 Applied Mathematics, 2010, 1, 481-488 doi:10.4236/am.2010.16063 Published Online December 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. AM Rotating Variable-Thickness Inhomogeneous Cylinders: Part I—Analytical Elastic Solutions Ashraf M. Zenkour Department of Mathematics, Faculty of Science, King AbdulAziz University, Jeddah, Saudi Arabia Department of Mathematics, Faculty of Science, Kafrelsheikh University, Egypt E-mail: zenkour@gmail.com Received June 2, 2010; revised December 12, 2010; accepted December 16, 2010 Abstract In this paper, an analytical solution for the rotation problem of an inhomogeneous hollow cylinder with variable thickness under plane strain assumption is developed. The present cylinder is made of a fiber- reinforced viscoelastic inhomogeneous orthotropic material. The thickness of the cylinder is taken as parabolic function in the radial direction. The elastic properties varies in the same manner as the thickness of the cylinder while the density varies according to an exponential law form. The inner and outer surfaces of the cylinder are considered to have combinations of free and clamped boundary conditions. Analytical solutions are given according to different types of the hollow cylinders. An extension of the present solutions to the viscoelastic ones and some applications are investigated in Part II. Keywords: Rotating, Inhomogeneous Cylinders, Orthotropic, Variable Thickness and Density 1. Introduction The rotation problem of inhomogeneous cylinder has been important applications, particularly in mechanical engineering, aerospace industry, underwater vehicles and biomechanics. The pertinent literature on the investiga tion of stresses and displacements in an inhomogeneous hollow circular cylinder may be reviewed here. The plane strain problem of a rotating inhomogeneous orthotropic hollow cylinder is solved by Senitskii . Horgan and Chan  analyzed two-dimensional plane stress/strain deformations by assuming Youngs modulus to be a power law function of the radial direction of the cylinder and constant Poisson's ratio. Vasilenko and Klimenko,  have analyzed the stress state of a rotating cylinder, inhomogeneous in the radial direction, having one plane of elastic symmetry and loaded with centrifugal forces. Rooney and Ferrari  have examined the tension, ben- ding, and flexure of cylinders with functionally graded (FG) cross-section. The effect of inhomogeneity of elastic properties and density in the circumferential direction on the distribution of stress and displacement in orthotropic cylindrical panels using load in the axial direction is investigated by Grigorenko and Vasilenko . Oral and Anlas  have analyzed the effect of continuous in- homogeneity on the stress distribution in an anisotropic cylinder. Pan and Roy  have solved a plane-strain problem for a FG cylinder by dividing it into several homogeneous cylinders. Tutuncu  has gave the power series solution for stresses and displacements in FG cylinders with exponentially-varying elastic modulus through the radial direction. Theotokoglou and Stam- pouloglou  have studied axisymmetric problems for radially inhomogeneous circular cylinders. The effect of varying Poisson's ratio on deformation fields in FG cylinders has been investigated by Mohammadi and Dryden . Li and Peng  have analyzed axisy- mmetric deformations of FG hollow cylinders and disks with arbitrarily varying material properties. In recent years considerable attention has been given to solutions for the cylinders with variable thickness. Variable-thickness hollow cylinder is a common struc- ture type which can be used in some applications in- volving aerospace, submarine structures, nuclear reactors as well as chemical pipes. Grigorenko and Rozhok  have studied the stress problem for non-circular hollow cylinder with variable thickness under uniform and local loads. Zenkour  has established the stresses in a rotating variable-thickness orthotropic cylinder cont- aining a solid core of uniform-thickness. Also, Zenkour  has analytically investigated the behavior of com- posite circular cylinders subjected to internal and ex- A. M. ZENKOUR Copyright © 2010 SciRes. AM 482 ternal surface loading. The cylinder consists of a number of homogeneous ply groups of axially variable thickness. Duan and Koh  have derived analytical solutions for axisymmetric transverse vibration of cylindrical shells with thickness varying monotonically in arbitrary power form due to forces acting in the transverse direction. Nie and Batra  have studied plane-strain static defor- mations of a cylinder with elliptical inner and circular outer surfaces composed of a material that is polar-or- thotropic and its moduli vary exponentially in the radial direction. In this paper, the rotating fiber-reinforced viscoelastic hollow cylinder is analytically studied. The thickness of the cylinder, the elastic properties and density are taken to be functions in the radial coordinate. The governing second-order differential equation is derived and solved with the aid of some hypergeometric functions. The dis- placement and stresses for rotating variable-thickness inhomogeneous orthotropic hollow cylinder subjected to various boundary conditions are obtained. Special cases of the studied problem are established. 2. Formulation of the Problem Consider an elastic hollow cylinder made of an inhomo geneous, orthotropic material and rotates about its axis. The cylindrical coordinates (, ,)rz are chosen su ch th a t the axial coordinate z coinciding with the axis of rota- tion, r is the radial coordinate. Assuming the cylinder is symmetric with respect to the zaxis, we have only the radial displacement u which is independent of the circumferential coordinate . Furthermore, in the planes perpendicular to the zaxis in plane strain, u is a fun- ction of r alone. Consequently, the Cauchy's relations under considerations can be written in the following form: =,=,= == =0,rrzz rrzzdu udr r  (1) where ij are the strain components. From the generalized Hooke's law and using the above geometric relations, we can obtain the stress components for an orthotropic cylinder in the following form: 11 12122213 23=,==,===0rrzzr rz zdu uduucc ccdr rdrrdu uccdr r (2) where ijc are the elastic properties. Let us assume now that the thickness h of the cylinder varies in the radial direction in a parabolic form given by:  0=1/,0 <1,>0,khrhnr bnk (3) where 0h is the thick ness at the axis of the cylinder, n and k are geometric parameters and b is the external radius of the cylinder. The parameter k determines the shape of the thickness profile while n determines the thickness at the surface of the cylinder relative to 0h. For three sets of geometric parameters n and k, the dimensionless thickness 0/hh as a function of the dimensionless radius /rb is described by the profiles shown in Figure 1 for =5ba in which a is the inner radius of the cylinder. In Figure 1(a) the thickness profile is concave for <1k while in Figures. 1(b) and 1(c) it is conv ex for >1k. Furthermore, the thickness of the cylinder is linearly decreasing by setting =1k. As the effect of thickness variation of rotating cylin- ders can be taken into account in their equilibrium equation, the theory of the cylin d ers of variable th ickness can give excellent results as that of the uniform thickness Figure 1. Parabolic cylinder profiles: (a) k = 0.6; n = 0.8; (b) k = 2.5; n = 0.8; (c) k = 2.5; n = 0.4. h / ho h / ho h / ho r / b r / b r / b A. M. ZENKOUR Copyright © 2010 SciRes. AM 483cylinders as long as they meet the assumption of plane strain. After considering this effect, the equilibrium equation of rotating cylinder with variable thickness can be written as: 22=0,rrdhrhh rdr (4) where  is the constant angular velocity and  is the density of the cylinder material. We characterize the elastic properties ijc and the ma- terial density  of inhomogeneous cylinder by: /0=1/ ,=,kkmrbij ijcnrb e (5) where ij and 0 are the values of ijc and  in the homogeneous case, respectively, and m is a rational number. 3. Elastic Solution Substituting from Equation (2) into Equation (4) with the aid of the expressions given in Equation (5) and the cylinder profile given in Equation (3), we can get the following confluent hypergeometric differential equation for the radial displacement ()ur : 23 (/)22202112 (/)2(/)1=0,1(/)1(/) 1(/)kmr bkkkkkred unk rbdunk rbrr udrdrn rbn rbnr b     (6) where 22 1112 11=/,=/. (7) Introducing the dimensionless radius =/rrb in Equation (6), then its general solution can be written as (1/2/ )(1/2/ )112 2ˆˆ()=()() (),kkurnCPrnCP rRr (8) where 1ˆC and 2ˆC are arbitrary integration constants and ()1()2=,,,,=1,1,2,,kkPrr MijnrPr r Mijnr   (9) in which 222=1,=1,=1,=2 .ij kkkkk   (10) Note that, the function (,, ,)Mz is the generalized hypergeometric function defined by : =0(,, ,)=,=,!qqq kqqzMzznrq  (11) where q, for example, is the Pochhammer's symbol given by ()=(1)(2)(1)=,()qqq   (12) in which  represents Gamma function. It is to be noted that, for real values of the upper parameters  and , and non-zero real value of the lower parameter  the generalized hypergeometric function (,, ,)Mz converges for ||<1z. The particular solution ()Rrfor Equation (8) is obtaine d u sing vari ation of parame t e r s as 112 2()=() ()()(),RrU rPrUrPr (13) where 211200()() ()()()=, ()=,() ()rrPf PfUrdUrd  (14) in which 23011()= ,1kmrkbrefr nr (15) and ()r is the Wronskian given by 2112() ()()= ()().dPrdP rrPrPrdr dr (16) Therefore,   2223 210120011= ,11kkmmrrkkeM eMbRrrM rdrMrdnF nF     (17) where  2112 12=2.dMrdM rFrrM rMrMrMrdr dr (18) Note that, the first derivative of the general hypergeo- metric function is given by:  , ,,=1,1,1,.ddzMzM zdr dr   (19) A. M. ZENKOUR Copyright © 2010 SciRes. AM 484 Consequently, the exact general solution for the radial displacement can be written as   1112 22=,urrMrCF rrMrCFr   (20) where  1/2 /1/2 /112 2ˆˆ=,=,kkCn CCn C (21) and   223 201011223 102011=,1=.1kmrkkmrkeMbFr dnFeMbFr dnF (22) Substituting from Equation (20) into Equation (2) yields the radial, circumferential and axial stresses for the rotating variable thickness and density inhomogen- eous orthotropic hollow cylinder in the following form:    ()111 2221111111211 122111112112=1rrkrnrdMM dFdMMdFrCFMr CFMbdrr drdrrdr            (23)  () ()1112221112122212122121222122=1()( )krnrdMM dFdMMdFrCFMr CFMbdrrdrdrrdr             (24) () 1112221113132313 122131323132=1.zzkrnrdMM dFdMMdFrCFMr CFMbdrrdrdrrdr        (25) Note that, if ==0nm then 00()= ,=, =ij ijhrhc and the radial displacement given in Equation (20) for the rotating uniform thickness and density homogeneous orthotropic hollow cylinder is reduced to  233012 211=,9brur CrCr (26) also, the corresponding stresses in this case are given by:   1122211 12111 12211 1202111122212 22111 222122202111122213 23113 23213 23021113=,913=,931=.9rrzzrC rCrbrbrC rCrbrbrC rCrbrb       (27) In addition, for isotropic cylinder we have   11 2212 13231== ,=== ,112 112EE  (28) where E and  are Young's modulus and Poisson's ratio of the cylinder material. Using Equation (28) we find that the solution given in Equations (26) and (27) for the rotating uniform thickness and density homogeneous isotropic hollow cylinder takes the form:     23321022212 0222212 0223210112=,8112 32=,112 8112 12=,112 811122=.112 41rrzzCur CrbrrEErCC brbrErCC brbrErC brbE       (29) A. M. ZENKOUR Copyright © 2010 SciRes. AM 485The previous elastic solutions will be completed by calculating the integration constantsiCusing various bo- undary conditions on the surfaces of the hollow cylinder. 4. Rotation of Elastic Hollow Cylinders In the present section, we will obtain the elastic so lutions for the rotating hollow cylinder. For the present hollow cylinder, the solution requires that one boundary con- dition be satisfied at each surface. The radial stress must be vanished at the free surface (F) of the cylinder while the radial displacement must be equal to zero at the clamped surface (C) of the cylinder. 4.1. Free-Free (FF) Hollow Cylinder When the inner and outer surfaces (=,=rarb or =/=,=1)rabar of the cylinder are free of any traction, the boundary conditions are given by: =0= ,=0 =1.rrrrratraratr (30) Using the above conditions into Equation (23), the constants 1C and 2C are given by     12122223 121112121322111 221221111 212 1321121 2222311211 22122111=,11=,FS FSSSFaSFaSSSCSS SSFaSFaSS SFSFSSSCSS SS    (31) where  1111111112=MaSa Maa  21211 21112=MaSa Maa   13 111122=SaMaFaaMaFa (32) 2111 111121=1 1SMM 2211211122=1 1SM M 2311 1122=1111SMFMF  in which the prime () means differentiation with respect to r. The radial displacement and stresses for the rotating variable thickness and density inhomogeneous orthotro- pic hollow cylinder with free surfaces can be calculated from Equations (20), (23)-(25) and (32). The solution given in Equations (26) and (27) for the rotating uniform thickness and density homogeneous orthotropic hollow cylinder with free surfaces can be obtained with the help of the following constants:   322311 1201112111112322311 120211211111231=,931=9aabCaaaabCaa      (33) Also, the radial displacement and stresses given in Equation (29) for the rotating uniform thickness and density homogeneous isotropic hollow cylinder with free surfaces can be written as  2222302112 32=13281 12aurar brEr     222220232=181rr ararbr   222220232 12=1813 2ararbr  (34)  22220=132241zz rarb A. M. ZENKOUR Copyright © 2010 SciRes. AM 486 This is the well-known solution of the rotating uni- form thickness cylinder . 4.2. Clamped-Clamped (CC) Hollow Cylinder When the inner and outer surfaces =,=1rar of the cylinder are clamped, the boundary conditions are given by: =0= ,=0 =1.urat raurat r (35) From these conditions and Equation (20), the con- stants 1C and 2C are given by      112241114241 212141411142 411112214221414111 111=,1111111=,11MFMF S FaSFaSMCMSMSFaSFaS MMFMFSCMSMS   (36) where 141 412=,= .SaMaSaMa (37) With the h elp of Equation s (2 0), ( 23)-(25) and(36) ,w e can obtain the radial displacement and stresses for the rotating variable thickness and density inhomogeneous orthotropic hollow cylinder with clamped surfaces. The solution given in Equation (27) for the rotating uniform thickness and density homogeneous orthotropic hollow cylinder with clamped surfaces can be calculated with the aid of the following co nstants:   33323 32300122211 1111=,=.99aab aabCCaa aa       (38) Finally, the radial displacement and stresses given in Equation (29) for the rotating uniform thickness and density homogeneous isotropic hollow cylinder with clamped surfaces becomes      2222302222 2202222 220222 220112=1 ,8112=1 32,8112=1 12,81=12 .41rrzzaurar brErabrarrabra rrrarb      (39) 4.3. Free-Clamped (FC) Hollow Cylinder When the inner surface of the cylinder 1=Cr a is free of any traction and the outer surface =1r is clamped, the boundary conditions are given by: =0= ,=0 =1.rr ratraurat r (40) From Equations (40), (20) and (23), the constants 1C and 2C are given by    112212111212 1321211112111 212 13 1112211221111211 111=,11111(1)1=.11MFMF S FaSFaSSMCMSMSFaSFaSS MMFMFSCMSMS   (41) Substituting from these constants into Equations (20), (23)-(25), we can get the radial displacement and stresses for the rotating variable thickness and density inhomo- geneous orthotropic hollow cylinder with free inner and clamped outer surfaces. In addition, the solution for the rotating uniform thickness and density homogeneous orthotropic hollow cylinder with free inner and clamped outer surfaces can be obtained from Equations (26) and (27) with the help of the following constants: A. M. ZENKOUR Copyright © 2010 SciRes. AM 487322311 1211 12011121111 1211 12322311 1211 12021121111 1211123=93=9aabCaaaabCaa            (42) Also, the radial displacement and stresses given in Equation (29) for the rotating uniform thickness and density homogeneous isotropic hollow cylinder with free inner and clamped outer surfaces can be obtained in the following form:        422223022342 222 2022242 22202211212321 32=,81 12 121232132 12=32,81 12 121232132 12=811212rraaaurr brEaaraa abrraaraa abraa             2242220212 ,12 32=2.41 12zzrrarrba  (43) 4.4. Clamped-Free (CF) Hollow Cylinder When the inner surface of the cylinder =ra is clamped and the outer surface =1r is free of any traction, the bound ary conditions are given by: =0= ,=0 =1.rrurat raratr (44) From Equations (20), (23) and (44), the constants 1C and 2C are given by    1212222341114241 221221421 41114 24121121222 2314222 1421 4111=,11=FS FS SSFaS FaSSCSSSSFaSFaS SFSFSS SCSS SS   (45) The radial displacement and stresses for the rotating variable thickness and density inhomogeneous orthotr- opic hollow cylinder with clamped inner and free outer surfaces can be obtained from Equations (20), (23)-(25) and (45). Also, the solution given in Equation (27) for the rotating uniform thickness and density homogeneous orthotropic hollow cylinder with clamped inner and free outer surfaces can be calculated with the help of the following constants: 332311 12111201211111211 12332311 1211 1202211111211 123=,93=.9aabCaaaabCaa      (46) Finally, one can obtain easily the radial displacement and stresses given in Equation (29) for the rotating uniform thickne ss and densit y hom ogeneous is otropic hollow cylinder with clamped inner and free outer surfaces in the following form:      422 22302234222 2202224222 2022211232 1232=,81 12 1232 121232=32,81 12 1232 121232=8112 12rraaaurr brEaaraabarraaraabaraar          242220212 ,32 12=2.41 12zzrarrba  (47) A. M. ZENKOUR Copyright © 2010 SciRes. AM 488 5. Conclusions The rotation problem of a variable-thickness inhomogen- eous, orthotropic, hollow cylinder has been studied. Analytical solution for rotating variable-thickness, in- homogeneous, orthotropic, hollow cylinder subjected to different boundary conditions are derived. The displace- ment and stresses for rotating uniform-thickness, homo- geneous, isotropic, hollow cylinder are obtained as special cases of the investigated problem. In the second part of this paper we will present the corresponding viscoelastic solutions and some applications concerning the effects due to many parameters on the radial displace- ment and stresses. 6. References  Yu. É. Senitskii, “Stress State of a Rotating Inhomoge-neous Anisotropic Cylinder of Variable Density,” Inter-national Applied Mechanics, Vol. 28, No. 5, 1992, pp. 28-35.  C. O. Horgan and A. M. 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