The electro-elastic scattered fields of an SH-wave by an eccentric two-phase circular piezoelectric sensor in an unbounded piezoelectric medium

Mechanics of Materials 75 (2014) 1–12

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The electro-elastic scattered fields of an SH-wave by an eccentric two-phase circular piezoelectric sensor in an unbounded piezoelectric medium Hossein M. Shodja a,b,⇑, Hamid Jarfi a, Ehsan Rashidinejad a a b

Department of Civil Engineering, Sharif University of Technology, 11155-9313 Tehran, Iran Institute for Nanoscience and Nanotechnology, Sharif University of Technology, 11155-9161 Tehran, Iran

a r t i c l e

i n f o

Article history: Received 12 October 2013 Received in revised form 27 March 2014 Available online 5 April 2014 Keywords: DEMEIM Eccentric piezoelectric coating-fiber Eigenbody-force field Eigenelectric field SH-wave

a b s t r a c t The dynamic equivalent inclusion method (DEIM) which was first proposed by Fu and Mura (1983), in its original context has some shortcomings, which were pointed out and remedied by Shodja and Delfani (2009) who introduced the new consistency conditions along with the related micromechanically substantiated notion of eigenstress and eigenbody-force fields. However, these theories are bound to elastic media with isotropic phases. The present work extends the idea of the above-mentioned new DEIM to the dynamic electro-mechanical equivalent inclusion method (DEMEIM) for the treatment of the scattering of SH-waves by a two-phase circular piezoelectric obstacle bonded to a third phase piezoelectric matrix. All the three transversely isotropic media have the same rotational axis of symmetry and the same poling direction which are parallel to the axis of the coated fiber, but perpendicular to the direction of propagation of the incident SH-wave. In general, the nested circular media are considered to be eccentric, i.e., the core fiber has a coating with variable thickness. Realization of the nature of the behavior of the field quantities a priori and its appropriate implementation in to the new extended consistency conditions is a critical step to insure a rigorous mathematical framework. As it will be shown, the expansion of the Green’s function and the eigenelectric, eigenstress, and eigenbodyforce fields in terms of the eigenfunctions of the pertinent field equations rather than the commonly considered polynomials in the traditional equivalent inclusion method (EIM) leads to an accurate solution with high convergence rate. The exact analytical expression for the total scattering cross-section which is influenced by the piezoelectric couplings is derived. The effects of the piezoelectric couplings and the properties of the fiber, coating, and the matrix as well as the wave number on the electromechanical scattered fields are examined. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Piezoelectric composites exhibit higher sensitivity and lower mechanical losses than single phase piezoelectric ⇑ Corresponding author at: Department of Civil Engineering, Sharif University of Technology, 11155-9313 Tehran, Iran. Tel.: +98 21 66164209; fax: +98 21 66072555. E-mail address: [email protected] (H.M. Shodja). http://dx.doi.org/10.1016/j.mechmat.2014.03.013 0167-6636/Ó 2014 Elsevier Ltd. All rights reserved.

materials, and are widely used in smart materials and structures such as sensors, actuators and electro-mechanical transducers, micro generators, and ultrasonic biomechanical imaging devices. Piezoelectric composites made by addition of piezoelectric ceramic fibers to a matrix has been recognized to be quite suitable for ultrasonics. PZT fibers in epoxy matrix has excellent application as active fiber composites in aerospace vehicle structures. By application of

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coating technology to the fibers, the electro-mechanical properties of the piezoelectric composites may be improved remarkably. Devices incorporating piezoelectric composites are often subjected to dynamic loadings, and thus the study of wave propagation within such composites and the pertinent scattering phenomenon by coated fiber made of distinct piezoelectric materials is of great interest. Wave propagation and scattering phenomenon in elastic media has been the subject of numerous studies. Eringen and Suhubi (1975) addressed two and three dimensional scattering of elastic waves due to cylindrical and spherical inhomogeneities, by the method of wave function expansion. Waterman (1976) and Pao and Varatharajulu (1976) employed the so-called ‘‘matrix theory’’ to study the scattering of elastic waves. Barratt and Collins (1965) obtained the scattering cross-section associated with the circular and spherical obstacles in elastic media for plane harmonic waves. On the other hand Pao and Mow (1962) studied the scattering of plane compressional waves by a spherical obstacle. Mikata and Nemat-Nasser (1991) extended the work of Pao and Mow (1962) to study the interaction of harmonic wave with a dynamically transforming spherical inhomogeneous inclusion containing uniform eigenstrain field, and subsequently Michelitsch et al. (2003) presented the dynamic Eshelby tensors for ellipsoidal inclusions with constant eigenstrains. An inter-phase layer, due to its role in enhancement of certain physical and mechanical properties of composites, is often inserted between the inhomogeneity and the matrix. The effect of such a transition zone on the dynamic behavior and scattering have been previously addressed by several authors (Paskaramoorthy et al., 1988; Liu and Kriz, 1996; Selsil et al., 2001). Shindo and Niwa (1996) solved the problem of a fiber reinforced composite medium with interfacial layer subjected to anti-plane shear waves. Also Shindo et al. (1997) and Sato and Shindo (2001) studied the multiple scattering of elastic waves in fiberreinforced composites with graded interfacial layer. The dynamic equivalent inclusion method (DEIM) proposed by Fu and Mura (1983) provides an interesting treatment for the determination of the elastic scattered fields of an ellipsoidal inhomogeneity subjected to time harmonic waves. This treatment is the extension of the equivalent inclusion method (EIM) given by Eshelby (1957). The formulation of DEIM in its original form given by Fu and Mura (1983) has some shortcomings in employment of the notion of the homogenizing eigenstrain field. Shodja and Delfani (2009) remedied the shortcomings through introduction of the concept of eigenbody-force field and the pertinent consistency conditions. Moreover, they extended the theory to consider multi-inhomogeneity system with eccentricity. EIM, due to its robustness, has been employed to various elastostatic inhomogeneity problems pertinent to elastic materials by numerous authors; a rather thorough review of the subject up to 1996 is given by Mura (1988) and Mura et al. (1996). The methodology has also been utilized to treat the static inhomogeneity problems associated with piezoelectric materials (Fan and Qin, 1995; Jiang et al., 1997; Dunn and Wienecke, 1997; Xiao and Bai, 1999; Mikata, 2000; Shodja et al., 2010); the corresponding consistency conditions associated to the equivalency of the

inhomogeneity problem with that of inclusion problem, in addition to the equivalency of the stress fields of the two problems, also consist of the equivalency of their elastic displacement fields. This view of point has been referred to as the electro-mechanical equivalent inclusion method (EMEIM) by Shodja et al. (2010). To date, the elastodynamic problems pertinent to the piezoelectric inhomogeneity/coated inhomogeneity embedded in a piezoelectric matrix have not yet been solved using the dynamic EMEIM (DEMEIM). Shindo and Togawa (1999) considered the multiple scattering of antiplane shear waves in fibrous piezoelectric composites with sliding interfaces using wave function expansion. Levin et al. (2002) studied the propagation of electro-acoustic waves in a piezoelectric medium with randomly distributed cylindrical inhomogeneities utilizing the Green’s function method. Ma and Wang (2005) described the scattering of electro-elastic waves by an ellipsoid inhomogeneity embedded in an infinite piezoelectric medium by integral equations in which the kernels are obtained using the pertinent Green’s functions. Kamali and Shodja (2010) presented an analytical solution for the scattering of Pwaves by a piezoelectric particle with functionally graded piezoelectric interfacial layers in a polymer matrix. But the problem of greater generality associated with the scattering of SH-waves by a piezoelectric double-inhomogeneity (coating-fiber) bonded to a piezoelectric matrix has not yet been studied. For further generalization of the problem one may disband the symmetry of the problem by considering eccentric piezoelectric coating-fiber system. With due attention to the above discussions, the current work is devoted to the formulation of a robust analytical methodology, namely, the DEMEIM suitable for studying the scattering of SH-waves by an eccentric piezoelectric coated-fiber system embedded in a piezoelectric matrix. In Section 2, the problem definition and the fundamental equations are presented. Section 3 is devoted to the development of the consistency conditions needed for the DEMEIM. The analytical expressions for the scattered elastic displacement and electric potential fields are derived in Section 4. The analytical expression for the total scattering cross-section is given in Section 5, the descriptive numerical examples and discussions are provided in Section 6, and at last, the concluding remarks are given in Section 7.

2. Problem definition and the fundamental equations Consider a two-phase circular cylindrical fiber in which the core, X and its surrounding coating, W are made of different piezoelectric materials. This piezocomposite cylinder is embedded in another piezoelectric medium, D which is unbounded. The origin of the Cartesian coordinates (x1 , x2 , x3 ) and the corresponding polar coordinates (r, h) in x1 x2 -plane is fixed at the center of the core region X. The axis of the core coincides with the x3 -axis and is parallel to the axis of X [ W. The circular regions X and X [ W with radii, respectively, R0 and R1 are generally eccentric; thus, the coating thickness is variable. Assume that the center of X [ W is located at (D; 0; 0), D > 0 as shown in Fig. 1. The regions, X, W, and D, are made of

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different piezoelectric materials and are assumed to be transversely isotropic with the x3 -axis as the axis of rotational symmetry which coincides with their poling directions. The ascribed three-phase system is subjected to an incident anti-plane time harmonic shear (SH) wave which as shown in Fig. 1 is assumed to propagate in the x1 direction. Let denote the electric displacement vector by D, the electric potential function by /, and the piezoelectric and dielectric tensors by e and j, respectively. Then the piezoelectric constitutive relations may be written as:

Laplacian operator in cylindrical coordinates. Suppose that the property, P i  li ; qi ; ei , or ji belongs to the core, X for i = 1, the coating, W for i = 2, and the matrix, D if no index is attached. Then, DP in the above relations is defined as:

rij ¼ C ijkl ekl  ekij Ek ;

ð1aÞ

u3 ðx; tÞ ¼ ui3 ðx; tÞ þ us3 ðx; tÞ;

ð7aÞ

Di ¼ eijk ejk  jij Ej ;

ð1bÞ

/ðx; tÞ ¼ /i ðx; tÞ þ /s ðx; tÞ;

ð7bÞ

where C is the elastic moduli tensor, E is the electric field, r is the stress tensor, e is the strain tensor and is related to the displacement field, u as:

1 2

eij ¼ ðui;j þ uj;i Þ:

ð2Þ

The electric field can be obtained from the electric potential as:

Ei ¼ 

@/ : @xi

ð3Þ

In the absence of body forces and electric charge density, the equations of motion and charge equation of electro-statics, respectively, become:

rij;j ¼ qu€i ; i; j ¼ 1; 2; 3;

ð4aÞ

Dj;j ¼ 0;

ð4bÞ

j ¼ 1; 2; 3;

in which q is the mass density and the symbol ‘‘.’’ over a quantity indicates its time derivative. For the anti-plane problem, it can be shown that by utilization of Eqs. (1–3), Eqs. (4a) and (4b) reduce to:

lr2 u3 þ Dlr2 u3 þ er2 / þ Der2 / ¼ qu€3 þ Dqu€3 ;

ð5aÞ

er2 u3 þ Der2 u3  jr2 /  Djr2 / ¼ 0;

ð5bÞ

respectively, where for brevity e  e113 , j  j11 , and l ¼ C 4444 . In Eqs. (5a) and (5b), r2 is the two-dimensional

8 > < P1  P DP ¼ P 2  P > : 0

in X; in W;

ð6Þ

in D:

The total displacement field component, u3 and the total electric potential, / may be expressed as:

i

in which ui3 ðx; tÞ and / ðx; tÞ are the incident displacement and electric potential, respectively. The scattered displacement, us3 ðx; tÞ and electric potential, /s ðx; tÞ are induced due to the presence of the piezoelectric double-inhomogeneity X [ W in D. The scattered fields, us3 ðx; tÞ and /s ðx; tÞ are to be evaluated such that the total displacement and electric potential fields, u3 ðx; tÞ and /ðx; tÞ satisfy the governing differential equations, (5a) and (5b). 3. Dynamic electro-mechanical equivalent inclusion method (DEMEIM) In this section, the DEIM approach presented by Shodja and Delfani (2009), incorporating the exact and micromechanically justified notion of eigenestress and eigenbodyforce fields, has been extended to DEMEIM which takes into account the electro-mechanical couplings as well as the instrumental concept of the eigenfields including eigenstress, eigenbody-force, and eigenelectric fields. To this end, the piezoelectric double-inhomogeneity system shown in Fig. 1 is replaced by a piezoelectric double-inclusion system in which all the three constituent phases are made of the same piezoelectric material as the matrix, but the fiber and the coating regions must have appropriate distributions of eigenstress, eigenelectric, and eigenbody-force fields such that the stress and electric displacement fields as well as the gradient of the stress field of the two systems are identically equal.

Fig. 1. Piezoelectric double-inhomogeneity system subjected to SH-wave.

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For the ascribed piezoelectric double-inclusion problem, the coupled governing differential equations in terms of the displacement field u3 ðx; tÞ, and the electric potential field, /ðx; tÞ are: 2

2

lr u3 þ er / ¼ qu€3 þ er2 u3  jr2 / ¼ jEj;j ;

q3

þr

 3j;j ;

j ¼ 1; 2;

j ¼ 1; 2;

ð8aÞ ð8bÞ

where r3j , q3 , and Ej are the unknown eigenstress, eigenbody-force, and eigenelectric fields, respectively. The pertinent total displacement and total electric potential fields can be written as

u3 ðx; tÞ ¼ ui3 ðx; tÞ þ u3 ðx; tÞ;

ð9aÞ ð9bÞ

where u3 ðx; tÞ and / ðx; tÞ represent the scattered displacement and electric potential fields induced by the eigenfields in the piezoelectric double-inclusion system and must be identically equal to us3 ðx; tÞ and /s ðx; tÞ given by Eqs. (7a) and (7b), respectively. The incident anti-plane elastic wave and the induced incident electric potential can be expressed as

ui3 ðx; tÞ ¼ u0 expðiksh r cos h  ixtÞ; e

j

ð10Þ

u0 expðiksh r cos h  ixtÞ;

ð11Þ

pffiffiffiffiffiffiffi Respectively, where i = 1. Assuming time harmonic ixt ~ ~ ðxÞeixt and /ðx; tÞ ¼ /ðxÞe solutions, uðx; tÞ ¼ u , the governing Eqs. (8a) and (8b) for the unknown amplitudes ~ ~ ðxÞ and /ðxÞ u reduce to:

lr2 u3 þ er2 / þ qx2 u3 ¼ q3 þ r3j;j ; j ¼ 1; 2;

ð12aÞ

er2 u3  jr2 / ¼ jEj;j ;

ð12bÞ

j ¼ 1; 2;

in which the symbol ’’’’ over uðxÞ and /ðxÞ has been omitted. Also for convenience, in the subsequent developments whenever applicable, the amplitudes pertinent to the field quantities will be presented without this symbol. The equivalency of the stress field, gradient of the stress field, and electric displacement field associated to the piezoelectric double-inhomogeneity system and those associated with the double-inclusion system leads to the following coupled consistency conditions:

Dlu3;j ðxÞ  DeEj ðxÞ þ r3j ðxÞ ¼ 0;

j ¼ 1; 2;

Dqx2 u3 ðxÞ þ q3 ðxÞ ¼ 0; Deu3;j ðxÞ þ DjEj ðxÞ þ jEj ðxÞ ¼ 0;

The scattered displacement and electric potential due to the introduced homogenizing eigenfields can be obtained by the simultaneous solution of Eqs. (12a) and (12b) as:

Z us3 ðxÞ ¼  G33 ðx  x0 Þr3j;j ðx0 Þdx0 X[W Z  G33 ðx  x0 Þq3 ðx0 Þdx0 ZX[W  G34 ðx  x0 ÞjEj;j ðx0 Þdx0 ; /s ðxÞ ¼ 

Z

j ¼ 1; 2:



ð13cÞ

Indeed, once Eqs. (12a) and (12b) are solved for the total displacement and electric potential fields in terms of the unknown eigenfields, then upon substitution into the consistency conditions (13a)–(13c) the exact solutions to the unknown eigenfields enforcing the equivalency of the piezoelectric double-inhomogeneity and double-inclusion problems are obtained.

ð14aÞ



ðj ¼ 1; 2Þ;

ð14bÞ

G43 ðx  x0 Þr3j;j ðx0 Þdx0

Z

ZX[W X[W

G43 ðx  x0 Þq3 ðx0 Þdx0 G44 ðx  x0 ÞjEj;j ðx0 Þdx0 ;

in which Gij ðx  x0 Þ (i, j = 1, 2, 3, 4) correspond to the time harmonic electro-elastic Green’s functions. For i, j = 1, 2, 3, Gij ðx  x0 Þ represents the amplitude of the elastic displacement at point x and time t in the xi direction due to a unit impulsive force acting at point x0 and time t = 0 in the xj -direction. While for i = 1, 2, 3, Gi4 ðx  x0 Þ is the elastic displacement at point x and time t in the xi -direction due to a unit electric charge located at point x0 and time t = 0. Likewise, for j = 1, 2, 3, G4j ðx  x0 Þ is the electric potential at point x and time t due to a unit impulsive force applied at point x0 and time t = 0 in the xj -direction, and G44 ðx  x0 Þ represents the electric potential at point x and time t due to an applied unit electric charge at point x0 and time t = 0. The non-zero components of Gij ðx  x0 Þ (i, j = 1, 2, 3, 4) associated to the propagation of SH-waves in transversely isotropic homogeneous piezoelectric medium have been obtained by Levin et al. (2002):

G33 ðx  x0 Þ ¼ i

j

ð1Þ

4ðlj þ e2 Þ

H0 ðksh jx  x0 jÞ;

G34 ðx  x0 Þ ¼ G43 ðx  x0 Þ ¼ i

ð15aÞ

e ð1Þ H ðksh jx  x0 jÞ; 4ðlj þ e2 Þ 0 ð15bÞ

G44 ðx  x0 Þ ¼ i

ð13aÞ ð13bÞ

ðj ¼ 1; 2Þ;

X[W

X[W

/ðx; tÞ ¼ /i ðx; tÞ þ / ðx; tÞ;

/i ðx; tÞ ¼

4. The scattered elastic displacement and electric potential fields

e2 4ðlj

2þ

ð1Þ

je2 Þ

H0 ðksh jx  x0 jÞ þ

1 lnjx  x0 j; 2pj ð15cÞ

ð1Þ

in which H0 refers to the Hankel function of the first kind pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi of order zero, ksh ¼ x qj=ðlj þ e2 Þ denotes the wave number, and x is the frequency of the time harmonic incið1Þ dent wave. The functions H0 ðksh jx  x0 jÞ and lnðjx  x0 jÞ have the following representation:

ð1Þ H0 ðksh jx  x0 jÞ ¼

8 1 X > 0 > 0 0 > an Jn ðksh rÞHð1Þ > n ðksh r Þcos½nðh  h Þ; r < r ; < n¼0

1 X > > > an Jn ðksh r 0 ÞHnð1Þ ðksh rÞcos½nðh  h0 Þ; r > r 0 ; > : n¼0

ð16aÞ

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lnðjx  x0 jÞ ¼

8 1 X   > > 1 r n >  lnðr0 Þ þ cos½nðh  h0 Þ; r < r0 ; > n r0 < n¼1

1 X >   > 1 r0 n > cos½nðh  h0 Þ; > :  lnðrÞ þ n r

r > r0 ;

ðjÞ analytical formulations of the functions U ðjÞ n ðrÞ, V n ðrÞ, and Q(r) which are given in Appendix B depend on the position of the field point x. While, the displacement and the electric potential fields within the matrix become:

n¼1

ð16bÞ

us3 ðxÞ ¼

1 X  sn Hð1Þ n ðksh rÞcosðnhÞ; x 2 D  fX [ W [ W g;

in which a0 ¼ 1, and an ¼ 2 for n ¼ 1; 2; 3; . . ., and J n and are the Bessel and Hankel functions of the first kind of order n, respectively. Consider the following series expansion for the incident displacement and electric potential:

ui3 ðxÞ

¼ u0

1 X

an i J n ðksh rÞ cosðnhÞ;

ð17aÞ

1 X n an i Jn ðksh rÞ cosðnhÞ:

ð17bÞ

n¼0

e

/i ðxÞ ¼

j

u0

n¼0

Subsequently the unknown eigenfields, and Ei (i = 1, 2) may be expressed as: 

e

/s ðxÞ ¼

j



r3i (i = 1, 2), q3 ,

8 1n o X ð1Þ > ð3Þ ð4Þ > > fn ðrÞ;fn ðrÞ; fn ðrÞ cosðnhÞ; x 2 X; > <

n¼0 r31 ðxÞ; q3 ðxÞ; E1 ðxÞ ¼ X 1 n > >

> > :

  32 ðxÞ; E2 ðxÞ

r

o ð3Þ ð4Þ g ð1Þ n ðrÞ; g n ðrÞ;g n ðrÞ cosðnhÞ; x 2 W;

fnðjÞ ðrÞ ¼



¼

8 1n o X ð2Þ > ð5Þ > > f n ðrÞ; fn ðrÞ sinðnhÞ; > <

g ðjÞ n ðrÞ ¼ x 2 X;

n¼0 1 n >X

> > > :

o ð2Þ ð5Þ g n ðrÞ; g n ðrÞ sinðnhÞ; x 2 W;

n¼0

ð1Þ

ð5Þ

where fn ðrÞ, g n ðrÞ; :::; g n ðrÞ are the unknown functions to be determined. By utilizing Eqs. (15, 16, and 18), the terms appearing in the expression (14a) and (14b), respectively, for us3 ðxÞ and /s ðxÞ are integrated. In particular, a careful treatment of the involved integrals is required. To this end a precise and rigorous remedy is given in Appendix A. When the field point, x is inside one of the regions, X, W , or W , then us3 ðxÞ and /s ðxÞ will have the following forms:

  1 X n þ 1 d h ð1Þ ð2Þ U nþ1 ðrÞ þ U nþ1 ðrÞ 2U ð3Þ ðrÞ þ þ n r dr n¼0 i n  1 d h ð4Þ ð5Þ ð1Þ U n1 ðrÞ  þeU nþ1 ðrÞ þ eU nþ1 ðrÞ  r dr io ð2Þ ð4Þ ð5Þ U n1 ðrÞ þ eU n1 ðrÞ  eU n1 ðrÞ cosðnhÞ i d h ð1Þ ð4Þ þ U þ eU 0 cosðhÞ; ð19Þ dr 0

us3 ðxÞ ¼

/s ðxÞ ¼

 i d h ð4Þ V n ðrÞ þ V nð5Þ ðrÞ cos½ðn  1Þh r dr j n¼1 h 1  i X n d ð5Þ V ð4Þ  þ n ðrÞ  V n ðrÞ cos½ðn þ 1Þh r dr n¼1 e

n¼1

us3 ðxÞ 

1  X n

þ Q ðrÞ cos h;

þ

ð20Þ

respectively. Where regions W and W are defined in Appendix A as well. In the above expression the exact

r nþ1

ð5Þ Cð4Þ n  Cn



x 2 D  fX [ W [ W g:

ð22Þ

1

X

 ðjÞ m AðjÞ ; nm J m ðksh rÞ þ Bnm r

0 < r 6 R0 ;

ð23aÞ

m¼0

n¼0

ð18bÞ ð1Þ

1 X cos½ðn þ 1Þh

The unknown function, TðrÞ and the unknown coeffið5Þ cients, sn , Cð4Þ (n = 0, 1, 2, . . .) appearing in the n , and Cn expressions (21) and (22) are given in Appendix C. With due attention to the series expansions of the Green’s functions associated with the problem (Eqs. (15a) and (15b)), and the fact that the incident waves are given in terms of the Fourier–Bessel series (Eqs. (16a) and ð1Þ ð1Þ (16b)), the unknown functions fn ðrÞ, g n ðrÞ, . . ., and ð5Þ g n ðrÞ may be expressed in the following series forms:

ð18aÞ



us3 ðxÞ þ

þ TðrÞ cos h; n

ð21Þ

n¼0

Hð1Þ n

1

 X ðjÞ ð1Þ ðjÞ m ðjÞ m C ðjÞ ; nm J m ðksh rÞ þ Dnm Hm ðksh rÞ þ Enm r þ F nm r m¼0

R0 < r < RðhÞ;

ð23bÞ

ðjÞ ðjÞ ðjÞ ðjÞ ðjÞ in which AðjÞ nm , Bnm , C nm , Dnm , Enm , and F nm are the unknown coefficients. These unknowns may be calculated by two different strategies. In the first approach, the functions ðjÞ s U ðjÞ n ðrÞ and V n ðrÞ appearing in the expressions for u3 ðxÞ s and / ðxÞ (given by Eqs. (19) and (20)) are expanded in terms of Bessel, Hankel, and integer powers of r. subsequently, with the aid of the consistency Eqs. (13a)–(13c) ð1Þ ð1Þ the unknown functions fn ðrÞ, g n ðrÞ . . ., are calculated. In the second approach, suppose that for N terms of the series in expressions (18a) and (18b), the eigenfields are represented with satisfactory accuracies. This assumption gives ðjÞ ðjÞ rise to 10N functions fn ðrÞ and g n ðrÞ with n = 0, 1, . . ., N  1 and j = 1, 2, . . ., 5. Subsequently, if M terms of the series in Eqs. (23a) and (23b) are sufficient for the desired approximation of these functions, then it turns out that ðjÞ there will be 10N * M unknowns, AðjÞ nm and Bnm pertinent to ðjÞ ðjÞ the region X and 20N * M unknowns, C nm , DðjÞ nm , Enm , and ðjÞ F nm associated with the coating W. Thus, for the complete determination of the unknown coefficients, the consistency Eqs. (13a)–(13c) are written at 2N * M points inside X and at 4N * M points inside W to obtain 30N * MEqs. needed to solve for the 30N * M unknowns. ðjÞ ðjÞ Once the functions fn ðrÞ and g n ðrÞ are obtained the scattered displacement and electric potential in each of the inclusions and the matrix are calculated utilizing relations (19) and (20) and relations (21) and (22), respectively.

5. Total scattering cross-section The study of total scattering cross-section is of great interest in non-destructive evaluations. For the piezoelectric

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double-inhomogeneity problem under consideration, the scattering cross-section is conveniently calculated by the proposed equivalent piezoelectric double-inclusion method. Total scattering cross-section Rsh is defined as the ratio of the mean scattered energy in all directions to the mean energy of the incident wave per unit area normal to the direction of propagation. Associated with the incident wave in a piezoelectric medium, the average energy flux per unit area is

i 1 h  s /s Þ ; s  D sj  r  sij usj Þ þ ðDsi / hk_ i i ¼  ix ðrsij u i 4

ð24Þ

in which the symbol ‘‘’’ over a quantity indicates complex conjugate of the quantity. In one period, the amount of scattered energy across a closed surface, S is given by

K¼

Z

k_ i ni dA;

ð25Þ

S

where ni is the ith component of the radial unit vector. As the radius of the surface S tends to infinity (n ! 1) in view of Eqs. (22) and (38):

lim

Z

n!1 S



  s ni /s dA ¼ lim s  D Dsi ni / i

Z

n!1 S

inhomogeneity system and in addition, the high convergence rate of the eigenfunction expansions is demonstrated. In the fourth example, the scattered stress fields for both concentric and eccentric coating-fiber systems are compared. The elastic and electro-mechanical properties of the materials considered in this section are given in Tables 1 and 2, respectively. In all of the examples of this ðjÞ ðjÞ ðjÞ ðjÞ section the unknown coefficients AðjÞ nm , Bnm , C nm , Dnm , Enm , ðjÞ and F nm are determined by employing the second approach discussed in Section 4. 6.1. SH-waves scattered by a fiber in an elastic isotropic fibermatrix system: verification It is readily seen that, by omission of the piezoelectric effect from the developments of Sections 4 and 5 the formulations given by Sarvestani et al. (2008) are recovered; the expression for the scattered displacement field, us3 ðxÞ given by Eqs. (19) reduces to:

  1 i X n þ 1 d h ð1Þ ð2Þ U nþ1 ðrÞ þ U nþ1 ðrÞ 2U ð3Þ ðrÞ þ þ n r dr n¼0   i n  1 d h ð1Þ ð2Þ U n1 ðrÞ  U n1 ðrÞ cosðnhÞ   r dr d h ð1Þ i U cosðhÞ; þ dr 0

us3 ðxÞ ¼

 s s   s /s dA ¼ 0;  D Dr / r ð26Þ

which shows that there is no electric power flux in the direction of wave propagation. For an incident SH-wave, by virtue of Eqs. (10), (11), and (24) it can be shown that

1 hk_ i i ¼ ksh x 2



lþ

e2

j

ju0 j2 :

ð27Þ

Rsh ¼

After some manipulations and using asymptotic expansions for us3 ðxÞ and /s ðxÞ at infinity the expression for the total scattering cross-section becomes:

Rsh ¼

 2 1 4 X 1  Sn  : ksh n¼0 en u0 

in which the corresponding expression for U ðjÞ n ðrÞ, j = 1, 2, 3 are given in the work of Sarvestani et al. (2008). Subsequently, the total scattering cross-section RSH becomes:

ð28Þ

It can be observed from (28) that the expression for the total scattering cross-section is similar to those for the pure elastic case Sarvestani et al. (2008) but the parameter sn in (28) includes both the electro-mechanical effects. 6. Results and discussions For the demonstration of the applicabilities and efficacy of the proposed theory several descriptive examples are stated and solved. For verification of the formulations, the first example is devoted to the recovery of the analytical expression of the total scattering cross-section pertinent to the SH-waves scattered by an isotropic elastic long fiber in an unbounded isotropic matrix. This simple bench mark problem, previously considered by Eringen and Suhubi (1975) and Sarvestani et al. (2008), is reconsidered for comparison. The second example is devoted to concentric double-inhomogeneity system (coated fiber embedded in an infinite matrix) while in the third example, the scattering cross-section corresponding to a single inhomogeneity embedded in an unbounded matrix is calculated and compared with that of an eccentric double-

 2 1 4 X 1  Sn  : ksh n¼0 en u0 

For which the parameters si , i = 0, 1, 2, . . . are presented by Sarvestani et al. (2008), or can be obtained from Eqs. (40a)–(40c) in Appendix C of the present work provided that the piezoelectric terms are omitted. For further validation of the current analytical solution, the problem of a cylindrical SiC fiber embedded in an infinite Al matrix subjected to SH-waves is reexamined via the present theory. The dimensionless total scattering crosssection, Rsh =R0 versus the dimensionless wave number,

Table 1 Material properties for the considered elastic materials. Material property

Epoxy

Aluminum

SiC

PZT-4(E)

C44 (GPa) q (kg/m3) j * 1010 (F/m)

1.731 1202 0.38

26.5 2706 –

188.1 3181 –

25.6 7500 64.64

Table 2 The electro-mechanical properties for the considered piezoelectric materials. Material property

BaTiO3

PZT-5H

PZT-4

C44 (GPa) e15 (C/m2) j * 1010 (F/m) q (kg/m3)

43 11.6 112 5700

35.5 17 151 7500

25.6 12.7 64.64 7500

H.M. Shodja et al. / Mechanics of Materials 75 (2014) 1–12

7

Fig. 2. Variation of the dimensionless total scattering cross-section against dimensionless shear wave number.

ksh R0 obtained using DEMEIM compared with the result of Eringen and Suhubi (1975) who employed the wave function expansion approach. It can be observed that the results of the present study whit just the first six terms of the expansions (18a) and (18b) are in very good agreement with the previous solution given by Eringen and Suhubi (1975) (Fig. 2). 6.2. SH-waves scattered by a concentric piezoelectric coated fiber Consider a piezoelectric fiber made of BaTiO3 having a coating of arbitrary thickness made of PZT-5H. The coated-fiber system in which the coating and the fiber have circular cross-section with no eccentricity is embedded in an infinite PZT-4 matrix. An incident SH-wave propagating along the direction of the x1 -axis will be scattered by the coated-fiber system. In order to examine the piezoelectric effect of the PZT-4 matrix on the total scattering cross-section comparison is made to the result pertinent to the case where the matrix is made of PZT-4(E). PZT4(E) refers to the purely elastic material whose elastic properties are identical to those of PZT-4. For the dimensionless wave number, ksh R0 ¼ 1, as shown in Fig. 3 the variation of the dimensionless total scattering cross-section, Rsh =R0 has a nearly linear relation with the dimensionless thickness, h=R0 for both case of the PZT-4 matrix and PZT-4(E) matrix; it should be recalled that R0 is the radius of the core fiber. It is observed that the piezoelectric effect is quite significant; it causes a nearly uniform increase in the dimensionless total scattering cross-section as h=R0 varies between 0 and 0.2. 6.3. SH-waves with large wave number scattered by an eccentric piezoelectric coated fiber This example considers both the single- and doubleinhomogeneity systems in which the constituent phases are made of different piezoelectric materials. More specifically, three cases of: (1) BaTiO3 fiber, (2) PZT-5H fiber, and (3) BaTiO3 fiber with PZT-5H eccentric coating are

Fig. 3. The piezoelectric effect of PZT-4 matrix on the variation of the dimensionless total scattering cross-section as a function of the dimensionless thickness of the coating pertinent to the dimensionless wave number, ksh R0 ¼ 1 .

considered. In all the three cases, the surrounding matrix is made of PZT-4. For the case of the coated fiber the ratio of the radius of the coating to that of the core is R1 =R0 ¼ 1:5 and the dimensionless eccentricity is D=R0 ¼ 0:4 . It should be noted that for all the three cases the same value of R0 is used. The dimensionless wave number of the incident SHwave is allowed to vary from 0 to 3.4 (0 6 ksh R0 6 3:4). The variations of the dimensionless total scattering cross-section in terms of the dimensionless wave number for the three cases mentioned above are plotted in Fig. 4. From the magnified plot in Fig. 4, it is evident that, for wave numbers up to about ksh R0 ¼ 0:4, with the naked eye the values of Rsh =R0 for all the considered cases coincide. However, beyond ksh R0 ¼ 1 the difference between their values of Rsh =R0 begins to show and become increasingly

8

H.M. Shodja et al. / Mechanics of Materials 75 (2014) 1–12

Fig. 4. The differences between the variations of the total scattering cross-section increase with the dimensionless wave number. For the case of BaTiO3 fiber with PZT-5H eccentric coating R1 =R0 ¼ 1:5 and D=R0 ¼ 0:4 have been considered.

noticeable as ksh R0 increases. That is the effects of the electro-mechanical as well as the coating become more pronounced at larger values of the wave number. For demonstration of the convergence behavior of the solution, the trend of the convergence of Rsh =R0 with the considered number of terms, n in the series solution pertinent to ksh R0 ¼ 1 and 3.4 are shown in Fig. 5. The discrepancy between the value of Rsh =R0 corresponding to N = 5 and 6 is only about 1.4% for ksh R0 ¼ 1 and 2.2% for ksh R0 ¼ 3:4 which reveals the high convergence rate of the present methodology. 6.4. Effects of the coating eccentricity and the wave number on the scattered interface stresses Throughout this example the fiber, coating, and matrix are made of BaTiO3 , PZT-4, and Epoxy, respectively. Two cases of concentric and eccentric coatings are considered; for both cases the radius of the core fiber, R0 is the same. In the case of the concentric coating we take R1 =R0 ¼ 1:5 and in the case of the eccentric coating the dimensionless

eccentricity is D=R0 ¼ 0:4 . for the dimensionless wave number, ksh R0 ¼ 1, the distributions of the normalized stress components, s3r ¼ rs3r =ri31 and s3h ¼ rs3h =ri31 along the fiber-coating interface (just inside the fiber) are given in Figs. 6 and 7, respectively. These figures compare the corresponding interface stresses for two cases of concentric and eccentric coatings. As it is seen from Fig. 6, the absolute maximum of s3r along the interface for both the concentric and eccentric cases occurs at h ¼ 0 (located on the interface opposite to the one facing the incoming wave). s3r also has a local maximum at h ¼ p in both cases. The case of eccentric coating in comparison to the concentric coating leads to higher and lower values of s3r at h ¼ 0 and h ¼ p, respectively. The minimum value of s3r occurs at h ¼ 1:66 for the concentric case, and at h ¼ 1:58 for the eccentric case. s3h at h ¼ 0 and h ¼ p for both the concentric and eccentric coating as in Fig. 7. In the case of the concentric coating s3h attains its minimum at h ¼ 0:78 and its maximum at ¼ 2:36 . In the case of the introduced eccentricity, the minimum and the maximum values shift a little to the left, that is, to h ¼ 0:65 and h ¼ 2:07, respectively. It

Fig. 5. Trend of convergence of the dimensionless total scattering cross-section with the considered number of terms, n in the series solution.

H.M. Shodja et al. / Mechanics of Materials 75 (2014) 1–12

9

Fig. 6. Comparison of the variation of the normalized scattered stress component, s3r ¼ rs3r =ri31 along the fiber-coating interface just inside the fiber corresponding to two cases of the concentric (R1 =R0 ¼ 1:5 and D ¼ 0) and eccentric (D=R0 ¼ 0:4) coatings.

Fig. 7. Comparison of the variation of the normalized scattered stress component, s3h ¼ rs3h =ri31 along the fiber-coating interface just inside the fiber corresponding to two cases of the concentric (R1 =R0 ¼ 1:5 and D ¼ 0) and eccentric (D=R0 ¼ 0:4) coatings.

Fig. 8. Comparison of the variation of the normalized scattered stress component, s3r ¼ rs3r =ri31 along the fiber-coating interface just inside the fiber with R1 =R0 ¼ 1:5, for the case of eccentric coating (D=R0 ¼ 0:4) and different values of ksh R0 ¼ 0:5; 1; 1:5.

10

H.M. Shodja et al. / Mechanics of Materials 75 (2014) 1–12

should also be noted that the mentioned minimum has a larger absolute value than the mentioned maximum. The effect of the dimensionless wave number on the distribution of the radial stress, s3r along the fiber-coating interface just inside the fiber is demonstrated in Fig. 8. In this study, the coating and fiber are eccentric with D=R0 ¼ 0:4 and it is assumed that R1 =R0 ¼ 1:5. Three different dimensionless wave numbers ksh R0 ¼ 0:5; 1; and 1:5 have been considered. As it is seen, larger values of wave number (smaller wave lengths) result in larger value of the shear stress component, s3r , everywhere (0 6 h 6 p); the effect is particularly notable at and in vicinities of h ¼ 0 and h ¼ p. 7. Conclusions The present work offers a new micromechanicallybased approach which is applicable to the electro-elastic fields of SH-waves scattered by a transversely isotropic piezoelectric eccentric coated fiber embedded in an infinite transversely isotropic piezoelectric media, while the traditional wave-function expansion approach commonly used in the literature ceases to hold when the geometry of the obstacle is not symmetric. The present analytical formulation which, as demonstrated, is capable of treating piezoelectric eccentric coating-fiber ensemble introduces the notion of ‘‘dynamic electro-mechanical equivalent inclusion method’’ (DEMEIM) for the first time. As explored in several descriptive examples with various degrees of complexities, the proposed method produces reasonably accurate result with only a few terms of the series. It is also

noteworthy to mention that consideration of large wave numbers is made possible as well. Appendix A Let IX[W denote the integration over X [ W in expressions (12a) and (12b). To evaluate IX[W , the following sub-regions W and W are defined as shown in Fig. 9:

W ¼ fx : R0 < jxj < R1  Dg;

ð29aÞ

W ¼ fx : R1  D < jxj < R1 þ Dg:

ð29bÞ

In Fig. 9, point x associated with the polar coordinates (r, h) is the observation point, whereas point x0 associated with (r0 , h) is location of the applied unit impulse load or unit electric charge. Associated to each point x0 there is a point on the boundary of X [ W with position vector Rðh0 Þ. According to the location of x, the integration IX[W can be carried out conveniently as follows:

I X[W ¼

Z 2p Z

Z 2p Z

Z 2p Z 0

þ

Z 2p Z 0

Z 2p Z 0

jRðh0 Þj

R0

0

. . . dr dh0

r

0

. . . dr dh0 ;

x 2 X;

ð30aÞ

R0

0

I X[W ¼

0

. . . dr dh0 þ 0

0

þ

r

R0

0

. . . dr dh0 þ

Z 2p Z

0 jRðh0 Þj

0 0

. . . dr dh0 ;

r

0

. . . dr dh0

R0

x 2 W ;

ð30bÞ

r

Fig. 9. Definition of regions W and W for the purpose of the evaluation of the integral, IX[W . The boundaries of W and W are tangent to the boundary of W at h0 ¼ 0 and h0 ¼ p, respectively.

11

H.M. Shodja et al. / Mechanics of Materials 75 (2014) 1–12

I X[W ¼

Z 2p Z 0

 

R0

Z 2p Z

0

. . . dr dh0 þ

0

Z

Z 2p Z 0

Z 2p Z

0

h0 ðrÞ jRðh0 Þj

. . . dr dh0 þ

r

0

Z

0

. . . dr dh0 þ

0

h0 ðrÞ Z r

jRðh0 Þj

0

. . . dr dh0 R0 0

When jxj P R1 þ D, the integration simply becomes:

I X[W ¼

Rðh0 Þ

0



0

. . . dr dh ;

x 2 D  fX [ W [ W g;

ð31Þ Q ðrÞ ¼

Appendix B

Q ðrÞ ¼ ðjÞ The functions U ðjÞ n ðrÞ, V n ðrÞ, and Q(r) associated with Eqs. (22 and 23) in terms of the unknown functions, ðjÞ ðjÞ fn ðrÞ and g n ðrÞ (n = 0, 1, 2, . . . ; j = 1, 2, 3, 4, 5) for the field point ,x within each of the regions can be obtained by some manipulations as Eqs (32, 33, and 34), respectively:



lþ

e2

j

ð1Þ 4iU ðjÞ n ðrÞ ¼ pHn ðksh rÞ

Z

r

0 R0

Z þ pJ n ðksh rÞ

r

fnðjÞ ðr 0 ÞJ n ðksh r 0 Þr 0 dr

Q ðrÞ ¼

lþ

2

e

j

ð1Þ 4iU ðjÞ n ðrÞ ¼ pHn ðksh rÞ

þp

Z

R0

jRðh Þj

0

1 r 1 r

Z

0

n

0 0 0 0 ðr 0 Þ F ðjÞ n ðr ; h Þr dr dh

0

Z

 r 0 ð4Þ f0 ðr 0 Þr 0 dr ; R0

x 2 W ;

0

x 2 X;

0

ð4Þ

f0 ðr0 Þr0 dr þ

1 r

Z

R0

0

0

ð4Þ

f0 ðr0 Þr0 dr þ

0 0 0 F ðjÞ n ðr ; h Þ ¼ cosðnh Þ

Z

r R0

ð34aÞ

ð4Þ

g 0 ðr 0 Þr 0 dr

0

 ;

x 2 W ;

Z

r R

ð4Þ

g 0 ðr 0 Þr 0 dr

0



x 2 W :

0

0 0 0 F ðjÞ n ðr ; h Þ ¼ sinðnh Þ

0

jRðh Þj r

1 X 0 0 g ðjÞ m ðr Þ cosðmh Þ;

0 0 F ðjÞ n ðr ; h Þ

j ¼ 1; 3; 4;

ð35aÞ

j ¼ 2; 5:

ð35bÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðD cos h0 Þ  D2 þ ðR1 Þ2 ;

ð36Þ

m¼0

0 ð1Þ 0 0 0 0 0 F ðjÞ n ðr ;h ÞHn ðksh r Þr dr dh ;

x2W ;

Z R0 e l þ 4iU nðjÞ ðrÞ ¼ pHnð1Þ ðksh rÞ fnðjÞ ðr0 ÞJn ðksh r0 Þr0 dr0 j 0 Z r 0 þ pHnð1Þ ðksh rÞ g nðjÞ ðr 0 ÞJ n ðksh r0 Þr 0 dr

1 X 0 0 g ðjÞ m ðr Þ sinðmh Þ;

m¼1



ð32bÞ

0

r h0 ðrÞ Z jRðh0 Þj

0

In the above expressions, the functions (j = 1, 2, 3, 4, 5), Rðh0 Þ, and h0 ðrÞ are as follows:

0

0 0 0 g ðjÞ n ðr ÞJ n ðksh r Þr dr

Z pZ þ an J n ðksh rÞ

n

0 0 ðr 0 Þ g ðjÞ n ðr Þr dr

ð34cÞ 0

r R0

R0

0

ð1Þ 0 0 0 0 0 F ðjÞ n ðr ;h ÞHn ðksh r Þr dr dh ; x 2 X

fnðjÞ ðr 0 ÞJ n ðksh r 0 Þr 0 dr

0

Z

Hð1Þ n ðksh rÞ

r

0 ( Z ) h0 ðrÞ Z jRðh0 Þj 1 1 0 ð4Þ 0 0 0 0  F 0 ðr ; h Þr dr dh ; r p 0 r

ð32aÞ

jRðh0 Þj

Z

0

R0

0

Z pZ

r n n

ð34bÞ

0

0 0 fnðjÞ ðr 0 ÞHð1Þ n ðksh r Þr dr

Z pZ þ an J n ðksh rÞ

0

ð33cÞ

0

0

0

n

ðr 0 Þ fnðjÞ ðr0 Þr0 dr þ



ð30cÞ Z 2p Z

R0

Z 2 r n 0 n ðr 0 Þ F nðjÞ ðr0 ; h0 Þr 0 dr dh0 p n 0 r 0 Z Z 2 r n h0 ðrÞ jRðh Þj 0 n ðjÞ 0 0 0 0 0  ðr Þ F n ðr ; h Þr dr dh ; pn 0 r

x 2 W :

. . . dr dh0 ;

Z

2 rn pn

þ

h0 ðrÞ jRðh0 Þj

R0

rn n

4V ðjÞ n ðrÞ ¼

. . . dr dh0

R0

0

h0 ðrÞ Z r

r

jRðh0 Þj ¼ D cosðh0 Þ þ

2

h0 ðrÞ ¼ cos1

" # R21  D2  r 2 : 2r D

ð37Þ

R0

Z pZ

jRðh0 Þj

0 F nðjÞ ðr 0 ;h0 ÞHnð1Þ ðksh r 0 Þr 0 dr dh0 r Z h0 ðrÞ Z jRðh0 Þj 0 ð1Þ 0 0 0 0 0  an J n ðksh rÞ F ðjÞ n ðr ; h ÞH n ðksh r Þr dr dh 0 r Z h0 ðrÞ Z jRðh0 Þj 0 þ an Hnð1Þ ðksh rÞ F nðjÞ ðr 0 ; h0 ÞJ n ðksh r 0 Þr0 dr dh0 ; 0 r

þ an J n ðksh rÞ

Appendix C

0

x 2 W ; ð32cÞ

4V ðjÞ n ðrÞ ¼

n Z r

r n

þ

0

2 rn pn

n

Z pZ 0

jRðh0 Þj R0

n

r n

r

n

ðr0 Þ fnðjÞ ðr0 Þr0 dr

0

R0

1 TðrÞ ¼ r

"Z

R0

0

0 ð4Þ f0 ðr 0 Þr 0 dr

þ

1

p

Z pZ 0

jRðh0 Þj

R0

# 0 ð4Þ F 0 ðr 0 ; h0 Þr0 dr dh0

;

ð38Þ

0

0 0 0 0 ðr 0 Þ F ðjÞ n ðr ;h Þr dr dh ; x 2 X; ð33aÞ

Z r n r 0 n ðjÞ 0 0 0 0 n ðr0 Þ fnðjÞ ðr 0 Þr 0 dr þ ðr Þ g n ðr Þr dr n R0 0 0 Z Z 2 r n p jRðh Þj 0 n ðjÞ 0 0 0 0 0 þ ðr Þ F n ðr ; h Þr dr dh ; x 2 W ; pn 0 r ð33bÞ

rn 4V ðjÞ n ðrÞ ¼ n

Z

0

ðr0 Þ fnðjÞ ðr 0 Þr 0 dr þ

n Z R0

The expression for the unknown function, TðrÞ and the unknown variables, CðjÞ n (j = 4, 5) and sn (n = 0, 1, 2, . . .) in Eqs. (24) and (25) are of the following forms:

CnðjÞ ¼

1 2

Z

þ

R0 0

2

p

n

ðr 0 Þ fnðjÞ ðr 0 Þr 0 dr

Z pZ 0

jRðh0 Þj

R0

n

0

0

0 0 0 0 ðr 0 Þ F ðjÞ n ðr ; h Þr dr dh ;

ð39Þ

and

h i ð3Þ ð1Þ ð2Þ ð4Þ ð5Þ s0 ¼ 2P0 þ ksh P1 þ P1 þ eP1 þ eP1 ;

ð40aÞ

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H.M. Shodja et al. / Mechanics of Materials 75 (2014) 1–12

h i ð3Þ ð1Þ ð2Þ ð4Þ ð5Þ s1 ¼ 2P1 þ ksh P2 þ P2 þ eP2 þ eP2 h i ð1Þ ð4Þ  2ksh P0 þ eP0 ;

ð40bÞ

h i ð1Þ ð2Þ ð4Þ ð5Þ sn ¼ 2Pð3Þ n þ ksh Pnþ1 þ Pnþ1 þ ePnþ1 þ ePnþ1 h i ð1Þ ð2Þ ð4Þ ð5Þ  ksh Pn1  Pn1 þ ePn1  ePn1 ; n ¼ 2; 3; . . . ð40cÞ In which



lþ

e2

j

Z 4iPðjÞ n ¼ p

R0

0

þ an

fnðjÞ ðr 0 ÞJ n ðksh r 0 Þr 0 dr

Z pZ 0

jRðh0 Þj

R0

0

0

0 0 0 0 0 F ðjÞ n ðr ; h ÞJ n ðksh r Þr dr dh :

ð41Þ

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