 phn1 ,hn ðr Þ

n¼1 N X

rn phn1 ,hn ðrÞ

(6b)

n¼1

! N N N X X  2 00 X 2 C n55 phn1 ,hn ðr Þ 2 C n55 phn1 ,hn ðr Þ þ ðkbÞ C n33 phn1 ,hn ðr Þ W r W þ 2rW 0 n¼1

n¼1

 þ r 2 W 0 rW þ ikbrU þW

N X

" C n44 phn1 ,hn ðr Þ þikbU

n¼1

N  X

!0

C n55 phn1 ,hn ðr Þ

n¼1 N  X þ ikb rU 0 þ 2U C n55 phn1 ,hn ðr Þ

n¼1 N X

C n23 phn1 ,hn ðr Þ þ

n¼1

N X

#

n¼1

C n33 phn1 ,hn ðr Þ ¼ o2 r 2 W

n¼1

N X

rn phn1 ,hn ðrÞ,

(6c)

n¼1

where the superscript ()0 is the partial derivative with respect to r. In order to save space, the general case is not given here. In the following numerical examples, we assume if not specified otherwise that the wave propagating direction is coincident with crystal axis. To solve the coupled wave equation, the conventional orthogonal polynomial method expand the U(r), V(r), W(r) into three Legendre orthogonal polynomial series [21] U ðr Þ ¼

1 X m¼0

 where pim i ¼ 1,

2,

p1m Q m ðr Þ, V ðr Þ ¼

1 X

p2m Q m ðr Þ, W ðr Þ ¼

m¼0

 3 are the expansion coefficients and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2m þ1 2rhN a Pm Q m ðr Þ ¼ hN a hN a

1 X

p3m Q m ðr Þ,

(7)

m¼0

(8)

with Pm being the mth Legendre polynomial. Theoretically, m runs from 0 to N. In practice, the summation over the polynomials in Eq. (7) can be halted at some finite value m¼M, when higher order terms become essentially negligible. However, as is mentioned above, the conventional orthogonal polynomial method can only solve the multilayered structure when the material properties of two adjacent layers do not change significantly. Here, we improved the

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J. Yu et al. / Journal of Sound and Vibration 332 (2013) 2816–2830

orthogonal polynomial method so as to make it suitable for the general multilayered spherical plates. We expand field quantities of each layer into one Legendre polynomial:

 for the first layer Q 1m ðr Þ ¼

 for the second layer: Q 2m ðr Þ ¼

y  for the Nth layer: QN m ðr Þ ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2m þ 1 2rh1 a Pm h1 a h1 a

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2mþ 1 2 h2 þ h1 r Pm h2 h1 h2 h1 h2 h1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2m þ 1 2 hN þ hN1 r Pm hN hN1 hN hN1 hN hN1

(9a)

(9b)

(9c)

Moreover, in order to automatically incorporate into the calculation the interface continuity conditions relative to the components of the mechanical displacement, ua(a ¼1,2,3) (ur,uy,uj) are expanded as follows:

 In the first layer: u1a ¼

1 X

  pam,1 Q 1m ðr Þexp ikbj

(10a)

m¼0

with: 1 ¼ u1a ðr ¼ h1 Þ ¼ u1,h a

1 X

  pam,1 Q 1m ðr ¼ h1 Þexp ikbj

m¼0

 In the 2nd layer: 1 þ ðrh1 Þ u2a ¼ u1,h a

1 X

  pam,2 Q 2m ðr Þexp ikbj

(10b)

m¼0

with: 2 1 ¼ u1,h þ ðh2 h1 Þ u2a ðr ¼ h2 Þ ¼ u2,h a a

1 X

  pam,2 Q 2m ðr ¼ h2 Þexp ikbj

m¼0

 In the third layer: 2 u3a ¼ u2,h þ ðrh2 Þ a

1 X

  pam,3 Q 3m ðr Þexp ikbj

(10c)

m¼0

with: 3 2 ¼ u2,h þ ðh3 h2 Þ u3a ðr ¼ h3 Þ ¼ u3,h a a

1 X

  pam,3 Q 3m ðr ¼ h3Þexp ikbj

m¼0

and so on. n Substituting Eqs. (9) and (10) into Eq. (6) then multiplying by Q 1j n ðr Þ, Q 2j n ðr Þ      Q N j ðr Þ, with j running from 0 to M, respectively, integrating over r from a to hN and taking advantage of the orthonormality of the Legendre polynomials, gives the following system:

n

n j,m 2 n j,m 3 1 2 n j 1 Aj,m 11 pm,n þ A12 pm,n þ A13 pm,n ¼ o U Mm pm,n

(11a)

n

n j,m 2 n j,m 3 1 2 n j 2 Aj,m 21 pm,n þ A22 pm,n þ A23 pm,n ¼ o U Mm pm,n

(11b)

J. Yu et al. / Journal of Sound and Vibration 332 (2013) 2816–2830 n n

j,m

n

2821

n j,m 2 n j,m 3 1 2 n j 3 Aj,m 31 pm,n þ A32 pm,n þ A33 pm,n ¼ o U Mm pm,n

(11c)

Mjm

are the elements of a non-symmetric matrix. They can be obtained according to Eq. (6). where Aab ða, b ¼ 1,2,3Þ and Eq. (11) can be written as 2 38 9 2n j 38 1 9 n j,m n j,m A A12 n Aj,m > p1m,n > > Mm 0 0 13 > > > pm,n > = < 6 11 7> 6 7< 2 = 6 n Aj,m n Aj,m n Aj,m 7 p2 n j 26 7 pm,n : M 0 0 ¼ o 4 (12) 6 21 m,n m 22 22 7 5 > > 4 5> 3 > > n j ; ; :p > : p3 > n j,m n j,m > M 0 0 A31 n Aj,m A m,n m,n m 32 33 2 So, Eq. (12) yields  a form of the  eigenvalue problem. The eigenvalue o gives the angular frequency of the guided wave; i 1, 2 , 3 eigenvectors pm,n i ¼ allows the components of the particle displacement to be calculated. According to Vph¼ o/k, the phase velocity and group velocity can be obtained. The complex matrix Eq. (12) can be solved numerically making use of standard computer programs for the diagonalization of non-symmetric square matrices. 3N(Mþ1) eigenmodes are generated from the order M of the expansion and the number of layers N. Acceptable solutions are those eigenmodes for which convergence is obtained as M is increased. We determine that the eigenvalues obtained are converged solutions when a further increase in the matrix dimension does not result in a significant change in the eigenvalue.

3. Numerical results Based on the foregoing formulations, two computer programs in terms of both the conventional polynomial method and the improved polynomial method have been written using Mathematica to calculate the dispersion curves and stress profiles for the layered spherical plates. 3.1. Validation of the method by comparison with the exact solution from global matrix method Firstly, for validation, we calculated a two-layered metal structure. As is well known, the dispersion curves for a flat plate are the same than those for a curved plate with a very large ratio of radius to thickness. So, we use the polynomial method to calculate a two-layered metal spherical plate with Z ¼1000 to make a comparison with exact solutions of the corresponding flat plate, as shown in Fig. 2. The spherical plate is composed of steel (inner layer with 1mm thickness) and brass (outer layer with 1mm thickness). Their material constants are shown in Table 1. In Fig. 2, solid lines are the exact solutions of Lamb-like wave dispersion curves for the flat plate from the global matrix method [13], and dotted lines are the results from the conventional polynomial approach for the spherical plate. The solution of the improved polynomial approach is the same as the dotted lines. In order to save space, it is not shown here. As can be seen, for the two-layered metal spherical structure, both the conventional and improved polynomial approaches can obtain correct dispersion curves. Next, we show an example of an equal-thickness (1 m) three-layered sandwich structure with very dissimilar materials. The material constants are listed in Table 2. Fig. 3 shows the exact solution dispersion curve for the sandwich flat plate from the global matrix method [13] using solid lines, the solutions of the sandwich spherical plate with Z ¼1000 from the improved polynomial approach using dotted lines, and the solutions from the conventional polynomial approach using dashed lines. It can be seen that solid lines and dotted lines agree very well. Dashed lines exhibit serious differences from solid lines and dotted lines. This illuminates the validity of the improved polynomial approach.

12.0

vph (km/s)

10.0 8.0 6.0 4.0 2.0 0.0 0.0

1.0 2.0 Frequency (MHz)

3.0

Fig. 2. Phase velocity dispersion curves for the steel–brass two-layered structures; solid lines: the results of the flat plate from the global matrix method, dotted lines: the results of the spherical plate with Z ¼ 100 from the conventional polynomial method.

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J. Yu et al. / Journal of Sound and Vibration 332 (2013) 2816–2830

Table 1 The material properties of the bilayer structure [27]. C11

C13

C33

C55

r

282 162.6

113 81.3

282 162.6

84 40.7

7.932 8.4

Property Steel Brass

Units: Cij(109N/m2), r(103kg/m3).

Table 2 The material properties of the sandwich structure. Property

C11

C22

C33

C12

C13

C23

C44

C55

C66

r

Middle layer Top/bottom layer

281 28.1

349 34.9

294 29.4

126 18.9

84 12.6

88 13.2

108 12.96

132 15.84

131 15.72

3.59 1.795

Units: Cij(109N/m2), r(103kg/m3).

Vph (km/s)

15.0

10.0

5.0

0.0 0.0

1.0

2.0 3.0 Frequency (kHz)

4.0

Fig. 3. Phase velocity dispersion curves for the three-layered sandwich structures with very dissimilar materials: solid lines, the results of the flat plate from the global matrix method; dotted lines, the results of the spherical plate with Z ¼ 1000 from the improved polynomial method; dashed lines, the results of the spherical plate with Z ¼1000 from the conventional polynomial method.

The results of above comparisons are from a flat plat. Here, a numerical comparison for a spherical curved plate with a little ratio (Z ¼5) is made to show that the polynomial method can treat curved structures with a sufficient degree of accuracy. This example is from Towfighi and Kundu [6] by Fourier series expansion method. The thickness of the spherical plate is 1 mm. Its outer radius is 5 mm. The mass density is r ¼1580 kg/m3, and the constitutive matrix is [6] 9 2 8 38 e 9 T rr > 14:95 7:115 7:115 0:215 0 0 > > rr > > > > > > > > > > > > 6 7:115 45:9675 32:5075 28:3125 > > > T yy > eyy > > > > > 0 0 7 > > > > 6 7 > 6 > > > > > > > 7 = < T jj = 6 7:115 32:5075 45:9675 28:3125 < e 7 0 0 jj 7 ¼6 : 6 7 T 2 e > > > 0:215 28:3125 28:3125 32:3375 0 0 7> yj > > > > 6 > > yj > > > > > > 6 7 > > > > > T rj > 2erj > 4 0 0 0 0 5:27 1:46 5> > > > > > > > > > > > ; ; :T > : 2 e 0 0 0 0 1:46 5:27 ry ry Stiffness values are given in GPa. Corresponding phase velocity dispersion curves are presented in Fig. 4. As is apparent, our results agree well with those obtained by Towfighi and Kundu [6]. Fig. 5 gives the cases of other radius to thickness ratios, Z ¼100, Z ¼30 and Z ¼10. In Fig. 5(a), the ring frequency of the nodes 2 and 3 is marked. Above it, the wave characteristics of spherical plate are quite different from those of the flat plate and below it they are rather similar. In fact, the ring frequencies of every mode are different. And they increase sharply with the decrease of the radius to thickness ratio, as shown in Fig. 5(b) and (c). Thus, in Fig. 3, with Z ¼1000, the ring frequency is almost zero, and in Fig. 5(a) it is greater but still very low. 3.2. The convergency of the improved method This section makes an analysis of the convergence of the improved polynomial method. A sandwich spherical plate which is composed of silicon nitride (N) and steel (S) is taken as the numerical example to be calculated for M¼ 3, 4, 5 and 6.

J. Yu et al. / Journal of Sound and Vibration 332 (2013) 2816–2830

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Vph(Km/s)

12

8

4

0

1

2

3 f (MHz)

f (MHz)

4

5

Fig. 4. Dispersion curves for a spherical curved plate with Z ¼5; (a) the results of Towfighi and Kundu [6], (b) the authors’ results.

The material properties of silicon nitride are listed in Table 3. Fig. 6 is its Lamb-like wave dispersion curves. The ratio of outer radius to thickness is Z ¼10. The stacking sequence and thicknesses are S/N/S-1/2/1 (m). It can be seen that for M¼3 the first four modes are convergent (for the first four modes, the four kind of lines overlap); for M¼4 the first seven modes are convergent; and for M¼6 the first ten modes are convergent. So, we can conclude that at least the first M modes are convergent. As for the calculation time issue, when we calculate 500 wavenumbers, the consumed CPU time is 56 s for M¼3, 85 s for M¼4, 137 s for M ¼5. When we calculate 100 wavenumbers, the calculating time is 130 s for M ¼6. The used computer is a Dell laptop. Its CPU is i7-2360QM. Its frequency is 2 GHz. The memory is not an issue. Mathematica is a singlethreaded calculating software.

3.3. The case of the wave propagation direction being not coincident with crystal axis This section shows an example for which the angle between the wave propagation direction and the crystal axis is not zero. The example still concerns the equal-thickness sandwich spherical plate with very dissimilar materials, as for the example of Fig. 3, but with a different Z. In this example, Z ¼100. The angle is taken as 451, as shown in Fig. 7. Then, through the coordinate transformation, the material properties in the wave propagation direction can be obtained, as shown in Eq. (13), of which Eq. (13a) represents the top and bottom layers and Eq. (13b) represents the middle layer. Fig. 8 shows the phase velocity dispersion curves, which are quite different from those in Fig. 5(a) 9 2 8 38 e 9 T rr > 40:92 9:48 12:9 0 0 1:7 > > rr > > > > > > > > > > > > > 6 7 > > > > T e > > > 9:48 40:92 12:9 0 0 1:7 yy > yy > > > 6 7> > > > > > > > > 6 7 < T jj = 6 12:9 12:9 29:4 0 0 3 7< ejj = 6 7 ¼6  109 (13a) T yj > 2eyf > > 0 0 14:4 1:44 0 7 > > > > 6 0 7> > > > > > 6 > > > 7> > > > > T rj > 2erj > 4 0 0 0 1:44 14:4 0 5> > > > > > > > > > > > > : T ry ; 1:7 1:7 3 0 1:46 6:3 : 2ery ; 9 8 T rr > > > > > > > > > T yy > > > > > > > > = < T jj >

2

351:5

6 89:5 6 6 6 86 ¼6 6 0 T yj > > > > 6 > > > > 6 > > > > T rj > 4 0 > > > > > :T ; 17 ry

89:5

86

0

0

351:5 86

86 29:4

0 0

0 0

0

0

120

12

0

0

12

120

17

2

0

1:46

38 e 9 17 > > rr > > > > > > > > > eyy > 17 7 > 7> > > > 7> < 2 7 ejj = 7  109 : > 2eyj > 0 7 > 7> > > > 7> > > 2erj > 0 5> > > > > > > ; : 2 e 6:3

(13b)

ry

3.4. Dispersion curves for the multilayered spherical curved plates This section takes the common ceramics–metal layered structures as examples to show the influences of volume fraction and stacking sequence on the dispersion curves. Three sandwich spherical plates that are composed of silicon nitride (N) and steel (S) with Z ¼10 are studied. The stacking sequence and thicknesses are S/N/S-1/2/1 (m), S/N/S-1/1/1 (m) and S/S/N-1/1/2 (m). The obtained dispersion curves are shown in Fig. 9. Obviously, the influences of the volume fraction and stacking sequence on the dispersion curves are both significant.

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15.0

Vph (km/s)

10.0

5.0

0.0 0.0 1.0 2.0 Ring frequency Frequency (kHz)

3.0

4.0

1.0

2.0 3.0 Frequency (kHz)

4.0

1.0

2.0 3.0 Frequency (kHz)

Vph (km/s)

15.0

10.0

5.0

0.0 0.0

Vph(km/s)

15.0

10.0

5.0

0.0 0.0

4.0

Fig. 5. Phase velocity dispersion curves for the three-layered sandwich structures with very dissimilar materials: solid lines, the results of the flat plate from the global matrix method; dotted lines, the results of the spherical plate with (a) Z ¼100, (b) Z ¼30 and (c) Z ¼ 10.

3.5. Stress and displacement shapes Above all, we illustrate the validation of the improved approach and its superiority with respect to the conventional one through the stress shapes of the S/N/S-1/2/1 layered spherical plate with Z ¼2 and kh ¼4.4, as shown in Fig. 10, of which Fig. 10(a) is from the conventional polynomial method and Fig. 10(b) is from the improved polynomial method. For a layered plate, the boundary conditions explicit the continuity of the force acting on the common surface between the two layers (common means here belonging to both layers). If one wrote that physical condition, one obtained the continuity of the normal stresses (Trr and Trj). The non normal stresses (such as Tjj) are stresses acting on surfaces which are not common to both layers (they are common only by a line: only a line is belonging to both layers) and therefore there is no condition of continuity.

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Table 3 The material properties of silicon nitride[27]. Property Silicon nitride 9

2

3

C11

C13

C33

C55

r

380

120

380

130

2.37

3

Units: Cij(10 N/m ), r(10 kg/m ).

Fig. 6. Analysis of the convergence through dispersion curves for the S/N/S-1/2/1 layered spherical plates (a) modes 1–6, (b) modes 7–12.





Wave propagation direction

Crystal element

45°



r

Fig. 7. Schematic diagram of the angle between the wave propagation direction and the crystal axis is 451.

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14

Vph (Km/s)

12 10 8 6 4 2 0

1

2 f (KHz)

4

3

Fig. 8. Phase velocity dispersion curves for the three-layered sandwich spherical plate with very dissimilar materials when the angle between the crystal axis and the wave propagation direction is 451, Z ¼100.

Vph (Km/s)

12

8

4

0

2

4 6 fh (KHz-m)

8

10

0

2

6 4 fh (KHz-m)

8

10

0

2

6 4 fh (KHz-m)

8

Vph (Km/s)

12

8

4

Vph (Km/s)

12

8

4

10

Fig. 9. Phase velocity dispersion curves for the layered spherical plates with Z ¼ 10; (a) S/N/S-1/2/1, (b) S/N/S-1/1/1, (c) S/S/N-1/1/2.

1

Mode 1

Stress (arbitrary unit)

Stress (arbitrary unit)

J. Yu et al. / Journal of Sound and Vibration 332 (2013) 2816–2830

Tr

0.5

Trr 0 T

-0.5 1

1.2

1.4

1.6

1 Tr 0 Trr

-1 T

Mode 2

-2 1.8

2

1

1.2

Radial coordinate (m)

Stress (arbitrary unit)

Trr

-2

1.2

1.4 1.6 1.8 Radial coordinate (m)

Stress (arbitrary unit)

Stress (arbitrary unit)

Mode 3

T

Tr Trr

T

-1

2

1 Tr 0 Trr -1 T

Mode 2

-2

1.4 1.6 1.8 Radial coordinate (m) Stress (arbitrary unit)

1.2

2

-1

Mode 1

1

1.8

0

1

0

1.6

Tr

-3

1

1.4

Radial coordinate (m)

1

2

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2

1

1.2

1.4 1.6 1.8 Radial coordinate (m)

2

Tr 0 -1 Trr -2 Mode 3

T 1

1.2

1.4 1.6 1.8 Radial coordinate (m)

2

Fig. 10. Stress distributions for the S/N/S-1/2/1 layered spherical plate with Z ¼ 2 and kh ¼ 4.4; (a) the results from the conventional polynomial method, (b) the results from the improved polynomial method.

It can be seen that even for the layered spherical plate without very dissimilar properties, the conventional polynomial approach can not give correct result. The obtained normal stresses Trr and Trj are discontinuous at the interfaces. On the other hand, the improved polynomial approach overcomes this drawback. Then, the displacement shapes at a large wavenumber (kh¼40.4) are discussed through the S/N/S-1/2/1 and S/S/N-1/1/2 layered spherical plate with Z ¼2 and with Z ¼10, as shown in Figs. 11 and 12. We can see that for the S/N/S-1/2/1 spherical plate, when Z is large, the displacement usually distribute either on the outer layer or the inner layer (the material of both layers is steel). But when Z is small, the displacement only distribute on the outer layer although the materials of inner and

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0.2

1.5

1

0

1

0.5 0

0.5

-0.2

-0.5

0

-0.4

36

37

38

39

40

36

37

38

39

40

1.5 1

0.4

0.5

0.2

0

0

-0.5 5

6 Mode 1

7

8

37

38

39

40

4

5

6 Mode 3

7

8

0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4

-0.2 4

36

4

5

6 Mode 2

7

8

Fig. 11. Displacement distributions for the S/N/S-1/2/1 layered spherical plate at kh¼ 40.4; dashed line: ur, solid line: uf; abscissa: radial coordinate (unit: m), ordinate: mechanical displacement (arbitrary units); (a) with Z ¼10, (b) with Z ¼2.

0.3 0.2 0.1 0 -0.1 -0.2 -0.3

0.2 0 -0.2 -0.4 -0.6 -0.8 -1

1.5 1 0.5 0 36

37

38

39

40

36

37

38

39

40

0.4

0.6

0

0.2

-0.2

0

-0.4 4

5

6 Mode 1

7

8

37

38

39

40

4

5

6 Mode 3

7

8

0.2 0.1 0 -0.1 -0.2 -0.3

0.2

0.4

36

4

5

6 Mode 2

7

8

Fig. 12. Displacement distributions for the S/S/N-1/1/2 layered spherical plate at kh¼ 40.4; dashed line: ur, solid line: uf; abscissa: radial coordinate (unit: m), ordinate: mechanical displacement (arbitrary units); (a) with Z ¼10, (b) with Z ¼2.

outer layers are still steel. For the S/S/N-1/1/2 spherical plate, when Z is large, the displacement usually distribute on the inner and middle layers (the material of both layers is still steel). But when Z is small, the displacement mainly distribute on the middle layer (the material of the outer layer is steel and that one of the inner layer is nitride). Making a comprehensive understanding among these figures, we can find two phenomena: (1) for a layered spherical plate with a large ratio Z, the displacement usually distribute on the steel layer, whatever the position of the steel. The reason lies in that the wave speed of steel is lower than that of N. The high frequency guided waves usually propagate in the layer with lower wave speed. (2) For a layered spherical plate with a small ratio Z, besides the relative wave speed of the layer materials, the stacking sequence is also important for the displacement distributions: the distributions tend to move towards the outer layer. 4. Conclusions Considering the intrinsic limitations of the conventional orthogonal polynomial method, this paper proposes an improved orthogonal polynomial method to make it suitable to solve the layered spherical curved plate whatever

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the dissimilarities of the layer material properties. Through the numerical analysis, we can draw the following conclusions: (a) Numerical comparisons on dispersion curves and stress profiles verified that the improved method is more competent in solving the guided wave propagation in multilayered curved structures. (b) For a layered spherical plate with a large ratio Z, the high frequency guided wave usually propagate on the layer with lower wave speed; for a layered spherical plate with a small ratio Z, besides the factor of wave speed, the position of the layer is also important.

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