Analytical Solution for Radial Deformations of Functionally Graded Isotropic and Incompressible Second-Order Elastic Hollow Spheres

Transcription

1 J Elast (01) 107: DOI /s Analytical Solution for Radial Deformations of Functionally Graded Isotropic and Incompressible Second-Order Elastic Hollow Spheres G.L. Iaccarino R.C. Batra Received: November 010 / Published online: 9 July 011 Springer Science+Business Media B.V. 011 Abstract We analytically analyze radial expansion/contraction of a hollow sphere composed of a second-order elastic isotropic incompressible and inhomogeneous material to delineate differences and similarities between solutions of the first- and the second-order problems. The two elastic moduli are assumed to be either affine or power-law functions of the radial coordinate R in the undeformed reference configuration. For the affine variation of the shear modulus μ the hoop stress for the linear elastic (or the first-order) problem at the point R = (R ou R in (R ou +R in )/) 1/ is independent of the slope of the μ vs. R line. Here R in and R ou equal respectively the inner and the outer radius of the sphere in the reference configuration. For μ(r) R n for the linear problem the hoop stress is constant in the sphere for n = 1. However no such results are found for the second-order (i.e. materially nonlinear) problem. Whereas for the first-order problem the shear modulus influences only the radial displacement and not the stresses for the second-order problem the two elastic constants affect both the radial displacement and the stresses. In a very thick homogeneous hollow sphere subjected only to pressure on the outer surface the hoop stress at a point on the inner surface depends upon values of the two elastic moduli. Thus conclusions drawn from the analysis of the first-order problem do not hold for the second-order problem. Closed form solutions for the displacement and stresses for the first-order and the second-order problems provided herein can be used to verify solutions of the problem obtained by using numerical methods. Keywords Analytical solutions Functionally graded material Sphere Second-order isotropic elastic material RCB dedicates with deep respect this work to the memory of Professor D.E. Carlson. G.L. Iaccarino Department of Constructions and Mathematical Methods in Architecture University of Naples Federico II via Forno Vecchio Naples Italy R.C. Batra ( ) Department of Engineering Science and Mechanics M/C 019 Virginia Polytechnic Institute and State University Blacksburg VA 4061 USA rbatra@vt.edu

2 180 G.L. Iaccarino R.C. Batra Mathematics Subject Classification (000) 74E5 74B5 45B5 1 Introduction Functionally graded materials (FGMs) are inhomogeneous and are usually comprised of two constituents whose volume fractions vary smoothly either in one or in two or in all three directions. Most works presented in the literature consider material properties varying smoothly only in one direction. Also with very few exceptions problems studied have been for linear elastic isotropic materials with material properties assumed to vary according to either a power law or an exponential relation. There is enormous literature on linear problems for FGMs and it is almost impossible to review it here. The reader is referred to the review paper of Byrd and Birman [1] for a summary of some of the literature. Nonlinear problems for FGMs studied in [ 10] are for rubberlike materials that are assumed to be isotropic incompressible and hyperelastic. Ono [11] Ikeda [1 1] and Ikeda et al. [14] have developed rubbers with continuous spatial variation of the chemical and mechanical properties. The constitutive relation for an incompressible material involves hydrostatic pressure that cannot be determined from the deformation gradient but is found as a part of the solution of the problem. The presence of the hydrostatic pressure generally facilitates finding an analytical solution of a problem. Rivlin [15] amongst others has shown that the analysis of problems for nd-order elastic materials can reveal effects of material nonlinearities that are representative of the behavior of general nonlinear elastic materials. We note that linear kinematic relations are used in deriving the constitutive relation for a nd-order elastic material hence effects of geometric nonlinearities are not considered. Here we study the radial expansion/contraction of a FG sphere made of an incompressible nd-order elastic isotropic material with the two material parameters smoothly varying in the radial direction. It extends our work on FG cylinders [4] to FG spheres; however the analysis presented here for the first-order (linear elastic) inhomogeneous problem with the shear modulus varying with the radius according to power law is more extensive than that given heretofore. The approach followed here is similar to that employed in [4 16] for analyzing problems for nd-order elastic materials. We note that problems for linear elastic FG spheres with the moduli varying according to power law have been analytically studied by Tutuncu and Ozturk [17] who assumed that the quadratic characteristic equation has two real and distinct roots. They plotted through-thethickness variation of stresses for a FG sphere normalized by the corresponding ones for a sphere made of a homogeneous material. Stresses in a FG sphere made of an isotropic linear thermoelastic material have also been studied in [18 ]. Problem Formulation We study radial deformations of a sphere made of a FG isotropic and incompressible ndorder elastic material subjected to uniform pressures p in and p ou on its inner and outer surfaces. We use spherical coordinates (rθφ) with the origin at the sphere center to describe deformations of a point with r = R in the undeformed reference configuration. We assume that material properties are continuous functions of R a material point undergoes radial displacement u(r) = r R denote du/dr by u (R)andset I (1) = u + u/r I () = (u ) + (u/r). (.1)

3 Analytical Solution for Radial Deformations of Functionally Graded 181 For a nd-order incompressible material u must satisfy I (1) + I (1) I () = 0. (.) For radial expansion/contraction of a sphere the constitutive relation for a nd-order elastic isotropic and incompressible material is σ rr = p + μu + (μ + α)(u ) σ θθ = σ φφ = p + μ u R + (μ + α) ( u R ) (.) σ rθ = σ θφ = σ φr = 0. Here σ is the Cauchy stress tensor p the hydrostatic pressure not determined from the deformation field μ the shear modulus and α the nd-order elastic constant of the material. Both μ and α have units of stress and are presumed to be functions of R. Physical components of the first Piola-Kirchhoff stress tensor T are given by T = p ( 1 u + (u ) ) + μ (u 1 ) (u ) + α(u ) ( T = T = p 1 u ( ) u ) ( u R + + μ R R 1 ( ) u ) ( ) u + α. R R (.4) The equilibrium equation and boundary conditions governing static radial deformations of a sphere are T + (T T ) = 0 R in <R<R ou R r r T = p in on R = R in T = p ou on R = R ou. R in R ou (.5) Thus T at R = R in and at R = R ou depends respectively upon the radial displacements of a point on the inner and the outer surfaces of the sphere. We non-dimensionalize stresses σ and T the pressure p and material parameters μ and α by μ ou = μ(r ou )andu r and R by R ou. Henceforth we employ non-dimensional variables and use for them the same symbols as those used for dimensional variables. Let ε = max(p in p ou ) 1. With the assumption that p in >p ou ε= p in. Assuming that u p T and T are analytic functions of ε we expand them in terms of Taylor series in ε. Recalling that for ε = 0u= p = 0 we get u = εu (1) + ε u () p= εp (1) + ε p () (.6) T = ε + ε T () T = ε + ε T () up to nd-order terms in ε. Substitution from (.6) into(.) (.4) and(.5) and equating terms of order ε and ε on both sides of resulting equations we arrive at the following set of equations: u (1) + u(1) R = 0 [u () + u() R ] [ (u (1) ) + ( u (1) R ) ] = 0 (.7)

4 18 G.L. Iaccarino R.C. Batra where (1) (T + ) = 0 T () R () (T + T () ) = 0 (.8) R = p(1) in on R = R in = p(1) ou on R = R ou T () = p() in on R = R in T () = p() ou on R = R ou p (1) in = 1 p() in = p u (1) (R in ) in R in p (1) ou = p ou p in p () ou = p ou p in u (1) (R ou ) R ou. (.9) (.10) Equations (.7) 1 (.8) 1 and (.9) 1 govern deformations of the 1st-order problem or equivalently the problem in the linear elasticity theory and (.7) (.8) and (.9) 4 are used to solve the nd-order problem. Relations in (.10) imply that pressures to be applied on the inner and the outer surfaces for the nd-order problem depend upon the solution of the 1st-order problem. Thus the nd-order problem can only be analyzed after the 1st-order problem has been solved. Sphere made of Homogeneous Material For a sphere made of a homogeneous material μ and α equal constants b 1 and b respectively. Proceeding in a way similar to that in [19 ] we get the following for the solution of the problem. u (1) (R) = ( p ou + p in )Rou R in 1 4p in b 1 (Rou R in ) R p (1) (R) = p our ou p inr in p in (R ou R in ) = p in(r R ou )R in p ou(r R in )R ou R p in (R ou R in ) = = p in(r + R ou )R in p ou(r + R in )R ou u () (R) = (p ou p in ) Rou [ R in pin (R ou R ou R in R in ) 16R 5 b1 p () (R) = (p ou p in ) R ou R in p in (R ou R in ) R p in (R ou R in ) ; (.1) [ 11b 1 + b + R ou R in 16b1 16b 1 R 6 ] + (11b 1 + b )(Rou + R in ) 64R b1 ( 1 + b )] b 1 T () = p() (R) + 10A 1 b 1 R 6 4b 1 R A T () = T () = p() (R) A 1 b 1 R 6 + b 1 R A A 1 = (p ou p in )R ou R in 4b 1 p in (R ou R in ) A = (11b 1 + b )(p ou p in ) R in (1 + R in /R ou ) 64b 1 p in (1 R in /R ou ). (.)

5 Analytical Solution for Radial Deformations of Functionally Graded 18 The complete solution of the nd-order problem is obtained by substituting into (.6) expressions for the 1st-order and the nd-order quantities. With our non-dimensionalization b 1 = 1 but we still use b 1 to indicate which quantities depend upon it. It follows from (.6) that for the 1st-order problem i.e. for a linear elastic sphere values of the radial displacement u and stresses T and T equal ε = p in times their values listed in (.1) and agree with those given in [19]. When p in = p ou = p i.e. the pressures on the inner and the outer surfaces are equal we get T = T = T = p. Thus the state of stress at every point of the sphere is a hydrostatic pressure. Whereas the 1st-order elastic constant b 1 affects only the 1st-order displacement the ndorder elastic constant b affects both the nd-order displacement and the nd-order stresses. Note that in the 1st-order theory the hydrostatic pressure is a constant but in the nd-order theory the hydrostatic pressure varies with the radius R. In the limit of β R ou /R in 1 (.6) 4 (.1) and(.) give lim = lim (εt + β β ε T () ) = p ou (p ou p in ) R in R [ 1 (b ] 1 + b )p ou (11b 1 + b )p in 16b1 (b 1 + b ) (p 8b1 ou p in ) R6 in R. (.) 6 Thus in a very thick hollow sphere the circumferential stress T (R in) at the inner surface is given by T (R in) = p ou (p ou p in ) [ 1 (b ] 1 + b )p ou (11b 1 + b )p in 16b1 (b 1 + b ) (p 8b1 ou p in ). (.4) For p in = 0andp ou > 0 we get T (R in) = p ou (b 1 + b ) p b1 ou and the stress concentration factor at the inner surface of a very thick sphere defined as T (R in)/p ou increases from 1.5 by the consideration of nd-order effects provided that (b 1 + b )>0. However for p ou < 0 the consideration of the nd-order deformations decreases the stress concentration factor at the inner surface. For a thin sphere we set R ou = R in + t where t is the sphere thickness and compute results in the limit of t/r in = 0. For p ou = 0 the circumferential stress T 0 is given by [ T 0 = lim T = p inr in t/r in 0 t b 1 + b p 16b1 in + p inr in 6tb 1 ]. (.5) Recall that in the linear theory p inr in equals the hoop stress in a thin sphere. The first term t on the right hand side of (.5) is the hoop stress according to the linear theory and the second and the third terms represent contributions from the consideration of the nd-order effects. Whereas the third term is always positive the sign of the second term depends upon that of (11b 1 + b ).

6 184 G.L. Iaccarino R.C. Batra For p ou = 0 (.1) and (.1) 4 simplify to = p (R Rou )R in in R (Rou R in ) = (R + Rou )R in R (Rou R in ) p in. (.6) Thus in a sphere made of a linear elastic homogeneous and isotropic material the radial stress is compressive and the hoop stress is tensile. For R in /R ou <(1/4) 1/ 0.6 T > (1) when R>0.6R ou and T = (1) at R 0.6R ou ; (.7) for 0.6 R in /R ou < 1 T > (1) throughout the sphere. Further discussion on the signs of (1) and T is given in Appendix A. 4 Sphere Material Functionally Graded 4.1 Affine Variation of the Two Moduli We assume that μ and α vary affinely in the radial direction i.e. μ(r) = b 1 (1 + nr) α(r) = b (1 + mr) (4.1) where b 1 b nand m are constants. Omitting details the solution of the problem is u (1) (R) = Ā1 R = [ Ā + Ā 1 b 1 ( n R Ā 1 = (p ou p in )R ou R in b 1 D R + (1 + nr) [ ( n = φφ = (1 + nr) Ā + Ā 1 b 1 R R )] )] Ā = p inrin ( + nr ou) p ou Rou ( + nr in) D [ D = p in R in ( + nr ou ) Rou ( + nr in) ] ; (4.) u () (R) = (p ou p in ) Rou [ R in R ou R in + 11b ] 1C + b C 4b1 p in R C 4 R 10b 1 C 5 T () (R) = B 1R + B R + B R 5 + B 4 R 6 + B 5 (4.) T () (R) = B R B R 5 B 4 R 6 + B 5 and expressions for constants are given in Appendix B. As before the complete solution of the nd-order problem is obtained by substituting into (.6) expressions for the 1st-order and the nd-order quantities.

7 Analytical Solution for Radial Deformations of Functionally Graded 185 For a linear elastic hollow FG sphere subjected to internal pressure only we get the following for the radial and the circumferential stresses. = p in R (R R ou )[n ou (R + R ou ) + (R + ou + R ou )] (R ou R in )[(R ou + R our in + R in ) + nr our in (R ou + R in )] = p in [R ( + nr ou ) + Rou ]R in R [Rin ( + nr ou) Rou ( + nr in)]. (4.4) For n 0 (4.4) 1 implies that T < 0anddT /dr > 0. Thus the radial stress is compressive for all values of R and is a monotonically increasing function of R or said differently the magnitude of T monotonically decreases from p in at R = R in to zero at R = R ou.for two distinct values of n i.e. n 1 n (R n 1) (R n ) = 6p in(n 1 n )(R R ou )(R R in )(R + R in + R ou )Rin R ou R (R ou R in )E (4.5) (R n 1) (R n ) = p in (n n 1 )( R + Rou R in + R ou Rin ) R (R ou R in )E where E = p in [Rin + R ou( + n R in )(R ou + R in )][Rin + R ou( + n 1 R in )(R ou + R in )]. (4.6) Thus (R n 1) (R n )>0 provided that n 1 <n and is independent of n only for R = R in and R = R ou i.e. at the inner and the outer surfaces which follows from the boundary conditions. However ( Rn 1 ) = ( Rn ) for R = ( ) Rou R in (R ou + R in ) 1/. (4.7) That is the hoop stress at the point R = R is independent of the value assigned to n. One can check that R in < R<R ou.forn 1 <n (R n 1) (R n ) > (<)0 forr < (>) R. For n 0 it follows from (4.4) that (1) > 0. Since dt /dr < 0 the first-order hoop stress is a monotonically decreasing function of R. Forn<0wesetk = n and get the following: for 1/R ou <k< k is tensile for R in <R< R compressive for R <R<R ou and vanishes at R = R. When k <k<c/r ou is tensile for R <R<R ou andcompressive for R in <R< R.Here R = R ou /(kr ou ) 1/ andc = (R ou /R in ) +. For other values of k (1) is tensile. By analyzing the sign of dt /dr wefindthatthefirst-order hoop stress is a monotonically increasing function of R when k > k but is a monotonically decreasing function of R for k< k with k = (Rou + R our in + Rin )/R our in (R in + R ou ). One can show that 1/R ou < k<1/r in. For R ou /R in = 5 and four different positive values of n we have plotted in Fig. 1 the variation with R/R in of and T (1).ForR ou/r in = 5 R =.47R in. With an increase in the value of n from 0 to.5 the maximum value of the hoop stress at the inner surface decreases from 0.5p in to 0.14p in.forr< R the hoop stress depends noticeably upon the value of n but for R> R values of the hoop stress are insensitive to the value assigned to n. Thus the maximum hoop stress at the inner surface can be decreased by suitably grading the value of the shear modulus in the radial direction. At R = R computed values of the are independent of the value of n as derived analytically above. It is clear that by suitably assigning a negative value to n one can have tensile and compressive (1) at an interior point of the sphere. Values of T strongly depend upon the value of hoop stress

8 186 G.L. Iaccarino R.C. Batra Fig. 1 (Color online) For R ou /R in = 5 and different values of n the variation of the hoop stress and the radial stress through the thickness of the hollow sphere; (a) black n = 0; red n = 0.5; blue n = 1.5; green n =.5; (b) black n = 0.1; blue n = 1; green n = ; (c) black n = 0.1; blue n = 0.5; red n = 0.8; green n = 1 n throughout the sphere thickness but those of vary rapidly with n only in the region 1 R/R in Power Law Variation of the two Moduli We assume that μ(r) and α(r) are given by μ(r) = b 1 R n α(r)= b R m where m and n are constants and b 1 = μ(r in )/Rin n b = α(r in )/Rin m. Expressions for displacements and stresses for n = are different from those for n. Accordingly we give below the solution for several integer values of n and m and list in Appendix B expressions for various constants appearing in the solutions.

9 Analytical Solution for Radial Deformations of Functionally Graded Solution of the 1st-order problem The solution of the 1st-order problem for n is u (1) (R) = (n )(p ou p in )R ou R in 1R b 1 p in (R ou Rn in Rn ou R in ) = p ou p in + (p ou p in )( R n R ou + R R n ou )R in R p in (R n ou R in R ou Rn in ) (4.8) = = p ou p in + (p ou p in )((1 n)r n R ou + R R n ou )R in R p in (R n ou R in R ou Rn in ) and that for n = andp ou = 0is u (1) (R) = 1 R 1 1b 1 ln(r ou /R in ) (R) = ln(r/r ou) ln(r ou /R in ) (R) = (1 + ln(r/r ou)). ln(r ou /R in ) (4.9) We note that the solution for n = is not given in [19]. 4.. Solution of the nd-order problem For the nd-order problem the solution for n 6; m 6isgivenbelow. u () (R) = (n ) (p ou p in ) Rou R in 144b1 p in (Rn ou R in R ou Rn in ) { [ (n )[(n 6)b ( Rou m R6 in + R6 ou Rm in ) + 11(m 6)b 1( Rou n R6 in + R6 ou Rn in )] ] (m 6)(n 6)b 1 (Rou n R in R ou Rn in ) R R ou R in R 5 } T () (R) = H 1R + H R 6+m + H R 6+n + H 4 R +n + H 5 T () (R) = H 1 R + H (m 4) R 6+m + H (n 4) Expressions for constants H i s are given in Appendix B. The solution for m = n = andp ou = 0is: R 6+n + H 4(n 1) R +n + H 5. (4.10) u () 1 (R) = 144R 5 (b 1 ln(r ou /R in )) + (11b 1 + b )( Rou + R in ) 864R (b 1 ln(r ou /R in )) T () = H { (11b 1 + b )R [ ln(r/r ou )Rou ln(r/r ] in)rin + Rou R in ln(r ou/r in ) [ ]} ( ln(r/r ou ))b 1 + b (4.11) T () = H { (11b1 + b )R [ (1 + ln(r/r ou ))Rou (1 + ln(r/r in))rin] + R ou R in ln(r ou/r in )[(1 1 ln(r/r ou ))b 1 b ] }

10 188 G.L. Iaccarino R.C. Batra where 1 H = 7b1 R (R in R ou ln(r ou /R in )). (4.1) The complete solution of the problem for m = n = 6p ou = 0isgivenbelow. u(r) = p in 4b 1 R (Rou R in ) + ln(r ou/r in )(11b 1 + b )pin pin R b1 (R ou R in ) 16b1 R5 (Rou R in ) T = p in(r Rou ) Rou + G [ b R { R ln(r ou /R in ) R R ou ln(r/r in) Rin ln(r/r ou) } in { + b 1 4R 6 ou + Rou( ( R 4 ln(r/r in ) ) ) + 4Rin + R ( R ln(r ou /R in ) + ( 4 ln(r ou /R) ) }] Rin (4.1) T = p in(5r Rou ) (Rou R in ) + G [ b R { 5R ln(r ou /R in ) Rou + R in R ou ln(r/r in) Rin ln(r ou/r) } { + b 1 4R 6 ou Rou( ( R ln(r/r in ) ) ) + 4Rin + R ( 165R ln(r ou /R in ) + Rin( 5 66 ln(rou /R) ))}] where pin G = 8R b1 (R ou R in ). (4.14) For n = m= 6 we give below the solution of the problem. u (1) (R) = B 1 R (R) = B + 4 B 1 b 1 (lnr 1) (R) = B + B 1 b 1 (6lnR + 1) (4.15) u () (R) = B 1 R 5 + B R = (p ou p in ) { 11R B (p ou p in ) [ ln(r/r ou )Rou ln(r/r ] in)rin 4 T () T () + R ou R in ln(r ou/r in ) [( ln(r/r in ) ) p ou + ( 11 1 ln(r/r ou ) ) p in ]} (4.16) = (p ou p in ) { 11R (p ou p in ) [( 1 + ln(r/r ou ) ) Rou B ( 1 + ln(r/r in ) ) ] Rin 4 R ou R in ln(r ou/r in ) [( ln(r/r in ) ) p ou + ( 1 1 ln(r/r ou ) ) p in ]}. Expressions for constants B i s are listed in Appendix B.

11 Analytical Solution for Radial Deformations of Functionally Graded 189 Fig. (Color online) For R ou /R in = 5 and different values of n the variation of the hoop stress through the thickness of the hollow sphere. Black n = 0; red n = 5; blue n =.5 In [19] it was proved that the hoop stress in a FG sphere is uniform if n = 1. Accordingly we list below the solution for m = n = 1 and p ou = 0. u (1) (R) = 1 6b 1 R u () (R) = R 6b 1 R ou R in Rou (R) = (R Rou )R in R in R (Rou R in ) (R) = Rou R in ( D 1 + D Rou ) R T () = D 1R + D R + D R 5 + D 4 (4.17) T () = D R D R 5 + D 4. Expressions for constants D i s are given in Appendix B. For a linear elastic sphere it follows from (4.8) that when n = 1 = = p ou + (p ou p in ) p in (1 Rou /R in )p ; (4.18) in thus T and T are constants throughout the sphere; (4.17) is a special case of (4.18) for p ou = 0. In a very thick sphere R ou /R in 1T p ou. That is in a very thick sphere with the shear modulus proportional to the radius R the hoop stress nearly equals the negative of the pressure applied on the outer surface of the sphere. For a very thin sphere with p ou = 0R ou = R in + tt/r in 1 and we recover the classical result that T = T = (p in R in )/t. Thus the linear gradation of material properties in the radial direction has no effect on the surface tension in a very thin sphere made of a linear elastic FG material. It follows from (4.17) that for n = 1 varies with R. Equation (4.18) implies that for p ou Rou = p inrin T(1) = 0 throughout the sphere. For a hollow sphere with p ou = 0andR in /R ou = 1/5 we have plotted in Fig. for different values of n vs. R/R in. Forn = 0and 5 (which are less than 1) the hoop stress is tensile everywhere. For n =.5 (which is greater than 1) the hoop stress is compressive for 1 <R/R in < R/R in (=.81) tensile for R<R<R ou and vanishes at R = R. It is evident from (4.17) 6 that even though the first-order hoop stress is constant through the sphere thickness the nd-order hoop stress varies with R. R in

12 190 G.L. Iaccarino R.C. Batra 5 Numerical Results 5.1 Sphere Made of Homogeneous Material In order to highlight differences between results for the 1st-order and the nd-order problems we consider a hollow sphere with R ou /R in = 5p ou = 0 and made of a homogeneous nd-order elastic material with μ = 05. kpaα = 77.4 kpa; these values of μ and α are for the rubber tested in []. We have plotted in Fig. T /p in and T /p in as functions of R/R in. Furthermore in order to illuminate the nd-order effects we have plotted in Fig. 4 T () and T () vs. R/R in. It should be clear from (.9)thatT () and hence T depend upon the radial displacement of the inner surface. That is why T for the nd-order problem does not equal p in on the inner surface of the sphere. Fig. (Color online) For R ou /R in = 5 variation through the sphere thickness of the radial and the hoop stress magnitude for the linear (black curve) and the nd-order elastic materials (red curve) Fig. 4 For R ou /R in = 5 plots of T () (1) () (1) /T and T /T vs. R/R in

13 Analytical Solution for Radial Deformations of Functionally Graded 191 Fig. 5 (Color online) For R ou /R in = 5 through the thickness variation of the radial stress for spheres made of 1st-order (black curve)and nd-order (red curve) elastic materials (a) n = 1 (b) n = 0.5 It is clear from results exhibited in Fig. 4 that the consideration of nd-order effects noticeably influences stress distribution in the sphere. We should add that results depend upon the relative values of the two elastic moduli μ and α. 5. Functionally Graded Material 5..1 Affine Variation of the Two Moduli In order to study the effect of material inhomogeneity on the structural response we have analyzed for different values of n the through the thickness variations of the radial and the hoop stresses. Results exhibited in Fig. 5 for a sphere with R ou /R in = 5 reveal that for n = 0.5 and n = 1 the radial stress for a nd-order elastic material differs noticeably from that for a linear (1st-order) elastic material. In Fig. 6 we have displayed through the thickness variation of the hoop stress corresponding to n = 0.5 andn = 1forR ou /R in = 5. For n = 0.5 the two curves intersect each other at R =.8 while for n = 1 they intersect each other at two points: R = 1. and R =.11. Power Law Variation of the Two Moduli For the case of a power law variation of material moduli the stress fields for spheres made of linear elastic materials have the same shape as those for nd-order elastic materials and differ only quantitatively. For this reason we omit their graphs.

14 19 G.L. Iaccarino R.C. Batra Fig. 6 (Color online) Through the thickness variation of the hoop stress for spheres made of 1st-order (black curve) and nd-order elastic materials (red curve) 6 Conclusions We have studied radial expansion/contraction of a hollow sphere with the inner and the outer surfaces subjected to pressures and the sphere material modeled as isotropic incompressible and second-order elastic with the two material moduli smoothly varying in the radial direction either according to a power-law relation or an affine relation. Whereas for a linear variation of the shear modulus the hoop stress is uniform through the sphere thickness for a linear elastic material it is not so for a second-order elastic material. Thus results for the nd-order problem cannot be deduced from those for the first-order problem. For the affine variation of the shear modulus there exists a point in the sphere where the hoop stress is independent of the slope of the shear modulus vs. the radius curve. By suitably varying the material moduli in the radial direction one can control the sign of the radial and the hoop stresses. The challenging inverse problem of finding the spatial distribution of material moduli (i.e. material tailoring) to achieve a desired through-the-thickness distribution of either the circumferential stress or the in-plane shear stress is addressed in [ 4 7]. The radial expansion as well as inversion of a functionally graded cylinder made of a Mooney-Rivlin material has been analyzed in [8]. Acknowledgements RCB s work was partially supported by the Office of Naval Research grants N and N to Virginia Polytechnic Institute and State University with Dr. Y.D.S. Rajapakse as the program manager. Views expressed in this paper are those of authors and neither of the funding agency nor of author s institutions. Appendix A: Discussion of signs of (1) and T Signs of Stresses for the Affine Variation of the Moduli when n<0 stresses when n<0 and set k = n so that k>0. Let f(rk)= (R R ou )P (R k) P(Rk)= ( kr ou )R(R + R ou ) + Rou. We now analyze (A.1) We note that f(rk)determines the sign of the numerator for in (4.4) 1. The expression of P(Rk)implies that f(rk) 0fork /(R ou ) and f(rk)= 0forR = R ou.wefirst consider the case of k>/(r ou ). Using the Descartes rule of signs the quadratic equation

15 Analytical Solution for Radial Deformations of Functionally Graded 19 P(Rk)= 0 has one positive root R + and one negative root R.SinceR cannot be negative we discard the negative root and study further the case of the positive root: R + = R ou[ A(kR ou + ) A] A= kr ou > 0. (A.) A The only case of interest here is R in R + R ou.forr + R in one can show that /(R ou )<k k with k = ( Rou + R ) our in + Rin R ou R in (R in + R ou ) (A.) and 1/R ou < k <1/R in. One can similarly prove that for R + R ou k 1/R ou. Thus for 1/R ou k kweget f (Rk) 0 when R + R R ou f(rk)<0 whenr in <R<R +. (A.4) For k< k the denominator in the expression (4.4) 1 for is positive. By combining the afore-stated discussion of the signs of the numerator and the denominator in the expression (4.4) 1 for we conclude that for 1/R ou k< k is tensile for R + <R<R ou compressive for R in <R<R + and equals zero for R = R +. For other values of k is compressive. Differentiation of both sides of (4.4) 1 with respect to R gives d dr = 6(1 kr)p in R in R ou R 4 [ R ou ( kr in) R in ( kr ou) ]. Since the denominator is positive for k< k we conclude that (A.5) d dr > 0 fork> 1 R ou (A.6) and increases for R<1/k decreases for R>1/k and is the maximum at R = 1/k. For R ou /R in = 5 and different negative values of n we have plotted in Fig. 1(b) (c) the variation with R/R in of. Signs of Stresses for the Power Law Variation of the Moduli We investigate the sign of (1) and T for n 1 6 when the sphere is loaded on the inner surface only (p ou = 0). We first note that the function f(r)= R n R in R R ou (A.7) is always positive. It is a monotonically increasing (decreasing) function of R for n > (<). Values of R for which f(r)is maximum and minimum and the sign of the term (Rou n R n in ) are summarized in Table 1. Thus (1) < 0 throughout the thickness of the hollow sphere and T (R ou) = 0asrequired by the boundary condition at R = R ou. Furthermore since d dr = (n )p in (Rou n R n in ) Rn 4 (A.8)

16 194 G.L. Iaccarino R.C. Batra Table 1 Values of R where f(r)is either maximum or minimum and the sign of Rou n Rin n R for f(r)maximum R for f(r)minimum Sign of Rou n Rin n n> R ou R in positive n< R in R ou negative Fig. 7 (Color online) For R ou /R in = 5 and different values of n variation of the radial stress through the thickness of the hollow sphere. Fuchsia line: n =.5; red line: n = 1.5; black line: n = 0; blue line: n = 1.5; green line: n =.5; dashed black line: n = 5; dashed red line: n = 6 is a monotonically decreasing function of R through the sphere thickness. For R ou /R in = 5 we have plotted in Fig. 7 vs. R for different values of n. It is clear that for a given value of R increases with an increase in n. When n>andp ou = 0 we derive from (4.8) the following result 0 Thus 0 if and only if R R where if and only if ((n 1)Rn Rou n ) 0. (A.9) R = β(n)r ou ( ) 1 n β(n)=. n 1 (A.10) We note that R<R ou andr in < R if and only if R in /R ou <β(n).forr in /R ou <β(n) 0 when R R<R ou 0 when R (A.11) in <R R. If R in /R ou >β(n)then > 0forR in R R ou. When 1 <n<andp ou = 0 we again conclude from (4.8) that 0 ( ) (n 1)R n Rou n 0 (A.1) which implies the following. If R in /R ou <β(n)then 0for R R<R ou and 0forR in <R R.ForR in /R ou β(n) > 0 throughout the sphere thickness. For n<1andp ou = 0 (4.8) implies that > 0forR in R R ou. Differentiation of (4.8) gives d dr = p in (n 1)(n ) (R n ou R n in ) Rn 4 (A.1) whichisalwaysnegativeforn<1. Thus is a monotonically decreasing function of R.

17 Analytical Solution for Radial Deformations of Functionally Graded 195 Appendix B Expressions for constants in (4.): B 1 = n(p ou p in ) Rou R in (11b 1C + b C ) 0pin C 5b1 B = (p ou p in )Rou R in [b 1C 6 b (p ou p in )C ] 10pin C 5b1 B = (nb 1 + mb )(p ou p in ) Rou 6 R6 in 10pin C 4b1 B 4 = (b 1 + b )(p ou p in ) R 6 ou R6 in 4p in C 4b 1 Expressions for constants in (4.): B 5 = (p ou p in ) Rou R in C 7 0pin C. 5b1 C 1 = R 5 ou + R4 ou R in + R ou R in + R ou R in + R our 4 in C = 5R 5 in + C 1 (5 + 6nR in ) C = 5R 5 in + C 1 (5 + 6mR in ) C 4 = [ R in ( + nr ou) R ou ( + nr in) ] C 5 = (R ou R in ) [ R in + R ou (R in + R ou )( + nr in ) ] C 6 = { 55R 5 ou + 11R ou R in (5 + 6nR ou )(R ou + R in ) + R ou R in (5 + nr ou) (R ou + R in ) R 4 in [ 5 + 6nR ou ( + 5nR ou )] } p in + { 55R 5 in R ou R in (5 + nr in) 11R ou R in (5 + 6nR in)(r ou + R in ) + (R ou + R in ) R 4 ou [ 5 + 6nR in ( + 5nR in )] } p ou C 7 = 5 (11b 1 + b )( + nr ou ) R ou + [b (5 + 6mR ou ) + 11b 1 (5 + 6nR ou )] [ ( + nr ou )(R ou + R in ) + nr in] Rin. Expressions for constants in (4.10): H 1 = (n )(p ou p in )R ou R in ( p inr n ou R in + p our ou Rn in ) 6b 1 p in (Rn ou R in R ou Rn in ) H = (n ) b (p ou p in ) Rou 6 R6 in 4b1 p in (m 6)(Rn ou R in R ou Rn in ) H = (n )(7n 9)(p ou p in ) Rou 6 R6 in 4(n 6)b 1 pin (Rn ou R in R ou Rn in ) (n ) (p ou p in ) Rou H 4 = R in 4b1 p in (m 6)(n 6)( Rn ou R in + R ou Rn in ) [(n 6)b ( Rou m R6 in + R6 ou Rm in ) + 11(m 6)b 1( Rou n R6 in + R6 ou Rn in )] (n ) (p ou p in ) Rou H 5 = R in 4b1 p in (m 6)(n 6)( Rn ou R in + R ou Rn in ) [11(m 6)b 1 Rou n Rn in (R ou R in ) + (n 6)b (Rou +n Rm in Rm ou R+n in )].

18 196 G.L. Iaccarino R.C. Batra Expressions for constants in (4.15)and(4.16): (p ou p in ) B 1 = 1b 1 p in ln(r in /R ou ) B = (p ou p in ) (b 864b1 ln(r ou /R in ) + 11b 1 (R p in R ou )) in B = (p ou p in ) + (p in ln R ou p ou ln R in ) B 4 = 7R Rou p in ln(r ou /R in ) R in b 1pin (ln(r ou/r in )). Expressions for constants in (4.17): D 1 = (11b 1 + b )R ou R in (R 4 ou + R ou R in + R ou R in + R our in + R4 in ) 0b 1 (R ou R in ) (R ou + R in ) D = R ou R4 in b 1 (R ou R in ) D = (b 1 + b )R 4 ou R4 in 0b 1 (R ou R in ) D 4 = (11b 1 + b )R ou R in (R ou + R our in + R in ) 0b 1 (R ou R in ) (R ou + R in ). References 1. Byrd L.W. Birman V.: Modeling and analysis of functionally graded materials and structures. Appl. Mech. Rev (007). Batra R.C.: Finite plane strain deformations of rubberlike materials. Int. J. Numer. Methods Eng (1980). Batra R.C.: Material tailoring and universal relations for axisymmetric deformations of functionally graded rubberlike cylinders and spheres. Math. Mech. Solids (011). doi: / Batra R.C. Iaccarino G.L.: Exact solutions for radial deformations of functionally graded isotropic and incompressible second-order elastic cylinders. Int. J. Non-Linear Mech (008) 5. Bilgili E.: Modeling mechanical behavior of continuously graded vulcanized rubbers. Plast. Rubber Compos. (4) (004) 6. Bilgili E.: Functional grading of rubber tubes within the context of a molecularly inspired finite thermoelastic model. Acta Mech (004) 7. Saravanan U. Rajagopal R.C.: On the role of the inhomogeneities in the deformation of elastic bodies. Math. Mech. Solids (00) 8. Saravanan U. Rajagopal R.C.: A comparison of the response of isotropic inhomogeneous elastic cylindrical and spherical shells and their homogenized counterparts. J. Elast. 71() 05 (00) 9. Saravanan U. Rajagopal R.C.: Inflation extension torsion and shearing of compressible inhomogeneous annular cylinder. Math. Mech. Solids 10(6) (005) Bilgili E.: Controlling the stress-strain inhomogeneities in axially sheared and radially heated hollow rubber tubes via functional grading. Mech. Res. Commun (005) 11. Ono K.: J. Soc. Rubber Ind. Jpn (1999) 1. Ikeda Y.: Preparation and properties of graded styrene-butadiene rubber vulcanizates. J. Polym. Sci. Part B Polym. Phys (00) 1. Ikeda Y.: Graded styrene-butadiene rubber vulcanizates. J. Appl. Polym. Sci (00) 14. Ikeda Y. Kasai Y. Muralrami S. Kohjiya S.: Preparation and mechanical properties of graded styrenebutadiene rubber vulcanizates. J. Jpn Inst. Metals (1998) 15. Rivlin R.S.: The solution of problems in second order elasticity theory. J. Ration. Mech. Anal (195) 16. Iaccarino G.L. Marasco A. Romano A.: Signorini s method for live loads and second-order effects. Int. J. Eng. Sci (006) 17. Tutuncu N. Ozturk M.: Exact solutions for stresses in functionally graded pressure vessels. Compos. Part B Eng (001) 18. Obata Y. Noda N.: Steady thermal stresses in a hollow circular cylinder and a hollow sphere of a functionally gradient material. J. Therm. Stresses 17() (1994) 19. Batra R.C.: Optimal design of functionally graded isotropic and incompressible linear elastic cylinders and spheres. AIAA J (008)

19 Analytical Solution for Radial Deformations of Functionally Graded Poultangari R. Jabbari M. Eslami M.R.: Functionally graded hollow spheres under non-axisymmetric thermo-mechanical loads. Int. J. Press. Vessels Piping (008) 1. Eslami M.R. Babaei M.H. Poultangari R.: Thermal and mechanical stresses in a functionally graded thick sphere. Int. J. Press. Vessels Piping (005). Nie G.J. Zhong Z. Batra R.C.: Material tailoring for functionally graded hollow cylinders and spheres. Comp. Sci. Tech (011). Batra R.C. Muller I. Strehlow P.: Treloar s biaxial tests and Kearsley s bifurcation in rubber sheets. Math. Mechs. Solids (005) 4. Nie G.J. Batra R.C.: Material tailoring and analysis of functionally graded isotropic and incompressible linear elastic hollow cylinders. Comp. Struc (010) 5. Nie G.J. Batra R.C.: Stress analysis and material tailoring in isotropic linear thermoelastic incompressible functionally graded rotating disks of variable thickness. Comp. Struc (010) 6. Nie G.J. Batra R.C.: Exact solutions and material tailoring for functionally graded hollow circular cylinders. J. Elasticity (010) 7. Nie G.J. Zhong Z. Batra R.C.: Material tailoring for orthotropic elastic rotating disks. Comp. Sci. Tech (011) 8. Batra R.C. Bahrami A.: Inflation and eversion of functionally graded nonlinear elastic incompressible cylinders. Int. J. Non-Linear Mechs (009)