Transient elastodynamic response of finite and infinite solid cylinders

Applied Acoustics 73 (2012) 798–802

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Transient elastodynamic response of finite and infinite solid cylinders S.X. Liu, L.G. Tang ⇑, X.M. Xu Department of Marine Technology & Engineering, Xiamen University, Xiamen 361005, China Key Laboratory of Underwater Acoustic Communication and Marine Information Technology, Ministry of Education, Xiamen University, Xiamen 361005, China

a r t i c l e

i n f o

Article history: Received 9 October 2011 Accepted 15 February 2012 Available online 18 March 2012 Keywords: Elastodynamic response Solid cylinder Eigenfunction expansion

a b s t r a c t The orthogonal eigensolutions for the vibrations of an isotropic finite solid cylinder with a traction-free lateral boundary and rigid-smooth end boundaries are provided. The transient elastodynamic response of this solid cylinder is then constructed using the method of eigenfunction expansion and further extended explicitly and concisely to that of an isotropic infinite solid cylinder. The numerically evaluated analytical solution is shown to compare favorably with that by finite element method (FEM). The effect of external forces on the excitation of each guided wave mode can be quantitatively investigated on the basis of the present solution. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Guided elastic-wave-inspection technique for solid and hollow cylinders has received a plenty of attention in recent years because of its high efficiency and low cost. The free vibration of an isotropic infinite solid cylinder has been treated pioneerly by Pochhammer and Chree independently. Thus, the solid cylinder problem is sometimes known as the Pochhammer–Chree problem [1]. The detailed review of three dimensional dynamic analysis of the circular cylinders was presented by Soldatos [2]. The guided wave modes propagating axially in the solid cylinder can be categorized as torsional waves T(0, m), longitudinal waves L(0, m) and flexural waves F(n, m)(n, m = 1,2,3, . . .). There are also guided waves propagating circumferentially. Generally, some single pure mode is excited for defects detection because the more wave modes are excited, the more complex are the waves reflected from the defects. It is important to study the dynamic response of the solid cylinder in order to control the excitation of guided waves and employ them to carry out nondestructive evaluation (NDE). Pan et al. [3] have solved the three dimensional transient response problem of the solid cylinder by the method of integral transform and employed the laser ultrasonic technique to evaluate their theoretical results. Ditri and Rose [4] obtained the transient response solution of a hollow cylinder by using the complex reciprocity relation of elastodynamics and normal modes expansion technique. Ebenezer et al. [5] extended the method used by Hutchinson [6], Hutchinson and El-Azhari [7] to ⇑ Corresponding author at: Department of Marine Technology & Engineering, Xiamen University, Xiamen 361005, China. Tel.: +86 13696992450; fax: +86 592 2186397. E-mail address: [email protected] (L.G. Tang). 0003-682X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2012.02.017

study the free vibrations of the solid and hollow cylinders to determine the vibration response of the solid cylinder to arbitrary distributions of certain function on the boundaries. In this study, the eigenfunction expansion method is employed to study the transient elastodynamic response of the solid cylinder. Pao pointed out that this method is one of the most elegant but least applied methods dealing with elastodynamic problems [8]. The formulae obtained by this method are particularly suitable for analyses of the effects of body and surface forces on the transient elastodynamic response and efficient in their representation of guided elastic modes [9]. That the method is least applied is due to the difficulty of finding the eigenfunctions. Liu and Qu [10] have employed the method of eigenfunction expansion to analyze the transient wave propagation in a circular annulus subjected to transient excitation on its outer surface. In addition, Cheng and his colleagues investigated the 2D or 3D transient laser-generated guided wave propagation in orthotropic plates [11], two-layered plates [12], and pipes [13] using the method. Tang and Xu [14] investigated the transient torsional wave propagation in pipes using the method. In this paper, the orthogonality of the eigenfunctions, which correspond to the vibrations of the finite solid cylinder with traction-free lateral boundary and rigid-smooth end boundaries, is shown, and as a result the transient elastodynamic response of the cylinder is obtained by employing the method of eigenfunction expansion [1,15]. Then the transient elastodynamic response solution of the isotropic infinite solid cylinder is derived from that of the finite one, and the form of this solution is not only explicit but also concise. Furthermore, the analytical solution is evaluated numerically. The numerical results are compared with those obtained with the finite element method (FEM) and a fairly good agreement is found. Note that the effect of external forces on the

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excitation of each guided wave mode can be quantitatively investigated on the basis of the present solution.

Rznmk ðr; hÞ ¼ nnmk Anmk Z n ðcnmk rÞH1 ðnhÞ !

x2nmk

þd 2. Eigenvibration of an isotropic finite solid cylinder with rigidsmooth end boundaries

nnmk ¼ The free vibration of an isotropic homogeneous finite solid cylinder, as shown in Fig. 1, is governed by 2

LðuÞ  ðk þ lÞ 5 ð5  uÞ þ l52 u ¼ q

@ u @t2

in X;

ð1Þ

where k and l are the Lame constants, u(r, t) is the displacement field vector, and X is the cylinder body. The operator L is defined as L  (k + l)55  + l 52. Traction-free boundary condition is employed for the lateral surface, that is,

rrr jr¼a ¼ 0; rrh jr¼a ¼ 0 and rrz jr¼a ¼ 0;

ð2Þ

where a is the radius of the solid cylinder, as shown in Fig. 1. Eq. (2) is always used to obtain the dispersion equations of guided waves. If the length of the solid cylinder is 2l and rigid-smooth boundary conditions are employed for the end surfaces, we have

rzr jz¼l ¼ 0 and rzh jz¼l ¼ 0:

uz jz¼l ¼ 0;

L½unmk  ¼ qx

in X;

B1 ½unmk  ¼ 0 on R1 ;

ð5Þ ð6Þ

where {unmk} are the eigenfunctions and xnmk are the corresponding real and nonnegative eigenvalues [1,9], n denotes the circumferential order while m and k denote the orders of discrete frequency x and wave number n, respectively. Eqs. (5) and (6) show that the lateral boundary R1 is traction-free and the end boundaries R2 are rigid-smooth, respectively. The expressions of the three components of unmk are

urnmk ðr; h; zÞ

¼

uhnmk ðr; h; zÞ ¼ uznmk ðr; h; zÞ ¼

Rrnmk ðr; hÞ cosðnnmk zÞ; Rhnmk ðr; hÞ cosðnnmk zÞ; Rznmk ðr; hÞ sinðnnmk zÞ;

ð9Þ

where

n Rrnmk ðr; hÞ ¼ cnmk Anmk Z 0n ðcnmk rÞH1 ðnhÞ þ Bnmk Z n ðjnmk rÞH02 ðnhÞ r þ djnmk nnmk C nmk Z 0n ðjnmk rÞH3 ðnhÞ; ð10Þ Rhnmk ðr; hÞ ¼

n Anmk Z n ðcnmk rÞH01 ðnhÞ  jnmk Bnmk Z 0n ðjnmk rÞH2 ðnhÞ r d þ nnnmk C nmk Z n ðjnmk rÞH03 ðnhÞ; ð11Þ r

ð13Þ

ð14Þ

uhnmk ðr; h; zÞ ¼ Rhnmk ðr; hÞ sinðnnmk zÞ;

ð15Þ

Rznmk ðr; hÞ cosðnnmk zÞ;

ð16Þ

uznmk ðr; h; zÞ

¼

where

n Rrnmk ðr; hÞ ¼ cnmk Anmk Z 0n ðcnmk rÞH1 ðnhÞ þ Bnmk Z n ðjnmk rÞH02 ðnhÞ r ð17Þ  djnmk nnmk C nmk Z 0n ðjnmk rÞH3 ðnhÞ; Rhnmk ðr; hÞ ¼

n Anmk Z n ðcnmk rÞH01 ðnhÞ  jnmk Bnmk Z 0n ðjnmk rÞH2 ðnhÞ r d ð18Þ  nnnmk C nmk Z n ðjnmk rÞH03 ðnhÞ; r

Rznmk ðr; hÞ ¼ nnmk Anmk Z n ðcnmk rÞH1 ðnhÞ !

x2nmk

þd

nnmk ¼

C 2T

2n þ 1 p; 2l

 n2nmk C nmk Z n ðjnmk rÞH3 ðnhÞ;

n ¼ 0; 1; 2;    :

ð19Þ

ð20Þ

The functions and variables used in the above equations are defined as

Hi ðnhÞ ¼ Ein cosðnhÞ þ F in sinðnhÞ; 2 nmk

c

2 nmk ;

¼ k1 a

a2nmk ¼

ð7Þ ð8Þ

n ¼ 0; 1; 2;    ;

ð12Þ

urnmk ðr; h; zÞ ¼ Rrnmk ðr; hÞ sinðnnmk zÞ;

ð4Þ

B2 ½unmk  ¼ 0 on R2 ;

 n2nmk C nmk Z n ðjnmk rÞH3 ðnhÞ;

or

ð3Þ

The eigenvalue problem of the linear differential operator L is defined as: 2 nmk unmk

n p; l

C 2T

x2nmk

( k1 ¼

C 2L

2 nmk

j

 n2nmk ;

¼

b2nmk ¼

ð21Þ ð22Þ

x2nmk C 2T

a2nmk > 0 ; k2 ¼ 1; a2nmk < 0

1;

i ¼ 1; 2; 3;

k2 b2nmk ;

(

 n2nmk ;

1;

b2nmk > 0

1; b2nmk < 0

ð23Þ

;

ð24Þ

and Anmk,Bnmk,Cnmk,Ein and Fin in Eqs. (10)–(12), (17)–(19) and (21) are arbitrary constants. CL and CT in Eqs. 12,19,23 are irrotational and equivoluminal wave velocities, respectively. Zn in Eqs. (10)– (12) and (17)–(19) is the Bessel functions Jn if ki = 1 (i = 1 or 2). And it is the modified Bessel functions In if ki =  1 (i = 1 or 2). The parameter d in Eqs. (10)–(12) and (17)–(19) is introduced so that these terms are dimensionally uniform. If both two end boundaries of the solid cylinder are rigid or traction-free, the eigenfunctions of the longitudinal and flexural vibrations cannot be obtained. It is well known that the operator L is self-adjoint under the boundary conditions defined by Eqs. (2) and (3). Therefore, the eigenmodes {unmk} form an orthogonal set, that is,

Z

qunmk  upqj dV ¼

X



0;

n – p or m – q or k – j

M nmk ; n ¼ p and m ¼ q and k ¼ j ð25Þ

where Mnmk is the norm of unmk and

Mnmk ¼ Fig. 1. A solid cylinder with length 2l and radius a.

Z

l

l

Z 0

a

Z 2p 0

qunmk ðr; h; zÞ  unmk ðr; h; zÞrdrdhdz:

ð26Þ

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3. Transient elastodynamic response of finite and infinite solid cylinders The response of the isotropic homogeneous finite-length solid cylinder to an externally applied force density f(r, t) on the body X is governed by elastodynamic equation

LðuÞ  ðk þ lÞ 5 ð5  uÞ þ l52 u ¼ q

@2u  qfðr; tÞ in X: @t2

ð27Þ

We consider the solution to it which satisfies the following conditions: (a) the boundary condition at the lateral surface R1,

rðnÞ ða; h; z; tÞ ¼ fk 5 u þ l½5u þ ð5uÞT g  n ¼ sðh; z; tÞ;

ui ðzÞ ¼ pi ðzÞ þ qi ðzÞ; ð28Þ

(b) the boundary conditions at the end surfaces R2,

ð29Þ

_ 0Þ ¼ v 0 ðrÞ in X: uðr;

ð30Þ

According to the method of eigenfunction expansion, we know that the solution of Eq. (27) that satisfies Eqs. (28)–(30) is

uðr; h; z; tÞ ¼

X

1

,nmk cos xnmk t þ

nmk

 xnmk

Z

xnmk

t

nmk sin xnmk t 

Unmk ðsÞ sin xnmk ðt  sÞds unmk

ð31Þ

0

where

nmk

1

Unmk ðtÞ ¼

ð32Þ

þ

Z

ð34Þ

Mnnmk ¼

If u0 = 0, v0 = 0, and two end boundaries of the solid cylinder are rigid-smooth, i.e., n(r, h, t) = 0, 1(r, h, t) = 0,g(r, h, t) = 0, we have

nmk

þ

Z

Z t Z 0

qunmk  fðr; tÞdV

X

  unmk  sðh; z; tÞds sin xnmk ðt  sÞds unmk :

g fi ðr; h; tÞzfi ðzÞei

qunmk  fðr; tÞdV

X

  unmk  sðh; z; tÞds sin xnmk ðt  sÞds unmk Mnnmk ;

Z

Z 2p  

ð36Þ





q Rrnmk 2 þ Rhnmk

0

0

2

  2 þ Rznmk rdrdh;

ð42Þ

and

p l

ð43Þ

:

As the solid cylinder is of infinite length, namely, l ? 1, interval between two successive roots Mnnmk approaches zero. Therefore, in the limit of l approaching infinity, the summation over the index k in Eq. (41) will be replaced by the integral over the continuous eigen-value nnm, then we have

uðr; h; z; tÞ ¼

According to the analyses in Sections 2, we know there are two groups of eigenfunctions in different forms, i.e., Eqs. (7)–(9) and (14)–(16). The group to be employed is determined by the body and surface forces applied to the solid cylinder. If the densities of body and surface forces are 3 X

a

ð35Þ

R1

fðr; tÞ ¼

0

where

R2

1 Mnmk xnmk

xnmk

ð41Þ

M nmk x X R1 Z h i  z r 1ðr; h; tÞrðnÞ ðunmk Þ  unmk nðr; h; tÞ  uhnmk gðr; h; tÞ ds : þ

uðr; h; z; tÞ ¼

p R1

 Z Z  qunmk  fðr; tÞdV  unmk  sðh; z; tÞds

X

Z t Z

1 M1 nmk

ð33Þ

M1 nmk ¼

2 nmk

ð40Þ

X nmk

1 qu0  unmk dV; M nmk X Z 1 ¼ qv 0  unmk dV; M nmk X

ð39Þ

Apparently, pi(z) is an even function while qi(z) is an odd one. We can get the transient responses corresponding to pi(z) and qi(z) first, then the total response corresponding to ui(z). The procedure of seeking the transient elastodynamic response of the finite solid cylinder can be summarized as following. Firstly, a complex loads is decomposed into simple radial, circumferential and axial components which are even or odd with respect to z. Secondly, the response corresponding to each component can be obtained by using Eq. (35). Lastly, the total transient response combines all components according to the superposition principle. It is easy to derive from Eq. (35) that

uðr; h; z; tÞ ¼

Z

,nmk ¼

ui ðzÞ þ ui ðzÞ

; 2 u ðzÞ  ui ðzÞ : qi ðzÞ ¼ i 2

(c) the initial conditions,

uðr; 0Þ ¼ u0 ðrÞ;

ð38Þ

where

pi ðzÞ ¼

rzr ðr; h; l; tÞ ¼ nðr; h; tÞ; rzh ðr; h; l; tÞ ¼ gðr; h; tÞ; uz ðr; h; l; tÞ ¼ fðr; h; tÞ;

respectively, where e1, e2 and e3 are the unit vectors along the radial, circumferential and axial coordinate directions, respectively. The transient response corresponding to each component in Eqs. (36) and (37) should be computed first before obtaining the total response of the finite hollow cylinder. Let ui(z) represents zfi ðzÞ and zsi ðzÞði ¼ 1; 2; 3Þ. Eqs. (7)–(9) are employed if u1(z) or u2(z) is an even function, and Eqs. (14)–(16) are employed if u1(z) or u2(z) is an odd one. Eqs. (7)–(9) are employed if u3(z) is an odd function, and Eqs. (14)–(16) are employed if u3(z) is an even one. These conclusions are based on the symmetric or anti-symmetric property of transient displacements when the external forces are symmetric or anti-symmetric with respect to z = 0. If ui(z) is neither even nor odd, it can be expressed as

XZ 0

nm

þ

1

 Z t Z

1 M1 nm

p

xnm

0

qunm  fðr; tÞdV

X

  unm  sðh; z; tÞds sin xnm ðt  sÞds unm dnnm :

Z R1

ð44Þ Eq. (44) is the transient elastodynamic response of the isotropic homogeneous infinite solid cylinder.

i¼1

4. Examples and numerical results

and

sðh; z; tÞ ¼

3 X i¼1

g si ðh; tÞzsi ðzÞei

ð37Þ

In this section, we will compute the analytic transient response of an isotropic infinite solid cylinder and compare it with that obtained by using the finite element method (FEM). The geometri-

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Fig. 2. A infinite solid cylinder loaded by the surface force s (h, z, t), as shown by (46).

Fig. 5. Normalized radial displacement components.

Fig. 3. Group velocity curves for a steel solid cylinder with radius a = 0.01 m.

Fig. 6. Normalized circumferential displacement components.

The parameters in Eqs. (46)–(48) are h0 = p/6, z0 = 0.003 m, f0 = 80 kHz and Ts = 8/f0. Fig. 2 shows the infinite solid cylinder loaded by the surface force mentioned above. Substituting Eqs. (45)–(47) into Eq. (44), we obtain

uðr; h; z; tÞ ¼

XZ nm

Fig. 4. The finite element mesh of the solid cylinder employed in the FEM simulation.

Z

e2 ¼

fðr; tÞ ¼ 0;

TðtÞ ¼

sðh; z; tÞ ¼



hðtÞe1 ;

t 2 ½0; T S ; h 2 ½h0 ; h0 ;

0;

t > TS

z 2 ½z0 ; z0 

; ð46Þ

where h(t) is defined as



 2pt : hðtÞ ¼ sinð2pf0 tÞ 0:5  0:5 cos TS

ð47Þ

0

(

h0

cosðnhÞdh ¼

h0

cal and material parameters of the solid cylinder used in all simulations are radius a = 0.01 m, density q = 7.8  103 kg/m3, Young’s module E = 215.04 Gpa and Possion coefficient c = 0.28. We presume that no body force is applied to the solid cylinder, that is,

and the density of the surface force is

e1 e2 TðtÞ

Rznm ða; 0Þ

pM 1 nm xnm

unm dnnm ;

ð48Þ

where

e1 ¼

ð45Þ

1

Z

z0

cosðnnm zÞdz ¼ z0

2 sin nh0 n

2h0 ;

; n ¼ 1; 2; 3;    n¼0

2 sinðnnm z0 Þ; nnm

ð49Þ

ð50Þ

(Rt

hðsÞ sin½xnm ðt  sÞds; t < T S 0 : R TS hðsÞ sin½xnm ðt  sÞds; t P T S 0

ð51Þ

The waveform of h(t) consists of eight cycles, 80 kHz tonebursts modulated by a Hanning window. Then we can judge from the group velocity curves, as shown in Fig. 3, that only modes L(0, 1) and F(1, 1) are generated. Thus, it is enough to consider n = 0,1 and m = 1 when we compute the transient waveforms from Eq. (48). Fig. 4 shows the finite element mesh of the solid cylinder. The solid cylinder can be reduced as the model shown in Fig. 4 because the cylinder and the surface force applied to it are

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S.X. Liu et al. / Applied Acoustics 73 (2012) 798–802

solid cylinder with mixed end boundary conditions defined by Eq. (29). The reason of employing the mixed end boundary conditions is that we cannot obtain the eigenfuntions unmk when both two ends of the solid cylinder are rigid or traction-free. Though it may be impossible to apply the mixed boundary conditions and they are not of great practical importance, we do not care which kind of boundary condition is applied on the two ends when the length of the solid cylinder l ? 1. The transient elastodynamic response solution of the infinite solid cylinder derived here is not only explicit but also concise, and provides a guide for controlling the distribution and direction of the external forces so as to generate specified guided wave modes. Acknowledgments

Fig. 7. Normalized axial displacement components.

symmetric to the planes y = 0 and z = 0. Eight-noded three-dimensional solid elements are employed in the FEM simulation. The element size along z axis is 0.001 m. And the lateral surface is circumferentially divided into thirty uniform parts. The time-step is 0.05 ls in the explicit algorithm. Figs. 5–7 are the normalized transient radial, circumferential and axial displacement components of the point located on the lateral surface of the infinite solid cylinder at h = 300 and z = 0.8 m, respectively. And the solid lines are computed from analytical solutions while the small circles are those simulated with FEM. We can find that the transient waveforms obtained with above two different methods agree well. 5. Conclusions The eigenfunctions corresponding to the vibrations of the finite solid cylinder with a traction-free lateral boundary and rigidsmooth end boundaries are shown to be orthogonal and with which the transient elastodynamic response solution of it is obtained. The effects of external forces on the excited guided waves need to be studied in order to employ the guided waves technique for nondestructive evaluation of the solid cylinders. This is the motivation of the study here. Note that the goal of this study is to derive the explicit expression of the elastodynamic response of the infinite solid cylinder by the eigenfunction expansion method. We can realize this goal based on the response solution of the finite

The work was supported by the National Natural Science Foundation of China (Grant No. 10704064), and the Open Research Fund of State Key Laboratory of Acoustics, Chisnese Academy of Sciences (Grant No. SKL0A201104). We would like to thank K.Y. Fung (Hongkong Polytechnic University, China) for his kindly help in writing this paper. References [1] Eringen AC, Suhubi ES. Elastodynamics, vol.2. New York: Academic Press; 1975. [2] Soldatos KP. Review of three dimensional dynamic analyses of circular cylinders and cylindrical shells. Appl Mech Rev 1994;47:501–16. [3] Pan YD, Rossignol C, Audoin B. Acoustic waves generated by a laser point source in an isotropic cylinder. J Acoust Soc Am 2004;116:814–20. [4] Ditri JJ, Rose JL. Excitation of guided elastic wave modes in hollow cylinders by applied surface tractions. J Appl Phys 1992;72:2589–97. [5] Ebenezer DD, Ravichandran K, Padmanabhan C. Forced vibrations of solid elastic cylinders. J Sound Vib 2005;282:991–1007. [6] Hutchinson JR. Vibrations of solid cylinders. J Appl Mech 1980;47:901–7. [7] Hutchinson JR, EI-Azhari SA. Vibrations of free hollow circular cylinders. J Appl Mech 1996;53:641–6. [8] Pao YH. Elastic waves in solids. J Appl Mech 1983;50:1152–64. [9] Weaver RL, Pao YH. Axisymmetric elastic waves excited by a point source in a plate. J Appl Mech 1982;49:821–36. [10] Liu GL, Qu JM. Transient wave propagation in a circular annulus subjected to transient excitation on its outer surface. J Acoust Soc Am 1998;104:1210–20. [11] Cheng JC, Zhang SY. Quantitative theory for Laser-generated lamb waves in orthotropic thin plates. Appl Phys Lett 1999;74:2087–9. [12] Du J, Cheng JC. Modelling of laser-generated guided waves in bonded plates with a weak interface by the two-layer normal mode expansion method. Chin Phys Lett 2003;20:76–9. [13] Tang LG, Cheng JC. Numerical analysis on laser-generated guided elastic waves in a hollow cylinder. J Nondestruct Eval 2002;21:45–53. [14] Tang LG, Xu XM. Transient torsional vibration responses of finite, semi-infinite and infinite hollow cyliders. J Sound Vib 2010;329:1089–100. [15] Reismann H. On the forced motion of elastic solids. Appl Sci Res 1967;18:156–65.