Torsional wave dispersion in a finitely pre-strained hollow sandwich circular cylinder

Journal of Sound and Vibration 330 (2011) 4519–4537

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Torsional wave dispersion in a finitely pre-strained hollow sandwich circular cylinder S.D. Akbarov a,b,n, T. Kepceler a, M. Mert Egilmez a a b

Yildiz Technical University, Faculty of Mechanical Engineering, Department of Mechanical Engineering, Yildiz Campus, 34349 Besiktas, Istanbul, Turkey Institute of Mathematics and Mechanics of the National Academy of Sciences of Azerbaijan, 37041 Baku, Azerbaijan

a r t i c l e i n f o

abstract

Article history: Received 25 November 2010 Received in revised form 4 April 2011 Accepted 6 April 2011 Handling Editor: H. Ouyang Available online 4 May 2011

This paper studies torsional wave dispersion in a three-layered (sandwich) hollow cylinder with finite initial strains. The investigations are carried out within the scope of the piecewise homogeneous body model with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies. The mechanical relations of the materials of the cylinders are described through their harmonic potential. The analytical expression is obtained for the low wavenumber limit values of the torsional wave propagation velocity. The numerical results on the influence of the initial stretching or compression of the cylinders along the torsional wave propagation direction are presented and discussed. & 2011 Elsevier Ltd. All rights reserved.

1. Introduction The problem of wave propagation in pre-strained (pre-stressed) piecewise homogeneous elastic solid bodies is of interest in a number of physical and mechanical areas, such as geophysics, electrical devices, earthquake engineering, composite materials, ultrasonic non-destructive stress analysis of solids and others. Accordingly, a large number of investigations have been made in this field. Some of these are now considered briefly starting with a paper [1] in which the Rayleigh surface waves in a pre-stressed half-space were studied. It was assumed that a wave propagates in one of the principal directions and it was established that Rayleigh surface waves in a pre-stressed half-space are not dispersive, just as in the classical linear theory of elastodynamics. Torsional wave propagation in an initially stretched circular solid cylinder was studied in a paper [2]. In the monograph [3], some problems related to the dynamics of elastic media under initial stress, including the influence of gravity on Rayleigh waves, the fundamental properties of wave propagation under initial stress and many others, were analyzed. An attempt to investigate axisymmetric wave propagation in an initially twisted circular cylinder was done in a study [4]. Longitudinal wave propagation in initially stretched circular homogeneous cylinders was a subject of many investigations [5–7]. The interfacial Stoneley waves in pre-strained bodies were studied in papers [8–10]. The effect of pre-stress on the propagation and reflection of plane waves in a compressible elastic half-space was examined in paper [11]. Paper [12] studied the propagation of Lamb waves in a pre-strained sandwich plate from incompressible materials. In paper [13], an efficient method for calculating the wave speed of surface waves in pre-stressed elastic half-spaces was proposed. Papers [14,15] investigated the influence of initial stresses on wave propagation (dispersion) in composite structures from piezoelectric

n Corresponding author at: Yildiz Technical University, Faculty of Mechanical Engineering, Department of Mechanical Engineering, Yildiz Campus, 34349 Besiktas, Istanbul, Turkey. Tel.: þ 90 212 383 45 96; fax: þ 90 212 383 45 90. E-mail addresses: [email protected] (S.D. Akbarov), [email protected] (T. Kepceler).

0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.04.009

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materials. Analyses of the investigations relating to wave propagation in a pre-stressed viscoelastic medium are given in paper [16]. The influence of multiple kinds of initial stresses, which appeared as a result of an initial twisting of a beam with noncircular cross-section on the natural vibration, was investigated in paper [17]. Note that the general and exact theory of nonlinear elastic waves and of elastic waves in initially stressed elastic bodies are detailed in works [18–20]. The theory related to the dynamics of initially stressed bodies is called the Threedimensional Theory of Elastic Waves in Initially Stressed Bodies (TLTEWISB). The relations and equations of the TLTEWISB were obtained from the exact relations and equations of the nonlinear theory of elastodynamics by linearization with respect to the small dynamical perturbations. Investigations carried out by utilizing TLTEWISB related to wave propagation in a pre-stressed bi-material compound cylinder were considered. These investigations started with the work [21], in which longitudinal wave propagation in the pre-stressed bi-material compound cylinder was studied and it was assumed that the initial stretching was small. Note that a more detailed review of the work [21] and other related works are given in the paper [22]. Paper [23] extended the work reported in [21] where the initial strains were finite and the elasticity relations of the materials were assumed to be compressible and were given by the harmonic-type potential. With the same assumptions, the influence of the finite initial strains on the axisymmetric wave dispersion in a circular cylinder, embedded in a compressible elastic medium, was studied in [24]. In paper [25] the problem considered in [24] was investigated for the case where the materials of the components of the system were incompressible and the stress–strain relations for them were given through the Treloar potential. Numerical results, regarding the influence of the initial strains in the cylinder and embedded body on the wave dispersion were presented and discussed. Paper [26] investigated the dispersion relations of axisymmetric wave propagation in initially twisted bi-material compound cylinders. It was assumed that the constituents of the compound cylinder were isotropic and homogeneous and, in particular, it was established that as a result of the existence of the initial twisting, at least in one constituent of the considered compound cylinder, the axisymmetric longitudinal and torsional waves cannot propagate separately, i.e. the new type axisymmetric waves, which differ from the axisymmetric torsional and longitudinal waves, must appear. In papers [27–29] the axisymmetric torsional wave propagation in the initially uni-axially stretched bi-material compound cylinder was investigated. The elasticity relations for the components of the compound cylinder were obtained from the Murnaghan potential. At the same time, in these works it was assumed that the initial strains were small and initial strain–stress state was determined within the scope of the classical linear theory of elasticity. It should be noted that in all the foregoing investigations relating to the pre-stressed bi-layered compound cylinder it was assumed that complete contact conditions were satisfied with respect to the contact surface between the inner and outer cylinders. In paper [30] the problems considered in the papers [27–29] were examined for the case where the specified contact conditions were imperfect and the numerical results on the effects of this imperfection on the influence of the initial stresses on the wave propagation velocity are presented and discussed. Recently, in a paper [31], the torsional wave propagation in a pre-stretched multilayered solid cylinder was studied within the scope of the assumptions used in papers [27–29]. This completes the review of related investigations, from which it follows that the investigations related to wave propagation in pre-stressed bi-material compound cylinders were made within the scope of the following assumptions: (i) a bi-material compound cylinder is a bi-layered cylinder (except the paper [31]) and (ii) in the investigations related to the propagation of the torsional wave propagation it was assumed that the initial strains were small. According to assumption (i), the results [21,23–31] cannot be used for the many-layered hollow cylinders, for instance, threelayered (sandwich) hollow cylinders which have many applications in various key branches of modern industry. Moreover, according to assumption (ii), the results of the investigations [27–31] which relate to torsional wave propagation can be employed only for the compounded cylinders made from stiff materials. But these results are not suitable for the compounded cylinders made from high elastic materials such as elastomers, various types of polymers, etc. Taking the foregoing statements into account, the investigations carried out in the papers [27–29,31] for the torsional wave dispersion are extended in the present paper for three-layered (sandwich) hollow cylinders made from high elastic materials, that is for the case where the initial strains in the components of the sandwich hollow cylinder are finite and the magnitude of these are not restricted. In this case, as in [21,23–31], it is assumed that in each component of the sandwich hollow cylinder there exists only the homogenous normal stress acting on the areas which are perpendicular to the lying direction of the cylinders. The mechanical relations of the materials of the cylinders are described through the harmonic potential.

2. Formulation of the problem We consider the sandwich hollow circular cylinder shown in Fig. 1 and assume that in the natural state the radius of the internal circle of the inner hollow cylinder is R and the thickness of the inner, middle and outer cylinders are h(1), h(2) and h(3), respectively. In the natural state we determine the position of the points of the cylinders by the Lagrangian coordinates in the cylindrical system of coordinates Oryz. The values related to the inner, middle and external hollow cylinders will be denoted by the upper indices (1)–(3), respectively. Furthermore, we denote the values related to the initial state by an additional upper index 0.

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Fig. 1. The geometry of the three-layered hollow cylinder.

It is assumed that the cylinders have infinite length in the direction of the Oz axis and the initial strain–stress state in each component of the considered body is axisymmetric with respect to this axis and is also homogeneous. Moreover, it is assumed that the mentioned initial strain–stress states in the inner, middle and external hollow cylinders are determined through the following displacement fields: ðkÞ

uðkÞ,0 ¼ ðl1 1Þr, r urðkÞ,0

ðkÞ

uðkÞ,0 ¼ 0, y

uzðkÞ,0 ¼ ðl3 1Þz,

ðkÞ lðkÞ k ¼ 1,2,3 1 al3 ,

(1)

lðkÞ 1

(uðkÞ,0 ) z

ðkÞ (l3 )

is the displacement along the radial direction (along the direction of the Oz axis) and are the where elongation parameters. Such an initial stress field may be present with stretching or compression of the considered body along the Oz axis. The stretching or compression may be conducted for the inner, middle and the external hollow cylinders separately before they are compounded. However, the results, which will be discussed below, can also be related to the case where the inner, middle and external hollow cylinders are stretched or compressed together after the compounding. In this case, as a result of the difference of the radial and circumferential deformations of the inner, middle and external cylinders’ materials (similar to the deformations in the classical linear theory of elasticity which are determined through Poisson’s coefficient), inhomogeneous initial stresses acting on the areas which are parallel to the Oz axis may arise. Nevertheless, according to the well known mechanical consideration, the aforementioned inhomogeneous initial stresses can be neglected under corresponding investigations, in the cases where these stresses are less significant than those acting on the areas which are perpendicular to the Oz axis. Otherwise, it is necessary to take the mentioned inhomogeneous stresses into account, which may be a subject of other investigations. Consequently, in the present investigation we assume that only the normal stress (denoted by sðkÞ,0 zz ) acting on the areas which are perpendicular to the Oz axis, is different from zero. For the initial state of the cylinders, we associate the Lagrangian cylindrical system of coordinates O0 r0 y0 z0 and introduce the following notation: ðkÞ

r 0 ¼ l1 r,

ðkÞ

z0 ¼ l3 z,

ð1Þ

R0 ¼ l1 R,

k ¼ 2 for Rþ hð1Þ o r r Rþ hð1Þ þ hð2Þ ,

k ¼ 1 for R rr r Rþ hð1Þ ,

k ¼ 3 for R þ hð1Þ þ hð2Þ o r rR þ hð1Þ þ hð2Þ þ hð3Þ

(2)

The values related to the system of coordinates associated with the initial state below, i.e. with O0 r0 y0 z0 , will be denoted by an upper prime. Within this framework, let us investigate the axisymmetric torsional wave propagation along the O0 z0 axis in the considered body. We do this investigation by using the coordinates r0 and z0 in the framework of the TLTEWISB. We will follow the style and notation used in the monograph [20] and in papers [23,24]. Thus, we write the basic relations of the TLTEWISB for the case considered. These relations are satisfied within each constituent of the considered body because we use the piecewise homogeneous body model. The equations of motion are 2 @ 0ðkÞ @ 1 0ðkÞ 0ðkÞ @ Qr0 y þ 0 Qz0ðkÞ þ 0 ðQr0ðkÞ u0ðkÞ 0 y þ Qyr 0 Þ ¼ r y 0 @r @z r @t 2 y

(3)

The elasticity relations are 0ðkÞ Qz0ðkÞ 0 y0 ¼ o1331

@u0ðkÞ y , @z0

0ðkÞ Qr0ðkÞ 0 y ¼ o1221

Qy0ðkÞ ¼ o0ðkÞ 2121 r0

@u0ðkÞ u0ðkÞ y y o0ðkÞ , 1212 0 @r r0

@u0ðkÞ u0ðkÞ y y o0ðkÞ : 2112 0 @r r0

(4)

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0ðkÞ In (3) and (4) through Qy0ðkÞ , Qr0ðkÞ 0 y , and Qz0 y , perturbation of the components of the non-symmetric Kirchhoff stress tensor r0 0ðkÞ are denoted. The notation u0ðkÞ y shows the perturbations of the components of the displacement vector. The constants o2121 , 0ðkÞ 0ðkÞ 0ðkÞ o0ðkÞ 1221 , o2112 , o1212 and o1331 in (4) are determined through the mechanical constants of the kth cylinder’s materials and through the initial stress state. r0 (k) is the density of the kth material.

For an explanation of the foregoing equations and relations, according to Refs. [18–20], let us consider briefly some basic relationships of the large (finite) elastic deformation theory, and their linearization, which are used in the present investigation. 2.1. Some related relations of the nonlinear theory of elasticity for hyper-elastic bodies Consider the definition of the stress and strain tensors in the large elastic deformation theory. For this purpose we use the Lagrange coordinates r, y and z in the cylindrical system of coordinates Oryz. In this case, the physical components of Green’s strain tensor e~ in the Oryz coordinate system are determined by the physical components ur, uy and uz of the displacement vector u through the relation (A1) given in Appendix A. Consider the determination of the Kirchhoff stress tensor. The use of various types of stress tensors in the large (finite) elastic deformation theory is connected with the reference of the components of these tensors to the unit area of the relevant surface elements in the deformed or un-deformed state. This is because, in contrast to the linear theory of elasticity, in the finite elastic deformation theory, the difference between the areas of the surface elements taken before and after deformation must be accounted for in the derivation of the equation of motion and under satisfaction of the boundary conditions. According to the aim of the present investigation, we here consider two types of stress tensors denoted by q~ and s~ the components of which refer to the unit area of the relevant surface elements in the un-deformed state, but which act on the surface elements in the deformed state. The physical components s(ij) of the stress tensor s~ are determined through the strain energy potential U ¼ U(err,eyy,y,eyz) by the use of the following expression:   1 @ @ þ U, (5) sð ijÞ ¼ 2 @eðijÞ @eðijÞ where (ij) ¼rr, yy, zz, ry, rz, zy. The physical components of the stress tensor q~ are determined through the physical components of the stress tensor s~ and the displacement vector u by the expression (A2) given in Appendix A. The stress tensor q~ with components determined by expression (A2) is called the Kirchhoff stress tensor, but the stress tensor s~ is called the Lagrange stress tensor. According to the expressions (6) and (A2), the stress tensor s~ is symmetric, but the stress tensor q~ is non-symmetric. In this case the equation of motion is written as follows: qqrr @qyr @qzr 1 @ 2 ur @qry @qyy 1 @q @2 u þ þ ðqrr qyy Þ ¼ r 2 , þ þ þ ðqry þ qyr Þ þ zy ¼ r 2y , r r @r r@y @z @r r@y @z @t @t @qrz @qyz 1 @qzz @ 2 uz þ ¼r 2 : þ qrz þ r @r r@y @z @t

(6)

For determination of the stress–strain relations it is necessary to give the explicit expression for the strain energy potential U in expression (5). In the present paper, we will use the following expression for the potential U which was proposed in a paper [32] and was called the harmonic potential:

U ¼ 12le21 þ me2 ,

(7)

where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e1 ¼ 1 þ 2e1 þ 1 þ 2e2 þ 1 þ2e3 3, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 e2 ¼ 1 þ 2e1 1 þ 1 þ2e2 1 þ 1þ 2e3 1 :

(8)

In relations (7) and (8), l and m are material constants and ei (i¼1,2,3) are the principal values of Green’s strain tensor. Thus, with this we restrict ourselves to consideration of the definition of the stress and strain tensor, the determination of the relations between them, and the equation of motion in the finite elastic deformation theory. 2.2. Determination of the initial strains and stresses Substituting the expression (1) into the relation (A1) and supplying them with the corresponding upper indices we obtain the following initial strains: ðkÞ 2 ðkÞ 2 ðkÞ,0 ðkÞ,0 ðkÞ,0 ðkÞ,0 1 1 eðkÞ,0 ¼ eðkÞ,0 rr yy ¼ 2 ððl1 Þ 1Þ, ezz ¼ 2ððl3 Þ 1Þ, er y ¼ erz ¼ eyz ¼ 0:

(9)

, eðkÞ,0 and eðkÞ,0 coincide with It follows from (9) that in the initial state, the principal values of Green’s strain tensor eðkÞ,0 1 2 3 ðkÞ,0 ðkÞ,0 eðkÞ,0 , e and e , respectively. Consequently, substituting the expression (9) into the relations (7) and (8) we obtain the rr zz yy

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following expression for the strain energy potential in the initial state: ðkÞ ðkÞ ðkÞ 2 2 2 ðkÞ UðkÞ,0 ¼ 12lðkÞ ð2lðkÞ 1 þ l3 3Þ þ m ð2ðl1 1Þ þ ðl3 1Þ Þ:

(10)

According to the expression (9), the following relations can be written: @ @eðkÞ,0 rr

¼

@

¼

@eðkÞ,0 yy

1

@

@

,

¼

ðkÞ @eðkÞ,0 lðkÞ zz 1 @l1

1

@

ðkÞ lðkÞ 3 @l3

(11)

:

Using (10) and (11) we obtain the following expressions for the stresses in the initial state: ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

¼ ½l ð2l1 þ l3 3Þ þ2mðkÞ ðl3 1Þðl3 Þ1 , sðkÞ,0 zz ðkÞ

ðkÞ

ðkÞ,0 ðkÞ,0 srðkÞ,0 y ¼ srz ¼ szy ¼ 0,

ðkÞ

ðkÞ

ðkÞ

1 ðkÞ sðkÞ,0 ¼ sðkÞ,0 rr yy ¼ ½l ð2l1 þ l3 3Þ þ 2m ðl1 1Þðl1 Þ :

(12)

According to the problem statement, we can write that ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

1 ðkÞ,0 ðkÞ ¼ sðkÞ,0 ¼0 srr yy ¼ ½l ð2l1 þ l3 3Þ þ2m ðl1 1Þðl1 Þ

and from this we obtain "

lðkÞ 1 ¼ 2

lðkÞ

#" ðkÞ

ðl3 3Þ ðkÞ

m

2

lðkÞ

mðkÞ

!#1 þ1

(13)

:

Also, we obtain from (12), (1) and (A2) the following expressions for the Kirchhoff stress tensor in the initial state: ðkÞ

ðkÞ

ðkÞ,0 ¼ l3 szz , qðkÞ,0 zz

qðkÞ,0 ¼ l1 sðkÞ,0 ¼ 0, rr rr

ðkÞ

ðkÞ,0 qðkÞ,0 yy ¼ l1 syy ¼ 0,

ðkÞ,0 ðkÞ,0 ðkÞ,0 ðkÞ,0 ðkÞ,0 qðkÞ,0 yr ¼ qry ¼ qrz ¼ qzr ¼ qzy ¼ qyz ¼ 0:

(14)

It follows from the relations (1) and (14) that the Eq. (6) satisfies automatically the initial strain–stress state. 2.3. Determination of the relations related to the perturbation state Now we assume that the considered three-layered hollow cylinder with the foregoing initial strain–stress state has an ðkÞ ðkÞ ðkÞ additional small perturbation determined by the displacement vector with components uðkÞ r ¼ 0, uy ¼ uy ðr,z,t Þ and uz ¼ 0. Taking into account the smallness of the displacement perturbation, we linearize the relationships (A1), (5), (A2), (6)–(8) for the perturbed state in the vicinity of the appropriate values for the initial state and then subtract from them the relationships for the initial state. As a result, we obtain the equations of the TLTEWISB. As an example, in the case under consideration, as a result of the mentioned linearization we obtain the following expressions for perturbations of the components of Green’s strain tensor: ! ðkÞ ðkÞ uðkÞ 1 ðkÞ @uy 1 ðkÞ @uy ðkÞ ðkÞ ðkÞ ðkÞ y  , eðkÞ ery ¼ l1 ¼ l1 (15) , eðkÞ rr ¼ eyy ¼ ezz ¼ ezr ¼ 0: y z 2 2 @r r @z Perturbation of the components of the stress tensor s~ ðkÞ (denote them by capital letter SðkÞ ) is determined from the ðijÞ linearization of the relation (5). We do not consider here the detail of this linearization procedure, but note that as a result of this linearization the following expressions for SðkÞ are obtained: ðijÞ ! ðkÞ ðkÞ ðkÞ uy ðkÞ ðkÞ ðkÞ @uy ðkÞ @uy ðkÞ ðkÞ SðkÞ  , SðkÞ ¼ m ¼ m (16) , S rr ¼ Syy ¼ Szz ¼ Srz ¼ 0: ry zy @r r @z Taking into account the relation (16), we obtain from (A2) (given in Appendix A) the following expression for perturbation ðkÞ of the components of the Kirchhoff stress tensor q~ ðkÞ (denote them by capital letter QðijÞ ) which differs from zero: ðkÞ

ðkÞ ðkÞ QrðkÞ y ¼ Qyr ¼ l1 Sry ,

ðkÞ

ðkÞ ðkÞ,0 QzðkÞ y ¼ l1 Szy þ szz

@uðkÞ y , @z

ðkÞ

QyðkÞ ¼ l3 SðkÞ : z zy

(17)

ðkÞ ðkÞ,0 Thus, substituting ðqðkÞ,0 þ QðilÞ Þ, uðkÞ,0 , uðkÞ for q(ij), ur, uy and uz, respectively, in Eq. (6) we obtain r y ðr,z,tÞ and uz ðijÞ

@ ðkÞ @ ðkÞ 1 ðkÞ @2 Q þ Qzy þ ðQr0 y þ QyðkÞ Þ ¼ rðkÞ 2 uðkÞ : r0 @r ry @z r @t y

(18)

Multiplying Eq. (18) with ððl1 Þ2 l3 Þ1 and using the notation: ðkÞ

ðkÞ

ðkÞ ðkÞ ðkÞ 1 2 ðkÞ 1 r0ðkÞ ¼ rðkÞ ððlðkÞ , Qr0ðkÞ , 0 y ¼ Qr y ðl1 l3 Þ 1 Þ l3 Þ ðkÞ

ðkÞ 2 , Qz0ðkÞ 0 y ¼ Qzy ðl1 Þ

ðkÞ u0ðkÞ y ¼ uy ,

ðkÞ

r 0 ¼ l1 r,

ðkÞ

z0 ¼ l3 z:

(19)

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we obtain Eq. (3) from Eq. (18). Moreover, from the relationships (16), (17) and (19) we derive the following expressions 0ðkÞ 0ðkÞ for the components o0ðkÞ 1221 , o1212 and o1331 which enter the relation (4): 0ðkÞ 0ðkÞ 0ðkÞ o0ðkÞ 2121 ¼ o1221 ¼ o2112 ¼ o1212 ¼

mðkÞ lðkÞ 3

o0ðkÞ 1331 ¼

,

lðkÞ 1 ðkÞ l1 þ lðkÞ 3

ðkÞ

2mðkÞ l1 þ

1

lðkÞ 3

SðkÞ,0 33 :

(20)

Thus, torsional wave propagation in the sandwich hollow cylinder will be investigated by the use of Eqs. (3), (4) and (20). In this case we will assume that the following complete contact and boundary conditions are satisfied:       Qr0y ð1Þr0 ¼ lð1Þ R ¼ 0, u0y ð1Þr0 ¼ lð1Þ Rð1 þ hð1Þ =RÞ ¼ u0y ð2Þr0 ¼ lð1Þ Rð1 þ hð1Þ =RÞ , 1 1 1    0ð2Þ  Qr0ð1Þ ð1Þ =RÞÞ ¼ Qr y r 0 ¼ lð1Þ Rð1 þ ðhð1Þ =RÞÞ , y r0 ¼ lð1Þ Rð1 þ ðh 1 1    0ð3Þ  , Qr0ð2Þ ð1Þ ð2Þ ð2Þ ¼ Q ð2Þ   0 ð1Þ 0 r y r ¼ lð1Þ y r ¼ l1 Rð1 þ ðh =RÞÞ þ l1 h Rð1 þ ðhð1Þ =RÞÞ þ l1 hð2Þ 1     ¼ u0ð3Þ , u0ð2Þ y r0 ¼ lð1Þ y r 0 ¼ lð1Þ Rð1 þ ðhð1Þ =RÞÞ þ lð2Þ hð2Þ Rð1 þ ðhð1Þ =RÞÞ þ l1ð2Þ hð2Þ 1 1 1  0ð3Þ  (21) Qry r0 ¼ lð1Þ Rð1 þ ðhð1Þ =RÞÞ þ lð2Þ hð2Þ þ lð3Þ hð3Þ ¼ 0: 1

1

1

In this way, investigation of the considered wave dispersion problem is reduced to the study of the eigenvalue problem formulated through Eqs. (3), (4) and (20) and condition (21). Note that in the case where the initial strains are absent in the ðkÞ ðkÞ constituents in the cylinder, i.e. in the case where l1 ¼ l3 ¼ 1:0, the foregoing formulation coincides with the corresponding one proposed within the scope of the classical linear theory of elastodynamics. In general, whilst studying the mechanical problems related to the sandwich structures, various types of approximate theories are used to describe their motion. However, the applicability of these theories requires satisfaction of some conditions restricted to the variation range of the problem parameters. For example, in the case where (Echc/Eshs)o0.1 (where Ec(Es) is the modulus of elasticity of the core (skin) layer and hc(hs) is the thickness of the core (skin) layer) it is assumed that the core layer of the sandwich structure responds to the shear deformation only. In particular, in the noted cases, the Timoshenko type plate or shell theories are employed to describe the field equations of the sandwich plates or shells considering them as homogeneous structures with effective mechanical constants. Moreover, for the sandwich structures consisting of two thin relatively stiff skins and a soft core ply between them, the dynamics of the skin layers are described by the standard Kirchhoff theory but the dynamics of the core layer are described by the classical linear theory of elastodynamics (see papers [33,34] and other references listed in these papers). Note the foregoing approximate approaches relate to the sandwich structures with soft cores. But the field equations of the sandwich structures with stiff cores and thin skin layers are described within the scope of the Kirchhoff theory. Consequently, the applicability of the certain approximate plate or shell theories for describing the motion of the sandwich structures requires the aforementioned type restrictions on the problem parameters. However, in the present paper, formulation of the problem under consideration has been made without any restriction on the foregoing type problem parameters. At the same time, it should be noted that the results obtained within this formulation can be applied not only for the sandwich cylinders made from polymers, elastomers and other types of hyper-elastic materials, but also for the sandwich cylinders made from any type of isotropic elastic material with constant through-the thickness properties. 3. Solution procedure and obtaining the dispersion equation For solution of Eqs. (3) and (4) we use the following presentation: 0 0 u0ðmÞ y ðr ,z ,tÞ ¼ 

@ ðmÞ 0 0 C ðr ,z ,tÞ, @r 0

(22)

where the function C(m) in (22) satisfies the equation written below: " # 2 r0 @2 0 0ðmÞ 2 @ D þðx Þ 02  0 C¼0 o1221 @t 2 @z

(23)

where

D0 ¼

d2 1 d þ , dr 02 r 0 dr 0

ðx0ðmÞ Þ2 ¼

ðmÞ 2ðl3 Þ3 ðmÞ 2 ðmÞ ðmÞ ðl1 Þ ðl1 þ l3 Þ

(24)

It follows from the problem statement that the presentation:

CðmÞ ðr0 ,z0 ,tÞ ¼ cðmÞ ðr0 Þ expðiðkz0 otÞÞ

(25) (m)

holds. Thus, we obtain from (22) and (25) the following equation for the unknown function, c " !# 0ðmÞ 2 d2 1 d o 2 0ðmÞ 2 r þ  k ð x Þ  cðmÞ ðr0 Þ ¼ 0: 1 dr 02 r 0 dr 0 o0ðmÞ 1221

0

(r ): (26)

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4525

Introducing the notation !

ðsðmÞ Þ2 ¼

k2 ðx1 Þ2  0ðmÞ

r0ðmÞ o2 : o0ðmÞ 1221

(27)

The solution to Eq. (25) can be written as follows:

c0ðmÞ ðr 0 Þ ¼ AðmÞ J0 ðsðmÞ kr0 Þ þBðmÞ Y0 ðsðmÞ kr0 Þ if ðsðmÞ Þ2 4 0, m ¼ 1,2,3 c0ðmÞ ðr0 Þ ¼ AðmÞ I0 ðsðmÞ kr0 Þ þ BðmÞ K0 ðsðmÞ kr0 Þ if ðsðmÞ Þ2 o 0:

(28)

Using Eqs. (25), (28), (22), (20) and (4) we obtain the following dispersion equation for the condition (21): det:aij : ¼ 0,

i; j ¼ 1,2,3,4,5,6,

(29)

The expressions of aij are given in Appendix B. Thus, the dispersion equation for the considered torsional wave propagation problem has been derived in the forms presented in (29) and (B1), (B2) given in Appendix B.

4. Numerical results and discussions It follows from the expressions (12) and (13) that the initial stress–strain state in each constituent of the sandwich ðkÞ cylinder is determined through the parameters l3 and l(k)/m(k). Consequently, if we assume that the foregoing initial stress–strain state in the sandwich plate appears after compounding of the inner, middle and outer cylinders and the condition: ð2Þ ð3Þ lð1Þ lð1Þ =mð1Þ ¼ lð2Þ =mð2Þ ¼ lð3Þ =mð3Þ ¼ 1:5 3 ¼ l3 ¼ l3 , ð1Þ

ð2Þ

(30)

ð3Þ

takes place, then from Eqs. (20) and (27) it follows that l1 ¼ l1 ¼ l1 . In other words, for the case where relation (30) takes place, then the initial stress–strain state determined by Eqs. (12) and (13) is satisfied exactly. This statement verifies the reality of the considered initial stress–strain state in the sandwich hollow cylinder. Consider the dispersion relation c¼c(kR), where c( ¼ o/k) is the wave propagation velocity, k is the wavenumber, o is the frequency and R is the inner radius of the inner cylinder (Fig. 1) obtained from a numerical solution of the dispersion Eq. (29). For simplification of the discussions below we introduce the following notation: sffiffiffiffiffiffiffiffiffi " #1=2 ðmÞ 2l3 mðmÞ ðmÞ ðmÞ ðmÞ ðmÞ ðmÞ c20 ¼ ¼ c ð l Þ ¼ c (31) , c 3 2 2 20 m ðmÞ ðmÞ ðmÞ rðmÞ ðl2 Þ3 l1 ðl3 þ l2 Þ ðmÞ

Note that the expression for c2ðmÞ ðl3 Þ in (31) is obtained from the following relation: ðmÞ c2ðmÞ ðl3 Þ ¼

" #1=2 ðx0ðmÞ Þ2 o0ðmÞ 1221

r0ðmÞ

,

(32)

Þ2 are determined through Eqs. (20) and (27), respectively. where o0ðmÞ 1221 and ðx Now we consider the limit values of the torsional wave propagation velocity. Low wavenumber limit values as kR-0: As usual, to obtain this limit value, each term of the corresponding dispersion equation is expanded into series form for small values of kR-0 and only the limit values of the wave propagation velocity are taken into account. For the considered problem, by using power series expansions of the Bessel functions, retaining only the dominant term at kR-0 and doing the corresponding mathematical manipulations to calculate the determinant in Eq. (29) we obtain the following limit value for the torsional wave propagation velocity for the low wavenumber limit: " #1=2 ð1Þ 0ð1Þ ð2Þ 0ð2Þ ð3Þ 0ð3Þ ðmð1Þ =l2 Þðx1 Þ2 þ ðmð2Þ =l2 Þaðx1 Þ2 þðmð3Þ =l2 Þbðx1 Þ2 c ¼ , (33) ð1Þ c20 mð1Þ þ mð2Þ aðc2ð1Þ =c2ð2Þ Þ2 þ mð3Þ bðc2ð1Þ =c2ð3Þ Þ2 0ðmÞ

where ðl2 Þ4 ðZð2Þ Þ4 ðl2 Þ4 ðZð1Þ Þ4 ð2Þ

a¼ 0ðkÞ

ð1Þ

ð1Þ ðl2 Þ4 ðð ð1Þ Þ4 1Þ

Z

ðl2 Þ4 ðZð3Þ Þ4 ðl2 Þ4 ðZð2Þ Þ4 ð3Þ

,

b¼

ð2Þ

ð1Þ ðl2 Þ4 ðð ð1Þ Þ4 1Þ

Z

:

(34)

Note that the values of x1 and Z(k) (k ¼1,2,3) are calculated through the expressions (27) and (B2) given in Appendix B. In the case where m(3) ¼0 the expression (33) transforms to the corresponding asymptotic expression given below for the bilayered finitely pre-strained hollow cylinder: " #1=2 ð1Þ 0ð1Þ ð2Þ 0ð2Þ ðmð1Þ =l2 Þðx1 Þ2 þ ðmð2Þ =l2 Þaðx1 Þ2 c ¼ : (35) ð1Þ c20 mð1Þ þ mð2Þ aðc2ð1Þ =c2ð2Þ Þ2

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ðkÞ

In the case where l3 ¼ l1 ¼ 1:0 (k¼1,2), i.e. in the case where the initial strains are absent in the bi-layered cylinder, the expression (35) transforms to the following one: "

c ð1Þ c20

¼

mð1Þ þ mð2Þ a1 ð1Þ m þ mð2Þ a1 ðc2ð1Þ =c2ð2Þ Þ2

#1=2 (36)

where a1 ¼ ajlð2Þ ¼ lð1Þ ¼ 1:0 . 1

1

Note that the expression (36) coincides with the corresponding one obtained in the paper [35]. Moreover, note that the asymptotic expression (35) is the generalization for the finite initial strain state of the corresponding one obtained in paper [27]. It needs to be remembered that in the paper [27], the aforementioned expression is obtained for the case where the initial strains are small and determined within the scope of the classical linear theory of elasticity. ðkÞ ðkÞ At the same time, in the case where l3 ¼ l1 ¼ 1:0(k¼1,2,3), i.e. in the case where the initial strains are absent in the layers of the considered three-layered (sandwich) hollow cylinder, the asymptotic expression (33) transforms to the corresponding one obtained within the scope of the classical linear theory of elastodynamics. Consequently, the expression (33) is also a new one for the classical linear theory of elastodynamics. High wavenumber limit values as kR-N: Asymptotic analyses of the dispersion Eqs. (29) and (B1) given in Appendix B show that the high limit value of the torsional wave propagation velocity, i.e. the limit velocity as kR-N is equal to ð1Þ ð2Þ ð3Þ minfc2ð1Þ ðl3 Þ; c2ð2Þ ðl3 Þ; c2ð3Þ ðl3 Þg, and in turn, the following relation holds: ð1Þ

ð2Þ

ð3Þ

c-minfc2ð1Þ ðl3 Þ; c2ð2Þ ðl3 Þ; c2ð3Þ ðl3 Þg as kR-1:

(37)

ðkÞ

The values of c2ðkÞ ðl3 Þ (k¼1,2,3) are determined by the expression (35). Note that the relation (37) is also confirmed by well-known physical–mechanical considerations. Numerical results obtained by the numerical solution to the dispersion Eq. (29): This solution is found by employing the ‘‘bi-section method’’ algorithm using the PC. We attempt to verify the numerical algorithm used and the corresponding programs. To the best knowledge of the authors, no suitable numerical results have been obtained for torsional wave propagation (dispersion) in a three-layered hollow cylinder with which we can compare the numerical results obtained from the solution to the dispersion Eq. (29). In general, there are only a few investigations, such as [36,37] in which the torsional wave propagation in the three-layered composite cylinder is studied within the scope of the classical linear theory of elastodynamics. Moreover, in papers [36,37] it was assumed that the materials of the skin layers are orthotropic and that the material of the core layer is isotropic. We cannot verify the algorithm used in the present paper with the results given in the papers [36,37] because in the present investigation we assume that the materials of the layers of the cylinder are isotropic. Therefore, we verify the algorithm used in the present paper with the results obtained in the papers [31,35]. Note that in the paper [35], the torsional wave propagation in a bi-layered hollow cylinder was studied within the scope of the linear theory of elastodynamics. But in the paper [31], as noted in Section 1, the torsional wave propagation in the many-layered (as well as in the three-layered) solid cylinder with small initial strains, was studied. At first, we suppose that mð3Þ ¼ mð2Þ and attempt to compare the results obtained from the solution to the dispersion equation (29) with the corresponding results given in the paper [35]. As in the paper [35], we assume that m(1)/m(2) ¼1, r(2)/r(1) ¼0.5, ð2Þ Þ and 2h(2)/L, where L is the waveh(1)/R¼0.2 and (h(2) þh(3))/R¼0.2 and consider the dependence between ohð2Þ =ðpc20 (2) (3) (2) length. Note that in [35] instead of the notation (h þh )/R(¼ 0.2), the notation h /R(¼ 0.2) is used. Thus, within the framework of the foregoing assumptions we consider the dispersion curves obtained from the solution to the dispersion equation (29) for ð1Þ

ð2Þ

ð3Þ

various values of the elongation parameter l3 ð ¼ l3 ¼ l3 ¼ l3 Þ. The curves for the lowest three modes are given in Fig. 2 from which it follows that the results obtained in the case where l3 ¼1 (i.e. where the initial strains are absent in the constituents of the cylinder) coincide with the corresponding ones given in the paper [35]. At the same time, it can be seen from Fig. 2 that, in general, the initial stretching (compression) of the compounded bi-material hollow cylinder causes an increase (a decrease) in the torsional wave propagation velocity. This conclusion holds strongly for the first mode. But for the second and third modes the foregoing conclusion is violated for the cases where kR-0. Consequently, there exists such values of kR before (after) which the initial stretching (compression) of the cylinder causes a decrease (an increase) in the velocity of the torsional wave propagation in the bi-layered hollow cylinder. This statement will also be discussed below for the three-layered cylinder. Now, we compare the results obtained from the solution to the dispersion equation (29) with the corresponding results given in the paper [31]. We introduce the notation R1 ¼Rþh(1) and, as in [31], assume that rð1Þ ¼ 7:20 g=cm3 , m(1) ¼3.38 MPa, rð2Þ ¼ 7:20 g=cm3 , m(2) ¼4.47 MPa, rð3Þ ¼ 7:795 g=cm3 , m(3) ¼7.75 MPa and hð2Þ =R1 ¼ hð3Þ =R1 ¼ 0:5. Consider the dispersion curves given in Fig. 3a–c which are constructed for the first (Fig. 3a), second (Fig. 3b) and third (Fig. 3c) modes under various values of hð1Þ =R1 in the cases where the initial strains are absent in the layers. Note that in Fig. 3a–c the corresponding dispersion curves obtained in [31] are for the system consisting of ‘‘solid cylinderþhollow cylinder with thickness h(2) þhollow cylinder with thickness h(3)’’. Also, consider the comparison of the results obtained in the present work with the corresponding results obtained in the paper [31] in the case where in the constituents of the sandwich cylinder there exist ðkÞ

small initial strains. As in [31], we introduce a parameter c also a parameter Z

ð1Þ ¼ ðcjc 4 0 cjc ¼ 0 Þ=c20

ðkÞ ¼ sðkÞ,0 and assume that c(1) ¼ c(2) ¼ c(3)(¼ c). We introduce zz =m

and estimate the influence of the small initial strains on the wave propagation

S.D. Akbarov et al. / Journal of Sound and Vibration 330 (2011) 4519–4537

4527

Fig. 2. The comparison of the numerical results obtained by applying of the algorithm used in the present paper with the corresponding results obtained in the paper [35].

Fig. 3. The comparison of the numerical results obtained by application of the algorithm used in the present paper with the corresponding results obtained in the paper [31]: dispersion curves for the first (a), second (b) and third (c) modes and graphs illustrated the influence of the small initial strains on the wave propagation velocity in the first mode in the cases where c ¼ 0.004 (d), 0.008 (e) and 0.012 (f).

velocity through that. The graphs of the dependencies between the parameter Z and kR in the first mode for various h(1)/R1 are given in Fig. 3d–f for the cases where c ¼0.004, 0.008 and 0.012, respectively. According to the well known mechanical consideration, with hð1Þ =R1 the results obtained in the present paper, i.e. the results obtained for the hollow sandwich cylinder must approach the corresponding results obtained for the system consisting of ‘‘solid cylinderþhollow cylinder with thickness h(2) þhollow cylinder with thickness h(3)’’ which was

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examined in the paper [31]. Analyses of the graphs given in Fig. 3 show that the numerical results obtained confirm this mechanical consideration. This statement also proves the validity of the algorithm used in the present investigation. The results discussed above indicate that the analytical and numerical solution methods used in the present investigation are correct. Now we consider the numerical results regarding the dependence between c=cð2Þ 20 and kR obtained for the sandwich cylinder. We consider the case where hð1Þ ¼ hð3Þ , rð1Þ ¼ rð3Þ , rð2Þ =rð1Þ ¼ rð2Þ =rð3Þ ¼ 1, mð1Þ ¼ mð3Þ and mð2Þ 4 mð1Þ ð ¼ mð3Þ Þ. Unless ð1Þ

ð2Þ

ð3Þ

otherwise specified, we will also assume that l =mð1Þ ¼ l =mð2Þ ¼ l =mð3Þ ¼ 1:5.

lð1Þ 3

lð2Þ 3

ð3Þ ¼ l3 .

ð1Þ

ð2Þ

ð3Þ

First, we consider the case where ¼ The notation l3 ð ¼ l3 ¼ l3 ¼ l3 Þ is introduced and the influence of the elongation l3 which characterizes the initial strains in the cylinder on the dispersion curves obtained for various values of the problem parameters is studied. Fig. 4 shows the dispersion curves obtained for the first lowest mode in the cases where mð2Þ =mð1Þ ¼ mð2Þ =mð3Þ ¼ 2 (Fig. 4a), 5 (Fig. 4b) and 10 (Fig. 4c) for various values of the parameter l3 under hð1Þ =R ¼ hð3Þ =R ¼ 0:1, h(2)/R¼ 0.4. It follows from these results that the initial stretching (compression) of the layers of the cylinder causes an increase (a decrease) in the torsional wave propagation velocity. Direct verification of the ð2Þ data shows that the results given in Fig. 4 approach the corresponding values of c=c20 calculated by the expression (33) as ð2Þ approach c2ð1Þ ðl3 Þð ¼ c2ð3Þ ðl3 ÞÞ as determined by the kR-0. At the same time, these results show that the values of c=c20 expression (21) as kR-N. Moreover, according to the graphs given in Fig. 4, it can be concluded that the change in ð2Þ ð2Þ with respect to kR, i.e. the values of the derivative ðdðc=c20 ÞÞ=ðdðkRÞÞ increase significantly velocity of the values of c=c20

with an increase in the values of mð2Þ =mð1Þ . This statement is more clearly illustrated by the graphs given in Fig. 5 which are constructed for the first mode in the cases where l3 ¼ 0.6 (Fig. 5a), l3 ¼1.0 (Fig. 5b) and l3 ¼1.4 (Fig. 5c) under h(2)/R¼0.4, hð1Þ =R ¼ hð3Þ =R ¼ 0:1, for various values of mð2Þ =mð1Þ . Note that in the case where mð2Þ =mð1Þ ¼ 1 (i.e. where the materials of the layers of the cylinder are the same) the first lowest mode of the torsional wave is non-dispersive. This fact is well-known in the classical linear theory of elastodynamics and as follows from the results given in Fig. 5 it also holds for the torsional wave propagation in the pre-strained homogeneous hollow cylinder.

Fig. 4. The influence of the initial strains on the dispersion curves of the first mode constructed in the cases where ðmð2Þ =mð1Þ Þ ¼ ðmð2Þ =mð3Þ Þ ¼ 2 (a), 5 (b) and 10 (c) under ðhð1Þ =RÞ ¼ ðhð3Þ =RÞ ¼ 0:1, ðhð2Þ =RÞ ¼ 0:4.

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Fig. 5. The influence of the ratio mð2Þ =mð1Þ ð ¼ mð2Þ =mð3Þ Þon the dispersion curves of the first mode constructed in the cases where l3 ¼ 0.6 (a), 1.0 (b) and 1.4 (c) under ðhð1Þ =RÞ ¼ ðhð3Þ =RÞ ¼ 0:1, hð2Þ =R ¼ 0:4.

ð2Þ The results given in Fig. 5 also show that the values of c=c20 decrease with mð2Þ =mð1Þ , after consideration of the case where ð2Þ Consequently, according to the mechanical consideration, the values of c=c20 must increase with h(2)/R, i.e. with the thickness of the middle layer of the cylinder. This prediction is confirmed by the graphs given in Fig. 6, which are constructed in the cases where l3 ¼ 0:6(Fig. 6a), l3 ¼ 1:0(Fig. 6b), l3 ¼ 1:4 (Fig. 6c) and l3 ¼ 1:8(Fig. 6d) for various values of h(2)/R under mð2Þ =mð1Þ ¼ 2, hð1Þ =R ¼ hð3Þ =R ¼ 0:1. Now, the numerical results relating to the second and third modes are considered. Fig. 7 shows the dispersion curves for these modes for the various values of l3 in the cases where mð2Þ =mð1Þ ¼ 2 (Fig. 7a), 5 (Fig. 7b) and 10 (Fig. 7c) under hð1Þ =R ¼ hð3Þ =R ¼ 0:1 h(2)/R ¼0.4. It follows from these results that, as in the case shown in Fig. 2 for a bi-layered cylinder, in the second and third modes the character (in the qualitative sense) of the influence of the initial strains on the wave propagation velocity depends on the values of the dimensionless wavenumber kR. So, there exists such a value for this wavenumber (denoted by (kR)n) after which (for the cases where kR4(kR)n), the character of the influence of the initial strains on the wave propagation velocity is similar to that which has been observed above for the first mode, i.e. the initial stretching (compression) causes an increase (a decrease) in the wave propagation velocity. However, in the cases where kRo(kR)n the discussed influence has a reverse character, i.e. the initial stretching (the initial compression) causes a decrease (an increase) in the wave propagation velocity in the second and third modes. For a clearer illustration of the ð2Þ foregoing statement we also consider the dispersion diagrams, i.e. the graphs of the dependencies between oR=c20 and kR for the second and third modes. These diagrams, constructed for the cases considered in Fig. 7a–c, are given in Fig. 8a–c, respectively. Moreover, the parts of these diagrams corresponding to the cases where 0okR o0.2 o(kR)n and which have also been constructed for the problem parameters shown in Fig. 7a–c are given in Fig. 9a–c, respectively. Note that the results given in Fig. 9 also show the influence of the initial strains in the constituents of the cylinder on the values of the cut-off frequency for the second and third modes. The graphs of the dependencies between the cut-off frequency and l3 are given in Fig.10 for various values of mð2Þ =mð1Þ , where hð1Þ =R ¼ hð3Þ =R ¼ 0:1, hð2Þ =R ¼ 0:4. It follows from the results given in Figs. 8 and 9 that, as noted above, under kRo(kR)n the wave propagation velocity of the second and third modes increases monotonically with decreasing l3, i.e. with initial compression. At the same time, ð2Þ ð1Þ c20 4c20 .

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Fig. 6. Dispersion curves for the first mode constructed for various values of hð2Þ =R in the cases where l3 ¼ 0.6 (a), 1.0 (b), 1.4 (c) and 1.8 (d) under mð2Þ =mð1Þ ¼ mð2Þ =mð3Þ ¼ 2, hð1Þ =R ¼ hð3Þ =R ¼ 0:1. ð2Þ under kRo(kR)n the initial stretching causes a decrease in the values of these velocities and the dependence between c=c20 and l3 is non-monotonic. Note that the values of (kR)n depend on the number of the mode and of the values of the problem parameters mð2Þ =mð1Þ , hð1Þ =Rð ¼ hð3Þ =RÞ,hð2Þ =R, etc. Therefore, it can be concluded that for fixed problem parameters and for the second and third modes in the case where kR¼(kR)n the initial strains in the cylinder do not influence the wave propagation velocity. Fig. 10 shows that under l3 o1 the values of the cut-off frequency of the second and third modes increase with decreasing l3, i.e. with initial compression of the cylinder. However, under l3 41 the values of the cut-off frequency become less than those obtained under l3 ¼1. ð1Þ

ð2Þ

ð2Þ

In all the previous investigations it was assumed that l =mð1Þ ¼ l =mð2Þ ¼ l =mð2Þ ¼ 1:5. Now we consider how the ð1Þ

ð2Þ

ð3Þ

change of the rate l =m ð ¼ l =m ¼ l =m Þ affects the behavior of the dispersion curves. For this purpose we examine the first mode only. First of all we note that, in the absence of the initial strains in all the constituents of the sandwich cylinder, ð1Þ

ð1Þ

ð2Þ

ð2Þ

ð3Þ

ð3Þ

the change of rate ðl =mð1Þ Þ ( ¼ l =mð2Þ ¼ l =mð3Þ ) does not influence the dependence between c=cð2Þ 20 and kR. This conclusion ðkÞ

also follows from the basic relationships given in Section 2 because under l3 ¼ 1:0 (k¼1,2,3) the mentioned relationships do ðkÞ

not contain the rate l =mðkÞ . In the cases where

lðkÞ 3 a1:0

ð1Þ

ð2Þ

ð3Þ

the influence of the rate ðl =mð1Þ Þ ( ¼ l =mð2Þ ¼ l =mð3Þ ) on the first ð1Þ

ð2Þ

ð3Þ

ð1Þ

ð2Þ

ð3Þ

mode is illustrated by the graphs given in Fig. 11 for the cases where l3 ¼ l3 ¼ l3 ¼ 1:4 (Fig. 11a) and l3 ¼ l3 ¼ l3 ¼ 0:8 (Fig. 11b) under ðm =m Þ ¼ 2:0. Thus, it follows from Fig. 11 that under initial stretching (compressing) of the constituents of ð2Þ

ð1Þ

ð1Þ

ð2Þ

ð3Þ

the sandwich cylinder the torsional wave propagation velocity increases (decreases) with ðl =mð1Þ Þ ( ¼ l =mð2Þ ¼ l =mð3Þ ). Note that the influence of the mentioned rate on the dispersion curves can be taken as the effect of the magnitude of the radial deformations of the inner, middle and external cylinders’ materials (similar to the deformations in the classical linear theory of elasticity which are determined through Poisson’s coefficient) which arise under the stretching or compressing of the cylinder along the Oz axis. We analyze also the difference between the initial strains of the layers of the cylinder on the dispersion curves. For this ð1Þ ð3Þ ð2Þ ð1Þ ð3Þ ð2Þ purpose consider the case where l3 ¼ l3 ¼ 1:0, l3 a1:0 (case 1), as well as the case where l3 ¼ l3 a1:0, l3 ¼ 1:0 ð1Þ ð2Þ ð3Þ (case 2). Assume that ðl =mð1Þ Þ ¼ ðl =mð2Þ Þ ¼ ðl =mð3Þ Þ ¼1.5 and consider the graphs given in Fig. 12 which show the

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4531

Fig. 7. The influence of the initial strains on the dispersion curves of the second and third modes constructed for the cases where mð2Þ =mð1Þ ¼ mð2Þ =mð3Þ ¼ 2 (a), 5 (b) and 10 (c) under hð1Þ =R ¼ hð3Þ =R ¼ 0:1, hð2Þ =R ¼ 0:4.

dispersion curves for the first mode in case 1 (Fig. 12a) and in case 2 (Fig. 12b) under ðmð2Þ =mð1Þ Þ ¼ 2:0. Note that in Fig. 12 ð1Þ ð2Þ ð3Þ the dispersion curves corresponding to the case where l3 ¼ l3 ¼ l3 a1:0 are also shown (dashed lines). To simplify the conclusion which follows from the results given in Fig. 12 we present the wave propagation velocity as a ð1Þ ð2Þ ð3Þ ð1Þ ð2Þ ð3Þ function of l3 , l3 , l3 , i.e. as c¼cðl3 , l3 , l3 Þ. According to this presentation, it follows from Fig. 12 that the following relations occur: ð2Þ

ð1Þ

ð3Þ

ð1Þ

ð2Þ

ð3Þ

ð1Þ

ð2Þ

ð3Þ

cð1, l3 ,1Þ rcðl3 ,1, l3 Þ rcðl3 , l3 , l3 Þ if l3 ¼ l3 ¼ l3 Z1:0, ð2Þ ð1Þ ð3Þ ð1Þ ð2Þ ð3Þ cð1, l3 ,1Þ Zcðl3 ,1, l3 Þ Z cðl3 , l3 , l3 Þ

if

lð1Þ 3

ð2Þ ¼ l3

ð3Þ

¼ l3 r1:0:

(38)

In other words, the initial stretching (compressing) of each constituent of the sandwich cylinder has its corresponding contribution to the increase or decrease of the torsional wave propagation velocity in this cylinder. Note that for the cases where the ‘‘time freezing’’ principle (according to which the ‘‘own’’ time is introduced for the perturbed state and this ‘‘own’’ time is independent of the ‘‘frozen’’ time of the initial state) is acceptable, the foregoing results can also be used for the estimation of the wave propagation velocity in the sandwich cylinder with viscoelastic layers. For this purpose, the following fact can be used: that under high (low) wavenumbers, the wave propagation velocity in the viscoelastic material close to the wave propagation velocity can be determined by the corresponding instantaneous (dilatational) values of the mechanical constants of this material. Consequently, the values of c obtained above for the high (low) wavenumbers can be used as the corresponding instantaneous (dilatational) torsional wave propagation velocities in the sandwich cylinder made from the corresponding viscoelastic layers and this statement can be used for determination of the corresponding loss factors. 5. Conclusions In the present paper, within the scope of the piecewise homogeneous body model with the use of the Threedimensional Linearized Theory of Elastic Waves in Initially Stressed Bodies, torsional wave dispersion in a three-layered (sandwich) hollow circular cylinder with finite homogeneous axisymmetric initial strains has been studied. The

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Fig. 8. Dispersion diagrams of the second and third modes constructed in the cases considered in Fig. 6a (a), b (b) and c (c).

corresponding dispersion equation was derived and analytical expressions (33) and (37) were found for the limiting values of the velocity at the lowest dispersive mode from this dispersion equation. An algorithm was developed to do numerical investigations and this algorithm was tested first on a known problem which had been investigated by other authors. The basic numerical investigations were made for the case where the materials of the inner and external hollow cylinders were the same and the materials of the middle hollow cylinders were stiffer than that of the external hollow cylinders. Concrete numerical results are presented basically for the case where the initial strains in the cylinders are equal to each other. According to these results the following concrete conclusions are indicated. – in the first lowest mode the initial stretching (compression) of the layers of the cylinder causes an increase (a decrease) in the torsional wave propagation velocity; – in the second and third modes the influence of the initial strains of the cylinder on the torsional wave propagation velocity depends (in the qualitative sense) on the values on the dimensionless wavenumber kR. In this case there exists such a value of kR( ¼(kR)n) before (after) which the initial stretching causes a decrease (an increase) but the initial compression causes an increase (a decrease) in the velocity of the torsional wave propagation; – under l3 o1, i.e. under initial compression the values of the cut-off frequency of the second and third modes increase with decreasing l3, but under l3 41 the noted values of the cut-off frequency become less than those obtained in the case where l3 ¼1, i.e. where the initial strains are absent in the considered sandwich cylinder. The results and the method of the present investigation can be used for determination and control of the noise of threelayered polymer pipes which are used to transfer various types of liquids. The initial strains in these pipes may arise as a result of corresponding manufacturing procedures or as a result of the change in the environmental temperature. Consequently, knowledge of the influence of the considered initial strains on the wave propagation velocity can be used, for example, for minimizing the noises which arise from fluid flowing in three-layered pipes. At the same time, the results of these investigations can be used for determination of the residual and applied stress in bi-layered high elastic materials under non-destructive stress analyses [20,38,39]. For this reason, the results obtained in the present paper can also be taken as a small contribution to the theoretical bases of non-destructive stress analyses of many-layered polymer

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4533

Fig. 9. The parts of the dispersion diagrams illustrated in Fig. 7a (a), b (b) and c (c).

Fig. 10. The influence of the initial strains on the ‘‘cut-off’’ frequency related the second and third modes.

(high-elastic) materials. In addition, the expression (33) obtained in the present paper for asymptotic values of the torsional wave propagation velocity can be directly used to determine the influence of the considered initial strains on the values of the effective shear modulus of the composite material consisting of three-layered hollow cylinders.

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ðkÞ

ð1Þ

ð2Þ

ð3Þ

Fig. 11. The influence of the rate l =mðkÞ on dispersion curves of the first mode under mð2Þ =mð1Þ ¼ mð2Þ =mð3Þ ¼ 2 in the cases wherel3 ¼ l3 ¼ l3 ¼ 1:4 (a) ð1Þ ð2Þ ð3Þ and l3 ¼ l3 ¼ l3 ¼ 0:8 (b).

Fig. 12. The influence of the difference of the initial strains in the constituents of the cylinder on dispersion curves of the first mode under ð3Þ ð2Þ ð1Þ ð3Þ ð2Þ mð2Þ =mð1Þ ¼ mð2Þ =mð3Þ ¼ 2 in the cases lð1Þ 3 ¼ l3 ¼ 1:0, l3 a1:0 (a) and l3 ¼ l3 a1:0, l3 ¼ 1:0 (b).

Appendix A First, we write the expression for calculation of the physical components of the Green’s strain tensor through the physical components of the displacement vector in the cylindrical system of coordinates. (     2 ) @ur 1 @ur 2 @uy 2 @uz þ err ¼ þ þ , 2 @r @r @r @r        1 @uy @ur uy 1 @ur @ur uy @u @uy ur @uz @uz þ þ þ y þ ery ¼   þ 2 @r 2 @r r@y r r@y r @r r@y r @r r@y    1 @ur @uz 1 @ur @ur @uy @uy @uz @uz þ þ þ þ , erz ¼ 2 @z 2 @r @z @r @r @z @r @z (      ) @u u 1 @ur uy 2 @uy ur 2 1 @uz 2 eyy ¼ y þ r þ  þ þ þ 2 , 2 r@y r r@y r r@y r @y r        1 @uz @uy 1 @ur @ur uy @u @uy ur 1 @uz @uz þ þ y þ , eyz ¼ þ  þ 2 r@y 2 @z r@y r r @y @z @z @z r@y r (  )    2 @u 1 @ur 2 @uy 2 @uz ezz ¼ z þ þ þ : 2 @z @z @z @z

(A1)

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4535

The expressions through which the physical components of the Kirchhoff stress tensor are expressed through the physical components of the Lagrange stress tensor and displacement vector are @ur @ur uy @ur Þþs ð ,  Þ þ srz @r  ry r@y r  @z @uy @uy ur @uy þ sry 1 þ þ srz , qry ¼ srr þ @r r@y r @z   @uz @uz @uz þ sry , þ srz 1 þ qrz ¼ srr @r r@y @z     @ur @ur uy @ur qyr ¼ syr 1þ þ syy þ syz ,  @r r@y r @z   @u @u ur @u þ syz y , qyy ¼ syr y þ syy 1 þ y þ @r r@y r @z   @uz @uz @uz þ syy , þsyz 1 þ qyz ¼ syr @r r@y @z     @ur @ur uy @ur qzr ¼ szr 1þ þ szy þ szz ,  @r r@y r @z   @u @u ur @u þ szz y , qzy ¼ szr y þ szy 1 þ y þ @r r@y r @z   @uz @uz @uz þ szy : þszz 1 þ qzz ¼ szr @r r@y @z qrr ¼ srr ð1 þ

Appendix B We write the expressions for calculation of the terms aij which enter the dispersion equation (29) ( ) ð1Þ J1 ðsð1Þ kl1 RÞ mð1Þ 1 ð1Þ ð1Þ ½ J0 ðsð1Þ kl1 RÞJ2 ðsð1Þ kl1 RÞ a11 ¼ ð1Þ if ðsð1Þ Þ2 4 0, ð1Þ l3 2 sð1Þ kl1 R ( ) ð1Þ I1 ð9sð1Þ 9kl1 RÞ mð1Þ 1 ð1Þ ð1Þ ð1Þ ð1Þ if ðsð1Þ Þ2 o0, a11 ¼ ð1Þ  ½ I0 ð9s 9kl1 RÞ þ I2 ð9s 9kl1 RÞ þ ð1Þ 2 l3 9sð1Þ 9kl1 R ( ) ð1Þ J1 ðsð1Þ kl1 RÞ mð1Þ 1 ð1Þ ð1Þ ð1Þ ð1Þ ½Y0 ðs kl1 RÞY2 ðs kl1 RÞ if ðsð1Þ Þ2 4 0, a12 ¼ ð1Þ ð1Þ l3 2 sð1Þ kl1 R ( ) ð1Þ K1 ð9sð1Þ 9kl1 RÞ mð1Þ 1 ð1Þ ð1Þ ð1Þ ð1Þ a12 ¼ ð1Þ  ½ K0 ð9s 9kl1 RÞ þK2 ð9s 9kl1 RÞ þ if ðsð1Þ Þ2 o 0, ð1Þ 2 l 9sð1Þ 9kl R 3

1

a13 ¼ 0, a14 ¼ 0, a15 ¼ 0, a16 ¼ 0, a25 ¼ 0, a26 ¼ 0, ð1Þ ð1Þ 2 ð1Þ ð1Þ 2 a21 ¼ sð1Þ J1 ðsð1Þ klð1Þ a21 ¼ 9sð1Þ 9I1 ð9sð1Þ 9klð1Þ 1 RZ Þ if ðs Þ 4 0; 1 RZ Þ if ðs Þ o 0, ð1Þ ð1Þ 2 a22 ¼ sð1Þ Y1 ðsð1Þ klð1Þ 1 RZ Þ if ðs Þ 4 0; ð1Þ ð1Þ Þ 23 ¼ s J1 ðs kl1 R ð1Þ ð2Þ ð2Þ ð1Þ Þ 24 ¼ s Y1 ðs kl1 R ð2Þ

a

ð2Þ

Z

ð2Þ 2

if ðs Þ 4 0;

ð1Þ ð1Þ 2 a22 ¼ 9sð1Þ 9K1 ð9sð1Þ 9klð1Þ 1 RZ Þ if ðs Þ o0, ð1Þ ð2Þ 2 a23 ¼ 9sð2Þ 9I1 ð9sð2Þ 9klð1Þ 1 RZ Þ if ðs Þ o 0,

Z if ðsð2Þ Þ2 4 0; a24 ¼ 9sð2Þ 9K1 ð9sð2Þ 9kl1 RZð1Þ Þ if ðsð2Þ Þ2 o0, ( ) ð1Þ J1 ðsð1Þ kl1 RZð1Þ Þ mð1Þ 1 ð1Þ ð1Þ ½J0 ðsð1Þ kl1 RZð1Þ ÞJ2 ðsð1Þ kl1 RZð1Þ Þ a31 ¼ ð1Þ if ðsð1Þ Þ2 4 0; ð1Þ l3 2 sð1Þ kl1 RZð1Þ ( ) ð1Þ I1 ð9sð1Þ 9kl1 RZð1Þ Þ mð1Þ 1 ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ a31 ¼ ð1Þ  ½I0 ð9s 9kl1 RZ Þ þ I2 ð9s 9kl1 RZ Þ þ , if ðsð1Þ Þ2 o 0, ð1Þ 2 l3 9sð1Þ 9kl1 RZð1Þ ( ) ð1Þ Y1 ðsð1Þ kl1 RZð1Þ Þ mð1Þ 1 ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ½Y0 ðs kl1 RZ ÞY2 ðs kl1 RZ Þ , if ðsð1Þ Þ2 40; a32 ¼ ð1Þ ð1Þ l3 2 sð1Þ kl1 RZð1Þ ( ) ð1Þ K1 ð9sð1Þ 9kl1 Rð1 þðhð1Þ =RÞÞÞ mð1Þ 1 ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ a32 ¼ ð1Þ  ½ K0 ð9s 9kl1 RZ Þ þK2 ð9s 9kl1 RZ Þþ , if ðsð1Þ Þ2 o0, ð1Þ 2 l3 9sð1Þ 9kl1 Rð1 þðhð1Þ =RÞÞ  9 8 ð1Þ J1 sð2Þ kl1 RZð1Þ = mð2Þ <1 ð1Þ ð1Þ ð2Þ ð1Þ ð2Þ ð1Þ ½J0 ðs kl1 RZ ÞJ2 ðs kl1 RZ Þ , if ðsð2Þ Þ2 4 0; a33 ¼  ð2Þ ð1Þ sð2Þ kl1 RZð1Þ ; l3 :2 ( ) ð1Þ I1 ð9sð2Þ 9kl1 RZð1Þ Þ mð2Þ 1 ð1Þ ð2Þ ð1Þ ð2Þ ð1Þ ð1Þ a33 ¼  ð2Þ  ½I0 ð9s 9kl1 RZ Þ þ I2 ð9s 9l1 kRZ Þ þ , if ðsð1Þ Þ2 o0; ð1Þ 2 9sð2Þ 9kl RZð1Þ l a

3

ð1Þ

1

(A2)

4536

S.D. Akbarov et al. / Journal of Sound and Vibration 330 (2011) 4519–4537

8

a34 ¼  a34 ¼ 

mð2Þ <1 :2 lð2Þ 3

mð2Þ

½Y0 ðs

ð2Þ

ð1Þ ð1Þ kl1 R ð1Þ ÞY2 ðsð2Þ kl1 R ð1Þ Þ

Z

Z

 9 ð1Þ Y1 sð2Þ kl1 RZð1Þ =

(

lð2Þ 3

ð1Þ

sð2Þ kl1 RZð1Þ

;

, if ðsð2Þ Þ2 40;

) ð1Þ K1 ð9sð2Þ 9kl1 Rð1 þ ðhð1Þ =RÞÞÞ 1 ð1Þ ð1Þ  ½K0 ð9sð2Þ 9kl1 RZð1Þ Þ þ K2 ð9sð2Þ 9l1 kRZð1Þ Þþ ð1Þ 2 9sð2Þ 9kl Rð1 þ ðhð1Þ =RÞÞ 1

if ðsð2Þ Þ2 o0, a41 ¼ 0, a42 ¼ 0, ( ) ð1Þ J1 ðsð2Þ kl1 RZð2Þ Þ mð2Þ 1 ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ ð2Þ ½J0 ðs kl1 RZ ÞJ2 ðs kl1 RZ Þ , if ðsð2Þ Þ2 4 0; a43 ¼ ð2Þ ð1Þ l3 2 sð2Þ kl1 RZð2Þ ( ) ð1Þ I1 ð9sð2Þ 9kl1 RZð2Þ Þ mð2Þ 1 ð1Þ ð2Þ ð2Þ ð2Þ ð1Þ ð2Þ , if ðsð2Þ Þ2 o0, a43 ¼ ð2Þ  ½I0 ð9s 9kl1 RZ Þ þ I2 ð9s 9l1 kRZ Þ þ ð1Þ 2 l3 9sð2Þ 9kl1 RZð2Þ ( ) ð1Þ Y1 ðsð2Þ kl1 RZð2Þ Þ mð2Þ 1 ð1Þ ð1Þ ½Y0 ðsð2Þ kl1 RZð2Þ ÞY2 ðsð2Þ kl1 RZð2Þ Þ a44 ¼ ð2Þ , if ðsð2Þ Þ2 4 0; ð1Þ l3 2 sð2Þ kl1 RZð2Þ ( ) ð1Þ K1 ð9sð2Þ 9kl1 RZð2Þ Þ mð2Þ 1 ð2Þ ð2Þ ð1Þ ð2Þ , if ðsð3Þ Þ2 o0, a44 ¼ ð2Þ  ½ K0 ð9sð2Þ 9klð1Þ R Z Þ þ K ð9s 9 l kR Z Þ þ 2 1 1 ð1Þ 2 l3 9sð2Þ 9kl1 RZð2Þ ( ) ð1Þ J1 ðsð3Þ kl1 RZð2Þ Þ mð3Þ 1 ð1Þ ð1Þ ½J0 ðsð3Þ kl1 RZð2Þ ÞJ2 ðsð3Þ kl1 RZð2Þ Þ a45 ¼  ð3Þ , if ðsð3Þ Þ2 4 0; ð1Þ l3 2 sð3Þ kl1 RZð2Þ ( ) ð1Þ I1 ð9sð3Þ 9kl1 RZð2Þ Þ mð3Þ 1 ð2Þ ð3Þ ð1Þ ð2Þ a45 ¼  ð3Þ  ½ I0 ð9sð3Þ 9klð1Þ R Z Þ þI ð9s 9 l kR Z Þþ , if ðsð3Þ Þ2 o 0, 2 1 1 ð1Þ 2 l3 9sð3Þ 9kl1 RZð2Þ ( ) ð1Þ Y1 ðsð3Þ kl1 RZð2Þ Þ mð3Þ 1 ð1Þ ð1Þ ð3Þ ð2Þ ð3Þ ð2Þ ½Y0 ðs kl1 RZ ÞY2 ðs kl1 RZ Þ a46 ¼  ð3Þ , if ðsð3Þ Þ2 4 0; ð1Þ l3 2 sð3Þ kl1 RZð2Þ ( ) ð1Þ K1 ð9sð3Þ 9kl1 RZð2Þ Þ mð3Þ 1 ð1Þ ð3Þ ð2Þ ð3Þ ð1Þ ð2Þ a46 ¼  ð3Þ  ½K0 ð9s 9kl1 RZ Þ þ K2 ð9s 9l1 kRZ Þ þ , if ðsð3Þ Þ2 o 0, a51 ¼ 0, a52 ¼ 0, ð1Þ 2 l3 9sð3Þ 9kl1 RZð2Þ   ð2Þ ð2Þ 2 ð2Þ ð2Þ 2 a53 ¼ sð2Þ J1 ðsð2Þ klð1Þ a53 ¼ sð2Þ I1 ð9sð2Þ 9klð1Þ 1 RZ Þ if ðs Þ 4 0; 1 RZ Þ, if ðs Þ o0,   ð2Þ ð2Þ 2 ð2Þ ð2Þ 2 a54 ¼ sð2Þ Y1 ðsð2Þ klð1Þ a54 ¼ sð2Þ K1 ð9sð2Þ 9klð1Þ 1 RZ Þ if ðs Þ 4 0; 1 RZ Þ, if ðs Þ o 0, ð2Þ ð3Þ 2 a55 ¼ sð3Þ J1 ðsð3Þ klð1Þ 1 RZ Þ if ðs Þ 40;

ð2Þ ð3Þ 2 a55 ¼ 9sð3Þ 9I1 ð9sð3Þ 9klð1Þ 1 RZ Þ, if ðs Þ o 0,

ð2Þ ð3Þ 2 a56 ¼ 9sð3Þ 9K1 ð9sð3Þ 9klð1Þ 1 RZ Þ, if ðs Þ o 0, a61 ¼ 0, a62 ¼ 0, a63 ¼ 0, a64 ¼ 0 ( ) ð1Þ ð3Þ J1 ðsð3Þ kl1 RZð3Þ Þ m 1 ð1Þ ð1Þ ½J0 ðsð3Þ kl1 RZð3Þ ÞJ2 ðsð3Þ kl1 RZð3Þ Þ a65 ¼ ð3Þ if ðsð3Þ Þ2 4 0; ð1Þ l3 2 sð3Þ kl1 RZð3Þ ð3Þ

ð3Þ

a56 ¼ s Y1 ðs

ð1Þ kl1 R ð2Þ Þ,

Z

ð3Þ 2

if ðs Þ 40;

) ð1Þ I1 ð9sð3Þ 9kl1 RZð2Þ Þ 1 ð1Þ ð3Þ ð3Þ ð3Þ ð1Þ ð3Þ ½ I ð9s 9k l R Z Þ þ I ð9s 9 l kR Z Þ þ if ðsð3Þ Þ2 o0, 0 2 1 1 ð1Þ 2 lð3Þ 9sð3Þ 9kl1 RZð2Þ 3 ( ) ð1Þ Y1 ðsð3Þ kl1 RZð3Þ Þ mð3Þ 1 ð1Þ ð1Þ ð3Þ ð3Þ ð3Þ ð3Þ ½Y0 ðs kl1 RZ ÞY2 ðs kl1 RZ Þ a66 ¼ ð3Þ if ðsð3Þ Þ2 40; ð1Þ l3 2 sð3Þ kl1 RZð3Þ   9 8 ð1Þ K1 sð3Þ kl1 RZð2Þ = mð3Þ < 1 ð3Þ ð3Þ ð1Þ ð3Þ a66 ¼ ð3Þ  ½ K2 ð9sð3Þ 9klð1Þ if ðsð3Þ Þ2 o0,   ð1Þ 1 RZ Þ þ K2 ð9s 9l1 kRZ Þ þ sð3Þ kl RZð2Þ ; l : 2

a65 ¼

mð3Þ

(



3

(B1)

1

where

Zð1Þ ¼ 1þ

hð1Þ , R

ð2Þ

Zð2Þ ¼ 1 þ

hð1Þ l1 hð2Þ þ ð1Þ , R l1 R

ð2Þ

Zð3Þ ¼ 1þ

ð3Þ

hð1Þ l1 hð2Þ l1 hð3Þ þ ð1Þ þ ð1Þ : R l1 R l1 R

(B2)

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