A mixed semi analytical solution for functionally graded (FG) finite length cylinders of orthotropic materials subjected to thermal load

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1 Int J Mech Mater Des (2012) 8: DOI /s A mixed semi anaytica soution for functionay graded (FG) finite ength cyinders of orthotropic materias subjected to therma oad P. Desai T. Kant Received: 21 Juy 2011 / Accepted: 5 January 2012 / Pubished onine: 20 January 2012 Ó Springer Science+Business Media, B.V Abstract A simpified and accurate anaytica cum numerica mode is presented here to investigate the behavior of functionay graded (FG) cyinders of finite ength subjected to therma oad. A diaphragm supported FG cyinder under symmetric therma oad which is considered as a two dimensiona (2D) pane strain probem of thermoeasticity in (r, z) direction. The boundary conditions are satisfied exacty in axia direction (z) by taking an anaytica expression in terms of Fourier series expansion. Fundamenta (basic) dependent variabes are chosen in the radia coordinate of the cyinder. First order simutaneous ordinary differentia equations are obtained as mathematica mode which are integrated through an effective numerica integration technique by first transforming the boundary vaue probem into a set of initia vaue probems. For FG cyinders, the materia properties have power aw dependence in the radia coordinate. Effect of non homogeneity parameters and orthotropy of the materias on the stresses and dispacements of FG cyinder are studied. The numerica resuts obtained are aso first vaidated with existing iterature for their accuracy. Stresses and dispacements in axia and radia directions in cyinders having various /r i and r o /r i ratios parameter are presented for future reference. P. Desai (&) T. Kant Department of Civi Engineering, Indian Institute of Technoogy Bombay, Powai, Mumbai , India e-mai: payadesai79@gmai.com Keywords Functionay graded materias Numerica Integration Boundary vaue probems Thick cyinder List of symbos r, h, z Cyindrica coordinates u, v, w Dispacement components r r, r h, r z Norma stress components on panes norma to r, h, and z axis s zr Shearing stress component in cyindrica coordinates e r, e h, e z Unit eongations (norma strain) components in cyindrica coordinates c zr Shearing strain component in cyindrica coordinates C ij Materia constants for orthotropic materias a i Coefficient of therma expansion per degree centigrade for orthotropic materias T Temperature rise at any point in a cyinder m Poisson s ratio r i Inner radius of the cyinder r o Outer radius of the cyinder Length of the cyinder T m Initia reference temperature u; w Nondimensionaized dispacement components r r ; r h ; r z Nondimensionaized norma stress components s rz Nondimensionaized shearing stress component in cyindrica coordinates

2 90 P. Desai, T. Kant r Nondimensionaized radius R Mean radius (r o /r i )/2 1 Introduction The demand for improved structura efficiency in space structures and nucear reactors has resuted in the deveopment of a new cass of materias, caed functionay graded materias (FGMs). FGMs have become one of the major research topics in the mechanics of materias community during the past 15 years. The concept of FGMs was proposed in 1984 by materias scientists in the Sendai (Japan) area as a means of preparing therma barrier materias (Koizumi 1997). Continuous changes in the composition, microstructure, porosity, etc. of these materias resut in gradients in properties such as mechanica strength and therma conductivity. Thus, FGMs are heterogeneous materias, characterized by spatiay variabe microstructures, and thus spatiay variabe macroscopic properties are introduced to enhance materia or structura performance. Particuary, materia properties can be designed to vary continuousy aong structura geometry to prevent deamination and stress concentration in traditiona mutiayered structures. The basic concept is to mix ceramic and meta such that the materia properties continuousy vary from one constituent materia to the other. The spatiay variabe materia properties make FGMs chaenging to anayze. Before these materia devices are used in engineering design, it is very important that these are anayzed very accuratey. For such a reason, present study focuses the anaysis of functionay graded (FG) cyinders using the exact approach. The uniqueness of this approach is: it first requires agebraic manipuation of basic easticity equations ike equiibrium, strain dispacement and constitute equations. After this manipuation, this becomes the two point boundary vaue probem (BVP) which governs the behavior of finite ength cyinder which is pane strain two dimensiona probem in r, z pane and gives four first order simutaneous partia differentia equations. This can be expained by the foowing equation (Kraus 1967;Godberg eta.1965). y 0 ðrþ ¼AðrÞyðrÞþpðrÞ ð1þ In the domain, r 1 B r B r 2, where, y(r)isann-dimensiona vector of dependent variabes; dependent variabes in the present case can be described as y ¼ ðu; w; r r ; s rz Þ t. Choice of dependent variabes is an important task. The variabes which naturay appear on r = constant are chosen as dependent variabes; such variabes are caed intrinsic variabes. Remaining variabes are described as auxiiary dependent variabes which are dependent on intrinsic dependent variabes. A(r) is a coefficient matrix of partia differentia equations. p(r) isann-dimensiona vector of non homogeneous (oading) terms. For boundary conditions, any n/2 eements of y(r) are specified at the two termini edges; mixed type of boundary conditions can be specified in this type of formuation. Recenty, Desai and Kant (2011) have obtained accurate stresses in aminated finite ength cyinders subjected to thermo eastic oad using simiar numerica mode. Research resuts obtained thus far have demonstrated that FGMs have great potentia for improving materia/structura performance in many engineering appications precisey because of their spatiay graded heterogeneous microstructure. Some of the recent iterature reevant in this study is described as foows. Horgan and Chan (1999) investigated the effects of materia inhomogeneity in fundamenta boundaryvaue probem of inear inhomogeneous isotropic pressurized hoow cyinder. The resuts are iustrated using a specific radiay inhomogeneous materia mode for which expicit exact soutions are obtained. Chen et a. (2002) considered the axisymmetric thermoeastic probem of a uniformy heated, functionay graded isotropic hoow cyinder and proposed an anaytica form of soution. Ye et a. (2001) studied the one-dimensiona axisymmetric thermoeastic probem of a functionay graded transversey isotropic cyindrica she and presented usefu discussion and numerica resuts. Exact and expicit soution is derived. Tutuncu and Ozturk (2001) obtained cosedform soutions for stresses and dispacements in functionay graded cyindrica and spherica vesses subjected to interna pressure aone using the infinitesima theory of easticity. Jabbari et a. (2002) deveoped a genera anaysis procedure for tacking one-dimensiona steady-state therma stress probem of a hoow thick cyinder made of FGMs. Recent iterature survey foows wi focus on FG cyinders of orthotropic materias subjected to therma oads. Literature survey wi focus on FG cyinders of orthotropic materias subjected to therma oads.

3 A mixed semi anaytica soution for FG finite ength cyinders 91 Wu and Tsai (2011) have recenty presented the three-dimensiona (3D) couped anaysis of simpy supported FG and piezoeectric sandwich cyinders under eectro mechanica oads using modified pagano method. The modification in the origina pagano s method were repacement of dispacementbased formuation with mixed formuation and set of compex-vaued soutions of system equation is transferred to a corresponding set of rea-vaued soutions. A transfer matrix method is used to anayse the effect of ayers. Chen and Lin (2010) have anaysed FG spheres with arbitrary Young s moduus and constant Poisson s ratio. Again the method of transmission matrix is used to account the effect of ayers and continuity conditions are required in this paper Shariyat et a. (2011) deveoped anaytica and numerica eastodynamic soutions for ong thick waed functionay graded cyinders subjected to arbitrary dynamic and shock pressures. Liew et a. (2003) presented an anaysis of the thermomechanica behavior of hoow circuar cyinders of FGM. The soutions are obtained by a nove imiting process that empoys the soutions of homogeneous hoow circuar cyinders, with no recourse to the basic theory or the equations of nonhomogeneous thermoeasticity. Yas and Aragh (2010) have investigated 3-D steady-state response of simpy supported FG fiber reinforced cyindrica pane. Suitabe temperature and dispacement functions that identicay satisfy the simpy supported boundary conditions are used to reduce the thermoeastic equiibrium equations to a set of couped ordinary differentia equations (ODEs) with variabe coefficients, which can be soved by differentia quadrature method. Peng and Li (2010) presented a nove method for anayzing steady therma stresses in a functionay graded hoow cyinder. The therma and thermoeastic parameters are assumed to arbitrariy vary aong the radia direction of the hoow cyinder. The BVP associated with a thermoeastic probem is converted to a Fredhom integra equation. By numericay soving the resuting equation, the distribution of the therma stresses and radia dispacement is obtained. Afshar et a. (2011) studied a gass-ike (viscoeastic) functionay graded cyinder by using finite eement method to investigate the mechanica responses. A subroutine is deveoped by using ANSYS parametric design anguage (APDL) to simuate two noninearities, which are the variation of materia properties with respect to time and position. In this paper, governing easticity equations of a simpy (diaphragm) supported symmetric FG orthotropic cyinder are used to predict its behaviour under ongitudinay sinusoida therma oad. Materia constants are assumed to have a power-aw dependence on the radia coordinate. By assuming a goba anaytica soution in the ongitudina direction satisfying the two end boundary conditions exacty, the 2D probem is reduced to a 1D probem in the radia direction. The equations are reformuated to enabe appication of an efficient and accurate numerica integration technique for the soution of the BVP of a cyinder in the radia coordinate. To enabe appication of numerica integration, BVP of a cyinder is converted into a set of initia vaue probems (IVPs). The basic approach to convert a BVP into a set of IVPs is aso expained in the foowing sections. Finay, a comparison of the resuting stresses with the easticity pane strain soution of infinitey ong cyinder (Ye et a. 2001) is carried out for ratios of the inner radius to outer radius of 1.5 and 1.05 and for two ratios of ength to inner radius, viz., 2 and 100. Resuts are vaidated through comparison with those given by Ye et a. (2001). In addition, one dimensiona easticity equations of an infinitey ong axisymmetric cyinder are utiized to reformuate the mathematica mode suitabe for numerica integration. These equations are summarized in the Appendix. This has been done with a view to check and compares the resuts of the present formuation of finite ength cyinder under uniform interna/externa therma and mechanica oads, when the ength of the cyinder tends to infinity. 2 Mathematica mode Basic governing equations of a 2D probem of a cyinder in cyindrica coordinates (Fig. 1a) are: Equiibrium equations or r or þ os zr þ r r r h ¼ 0; r Strain dispacement reations os zr or þ or z þ s zr r ¼ 0 ð2aþ

4 92 P. Desai, T. Kant a b Diaphragm supported u = 0, σ = 0 z e r ¼ ou or r r o ru, r i θ e h ¼ u r e z ¼ ow c zr ¼ ow or þ ou ð2bþ Stress strains temperature reations for cyindricay orthotropic materia e r ¼ r r r h r z m hr m zr þ a r T; E r E h E z r r e h ¼ m rh þ r h r z m zh þ a h T; E r E h E z r r r h e z ¼ m rz m hz þ r z þ a z T; E r E h E z c rz ¼ s rz G rz ; ð2cþ Stresses in terms of strains can be written as foows r r C 12 C 13 0 e r a r T >< >= r h C ¼ 21 C 22 C 23 0 >< >= 6 7 e h a h T r >: z 4 C >; 31 C 32 C e z a z T >: >; s rz C 44 c rz ð2dþ where, m rh ¼ m hr E h E r ; T(r,z) T(r,z) m rz ¼ m zr E z E r ; zw, m zh ¼ m hz E h E z Graded materia u 0, σ 0 z = = Diaphragm supported Graded materia Fig. 1 a Coordinate system and geometry of cyinder. b Finite FG cyinder under sinusoida externa therma oading z ¼ E rð1 t hz t zh Þ ; C 12 ¼ E rðt hr þ t zr t hz Þ ; D D C 13 ¼ E rðt zr þ t hr t zh Þ ; C 22 ¼ E hð1 t rz t zr Þ ; D D C 32 ¼ E hðt zh þ t rh t zr Þ ; C 33 ¼ E zð1 t rh t hr Þ D D where D ¼ ð1 m rh m hr m hz m zh m zr m rz 2m hr m zh m rz Þ C 21 ¼ C 12 ; C 23 ¼ C 32 ; C 31 ¼ C 13 ; C 44 ¼ G ð2eþ It is assumed that a materia constants have a poweraw dependence on the radia coordinate, i.e., C ij ¼ C 0 ij nn ; a i ¼ a 0 i nn ð2fþ where n ¼ r r i ; Cij 0, and a i 0 are constants, and n is a inhomogeneity parameter or gradient index. Spatia variation of Poisson s ratio is of much ess practica significance than that of Young s moduus. Poisson s ratio is thus assumed to be a constant. This assumption, commony made in the iterature on FGMs, eads to considerabe mathematica simpification. It can be easiy proved that when the materia is isotropic and if n = 0 for the homogeneous case, without taking therma effect, resuts are same as given by Timoshenko and Goodier (1951) for pane strain easticity soution for Lame cyinder. Stresses in terms of dispacement components can be cast as foows: ou r r ¼ or a u rt þc 12 r a ow ht þc 13 a zt ou r h ¼C 21 or a u rt þc 22 r a ow ht þc 23 a zt ou r z ¼C 31 or a rt s rz ¼C 44 c rz ¼C 44 þc 32 ow or þou u r a ht þc 33 ow a zt ð2gþ and boundary conditions (Fig. 1b) in the ongitudina and radia directions are written in Eq. 3 as, at z ¼ 0; ; u ¼ r z ¼ 0; at r ¼ r i ; r o ; r r ¼ s rz ¼ 0 ð3þ in which is the ength, r i is the inner radius and r o is the outer radius of a hoow cyinder. Radia direction r is chosen to be a preferred independent coordinate. Four fundamenta dependent variabes, viz., dispacements, u and w and corresponding stresses, r r and s rz that occur naturay on a

5 A mixed semi anaytica soution for FG finite ength cyinders 93 tangent pane r = constant, are chosen in the radia direction. Circumferentia stress r h and axia stress r z are treated here as auxiiary variabes since these are found to be dependent on the chosen fundamenta variabes (Kant and Ramesh 1981). A set of four first order partia differentia equations in independent coordinate r which invoves ony fundamenta variabes is obtained through agebraic manipuation of Eqs. 2a 2g. These are, ou or ¼ r r þ a r T þ C 12 a h T u r ow or ¼ 1 s rz ou C 44 or r or ¼ os rz þr r r a ht r u r 2 os rz or ¼ s rz r C 31 or r C 31 þ C 32 C 12C 31 o þ C 13 a z T ow ; C 21 1 þ C 21C 12 C 22 þ C 21C 13 az T C 23 r 1 r þ C 33 C 13C 31 oða r TÞ a ht u r a z T ow o ow ð4aþ and the auxiiary variabes, ou r h ¼ C 21 or a u rt þ C 22 r a ow ht þ C 23 a zt ou r z ¼ C 31 or a u rt þ C 32 r a ow ht þ C 33 a zt ð4bþ A ongitudinay sinusoida variation of temperature is assumed as foows, Tðr; zþ ¼T m sin pz ð5aþ Variations of the four fundamenta dependent variabes which competey satisfy the boundary conditions of simpe (diaphragm) supports at z = 0, can then be assumed as, uðr; zþ ¼UðrÞ sin pz wðr; zþ ¼WðrÞ cos pz ; rrðr; zþ ¼rðrÞ sin pz s rz ðr; zþ ¼sðrÞ cos pz ð5bþ Substitution of Eqs. 5a and 5b in Eq. 4a, b and simpification resuting from orthogonaity conditions of trigonometric functions ead to the foowing four simutaneous ODEs invoving ony fundamenta variabes. These are, U 0 ðrþ ¼ rðrþ þ a r T m þ C 12 a h T m UðrÞ r þ C 13 a z T m þ p WðrÞ ; W 0 ðrþ ¼ 1 G sðrþ UðrÞ p r 0 ðrþ ¼ p sðrþþ C 21 rðrþ 1 r þ C 21C 12 ah C 22 r T m UðrÞ r 2 þ C 21C 13 az C 23 r T m þ p WðrÞ r s 0 ðrþ ¼ sðrþ p C 31 p rðrþ C 31 a r r T m þ C 32 C 12C 31 p a h T m p UðrÞ r þ C 33 C 13C 31 p a z T m þ p 2WðrÞ ð6aþ and the auxiiary variabes, r h ¼ C 21 rðrþþ C 21C 12 C 22 a h T m UðrÞ r þ C 13C 21 C 23 a z T m þ p WðrÞ sin pz r z ¼ C 31 rðrþþ C 31C 12 C 32 a h T m UðrÞ r þ C 13C 31 C 33 a z T m þ p WðrÞ sin pz ð6bþ 3 Numerica soution The above system of first order simutaneous ODEs (6a) together with the appropriate boundary conditions (3) at the inner and outer edges of the cyinder forms a two-point BVP. However, a BVP in ODEs cannot be numericay integrated as ony a haf of the dependent variabes (two) are known at the initia edge and numerica integration of an ODE is intrinsicay an IVP. It becomes necessary to transform the probem into a set of IVPs. The

6 94 P. Desai, T. Kant initia vaues of the remaining two fundamenta variabes must be seected so that the compete soution satisfies the two specified conditions at the termina boundary (Kant and Ramesh 1981). The Nth (N = 4 here) order BVP is transformed into a set of (N/2? 1) IVPs. ODEs are integrated from initia edge to fina edge using the initia vaues specified in Tabe 1. TheN/2? 1 soutions given in the Tabe 1 may be thought of as (i) one nonhomogeneous integration which incudes a the nonhomogeneous terms (e.g., oading) and the known N/2 quantities at starting edge, with the unknown N/2 quantities at the starting edge set equa to zero, (ii) N/2 homogeneous integrations which are carried out by setting the known quantities at the starting edge as zero and choosing the N/2 unknown quantities at starting edge as unit vaues in succession and deeting the non-homogeneous terms from the ODEs. The soutions at the termina boundary corresponding to the initia vaues are given in the right side coumns in Tabe 1. A inear combination of the (N/2? 1) soutions must satisfy the boundary conditions at the termina edge, i.e., Y 3;0 Y 4;0 þ Y 3;1 Y 3;2 X1 Y 4;1 Y 4;2 Y i;0 þ Y i;j X j ¼ Y i or or 1 X j ¼ Y i;j Yi Y i;0 X 2 ( ) ¼ Y 3 Y 4 ð7þ where i indicates the N/2 variabes consistent with the specified boundary vaues at termina edge, j refers to soution number and ranges from 1 to N/2, Y i is a vector of specified dependent variabes at the termina Tabe 1 Initia and integrated vaues IN Initia boundary Termina boundary Load term u w r r s rz u w r r s rz (S) 0 (S) Y 1,0 Y 2,0 Y 3,0 Y 4,0 I Y 1,1 Y 2,1 Y 3,1 Y 4,1 D Y 1,2 Y 2,2 Y 3,2 Y 4,2 D FI X 1 X 2 0 (S) 0 (S) C C C C I IN integration number, S specified, C correct vaue, FI fina integration, I incude, D deete boundary and X j is a vector of unknown dependent variabes at the starting edge. Finay, a non-homogeneous integration with a the dependent variabes known at the starting edge is carried out to get the desired resuts. Fourth order Runge Kutta agorithm with modifications suggested by Gi (1951) is used for the numerica integration of the IVPs. Fow chart for numerica integration is shown in the Fig Resuts and discussion Nondimensionaized parameters are defined as foows in Eq. 8: r ¼ r R ; R ¼ 1 ð 2 r o þ r i Þ; ðu; wþ ¼ 1 atr o ðu; wþ; TR 1 r r ; r h ; r z; s rz ¼ atr o r r ; r h ; r z; s rz TCo 11 ð8þ A hoow cyinder is anaysed by taking two r o /r i ratios, 1.05 and 1.5, which cover both thick and thin cases. Materia properties for transversey isotropic materia are taken as foows (Ye et a. 2001). Materia I C12 o =Co 11 ¼ 0:364; Co 13 =Co 11 ¼ 0:372; C33 o =Co 11 ¼ 1:002; ao 3 =ao 1 ¼ 1:010 Materia II C12 o =Co 11 ¼ 0:5; Co 13 =Co 11 ¼ 0:2; C33 o =Co 11 ¼ 3:2; ao 3 =ao 1 ¼ 2:6 Using above reations between the materia constants foowing properties are taken for numerica anaysis. Materia I C11 o ¼ 22: ; C12 o ¼ 8: ; C13 o ¼ 8: ; C21 o ¼ 8: ; C22 o ¼ 22: ; C23 o ¼ 8: ; C31 o ¼ 8: ; C32 o ¼ 8: ; C33 o ¼ 22: ; C o 44 ¼ 8: a r ¼ ; a h ¼ ; a z ¼ 5:

7 A mixed semi anaytica soution for FG finite ength cyinders 95 start s=s1 Path of integration s=s2 n Input no.of 1 st order ordinary differentia equtions n/2 no. of unkown components of the vector y (s1) n/2 BC s at s=s1 n/2 BC s at s=s2 Sove for n/2 unkown components of y(s1) by inear combination of n/2+1 soutions, y0(s1) and yj (s2), so obtained in conjunction with n/2 BC s at s=s2 Non homogeneous integration incude oad terms yes Integration=? no Non homogeneous integration incude oad terms Homogeneous integration deete oad terms Set now true vaues to a components of the vector y(s1) Set prescribed vaues of known components of y(s1) Set zero vaues of to known components of y(s1) Set zero vaues to remainng unkown components of y(s1) Set zero vaues to remaining unkown components of y(s1) Integration to s=s2 Set unit vaue to one of the unknown components of y(s1) at a time Store intermediate vaues of the vector y(s) within the interva s1<=s<=s2 Integrate to s=s2 output stop Store y(s2) for a integrations as y0(s2) and yj(s2): j=1,n/2 yes Integration<=n/2+1? no Fig. 2 Fowchart for numerica integration

8 96 P. Desai, T. Kant Materia II C11 o ¼ 22: ; C12 o ¼ 11: ; C13 o ¼ 4: ; C21 o ¼ 11: ; C22 o ¼ 22: ; C23 o ¼ 4: ; C31 o ¼ 4: ; C32 o ¼ 4: ; C33 o ¼ 71: ; C o 44 ¼ 8: a r ¼ ; a h ¼ ; a z ¼ 1: Numerica anaysis is carried out with both materia I and materia II, various inhomogeneity parameters n = 2, 1 and -2 and two /r i ratios 2 and 100. Figures 3, 4, 5, 6, 7, and 8 show the variations of basic fundamenta variabes as we as auxiiary variabes through thickness for r o /r i ratio of 1.5 signifying a thick cyinder and Figs. 9 and 10 show the variation of basic fundamenta variabe radia stress as we as auxiiary variabe hoop stress through thickness for r o / r i ratio Radia and hoop quantities are maximum at z = /2 whereas axia quantities are maximum at z = 0,. Tabe 2 shows vaues of radia, hoop and axia stresses for /r i = 2 and 100. These vaues are compared with those of Ye et a. (2001) for pane strain easticity soution for infinitey ong cyinder in Tabe 2.For/r i = 100, resuts are cose to those of Ye et a. (2001). Axia dispacement is constant over the thickness for both r o /r i ratios. Radia dispacement is inear through thickness; gives higher vaue for Fig. 3 Distribution of r r through thickness for r o /r i = 1.5 for a materia I and b materia II Fig. 4 Distribution of r h through thickness for r o /r i = 1.5 for a materia I and b materia II

9 A mixed semi anaytica soution for FG finite ength cyinders 97 Fig. 5 Distribution of r z through thickness for r o /r i = 1.5 for a materia I and b materia II positive n as compared to negative n. Paraboic variations of shear stress and radia stress are seen in both thin and thick cases. In thin cyinder, hoop and axia stresses have inear variation and noninear paraboic through thickness in thick cyinder. From Fig. 3a, it is seen that positive vaue of n produces positive magnitude of radia stress (n = 2, 1) whereas negative vaue of n =-2 gives negative radia stress for both /r i = 2, 100. Both /r i ratio foow simiar trend, whereas opposite behavior is seen for non homogeneity parameter n = 2 and n =-2. Figure 3b shows radia stress resuts for materia property II. There is no significant effect is seen in terms of trends of radia stress through thickness for both materia properties. Figure 4a gives noninear behavior of hoop stress for n = 2, 1, it gives variation from positive to negative (higher to ower) whereas n =-1 gives variation ower to higher vaues. From Fig. 4b it is seen that a positive hoop stress is obtained for a cases for materia II. Negative vaues of Fig. 6 Distribution of s rz through thickness for r o /r i = 1.5 for a materia I and b materia II nonhomogeneity give drastic behavior change in the magnitude. From, Fig. 5a, it is seen that there is a effect of /r i ratios on axia stresses unike for radia stresses. Simiary from, Fig. 6a, b, it is seen the effect of /r i ratios on stresses. Paraboic shear stresses are obtained for both cases. From Fig. 7a, it is seen that radia dispacement is inear. Effect of n is seen for n = 2, n =-1 and n =-2. n = 2 gives higher range of stresses, it decreases with n = 1 and graduay further decreases for n =-2. Simiar is the case for materia II (Fig. 7b). Constant vaues are obtained for axia dispacements as seen from Fig. 8a. Negigibe and ower vaues are obtained for /r i = 2 whereas higher vaues are obtained for /r i = 100. Same is the case for materia properties II (Fig. 8b). From Fig. 9a, b shows radia stresses for thin cyinder. For different n vaues, behavior of thin cyinder is same as thick

10 98 P. Desai, T. Kant Fig. 8 Distribution of w through thickness for r o /r i = 1.5 for a materia I and b materia II Fig. 7 Distribution of u through thickness for r o /r i = 1.5 for a materia I and b materia II cyinder. From Fig. 10a, b, it is seen that hoop stresses are ineary varying as compared to ratio thick cyinder. It is seen that non-homogeneity parameter n both positive and negative has greater effect on distribution of stresses and dispacements. Stresses can be triggered easiy with functionay graded mode presented according to the engineering design requirements. 5 Concusion An attempt is made here to anayze the FG cyinders which are subjected to eastostatic and temperature fieds through exact semi anaytica cum numerica approach which differs from conventiona approximate finite eement approach and is aso free from any assumptions in the theory. Resuts are very usefu when one is designing pressurized cyinders made up of FG materias subjected to therma oad. This approach can be appied to very thick cyinders. Technique is very convenient to obtain the stresses with an ease, since no separate integration is required to account the non homogeneity effect occurred due to gradation. This is an important feature of the proposed mode. Aso, it invoves mixed variabes in the derivations, both stresses and dispacements are obtained accuratey simutaneousy. Systematic deveopment of mathematica mode has significanty contributed in understanding the behavioura phenomenon of graded cyinders under extreme oading environment of therma oadings. Mathematica mode deveoped here is simpe in nature and easiy appicabe for the arge cass of she probems. Choice of fundamenta variabes is an important task for deveoping the mode. Current mode is appicabe to simpy diaphragm supported cyinder ony. Change in

11 A mixed semi anaytica soution for FG finite ength cyinders 99 Fig. 9 Distribution of r r through thickness for r o /r i = 1.05 for a materia I and b materia II Fig. 10 Distribution of r h through thickness for r o /r i = 1.05 for a materia I and b materia II Tabe 2 Comparison of non-dimensiona radia stress r r ðz ¼ =2Þ, hoop stress r h ðz ¼ =2Þ and axia stress r z ðz ¼ 0; Þ through thickness for /r i = 1.5, materia I and n = 2 for diaphragm supported eastic orthotropic cyinder under therma oad with easticity pane strain soutions given by Ye et a. (2001) Quantity r/r i Present numerica soutions finite ength cyinder /r i = 2 /r i = 100 r r ðz ¼ =2Þ r h ðz ¼ =2Þ r z ðz ¼ 0; Þ Anaytica easticity pane strain soution by Ye et a. (2001) and numerica soution for pane strain infinitey ong cyinder

12 100 P. Desai, T. Kant Fourier series expansion can be extended to tacke other boundary conditions that can be considered as imitation of the current mode. Numerica resuts presented for different r o /r i and /r i ratios wi be usefu for future reference and can be used as benchmark resuts. Appendix: 1D formuation for orthotropic cyinder under therma oading dr r dr þ 1 ð r r r r h Þ ¼ 0; e r ¼ ou or ; e h ¼ u r ; ð9þ r r ¼ ðe r a r T ÞþC 12 ðe h a h TÞ r h ¼ C 12 ðe r a r TÞþC 22 ðe h a h TÞ ; du r r ¼ C u dr 11a r T þ C 12 r C 12a h T du r h ¼ C 21 C u dr 21a r T þ C 22 r C 22a h T ; du dr ¼ r r þ a r T C 12 u r þ C 12 a h T; dr r dr ¼ r r C 21 1 þ u r r 2 C 22 C 21C 12 þ a ht C 21 C 12 C 22 r where, m rh ¼ m hr E r ; ¼ E h ð1 t rh t hr Þ ; t rh E h C 12 ¼ ð1 t rh t hr Þ ; C E h 22 ¼ ð1 t rh t hr Þ ; C 21 ¼ C 12 ; C ij ¼ Cij 0 nn ; a i ¼ a 0 i nn References E r ð10þ Afshar, R., Bayat, M., Lawani, R.K., Yau, Y.H.: Eastic behavior of gass-ike functionay graded infinite hoow cyinder under hydrostatic oads using finite eement method. Mater. Des. 32, (2011) Chen, Y.Z., Lin, X.Y.: An aternative numerica soution of thick-waed cyinders and spheres made of functionay graded materias. Comput. Mater. Sci. 48, (2010) Chen, W.Q., Ye, G.R., Cai, J.B.: Thermoeastic stresses in a uniformy heated functionay graded isotropic hoow cyinder. J. Zhejiang Univ. Sci. 3(1), 1 5 (2002) Desai, P., Kant, T.: Ton accurate stress determination in aminated finite ength cyinders subjected to thermoeastic oad. Int. J. Mech. Soids 6(1), 7 26 (2011) Gi, S.: A process for the step-by-step integration of differentia equations in an automatic digita computing machine. Proc. Camb. Phios. Soc. 47(Part 1), (1951) Godberg, J.E. Setur, A.V., Aspaugh, D.W.: Computer anaysis of non-circuar cyindrica shes. In: Proceedings of the IASS Symposium on She Structures in Engineering Practice, Budapest, Hungary (1965) Horgan, C.O., Chan, A.M.: The pressurized hoow cyinder or disk probem for functionay graded isotropic ineary eastic materias. J. East. 55, (1999) Jabbari, M., Sohrabpour, S., Esami, M.R.: Mechanica and therma stresses in a functionay graded hoow cyinder due to radiay symmetric oads. Int. J. Press. Vesses Pip. 79, (2002) Kant, T., Ramesh, C.K.: Numerica integration of inear boundary vaue probems in soid mechanics by segmentation method. Int. J. Numer. Methods Eng. 17, (1981) Koizumi, M.: FGM activities in Japan. Compos. Part B 28B, 1 4 (1997) Kraus, H.: Thin Eastic Shes. Wiey, New York (1967) Liew, K.M., Kitipornchai, S., Zhang, X.Z., Lim, C.W.: Anaysis of the therma stress behaviour of functionay graded hoow circuar cyinders. Int. J. Soids Struct. 40, (2003) Peng, X.L., Li, X.F.: Thermoeastic anaysis of a cyindrica vesse of functionay graded materias. Int. J. Press. Vesses Pip. 87, (2010) Shariyat, M., Nikkhah, M., Kazemi, R.: Exact and numerica eastodynamic soutions for thick-waed functionay graded cyinders subjected to pressure shocks. Int. J. Press. Vesses Pip. 88, (2011) Timoshenko, S., Goodier, J.N.: Theory of Easticity. McGraw- Hi, New York (1951) Tutuncu, K., Ozturk, M.: Exact soutions for stresses in functionay graded pressure vesses. Compos. Part B 32, (2001) Yas, M.H., Sobhani Aragh, B.: Three-dimensiona anaysis for thermoeastic response of functionay graded fiber reinforced cyindrica pane. Compos. Struct. 92, (2010) Ye, G.R., Chen, W.Q., Cai, J.B.: A uniformy heated functionay graded cyindrica she with transverse isotropy. Mech. Res. Commun. 28(5), (2001) Wu, C.-P., Tsai, T.-C.: Exact soutions of functionay graded piezoeectric materia sandwich cyinders by a modified Pagano method. App. Math. Mode. (2011). doi: / j.apm