Accepted Manuscript Dynamic Response of A Functionally Graded Tube Embedded in an Elastic Medium Due to Sh-Waves Hasan Faik Kara, Metin Aydogdu PII: DOI: Reference: S0263-8223(18)31559-9 https://doi.org/10.1016/j.compstruct.2018.08.032 COST 10082 To appear in: Composite Structures Received Date: Revised Date: Accepted Date: 27 April 2018 20 July 2018 14 August 2018 Please cite this article as: Kara, H.F., Aydogdu, M., Dynamic Response of A Functionally Graded Tube Embedded in an Elastic Medium Due to Sh-Waves, Composite Structures (2018), doi: https://doi.org/10.1016/j.compstruct. 2018.08.032 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. DYNAMIC RESPONSE OF A FUNCTIONALLY GRADED TUBE EMBEDDED IN AN ELASTIC MEDIUM DUE TO SH-WAVES Hasan Faik Karaa and Metin Aydogdub* a Trakya University, Engineering Faculty, Civil Engineering Department,22030, Edirne ,TURKEY b Trakya University, Engineering Faculty, Mechanical Engineering Department,22030, Edirne ,TURKEY Abstract: Dynamic response of a cylindrical tube surrounded by an unbounded elastic medium due to plane harmonic SH-Waves is studied. A two-dimensional mathematical model is considered. Cylindrical coordinates are used for convenience. The surrounding medium is assumed to be homogeneous, isotropic and linear elastic. The tube is assumed to be made of linear elastic Functionally Graded Materials (FGMs) such that shear modulus and shear wave velocity are assumed to change linearly from inner surface to outer surface. Material properties are constant along circumferential direction. It is assumed that the inner surface of the tube is traction-free and there is a welded contact between the tube and the surrounding medium. Governing equations are slightly different in the tube region and the unbounded region. Both of the governing equations are solved by applying Finite Fourier Transform in circumferential direction. The exact solution series are presented in terms of Fourier-Bessel series in the unbounded region and power series in the tube region. The presented numerical results show that when the incoming wave lengths decrease, shear stresses at the tube increase significantly. It was shown that for the shorter incoming wave lengths, tubes made of FGMs are subjected to smaller shear stresses compared to the tubes homogeneously made of outer surface material of the FG cases. Keywords: Cylindrical Tubes, Functionally Graded Materials, SH-Waves Corresponding author, metina@trakya.edu.tr, 902842251295 1 I. Introduction Functionally graded materials (FGMs) were invented by a group of Japanese scientists in 1984. FGMs are two-component composites in which that material properties change gradually from one component to another in the desired direction. This structure of FGMs enables to benefit from different advantages of two components. Therefore, FGMs are highly convenient for many engineering applications. For example, it is possible to produce a cylindrical tube that can withstand extremely high temperatures without compromising structural integrity. Because of many other additional advantages, FGMs are widely used in many industries such as civil engineering, automotive and aerospace applications. Wave motion problems in FGMs are considered by many researchers to reveal the dynamic behavior of FGMs. Han et al. [1] investigated elastic waves in FGMs by using quadratic layer element method. A hybrid numerical method is presented by Han et al. [2] for analyzing transient waves in a cylinder made of FGM. SH-wave, also called secondary horizontal wave, is a type of elastic body wave together with P and SV-waves. Polarization of SH-waves (direction of particle motion) is perpendicular to the two-dimensional model. Scattered waves have also the same polarization and therefore, unlike P and SV-waves, SH-waves are uncoupled. As a result, only SH-waves needs to be considered in the complete analysis [29]. A computational method is presented by Han and Liu [3] to investigate SH-waves in functionally graded material (FGM) plates. Han et al. [4] studied characteristics of waves in a functionally graded cylinder. Shakeri et al. [5] studied vibration and radial wave propagation velocity in functionally graded thick hollow cylinder. Daros [6] presented a fundamental solution for SH-waves in a class of inhomogeneous anisotropic media. Hu et al. [7] studied the scattering of shear waves from a circular cavity buried in a semi-infinite slab of functionally graded materials. Scattering of SH waves by a cavity in an exponentially graded half-space is analysed by Martin [8]. Baron [9] studied propagation of elastic waves in an anisotropic functionally graded hollow cylinder in vacuum. Wave propagation of functionally graded material plates in thermal environments is studied by Sun and Luo [10]. Sun and Luo [11] also studied wave propagation and transient response of functionally graded material circular plates under a point impact load. A dynamic solution for the propagating viscoelastic waves in functionally graded material plates subjected to stress-free conditions is presented by Yu et al. [12]. By using Ritz method, 2 Uymaz et al. [13] analysed vibration of FGM plates with in-plane material inhomogeneity. Daros [14] developed a Green’s function for SH-waves in inhomogeneous anisotropic elastic solid with power-function velocity variation. Antiplane scattering of SH-Waves by a circular cavity in an exponentially graded half space is studied by Liu et al. [15]. Elastic wave propagation in functionally graded circular cylinders is studied by Dorduncu et. al. [16]. Filiz and Aydogdu [17] analyzed wave propagation in functionally graded nanotubes conveying fluid. By using a nonlocal strain gradient theory, Li et al. [18] analysed flexural wave propagation in small-scaled functionally graded beams. Qiao et al. [19] presented an analytical solution of characteristics of elastic waves in FGM spherical shells. Yang et al. [20] studied scattering of out-plane wave by a circular cavity near the right-angle interface in the exponentially inhomogeneous media. Aminipour at al. [21] developed a new model for wave propagation in functionally graded anisotropic doubly-curved shells. Propagation of axisymmetric waves in pressurized functionally graded elastomeric hollow cylinders is studied by Wu et al. [22]. There are also many analytical studies on scattering and diffraction of plane harmonic SHWaves by homogeneous cylindrical tubes, pipes or tunnels surrounded by homogeneous elastic mediums. By using wave function expansion method and image technique, an analytical closed-form solution of dynamic response of cylindrical tunnels to incident SHWaves is presented by Lee and Trifunac [23]. By using a similar approach, Balendra and Thambiratnam [24] presented the steady-state solution for two parallel underground tunnels of circular cross-section subjected to incident plane harmonic SH-waves in closed form. A closed-form solution of a cylindrical tunnel embedded in a quarter-space subjected to plane harmonic SH-Waves is presented by Kara [25]. An analytical solution of scattering of plane SH Waves induced by a semi-cylindrical canyon with a subsurface circular lined tunnel is obtained by Gao et al. [26]. Although analytical studies of response of cylindrical tubes surrounded by elastic mediums due to SH-waves are highly extensive, similar analytical studies that consider tubes made of FGM do not exist. A possible reason may be that the fundamental analytical solutions of cylindrical tubes subjected to SH-Waves had already been developed before the invention of FGMs. Because the application areas of FGMs have been expanding day by day since their invention, it is believed that developing an analytical mathematical model to analyze the dynamic response of cylindrical tubes made of FGM to SH-Waves would be a substantial contribution to the literature. 3 The aim of this study is to present an analytical closed-form solution of dynamic response of a cylindrical tube made of FGM surrounded by an unbounded medium due to plane harmonic SH-waves. Numerical results are also presented to demonstrate the dynamic behavior. 4 II. Model and Formulation of the Problem The cross-section of the two-dimensional model considered in this paper is shown in Figure 1. There is a cylindrical tube with inner radius ai and outer radius ao. The tube is surrounded by an unbounded, homogeneous, isotropic and linear elastic medium which is excited by out-of-plane harmonic incident SH-Waves with unit amplitude and angular frequency in +x direction. The surrounding medium is characterized by shear modulus s and shear wave velocity s. The tube is also linear elastic but shear modulus and shear wave velocity of the tube, represented by t andt, are linearly changing from the inner surface to outer surface of the tube as a function of r. t andt take the values i andi at inner surface and o ando at outer surface. The general form of the equation of motion of the problem is: [27] ji 2U i fi 2 x j t (1) Here, Tji represents the stress tensor, represents mass density of the material, fi represents components of the body force vector and Ui represents components of the displacement vector. In the considered problem, the system is excited by SH-Waves, so there are only out-ot-plane displacements and the only displacement component is Uz, which is the displacement component in the z direction. If the out-of-plane displacement function in the tube region is denoted by Ut, mass density of the tube material is denoted by t and the body force per unit mass is neglected, then Eq. (1) can be rewritten in terms of cylindrical coordinates as follows: (t ) 1 1 (t ) 2 rz (r , , t ) (rzt ) (r , , t ) z (r , , t ) t (r ) 2 U t (r , , t ) r r r t (2) where Trz(t) and Tz(t) are shear stress components inside the tube, defined by [29]: U t (r , , t ) r (3) 1 U t (r , , t ) r (4) (rzt ) (r , , t ) t (r ) (tz) (r , , t ) t (r ) 5 When Eqs. (3) and (4) are substituted into Eq. (2), there follows: U t ( r , , t ) 2U t (r , , t ) t (r ) t ( r ) t ( r ) U t (r , , t ) r r 2 r r 1 1 2 t ( r ) U t (r , , t ) t (r ) 2 U t (r , , t ) r r t ' (5) Since steady-state wave propagation is considered, time dependent part of the displacement can be separated as follows: Ut (r , , t ) ut (r , )eit (6) where, i is the imaginary unit. Substituting Eq. (6) into Eq. (5) gives: (r ) 2 ut (r , ) t (r ) 2 ut (r , ) t ut (r , ) r r r r 1 1 t ( r ) ut (r , ) 2 t (r )ut (r , ) 0 r r t ' ( r ) (7) Angular frequency () and shear wave velocity (t) in tube region are [29]: kt (r )t (r ) (8) t (r ) t (r ) / t (r ) (9) In identity (8), kt denotes wave number in the tube region. For convenience, a new auxiliary function, tr is introduced such that: tr (r ) 1 d (r ) t (r ) dr t (10) 2ut (r , ) 1 ut (r , ) 1 2ut (r , ) 2 2 kt (r ) ut (r , ) Substituting Eqs. (8), (9) and (10) into (7) leads to: 2 2 r r r r To simplify Eq. (11), shear modulus t and shear wave velocity t are assumed to change linearly with r, described by the following functions: t (r ) o (r / ao ) (12) t (r ) o (r / ao ) (13) In this case, expression (10) and the wave number (kt) take the following forms: 6 tr (r ) 1 d 1 d t ( r ) (r / ao ) 1/ r t (r ) dr o (r / ao ) dr o (14) k 1 1 o o k a /r o (r / ao ) o (r / ao ) o o (15) 2ut (r , ) 2 ut (r , ) 1 2ut (r , ) 2 2 ko ao / r ut (r , ) 0 2 2 r r r r (16) kt (r ) t ( r ) o (r / ao ) Eq. (11) becomes: The Finite Fourier Transform and its inverse are defined as: 1 ut ,n (r ) 2 2 in ut (r, )e d ut (r , ) u t ,n (r )ein (17) n 0 Finite Fourier Transform of Eq. (16) is: r 2 d 2ut ,n (r ) dr 2 2r dut ,n (r ) dr ko ao n2 ut ,n (r ) 0 2 (18) For: n ko ao n2 2 (19) Eq. (18) takes the form: r 2 d 2ut ,n (r ) dr 2 2r dut ,n (r ) dr nut ,n (r ) 0 (20) Here, the solution of Eq. (20) is assumed to be in power series form as: ut ,n (r ) r (21) Substituting solution function (21) into Eq. (20) yields: 2 n r 0 (22) In Eq. (22), the solution function r cannot be zero so the coefficient of the solution function must be zero so that Eq. (22) holds: 2 n 0 7 (23) The quadratic equation (23) has two solutions: (24) (25) 1 1 1 4 n / 2 2 1 1 4 n / 2 Substituting the roots given in Eqs. (24) and (25) into Eq. (21) provides two solution functions that both satisfy Eq. (20). A general solution would be a linear combination of these two functions as follows: ut ,n (r ) At1,n r 1 At 2,n r 2 At1,n r 1 A r 1 t 2, n 1 4 n /2 1 4 n /2 (26) Inverse transform of Eq. (26) gives the static part of displacement function in tube region: ut (r , ) ut ,n (r )ein n At1,n r 1 e 1 4 n /2 in n A t 2, n r 1 e 1 4 n /2 in (27) n In the homogeneous unbounded region, shear modulus (s) and wave number (ks) are constant and derivative of shear modulus is zero. As a result, governing equation for this region is slightly simpler than Eq. (11): 2us (r , ) 1 us (r , ) 1 2us (r , ) 2 ks2us (r , ) 0 2 2 r r r r (28) Here, us denotes static part of displacement function in the homogeneous region. Applying Finite Fourier Transform to Eq. (28) gives: r 2 d 2us , n ( r ) dr 2 r dus ,n (r ) dr rks n 2 us ,n (r ) 0 2 (29) For convenience, the following variable transformation is made: rks (30) d d d d ks dr d dr d (31) d2 d d d 2 d ks 2 dr d dr dr d (32) With this variable transformation, Eq. (29) takes the form: 8 2 d 2us ,n ( / ks ) d 2 dus ,n ( / ks ) d 2 n2 us ,n ( / ks ) 0 (33) The ordinary differential equation (33) is the well-known Bessel differential equation and the solution is described as [28]: us ,n ( / ks ) Asj ,n J n () Asy ,nYn () (34) Inverse variable transformation yields: us ,n (r ) Asj ,n J n (ks r ) Asy ,nYn (ks r ) (35) Here, Jn and Yn are the first and second kind of Bessel functions of order n. In our case, describing solution function (35) in terms of the first and second kind of Hankel functions is more convenient: us ,n (r ) Ash1,n H n(1) (ks r ) Ash 2,n H n(2) (ks r ) (36) where the first and second kind of Hankel functions are defined as [28]: H n(1) (ks r ) J n (ks r ) iYn (ks r ) (37) H n(2) (ks r ) J n (ks r ) iYn (ks r ) (38) Inverse transform of Eq. (36) is: us ( r , ) us,n (r )ein n Ash1,n H n(1) (ks r )ein n A sh 2, n H n(2) (ks r )ein (39) n Solution function (39) corresponds to the scattered waves from the tube. In this function, terms multiplied by Hn(1) and Hn(2) represents outgoing and incoming waves respectively. Due to Sommerfeld Radiation Condition, there will not be incoming waves from infinity, which will reduce Eq. (39) to: us ( r , ) A sh1, n H n(1) (ks r )ein (40) n In the unbounded homogeneous region, there are also incident SH-Waves with unit amplitude, angular frequency and +x direction. Incident waves are denoted by Ui and they can be easily described by the following function: Ui eiks ( xst ) eiks x eit ui eit 9 (41) By using identity (42) [28], spatial part of incident waves (ui) can be expressed in FourierBessel series form: e 1 z ( t 1/ t ) 2 t k (42) J k ( z) k ui (r , ) e in /2 J n (ks r )ein (43) n Resultant displacement function in the unbounded region is the sum of the scattered waves (us) and the excitation (ui): usT (r , ) us (r, ) ui (r, ) (44) Displacement function that corresponds to scattered waves from the cavity (us) has one set of unknowns (Ash1,n). Displacement function that corresponds to the waves inside the tube (ut) has two sets of unknowns (At1,n and At2,n). Totally three sets of unknowns will be determined from the following three boundary conditions: Zero stress at inner side of the tube: rz(t ) r ai , i ut r ai , 0 r (45) Continuity of stress and displacement at outer side of the tube: rz(t ) r ao , rz( s ) r ao , o ut r ao , s usT r ao , 0 r r ut r ao , usT r ao , 0 (46) (47) Substituting displacement functions (27) and (44) into boundary conditions (45), (46) and (47) yields: A t1, n n 1 1 4 n ai At 2,n 1 1 4 n ai / 2 1 1 1 4 n /2 10 / 2 ein 0 1 1 1 4 n /2 (48) A t n 1 t1, n s 1 4 n ao At 2,n 1 1 4 n ao H /2 1 1 1 4 n /2 /2 1 1 1 4 n /2 (49) ks in /2 e J n1 (ks ao ) J n1 (ks ao ) 2 Ash1,n (1) n 1 (k s ao ) H n(1)1 (k s ao ) A a 1 e in A a 1 t 2, n o 1 4 n /2 t1, n o n e in /2 0 1 4 n /2 J n (ks ao ) Ash1,n H (k s ao ) e (1) n in (50) 0 Because of periodicity conditions, Eqs. (48), (49) and (50) reduce to: At 2,n 1 1 4 a At1,n 1 1 4 n ai n / 2 /20 1 1 1 4 n /2 1 1 1 4 n /2 i t 1 1 1 4 /2 At1,n 1 1 4 n ao / 2 s 1 1 1 4 /2 A 1 1 4 a /2 t 2, n n H (51) n n o (52) (53) ks in /2 e J n1 (ks ao ) J n1 (ks ao ) 2 Ash1,n A (1) n 1 a 1 t1, n o (k s ao ) H n(1)1 (k s ao ) 0 A a 1 t 2, n o 1 4 n /2 1 4 n /2 ein /2 J n (ks ao ) Ash1,n H n(1) (ks ao ) 0 Unknown constants are determined by solving Eqs. (51), (52) and (53) simultaneously, as follows: Ash1,n 0 At1,n M 21 At 2,n M 31 M 12 M 22 M 32 M 13 M 23 M 33 1 0 k s ein /2 J n 1 (ks ao ) J n 1 (ks ao ) 2 in /2 e J n (ks ao ) where 11 (54) 1 1 1 4 n /2 1 1 1 4 n /2 M12 1 1 4 n ai M13 1 1 4 n ai M 21 /2 (55) /2 (56) ks H n(1)1 (ks ao ) H n(1)1 (ks ao ) 2 M 22 t 1 1 1 1 4 n ao 2 s M 23 t 1 1 1 1 4 n ao 2 s M 31 H n(1) (ks ao ) M 32 ao M 33 ao III. 1 1 4 n /2 1 1 4 n /2 (57) (58) (59) 1 4 n /2 1 4 n /2 (60) (61) (62) Numerical Results and Discussions Displacement functions (27) and (44) given in the previous section are out-of-plane displacements in complex form in the tube and the unbounded regions respectively. Maximum displacements in a full period (displacement amplitudes) at any point can be calculated by evaluating absolute value of these functions. Displacements as a function of time can also be obtained by multiplying Eqs. (27) and (44) by exp(-it) and calculating either real or imaginary part [29]. Stress components can also be derived from displacements as given in Eqs. (3) and (4). In this section, some numerical results are demonstrated for normalized problem parameters. Incident wave length (s) and inner radius of the tube (ai) are normalized with respect to tube outer radius (ao). Material coefficients of the tubes outer surface (o, o) are normalized with respect to material coefficients of the unbounded region (s, s). Plot ranges are in between =0o and =180o since the displacements or stress components are perfectly symmetric with respect to x axis. Since there is no study on two-dimensional diffraction and scattering of SH-Waves by a FGM tube embedded in an elastic unbounded space, the results obtained in this study cannot 12 be directly compared with the existing studies. However, there are many analytical studies on two-dimensional diffraction and scattering of SH-Waves by homogeneous tubes, tunnels or pipes (including but not limited to [23-26]). In these studies, none of the surrounding mediums is unbounded and therefore, there are additional reflected waves. These additional reflected waves significantly affect displacement variations. Consequently, a comparison with these results is also uninformative. In order to verify the results of this study, an analytical solution of a similar problem where the tube material is homogeneous, is developed. Displacement functions that correspond to incident waves (Eq. (43)) and scattered waves from the tube (Eq. (40)) remain in the same form. Displacement function in the homogeneous tube region is given below: ut( hom) (r , ) Ath1,n H n(1) (kt r )ein n A th 2,n H n(2) (kt r )ein (63) n where 0 Ash1,n H n(1) (k s r ) A th1,n s r r ao Ath 2,n (1) H n (ks ao ) 1 H n(1) (kt r ) r r a i o H (kt r ) r r a (1) n o (1) n H (kt ao ) H n(2) (kt r ) r r ai 0 H n(2) (kt r ) (64) in /2 J n ( k s r ) o s e r r r ao r ao in /2 H n(2) (kt ao ) e J n (ks ao ) Note that Eq. (63) is in exactly same form as Eq. (39) since the governing equations are the same. In Eq. (39), the terms that correspond to incoming waves are omitted because of Sommerfeld Radiation Condition. In the tube region, however, there are both incoming and outgoing waves, so, none of the terms is omitted. The solution function given in Eq. (63) is convenient to be compared with the solution function of a homogeneous cylinder embedded in an unbounded medium subjected to SH-Waves. Pao and Mow [29] presents the analytical displacement function in the homogeneous cylinder as: ut( P&M ) (r , ) Cn J n (kt r )Cos n n0 where 13 (65) Cn i n n k s s H n(1) (k s ao ) J n 1 (k s ao ) J n 1 (k s ao ) J n (k s ao ) H n(1)1 (k s ao ) H n(1)1 (k s ao ) kt t H n(1) (k s ao ) J n 1 (kt ao ) J n 1 (kt ao ) k s s J n (kt ao ) H n(1)1 (k s ao ) H n(1)1 (k s ao ) (66) and 1, n 0 2, n 1, 2,... n (67) As the inner radius of the homogeneous tube approaches to zero, the dynamic behavior of the homogeneous tube must be similar to the homogeneous cylinder. Figure 2 demonstrates this comparison. In this Figure, displacement amplitudes at cylinders and tubes outer surfaces are plotted. When ai/ao=0.05, displacement differences are barely distinguishable. For smaller values of ai/ao, the differences become unnoticeable. So, the analytical solution of the homogeneous tube given in Eq. (63) converges to the analytical solution of the homogeneous cylinder given in Eq. (66) presented by Pao and Mow when inner radius of the tube goes to zero. Figure 2 also demonstrate another comparison. According to the presented mathematical model for the FGM tube, when the tube thickness decreases (or inner radius ratio, ai/ao ,increases), the difference of material properties of outer and inner side of the tube also decrease. Therefore, for a thin FGM tube, the dynamic behavior must be similar to the homogeneous tube. In Figure 2, displacement amplitudes at homogeneous and FGM tubes outer surfaces are plotted for ai/ao=0.96. It can be seen that there is a perfect agreement with these results. Briefly, the results presented in this study for the homogeneous tube converges to the results of the analytical formulation of a homogeneous cylinder presented by Pao and Mow as the inner radius goes to zero. When inner radius ratio (ai/ao) approaches to 1, the results for the homogeneous and FGM tubes coinside. In Figures 3-14, rz stress amplitudes are given for three different outer shear modulus ratios (4, 8 and 12) and a fixed shear wave velocity ratio (1.6). Stress amplitudes in homogeneous cases are also included for a fixed shear modulus ratio, which is the average of inner and outer shear modulus ratios of FG cases (t=(i+o)/2). Shear modulus ratios of the homogeneous cases are chosen to be same as the minimum inner shear modulus ratio and the maximum outer shear modulus ratio among the FG cases, respectively. Within the range of used parameters, the stress distributions of FG cases are generally in between these homogeneous cases as expected. In general, peak stresses increase as the rigidities increase. In Figures 4 and 7, peak stresses are higher in some FG cases, but the 14 differences are small. In these two cases, incoming wave lengths are relatively long. In the rest of the Figures, peak stresses are always higher in the homogeneous cases with highest rigidity. There are also considerable increases in stress amplitudes when incoming wave lengths decrease because of increased spatial variations of displacements. As an example, when the wavelength fall in half, rz, approximately doubles for all thicknesses. These results show that the FG tubes are subjected to less rz shear stresses compared to the tubes uniformly made of outer surface material of the FG case for shorter wave lengths which cause significantly higher stresses. Fluctuations in stress distributions also increase for shorter wave lengths. In the most of the cases, the peak stresses occur at =180o. But in some cases (Figure 3 and 7), peak stresses occur at the other side of the tube (at =0o). It is observed that the difference between the compared cases increase by the thickness which also determine the material coefficients of the inner side of the FGM according to the proposed mathematical model. In Figures 15-26, z stress amplitudes are compared. z stresses are always zero at =0o and =180o because of the displacement symmetry. In between these points, the z stresses are higher than the rz stresses. Similar to the rz stresses, z stresses increase remarkably as incoming wave lengths decrease and stresses generally increase by rigidities. Stresses for FG cases are usually in between homogeneous cases. For relatively shorter incoming wave lengths, peak stresses usually occur in the most rigid homogeneous cases. The point where the peak stresses occur changes with incoming wave lengths and thicknesses. Same as rz stresses, fluctuations increase for shorter wave lengths. The difference between FG and homogeneous cases increase for higher thicknesses also here. It should be noted that actual stresses in FGM tube changes between positive and negative values of these stress amplitudes. Periodic behavior of stresses is an important effect on fatigue life FGM tubes. Tables 1-4 demonstrate the displacement amplitudes at tubes outer surface for various incident wave lengths. It can be seen that average displacement amplitudes usually increase as the incoming wave lengths increase. Fluctuation in the displacement amplitudes also increase as the incoming wave lengths decrease as expected. IV. Conclusions 15 Scattering and diffraction of plane SH-Waves by a functionally graded cylindrical tube surrounded by an unbounded homogeneous medium is studied. Governing equations of the two-dimensional wave propagation problem in each region are solved analytically and displacement functions are obtained in closed form. The presented closed-form solutions are very compact, so they are remarkably convenient for programming and can be used directly in many engineering applications. Presented results can also be very useful for verification of subsequent numerical studies that may consider nonlinearity. The solution technique proposed in this study can be easily applied to many similar problems. For instance, by using Addition Theorems, dynamic response of twin functionally graded tubes surrounded by an infinite or semi-infinite medium can be solved analytically. The numerical results presented in this study can be outlined as: 1) Stress distributions at a FG cylinder are usually in between homogeneous tubes made of outer surface and inner surface material of the FG tube. 2) The behavior of FG tubes are close to the homogeneous tubes when material coefficients of outer and inner side are close. Otherwise, the behaviors of FG cylinders are significantly different. These findings indicate the necessity of further research on dynamic response of FG tubes. 3) The decrease in incoming wave lengths result in remarkably higher shear stresses inside the FG tube. 4) In most of the cases, FG tubes are subjected to smaller shear stresses compared to the most rigid homogeneous cases, especially when the incoming wave lengths are short. 5) Locations of peak stresses change by the values of problem parameters and they are highly unpredictable. 6) Fluctuations in stress distributions increase as the incoming wave lengths decrease. REFERENCES [1] Han X, Liu GR, Lam KY, Ohyoshi T. A quadratic layer element for analyzing stress waves in FGMs and its application in material characterization. Journal of Sound and Vibration. 2000 Sep 14;236(2):307-21. [2] Han X, Liu GR, Xi ZC, Lam KY. Transient waves in a functionally graded cylinder. International Journal of Solids and Structures. 2001 Apr 1;38(17):3021-37. 16 [3] Han X, Liu GR. Effects of SH waves in a functionally graded plate. Mechanics Research Communications. 2002 Sep 1;29(5):327-38. [4] X. Han, G. R. Liu, Z. C. Xi, K.Y. Lam "Characteristics of waves in a functionally graded cylinder." International Journal for numerical methods in engineering 53.3 (2002): 653-676. [5] Shakeri M, Akhlaghi M, Hoseini SM. Vibration and radial wave propagation velocity in functionally graded thick hollow cylinder. Composite structures. 2006 Oct 1;76(12):174-81. [6] C. H. Daros "A fundamental solution for SH-waves in a class of inhomogeneous anisotropic media." International Journal of Engineering Science 46.8 (2008): 809-817. [7] C. Hu, X. Q. Fang, W. H. Huang. "Multiple scattering of shear waves and dynamic stress from a circular cavity buried in a semi-infinite slab of functionally graded materials." Engineering Fracture Mechanics 75.5 (2008): 1171-1183. [8] P. A. Martin "Scattering by a cavity in an exponentially graded half-space." Journal of Applied Mechanics 76.3 (2009): 031009. [9] Baron C. Propagation of elastic waves in an anisotropic functionally graded hollow cylinder in vacuum. Ultrasonics. 2011 Feb 1;51(2):123-30. [10] Sun D, Luo SN. Wave propagation of functionally graded material plates in thermal environments. Ultrasonics. 2011 Dec 1;51(8):940-52. [11] Sun D, Luo SN. Wave propagation and transient response of functionally graded material circular plates under a point impact load. Composites Part B: Engineering. 2011 Jun 1;42(4):657-65. [12] Yu JG, Ratolojanahary FE, Lefebvre JE. Guided waves in functionally graded viscoelastic plates. Composite Structures. 2011 Oct 1;93(11):2671-7. [13] Uymaz B, Aydogdu M, Filiz S. Vibration analyses of FGM plates with in-plane material inhomogeneity by Ritz method. Composite Structures. 2012 Mar 1;94(4):1398405. [14] Daros CH. Green’s function for SH-waves in inhomogeneous anisotropic elastic solid with power-function velocity variation. Wave Motion. 2013 Mar 1;50(2):101-10. 17 [15] Q. Liu, M. Zhao, C. Zhang, “Antiplane scattering of SH waves by a circular cavity in an exponentially graded half space.” International Journal of Engineering Science, 78, (2014): 61-72. [16] Dorduncu M, Apalak MK, Cherukuri HP. Elastic wave propagation in functionally graded circular cylinders. Composites Part B: Engineering. 2015 May 1;73:35-48. [17] S. Filiz, M. Aydogdu "Wave propagation analysis of embedded (coupled) functionally graded nanotubes conveying fluid." Composite Structures 132 (2015): 1260-1273. [18] Li L, Hu Y, Ling L. Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory. Composite Structures. 2015 Dec 1;133:1079-92. [19] Qiao S, Shang X, Pan E. Characteristics of elastic waves in FGM spherical shells, an analytical solution. Wave Motion. 2016 Apr 1;62:114-28. [20] Yang Z, Zhang C, Yang Y, Sun B. Scattering of out-plane wave by a circular cavity near the right-angle interface in the exponentially inhomogeneous media. Wave Motion. 2017 Jul 1;72:354-62. [21] Aminipour H, Janghorban M, Li L. A new model for wave propagation in functionally graded anisotropic doubly-curved shells. Composite Structures. 2018 Apr 15;190:91111. [22] Wu B, Su Y, Liu D, Chen W, Zhang C. On propagation of axisymmetric waves in pressurized functionally graded elastomeric hollow cylinders. Journal of Sound and Vibration. 2018 May 12;421:17-47. [23] V. W. Lee, M. D. Trifunac. "Response of Tunnels to Incident Sh-Waves." Journal of the Engineering Mechanics Division-Asce 105.4 (1979): 643-659 [24] T. Balendra, D.P. Thambiratnam, C.G. Koh, S.L. Lee. "Dynamic response of twin circular tunnels due to incident SH‐ waves." Earthquake engineering & structural dynamics 12.2 (1984): 181-201. [25] H. F. Kara, "A note on response of tunnels to incident SH-waves near hillsides." Soil Dynamics and Earthquake Engineering 90 (2016): 138-146. 18 [26] Y. Gao, X. Chen, N. Zhang, D. Dai, X. Yu. “Scattering of Plane SH Waves Induced by a Semicylindrical Canyon with a Subsurface Circular Lined Tunnel.” International Journal of Geomechanics. 2018 Mar 30;18(6):06018012. [27] R. C. Payton “Elastic wave propagation in transversely isotropic media” Vol. 4. Springer Science & Business Media, 2012. [28] M. Abramowitz, I.A. Stegun "Handbook of Mathematical Functions with Formulas, Graph, and Mathematical Tables." Applied Mathematics Series 55 (1965): 1046. [29] C.C. Mow, Y.H. Pao "The diffraction of elastic waves and dynamic stress concentrations." (1971). FIGURE CAPTIONS Figure 1: Geometry of the problem Figure 2: Displacement Amplitude at tubes and cylinders outer surface for s/ao=2, o/s=1.5 and o/s=10 Figure 3: rz Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=4 Figure 4: rz Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=2 Figure 5: rz Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=1 Figure 6: rz Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=0.5 Figure 7: rz Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=4 Figure 8: rz Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=2 19 Figure 9: rz Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=1 Figure 10: rz Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=0.5 Figure 11: rz Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=4 Figure 12: rz Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=2 Figure 13: rz Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=1 Figure 14: rz Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=0.5 Figure 15: z Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=4 Figure 16: z Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=2 Figure 17: z Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=1 Figure 18: z Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=0.5 Figure 19: z Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=4 Figure 20: z Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=2 Figure 21: z Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=1 Figure 22: z Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=0.5 Figure 23: z Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=4 Figure 24: z Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=2 Figure 25: z Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=1 Figure 26: z Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=0.5 20 Figure 1: Geometry of the problem Figure 2: Displacement Amplitude at tubes and cylinders outer surface for s/ao=2, o/s=1.5 and o/s=10 21 Figure 3: rz Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=4 Figure 4: rz Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=2 Figure 5: rz Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=1 22 Figure 6: rz Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=0.5 Figure 7: rz Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=4 Figure 8: rz Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=2 23 Figure 9: rz Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=1 Figure 10: rz Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=0.5 Figure 11: rz Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=4 24 Figure 12: rz Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=2 Figure 13: rz Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=1 Figure 14: rz Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=0.5 25 Figure 15: z Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=4 Figure 16: z Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=2 Figure 17: z Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=1 26 Figure 18: z Stress Amplitude at tubes outer surface for ai/ao=0.75, s/ao=0.5 Figure 19: z Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=4 Figure 20: z Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=2 27 Figure 21: z Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=1 Figure 22: z Stress Amplitude at tubes outer surface for ai/ao=0.5, s/ao=0.5 Figure 23: z Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=4 28 Figure 24: z Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=2 Figure 25: z Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=1 Figure 26: z Stress Amplitude at tubes outer surface for ai/ao=0.25, s/ao=0.5 29 Table 1: Displacement Amplitude at tubes outer surface for ai/ao=0.75, o/s=1.6 and o/s=8 θ= 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100° 110° 120° 130° 140° 150° 160° 170° 180° s/ao=4 1.218153 1.204439 1.163781 1.097877 1.01038 0.908191 0.803255 0.714128 0.66398 0.669311 0.72591 0.812751 0.907842 0.996258 1.070176 1.126705 1.165768 1.188457 1.195873 s/ao=2 0.973206 0.922322 0.776354 0.557975 0.331377 0.323084 0.569618 0.832862 1.024632 1.107547 1.073863 0.94523 0.768765 0.609413 0.531754 0.548774 0.608174 0.659233 0.678306 s/ao=1.5 0.688102 0.621512 0.437673 0.195951 0.220595 0.482936 0.691131 0.779322 0.723277 0.556808 0.409158 0.44663 0.566772 0.620808 0.582168 0.483103 0.37262 0.292502 0.26437 30 s/ao=1 0.715233 0.556847 0.168109 0.33927 0.661981 0.718211 0.505537 0.310209 0.615245 0.849159 0.758499 0.481381 0.482478 0.638759 0.610781 0.414292 0.187628 0.153835 0.202529 s/ao=0.75 0.584825 0.361854 0.156127 0.542195 0.527176 0.185532 0.422737 0.639843 0.463997 0.410676 0.695063 0.568586 0.294133 0.421468 0.407551 0.217987 0.072766 0.22639 0.300231 s/ao=0.5 0.481908 0.1026 0.442727 0.29406 0.34281 0.437653 0.21839 0.493464 0.220533 0.561158 0.438245 0.568668 0.435595 0.337594 0.26246 0.20713 0.345673 0.237038 0.299292 Table 2: Displacement Amplitude at tubes outer surface for ai/ao=0.75, o/s=1.6 and o/s=12 θ= 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100° 110° 120° 130° 140° 150° 160° 170° 180° s/ao=4 1.29263 1.276979 1.230366 1.153964 1.050267 0.923991 0.783622 0.644149 0.531341 0.481221 0.513622 0.605156 0.718403 0.828879 0.924435 0.999891 1.053622 1.08559 1.09618 s/ao=2 0.984711 0.931523 0.778255 0.544642 0.278494 0.232452 0.502463 0.772441 0.964396 1.044823 1.004558 0.860165 0.651746 0.442812 0.335611 0.393481 0.508667 0.595519 0.626539 s/ao=1.5 0.551038 0.496753 0.345948 0.137978 0.150438 0.367996 0.531736 0.594921 0.536675 0.377004 0.216866 0.265623 0.392712 0.445488 0.412536 0.329768 0.248646 0.207054 0.198787 31 s/ao=1 0.696482 0.541031 0.151876 0.323796 0.638703 0.682531 0.448156 0.200256 0.536251 0.764648 0.661704 0.340787 0.369138 0.585695 0.577656 0.365726 0.092684 0.189948 0.278451 s/ao=0.75 0.501829 0.30875 0.129515 0.462488 0.443503 0.120841 0.348825 0.527024 0.337934 0.275727 0.539487 0.41269 0.135824 0.313152 0.337272 0.203393 0.024068 0.222598 0.313721 s/ao=0.5 0.438615 0.090233 0.402868 0.257993 0.307319 0.384256 0.177845 0.4346 0.129455 0.467968 0.309136 0.461507 0.343321 0.296181 0.257448 0.192884 0.328207 0.153198 0.258657 Table 3: Displacement Amplitude at tubes outer surface for ai/ao=0.5, o/s=1.6 and o/s=8 θ= 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100° 110° 120° 130° 140° 150° 160° 170° 180° s/ao=4 1.291011 1.275305 1.228318 1.150584 1.043494 0.91012 0.756611 0.59483 0.448567 0.364768 0.390738 0.497831 0.628422 0.752191 0.856979 0.938512 0.995995 1.029982 1.041208 s/ao=2 0.94616 0.895442 0.748833 0.522996 0.25399 0.183764 0.461351 0.731688 0.921341 0.994337 0.938756 0.770335 0.529436 0.2818 0.201494 0.358934 0.5182 0.620998 0.65605 s/ao=1.5 0.43091 0.388933 0.272139 0.107303 0.105246 0.28336 0.420717 0.474865 0.420188 0.268898 0.121174 0.214206 0.31957 0.324998 0.236586 0.095513 0.04822 0.151928 0.189456 32 s/ao=1 0.632024 0.490642 0.134081 0.287053 0.572776 0.617531 0.411155 0.157073 0.481551 0.700766 0.576385 0.189585 0.286221 0.487351 0.41276 0.283819 0.506792 0.776897 0.882732 s/ao=0.75 0.598331 0.45319 0.32182 0.524355 0.554421 0.415966 0.526205 0.69269 0.552602 0.251342 0.163941 0.255653 0.68955 0.85439 0.678128 0.468606 0.50493 0.648837 0.717213 s/ao=0.5 0.202543 0.120168 0.138951 0.229286 0.303313 0.067994 0.471361 0.315389 0.306326 0.678814 0.331885 0.842008 0.601984 0.487994 0.464441 0.123711 0.391839 0.189825 0.140725 Table 4: Displacement Amplitude at tubes outer surface for ai/ao=0.5, o/s=1.6 and o/s=12 θ= 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100° 110° 120° 130° 140° 150° 160° 170° 180° s/ao=4 1.255664 1.239465 1.191057 1.111094 1.000963 0.863295 0.702828 0.528137 0.357137 0.240776 0.273677 0.410748 0.564393 0.705543 0.824593 0.917957 0.984548 1.02433 1.03755 s/ao=2 0.893182 0.843934 0.70168 0.482367 0.215338 0.14236 0.42109 0.675747 0.853019 0.922411 0.872318 0.714217 0.480511 0.223545 0.150248 0.352197 0.531115 0.645377 0.684425 s/ao=1.5 0.302584 0.272584 0.189179 0.07049 0.071182 0.197569 0.292999 0.3295 0.288266 0.175549 0.062314 0.15379 0.235249 0.238273 0.169442 0.060199 0.053103 0.133858 0.163236 33 s/ao=1 0.562083 0.435648 0.11525 0.257675 0.510429 0.54153 0.344406 0.096334 0.412866 0.608694 0.505496 0.161386 0.256432 0.45893 0.3986 0.206457 0.370472 0.636638 0.742277 s/ao=0.75 0.501687 0.411113 0.307337 0.380594 0.383998 0.340344 0.480103 0.596578 0.477827 0.249466 0.130093 0.172446 0.510228 0.627047 0.486704 0.312377 0.307965 0.401502 0.450574 s/ao=0.5 0.059595 0.078622 0.079163 0.138231 0.26864 0.011845 0.384868 0.285977 0.241082 0.546909 0.218147 0.633203 0.436422 0.348327 0.368591 0.137592 0.33956 0.097768 0.105755

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