Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 173 (2017) 1959 – 1966 11th International Symposium on Plasticity and Impact Mechanics, Implast 2016 Shear wave propagation in a cylindrical Earth model Moumita Mahantya*, Amares Chattopadhyayb, Abhishek Kumar Singhc a,b,c Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad-826004, India Abstract In the present paper the study of shear wave propagation in a cylindrical fibre-reinforced earth model has been discussed in the presence of radial inhomogeneity The dispersion equation of shear wave propagation has been derived by using Debye asymptotic expansion for upper layer and lower layer to be heterogeneous and homogeneous respectively The oncoming results are verified with the classical result of Love wave in the absence of reinforcement and inhomogeneity parameter The variation of dimensionless phase velocity has been plotted against dimensionless wave number to show the effect of inhomogeneity parameter, reinforcement and the ratio of radii © Authors Published by Elsevier Ltd This Ltd is an open access article under the CC BY-NC-ND license The Authors Published by Elsevier ©2017 2016The (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the organizing committee of Implast 2016 Peer-review under responsibility of the organizing committee of Implast 2016 Keywords: Shear wave; Cylindrical Co-ordinate; Fibre-reinforced medium; Inhomogeneity; Debye asymptotic expansion; Dispersion Equation Introduction The dynamical problems on the propagation of horizontally polarized shear waves (SH waves) in anisotropic media have great geophysical importance because they help to investigate the structure of the earth In particular, the propagation of elastic waves in fiber-reinforced media plays a very important role in construction sectors and geomechanics Reinforced composites are widely used in various fields such as aviation, space, construction, medical service and the sports and leisure fields The characteristic property of a fibre-reinforced material is that the components of the material act together as a single anisotropic unit as long as they remain in an elastic condition (i.e there is no relative displacement between the two components of the material) The earth’s crust exhibit reinforcement in nature as it contains some hard and soft rocks or materials Many research works have been done to prove that the Earth’s crust may contain some hard or soft materials that may exhibit fibre-reinforcement Spencer [1] introduced the constitutive equation for a fibre-reinforced linearly anisotropic material in preferred direction The concept of introducing a continuous reinforcement at every point of * Corresponding author Tel +91-9852156012 E-mail address: mahantymoumita@gmail.com 1877-7058 © 2017 The Authors Published by Elsevier Ltd This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the organizing committee of Implast 2016 doi:10.1016/j.proeng.2017.01.314 1960 Moumita Mahanty et al / Procedia Engineering 173 (2017) 1959 – 1966 an elastic solid was given by Belfield et al [2] Then Verma and Rana [3] discussed the rotation of a circular cylindrical tube which is reinforced by fibres lying along helices Chattopadhyay et al [4] have studied the propagation of Love waves on a cylindrical earth model Chattopadhyay and Choudhury [5] have analyzed the propagation of shear waves in self-reinforced medium In the present work a modified dispersion equation has been obtained when the upper layer is heterogeneous and lower layer is considered as homogeneous Here the radial variation of both rigidity and density for the upper media has been taken Also, comparative study has been made through graphs to find the effect of reinforcement, the ratios of the radii of the inner and outer media, inhomogeneity Formulation and solution of the problem Let r,T , z be the cylindrical coordinates of a point inside the model earth, and the radius of its inner and outer surfaces are defined by r a and r b respectively, h b - a be the thickness (Fig.1) The z-axis is taken along the axis of the cylinder and the propagation of a wave over the cylindrical surface is symmetric about the z-axis b O a Fig.1: Geometry of the problem The constitutive equation for a fibre-reinforced material whose preferred direction is that of a unit vector a is (Spencer [1]) W ij O ekk G ij 2PT eij D ak am ekmG ij ekk a j PL - PT ak ekj a j ak eki E ak am ekm a j ; i, j , k , m 1, 2,3 (1) where W ij are components of stress, eij are components of infinitesimal strain, and the components of a, which are referred to rectangular Cartesian co-ordinates xi The vector a may be a function of position Indices take the value 1, and 3, and the repeated suffix summation convention is adopted The coefficients O , PL , PT , D and E are all elastic constant with the dimension of stress The components of the stress tensor W is W rr ,W rT ,W rz ,W TT ,W zz and the infinitesimal strain tensor e is err , erT , erz , eTT , ezz For fibre-reinforcement the vector a is everywhere directed in tangential T direction So a has component 0,1,0 in cylindrical coordinate Here we consider the lower layer r a as homogeneous and the upper layer a d r d b as inhomogeneous layer and the inhomogeneity is taken by PL2 Đrà â l P Lc ă , PT b Đrà â l PTc ă , U b Đrà â l U 2c ă b (2) In absence of body force the equation of motion in terms of stress components in cylindrical coordinate can be written as 1961 Moumita Mahanty et al / Procedia Engineering 173 (2017) 1959 – 1966 w ur ½ ,° wt ° ° wW rT wW TT wW T z 2W T z w 2u ° U 2T , ¾ (3) wr r wT wz r wt ° ° wW rz wW T z wW zz W rz w 2u ° U 2z wr r wT wz r wt ¿° Where ur , uT , uz are the displacement components in r,T , z direction respectively and the stress tensors are given in the following form wW rr wW rT wW rz W rr - W TT wr r wT wz r W rr Merr NeTT Lezz ,W rT M O 2PT , N Q O E 4P L - 2PT U 2P L erT ,W rz 2PT erz , ½ ° ¾ W TT Nerr QeTT Nezz ,W T z 2P L eT z ,W zz Lerr NeTT Mezz °¿ where M , L, N and Q are all elastic constants given by (4) O D , L O , ẵ ắ (5) The strain displacement relationship for cylindrical coordinate is imposed by ª wuT uT wur º , erz r r wT ằẳ ôơ wr err wur , erT wr eTT wu º 1ª ur T ằ , eT z wT ẳ r ôơ ê wu z wur ẵ , ôơ wr wz ằẳ ắ wu z ê wu z wuT º , ezz ° wz ằẳ wz ôơ r wT For shear wave propagation ur uT (6) u r,T , t and 0, uz components in r,T , z direction respectively w { 0, where wz ur , uT , uz are the displacement Substituting (4), (5) and (6) in equation (3) the only non vanishing equation of motion for lower layer r a is r2 w u1z w u1z wu1 f12 r z 2 wr wr wT r w u1z E12 wt (7) where u1z is displacement of the lower layer along z direction, PL PT PT and PT , U1 are rigidity and U1 f12 , E12 1 density respectively Substituting uz r2 U z r eipt cos nT in (7), it takes the form d 2U z (r ) dU z (r ) § r p 2 à r ă - n f1 áU z (r ) dr dr â E1 ¹ (8) Again the non vanishing equation of motion for the upper inhomogeneous layer a d r d b is given by r2 2 w u z2 wu z2 w uz f r l wr wr wT r w u z2 E 22 wt where uz2 is displacement components for the upper layer in (9) z direction, f 22 PLc PTc , E 22 PTc U 2c ,and PTc , U c are rigidity 1962 Moumita Mahanty et al / Procedia Engineering 173 (2017) 1959 – 1966 and density of the upper medium respectively Substituting uz2 eipt cos nT r l / 2U z2 r , equation (10) takes the form d 2U z2 r dU z2 r § r p § 2 l · · ăă ă f n ¸ ¸¸ U z r (10) dr ¹¹ dr © © E2 where n is positive integer The solution of equation (8) is given by § pr · Đ pr à U 1z r CJ n f ă ¸ , for r a where C is an arbitrary constant and J n f ă is the Bessels function of order â E1 â E1 ¹ nf1 Thus the solution for the layer r a is given by r2 r 1 § pr · Ceipt J n f ¨ ¸ cos nT , for r a â E1 For the layer a d r d b the solution of (10) is u1z (11) eipt cos nT r u z2 l n2 f 22 where q ª AH q 1 BH q º , for a d r d b ẳ (12) l2 , H n and H n are Hankel function of 1st and 2nd kind of order n and A and B are arbitrary constant H n z J n z iYn z , H n z J n z iYn z 2.1.Boundary Condition For this problem the boundary conditions are given by ½ a° ° ° l wu z ° § r · wu z c PT ă on r a ắ PT wr w b r â l ° § r · wu z on r b PTc ă â b wr ¿ Using the boundary condition (13) and eliminating A, B, C , we get u1z u z2 on r (13) ê Đ J nfc ă â Đ ô J nf ă â ơô PT Đ b Ãl p ôô PTc ăâ a áạ E1 ô 1 1 pa à áằ E1 ằ pa à ằ áằ E1 ẳằ N1 D1 (14) where N1 ê Đ pa Ã Đ pb à § pb · § pa · º 2alp ª § pb · § pa · § pa · § pb à l2 ôJq ă ôJq ă Yq ă Jq ă Yq ă áằ Yqc ă J qc ă Yq ă áằ â E â E ẳằ E ơô â E â E â E â E ẳằ ơô â E â E § pa · § pb · º 4abp 2blp ê Đ pb Ã Đ pa à ô J qc ¨ ¸ Yq ¨ ¸ Jq ¨ ¸ Yqc ă áằ E ôơ â E © E ¹ E 22 © E ¹ â E ằẳ ê Đ pa Ã Đ pb Ã Đ pb Ã Đ pa à ô J qc ă Yqc ă J qc ă Yqc ă áằ , ôơ â E ¹ © E ¹ © E ¹ © E ằẳ 1963 Moumita Mahanty et al / Procedia Engineering 173 (2017) 1959 1966 ê Đ pb · § pa · § pa · § pb · 2bp ê Đ pa Ã Đ pb Ã Đ pb Ã Đ pa à l ôJq ă ôJq ¨ ¸ Yq ¨ ¸ Jq ¨ ¸ Yq ¨ ¸» ¸ Yqc ¨ ¸ J qc ă Yq ă áằ â E â E ẳằ E ơô â E â E â E â E ẳằ ơô â E © E ¹ D1 For the left hand side of Eq (14), following Debye asymptotic expansion for function of large order and argument (cf Watson [6]) is used exp nf1 D D J nf nf1 sec hD 2S nf1 D (15) E where 1Ã Đ *ăr f Ư â Đ Ã r *ă â2ạ E 2m Dr nf1 D , D0 r 1, D1 - coth D , D2 24 For D be any fixed positive number, taking nf1 sec hD J nfc nf1 sec hD exp nf1 D D 2S nf1 D 77 385 coth D coth D 128 576 3456 pa , we get sinh D E1 nf1E1 Đ p2 a2 à ă1 2 , pa â n f1 E1 sinh D (16) For the right hand side of Equation (14), expansion for J q q sec E and Yq q sec E will be used, which are respectively given by · S Đ Ã ẵ cos ă q tan E q E , â ắ Đ Ã Đ S à Yq q sec E ~ ă sin ă q tan E q E â S q tan E â Đ J q q sec E ~ ă â S q tan E (17) pb where E is fixed positive acute angle and q is large Assuming q sec\ tan I \ I sin\ where E2 pa , q sec I E2 we get ½ ° G a , tan\ , tan\ tan I , ° 2 2 G a 1 G a 1 ° ° °° h qh 1 G a ,sin I G 2a2 1 ,¾ , Q q tan\ tan I q \ I a G2a2 1 a Ga ° ° ° § h G2a2 · ° 2 ¸ G a ¨1 ° 2 ¨ a G a 1 Ga â p G qE2 2 h G a 1 a 2 G2a2 h a The derivatives of (17) with respect to E are given by G2a2 (18) 1964 Moumita Mahanty et al / Procedia Engineering 173 (2017) 1959 1966 Đ Ã S Đ Ãẵ J qc q sec E ~ ă sin E sin ă q tan E q E ¸ ,° â â S q tan E ắ § · S § ·° Yqc q sec E ~ ă sin E cos ă q tan E qE â â S q tan E ¹ Using (17), (18) and (19), we have Yq q sec I J q q sec\ J q q sec I Yq q sec\ ~ S q tan I tan\ (19) (20) sinQ , Yq q sec I J qc q sec\ Yqc q sec\ J q q sec I ~ (21) sin\ cosQ , S q tan I tan\ Yqc q sec I J q q sec\ J qc q sec I Yq q sec\ ~ S nf tan I tan\ (22) sin I cosQ , Yqc q sec I J qc q sec\ J qc q sec I Yqc q sec\ ~ S q tan I tan\ Writing c pa n , n a (23) sin I sin\ sinQ k and using (15), (16), (18), (20), (21), (22), and (23) the dispersion equation takes the form ê Đ ô ă 2 ô Đ l à c2 tan ô kh ă f 22 ă 4n ă Đ l ô â ă E2 ă f2 ă ô 4n â â ôơ à á 1 à á áá ạ ằ » » » » »¼ N2 D2 (24) where N , D2 are given in appendix A This is the dispersion equation of shear wave propagation in a cylindrical reinforced model when the upper layer is heterogeneous and lower layer is homogeneous Particular Case: When both the layers are isotropic and homogeneous i.e l and PL PT , PLc PTc , then equation (25) takes the form as § § · 2à c tan ă kh ă 1á ă â E2 áạ â Đ c2 à ă1 PT â E1 PTc Đ c à ă 1á â E2 The above equation is the dispersion equation of Love wave propagation in isotropic homogeneous layer lying over 1965 Moumita Mahanty et al / Procedia Engineering 173 (2017) 1959 – 1966 an isotropic homogeneous half-space Numerical discussion and Graphical representation The numerical calculations are done to illustrate the theoretical results obtained earlier The graphs have been plotted using the following data for fibre-reinforced lower and upper layer For the lower fibre-reinforced layer (Markham [7]) PL1 5.66 u109 N / m2 , PT1 2.46u109 N / m2 , U1 7800kg / m3 For the upper fibre-reinforced layer (Hool and Kinne [8]) PL0 7.07 u109 N / m2 , PT 3.5u109 N / m2 , U0 1600kg / m3 The effect of reinforcement and the ratio of the radius of upper and lower layer on the propagation of shear wave in a fibre- reinforced layer in cylindrical co-ordinate have been represented by means of graphs The figures represent § c · the variation of non-dimensional velocity ă with respect to dimensionless wave number kh â E2 2.2 curve1: l=0.0 curve2: l=0.5 curve3: l=1.0 1.85 2.1 1.8 c/bita2 c/bita2 1.9 1.75 curve1: l=0 curve2: l=0.5 curve3: l=1 1.9 1.7 1.8 3 1.65 1.6 1.1 1.7 1.15 1.2 1.25 1.3 1.35 1.4 1.6 kh Fig:2 Variation of kh with Đc à ă E 2ạ â 1.05 1.1 1.15 1.2 1.25 1.3 1.35 for different Fig:3 Variation of inhomogeneity parameter in absence of reinforcement kh with Đc à ă E 2ạ â for different values of inhomogeneity of parameter in presence of reinforcement 2.2 curve1: bba=1.3 curve2: bba=1.31 curve3: bba=1.32 2.1 c/bita2 1.9 1.8 1.7 1.6 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 kh Fig:3 1.4 kh Variation of kh with Đc à ă E © 2¹ for different values of bba (=b/a) 1966 Moumita Mahanty et al / Procedia Engineering 173 (2017) 1959 – 1966 Conclusion In present work an analytical approach has been done for the propagation of shear wave in fibre-reinforced media in cylindrical co-ordinate, not attempted till now The dispersion equation has been obtained for cylindrical fibrereinforced media Non-homogeneity, ratio of radii and reinforcement have been found to have a remarkable effect on the propagation of shear waves The phase velocity always increases with increase of the ratio of radii and decreases when inhomogeneity parameters increase Acknowledgement The authors convey their sincere thanks to Indian School of Mines, Dhanbad for facilitating us with best research facility and also thanks to DST India to provide a Inspire research fellowship to Miss Moumita Mahanty Reference [1] A.J.M Spencer, Deformations of fibre reinforced material London Oxford University, (1972) [2] A.J Belfield, T.G Rogers, A.J.M Spencer, Stress in elastic plates reinforced by fibres lying in concentric circles, J Mech Physic Solids 31, 25-54 (1983) [3] P.D.S Verma, O.H Rana, Rotation of a circular cylindrical tube reinforced by fibres lying along helices, Mech Mater 2, 353-359 (1983) [4] A Chattopadhyay, N.P Mahata, Propagation of Love waves on a cylindrical earth model J Acoust Soc Am 74, 286-293 (1983) [5] A Chattopadhyay, S Chaudhury, Magnetoelastic shear waves in an infinite self- reinforced plate Int J Numer Anal Methods Geomechanics 19, 289-304 (1995) [6] G.N Watson, Theory of Bessel Function Cambridge University Press pp 234, 358, 365 and 243 (1922) [7] M.F MarkhaM, Measurement of elastic constants of fibre- reinforced by ultrasonic Composite, 1, 145-149 (1970) [8] G.A Hool, W.S Kinne, Reinforced concrete and masonary structure, McGraw- Hill, New York, (1924) Appendix A N2 D2 § ¨ ¨ c2 ¨ · § l2 · ăĐ E 22 ă f 22 á ăă n c h â 4qnf1 ă ă 1á ă a Đ Ã l ă E2 f Đ Ã ă ăâ ăâ 4n áạ ă c2 ă ă 1á ă ă 2Đ l2 à E f ă 2ă á ăă n â â â Đ c2 à 2nf1l ă1 2 â E1 f1 à á c2 Đ l2 à E 22 ă f 22 l 2 Đ P à n c Đbà T1 â u ă u 2lq , u ă1 2 c E P f a â T2 1 â Đ Ã ă c2 ă 1á ă 2Đ l2 à E f áá ă 2ă á n â â ạ Đ ă ă ă ăĐ ă PTc Đ b Ãl PTc Đ b Ãl ă ă c2 ă l 4 ă q ăă PT â a PT â a ă Đ l2 ă ăă E ă f 4n ăâ â ă ă ă â 2 1 · ¸ ¸ 1 · ¸ ¸ áá ạ c Đ l2 à E ă f2 4n â 2 h aĐ ă c2 ă ă Đ l2 ăă E ă f 4n â â à 1á à ¸¸ ¹ ¹ · ¸ ¸ ¸ ¸ Đ ă c2 uă ă l2 2Đ ăă E ă f 4n â â á à 1á à á áá ¹ ... reinforced by fibres lying along helices Chattopadhyay et al [4] have studied the propagation of Love waves on a cylindrical earth model Chattopadhyay and Choudhury [5] have analyzed the propagation. .. surfaces are defined by r a and r b respectively, h b - a be the thickness (Fig.1) The z-axis is taken along the axis of the cylinder and the propagation of a wave over the cylindrical surface... the inner and outer media, inhomogeneity Formulation and solution of the problem Let r,T , z be the cylindrical coordinates of a point inside the model earth, and the radius of its inner and