Stability of Long Transversely-Isotropic Elastic Cylindrical Shells Under Bending

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International Journal of Solids and Structures 47 (2010) 10–24

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International Journal of Solids and Structures

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Stability of long transversely-isotropic elastic cylindrical shells under bending

Sotiria Houliara, Spyros A. Karamanos *

Department of Mechanical Engineering, University of Thessaly, 38334 Volos, Greece

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Article history: The present paper investigates buckling of cylindrical shells of transversely-isotropic elastic material sub- Received 4 December 2008 jected to bending, considering the nonlinear prebuckling ovalized conﬁguration. A large-strain hypoelas- Received in revised form 1 September 2009 tic model is developed to simulate the anisotropic material behavior. The model is incorporated in a Available online 11 September 2009 ﬁnite-element formulation that uses a special-purpose ‘‘tube element”. For comparison purposes, a hyperelastic model is also employed. Using an eigenvalue analysis, bifurcation on the prebuckling oval- Keywords: ization path to a uniform wrinkling state is detected. Subsequently, the postbuckling equilibrium path is Stability traced through a continuation arc-length algorithm. The effects of anisotropy on the bifurcation moment, Buckling the corresponding curvature and the critical wavelength are examined, for a wide range of radius-to- Postbuckling Imperfection sensitivity thickness ratio values. The calculated values of bifurcation moment and curvature are also compared Anisotropy with analytical predictions, based on a heuristic argument. Finally, numerical results for the imperfection Finite elements sensitivity of bent cylinders are obtained, which show good comparison with previously reported asymp- Cylindrical shell totic expressions. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction rh0 of the ovalized section (often referred to as ‘‘equivalent cylinder”). This buckling condition can be expressed as follows

The application of bending loading in long cylinders causes rx ¼ rcr, and using classical Donnell shell buckling equation (Brush cross-sectional distortion, resulting in loss of bending stiffness and Almroth, 1975), one readily obtains: and a nonlinear moment–curvature equilibrium path. This phe- E t nomenon, referred to as ‘‘ovalization” or ‘‘Brazier effect”, was first rx ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ 2 r analyzed by Brazier (1927) for the case of elastic isotropic cylin- 3ð1 m Þ h0 ders. Reissner (1959) also investigated ovalization response for where 1=rh0 is the deformed (ovalized) cross-sectional curvature at both initially straight and curved tubes, taking under consideration the center of the buckling zone, E is Young’s modulus and m is Pois- the effects of pressure. In addition to cross-sectional ovalization, son’s ratio. In subsequent works (Stephens et al., 1975; Fabian, the increased axial stress at the compression zone of the ovalized 1977; Ju and Kyriakides, 1992; Karamanos, 2002; Houliara and cylinder causes bifurcation instability (buckling) in a form of longi- Karamanos, 2006), more elaborate solutions for determining the tudinal wavy-type ‘‘wrinkles”. This often occurs before a limit mo- bifurcation point on the ovalization path of isotropic elastic cylin- ment is reached on the moment–curvature ovalization path and ders have been reported, which verified the good accuracy of the constitutes a buckling problem associated with a highly nonlinear simplified formula (1) proposed in Axelrad (1965). In addition, the ovalized prebuckling state, where the compression zone of the de- corresponding buckling half wavelength of the isotropic cylinders formed cylindrical shell wall has a double curvature of opposite was found to be very close to the value of the half wavelength for sign in the longitudinal and in the hoop direction. the axisymmetric buckling mode of the uniformly-compressed An approximate, yet quite efficient method to determine the axi equivalent cylinder Lhw, when radius-to-thickness ratio is increased bifurcation point on the ovalized prebuckling moment–curvature ðr=t !1Þ. path has been proposed by Axelrad (1965) for long isotropic elastic . pffiffiffiffiffiffiffiffi ÂÃÀÁ tubes, considering a rigorous ovalization analysis. Axelrad assumed axi 2 1=4 Lhw ¼ p rh0t 12 1 m ð2Þ that for a bent cylinder buckling would occur when the local com-

pressive longitudinal stress rx at any point around the tube cir- The initial post-bifurcation behavior of isotropic elastic cylinders cumference reaches the critical stress rcr of an axially- under bending has been investigated in Fabian (1977), using an compressed circular cylinder with radius equal to the local radius asymptotic analysis. It was demonstrated that the bifurcation mode is symmetric, followed by an unstable initial postbuckling behavior. * Corresponding author. Tel.: +30 24210 74008; fax: +30 24210 74012. Numerical results in Ju and Kyriakides (1992) veriﬁed the initially E-mail address: [email protected] (S.A. Karamanos). unstable postbuckling behavior of a bent elastic cylinder.

0020-7683/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2009.09.004 S. Houliara, S.A. Karamanos / International Journal of Solids and Structures 47 (2010) 10–24 11

Furthermore, the results from the more refined numerical formula- anisotropy on the buckling moment, the corresponding curvature, tion in Karamanos (2002) and Houliara and Karamanos (2006) also the buckling wavelength, and the shape of the buckling modes are demonstrated the unstable nature of the initial postbuckling examined, considering a wide range of radius-to-thickness ratio response, in the form of a ‘‘snap-back”. values ðr=tÞ. Furthermore, using an ‘‘arc-length” continuation tech- The increasing use of composite materials in engineering appli- nique, postbuckling paths are traced for different levels of anisot- cations has motivated many studies in the elastic stability of aniso- ropy, and the sensitivity of cylinder response in the presence of tropic shells. Kedward (1978) enhanced Brazier’s solution initial imperfections is examined. Finally, an analytical approach, considering a different modulus for the longitudinal and the hoop based on the concept of ‘‘equivalent cylinder”, is presented leading direction, and obtained a closed-form expression for the ovaliza- to closed-form expressions for the buckling moment, curvature tion and the moment–curvature path. Spence and Toh (1979) and the corresponding wavelength, which are compared with the extended Reissner’s formulation (Reissner, 1959) to account for corresponding numerical results. orthotropic material behavior, using nonlinear finite deflection thin shell theory. These numerical results were compared with 2. Numerical formulation and solution experimental test data from steel and ‘‘melenex” cylinders. Stock- well and Cooper (1992) presented a direct extension of Reissner’s The nonlinear formulation adopted in the present work was isotropic cylinder formulation (Reissner, 1959) to obtain a introduced in its general form by Needleman (1982). It has been closed-form expression for the moment–curvature relationship, employed for the nonlinear analysis of relatively thick elastic–plas- and the analytical results were compared with numerical results tic offshore tubular members (Karamanos and Tassoulas, 1996) from a commercial finite element program. Libai and Bert (1994) and, more recently, for the elastic stability of thin-walled isotropic used thin-shell theory and a mixed variational principle to investi- cylinders under bending and pressure (Karamanos, 2002; Houliara gate the nonlinear ovalization behavior of anisotropic cylinders, and Karamanos, 2006). and reported solutions for long, medium-length and short cylin- The cylinder is considered as an elastic continuum. A net of ders, including the effects of end boundary conditions. Further- coordinate lines embedded in and deforming with the continuum more, a closed-form expression for the moment–curvature is employed. The embedded coordinates are denoted by ovalization path was derived for infinitely long cylinders. Corona ni i 1; 2; 3 , where coordinate lines n1 and n2 are directed in and Rodrigues (1995) presented numerical results for orthotropic ð ¼ Þ the hoop and in the axial direction of the cylinder, respectively, cylinders with cross-ply layers. Using nonlinear ring theory, the and for a constant value of n3, they define a shell lamina, whereas prebuckling ovalization solution was determined. Bifurcation was the coordinate line n3 is initially directed through the shell thick- detected through a perturbation technique, based on non-shallow ÀÁ ness. Therefore, x x n1; n2; n3; t is the position vector of the cylindrical shell kinematics. Tatting et al. (1996) analyzed long ÀÁ¼ material point n1; n2; n3 in the current (deformed) configuration anisotropic cylinders with symmetric lay-ups through a finite-dif- ÀÁ at time t. The position of the material point n1; n2; n3 at t 0in ference solution of semi-membrane shell equations. The final mo- ¼ the reference (undeformed) configuration is denoted by ment–curvature expression was identical to the one reported by ÀÁ X X n1; n2; n3 . At any material point, the covariant base vectors Reissner (1959) for isotropic tubes, and its solution was found to ¼ in the reference and in the current configuration, which are tan- compare well with the simplified Kedward solution (Kedward, gent to the coordinate lines, are denoted by G oX=oni 1978). Using the Kedward solution for the prebuckling path, bifur- i ¼ ¼ X and g ox=oni x , respectively, where the comma denotes cation was examined, considering the simplified engineering ;i i ¼ ¼ ;i partial differentiation with respect to ni. Furthermore, Gk and gk hypothesis of ‘‘equivalent cylinder” proposed in Axelrad (1965), denote the contravariant (reciprocal) base vectors in the reference as well as the effects of internal or external pressure, taking into and current configuration, respectively. account the influence of laminate stacking sequence. It was found that bifurcation occurs before a limit point is reached on the ovalization path. Harursampath and Hodges (1999) developed an en- 2.1. Governing equations hanced beam model accounting for cross-sectional deformation, in terms of trigonometric series expansion, to analyze long, thin- Deformation is described by the rate-of-deformation (stretch) walled cylinders of anisotropic materials. Employing one term of tensor d, which is the symmetric part of the velocity gradient L, series expansion, they provided closed-form expressions for the whereas the anti-symmetric part of L is the continuum spin x.It ovalization path and the corresponding stresses. Using this simpli- can be shown that the covariant components of the rate-of-defor- fied solution, limit-moment instability, local buckling and material mation tensor are: ply failure were examined. Recently, Wadee et al. (2007) using an ÀÁÀÁ ÀÁ analytical formulation and a second-degree trigonometric series 1 m m dkl ¼ V m=l G gk þ V m=k G gl ð3Þ solution in the hoop direction, obtained results for anisotropic cyl- 2 inders, in terms of the ultimate moment and the buckling where V m=l is the covariant derivative of the velocity vector compo- wavelength. nents with respect to the reference basis. The present paper investigates bifurcation instability and post- Equilibrium is expressed through the principle of virtual work, buckling behavior of long elastic cylinders under bending loading, considering an admissible displacement field du. For a continuum using a numerical formulation and solution methodology, with occupying the region V0 and V in the reference and in the current special emphasis on material modeling. The material of the cylin- configuration, respectively, and with boundary B in the deformed der is considered to be transversely-isotropic, where the anisot- configuration, the principle of virtual work is expressed as: ropy axis is initially directed in the longitudinal cylinder Z Z direction. A hypoelastic model is developed for describing the dU Gi g kj dV du tdB Mdh 4 mechanical behavior of transversely-isotropic materials, through i=j k s 0 ¼ q þ ð Þ V0 B a stress rate co-rotational with the anisotropy axis. Furthermore, a hyperelastic model based on a quadratic free energy function is where t is the surface traction, M is the moment, dh is the variation also considered for comparison purposes. The material models of end rotation, sij are the contravariant components of the are embedded in a special-purpose nonlinear finite element formu- Kirchhoff stress tensor s that is parallel to the Cauchy stress lation, which accounts for the presence of pressure. The effects of r ðsdV 0 ¼ rdVÞ and 12 S. Houliara, S.A. Karamanos / International Journal of Solids and Structures 47 (2010) 10–24

oðduÞ the components Sijkl of stress tensor S, with respect to the reference dUi=j ¼ Gi ð5Þ onj base, are related to the components of Lagrange–Green strain tensor E, also expressed with respect to the reference base, as follows: Following classical shell theory the traction component normal to ij ijkl any shell lamina is imposed to be zero at any stage of deformation. S ¼ D Ekl ð11Þ Considering that the traction is 1 rg3, where kg3k is the magnitude kg3k ÀÁwhere Dijkl are the components of a fourth-order rigidity tensor D 3 1 3 3 of g , and its component normal to the lamina is 2 r g g , kkg3 that depends exclusively on the material constants and is indepen- 1 33 33 dent of deformation. Therefore, differentiation of (11) with respect which is equal to 2 r , one obtains r ¼ 0, or equivalently, kkg3 to time results in the following rate equation: since the Kirchhoff stress is parallel to the Cauchy stress, s33 ¼ 0. _ ij ijkl _ In the context of hypoelasticity, a constitutive relation of the S ¼ D Ekl ð12Þ following form is used: From continuum mechanics, ij ijkl s_ ¼ R d ð6Þ T kl E_ ¼ F dF ð13Þ where the dot denotes differentiation with respect to time and Rijkl therefore, for the case of the convected coordinates ðg ¼ FG Þ one are the components of the instantaneous moduli, to be discussed in i i can readily show that the rate of the covariant components E_ of a subsequent section. ij the Lagrange–Green strain tensor E, expressed in the initial (refer- Alternatively, a hyperelastic constitutive equation can be used, ence) conﬁguration, are equal to the covariant components d of which expresses the second Piola–Kirchhoff stress tensor S, ij the rate-of-deformation tensor d, expressed in the deformed through an free-energy function. From continuum mechanics, s conﬁguration: and S are related as follows:

T E_ ¼ d ð14Þ s ¼ FSF ð7Þ ij ij Therefore, expressing s and S in terms of their components in the The constitutive models and their implementation are described in deformed and reference (initial) conﬁguration, respectively: detail in a following section.

ij ij T s ðgi gjÞ¼F½S ðGi GjÞF ð8Þ 2.2. Discretization and taking into account that for convected coordinates g ¼ FG , one i i The cylinder is discretized through a three-node ‘‘tube element” results in (Fig. 1), as introduced in Karamanos and Tassoulas (1996). This ele- sij ¼ Sij ð9Þ ment combines longitudinal (beam-type)ÀÁ with cross-sectional deformation. Convected coordinates n1; n2; n3 , which are denoted ij In other words, the contravariant components s of the Kirchhoff as ðh; f; qÞ, are located in the hoop, axial and radial direction in the stress tensor with respect to the current deformed configuration, reference configuration. Nodes are located along the cylinder axis, ij can be correlated to the contravariant components S of the second which lies on the plane of bending, and each node possesses three Piola–Kirchhoff stress tensor with respect to the initial undeformed degrees of freedom (two translational and one rotational). A refer- configuration. Using expression (9), the principle of virtual work ence line is chosen within the cross-section at node (k) and a local can also be expressed with respect to the components of the second Cartesian coordinate system is defined, so that the x; y axes define Piola–Kirchhoff stress tensor as follows: the cross-sectional plane. The orientation of node (k) is defined by Z Z ðkÞ ðkÞ ðkÞ the position of three orthonormal vectors ex ; ey and e . For in- k ij z dUk=j G gi S dV 0 ¼ du tdBq þ Mdh ð10Þ plane (ovalization) deformation, fibers initially normal to the refer- V B 0 ence line remain normal to the reference line. Furthermore, those For a hyperelastic constitutive model with a free-energy function fibers may rotate in the out-of-plane direction by angle cðhÞ. Using that is quadratic in terms of the Lagrange–Green strain tensor E, quadratic interpolation in the longitudinal direction, the position

Fig. 1. ‘‘Tube” element. S. Houliara, S.A. Karamanos / International Journal of Solids and Structures 47 (2010) 10–24 13 vector x(h,f,q) of an arbitrary point at the deformed configuration 3. Constitutive material models is: Transversely-isotropic material behavior is considered. This is a X3 hiÀÁspecial case of anisotropic material behavior, where the material ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ xðh; f; qÞ¼ x þ r ðhÞþqn ðhÞþqcðhÞez N ðfÞ ð15Þ possesses at every point a single preferred direction. The anisot- k¼1 ropy axes in the initial and in the deformed configurations are depicted in Fig. 2. In the initial configuration the anisotropy axis is where xðkÞ is the position vector of node (k), rðkÞðhÞ is the position of aligned in the axial direction of the undeformed cylinder, which the reference line at a certain cross-section relative to the corre- is the direction of the covariant base vector G2. Therefore, the cor- sponding node (k), nðkÞðhÞ is the ‘‘in-plane” outward normal of the responding direction unit vector is expressed as follows: reference line at the deformed configuration and NðkÞðfÞ is the corresponding Lagrangian quadratic polynomial. Using nonlinear ring 1 N ¼ G2 ð20Þ theory (Brush and Almroth, 1975), vector functions kG2k rðkÞðhÞ and nðkÞðhÞ, can be expressed in terms of the radial, tangential In the deformed configuration, the direction of the anisotropy axis is and out-of-plane displacements of the reference line, denoted as assumed to follow the direction of the local covariant base vector g , wðhÞ; ðhÞ; uðhÞ, respectively. Functions wðhÞ; ðhÞ; uðhÞ and cðhÞ 2 v v so that the corresponding direction unit vector is: are discretized as follows: X X 1 n ¼ g2 ð21Þ wðhÞ¼a0 þ a1 sin h þ an cos nh þ an sin nh ð16Þ kg2k Xn¼2;4;6;... Xn¼3;5;7;...: In the above equations, kG2k and kg2k are the magnitudes of vectors ðhÞ¼a cos h þ b sin nh þ b cos nh ð17Þ v 1 n n G2 and g2, respectively. X n¼2;4;6;... X n¼3;5;7;...: uðhÞ¼ cn cos nh þ cn sin nh ð18Þ 3.1. Hypoelastic model n¼2X;4;6;... n¼3;5X;7;...: cðhÞ¼ cn cos nh þ cn sin nh ð19Þ From continuum mechanics, the rate of change of unit vector n n¼0;2;4;6;... n¼1;3;5;7;...: in the deformable cylinder is given by the following expression:

n_ ¼ xan ð22Þ Coefﬁcients an; bn refer to in-plane cross-sectional deformation, and express the ovalization of the cross-section, whereas cn; cn refer to where the second order tensor xa is expressed as: out-of-plane cross-sectional deformation, expressing cross-sec- x ¼ x þ dðn nÞðn nÞd ð23Þ tional warping. a and is the tensor product of two vectors. Based on the above, the following hypoelastic constitutive equation is considered:

2.3. Numerical implementation s ¼ Dd ð24Þ For the purposes of the present study, a 16th degree expansion where s is a rate of Kirchhoff stress, co-rotational with the anisot- is used for wðhÞ; ðhÞ; uðhÞ and cðhÞ (considering n 6 16 in Eqs. v ropy direction vector n, defined as follows: (16)–(19)), and four ‘‘tube-elements” per half wavelength have found to be adequate. Regarding the number of integration points, s ¼ s_ þ sxa xas ð25Þ 23 equally spaced integration points around the half-circumfer- ence, three in the radial (through the thickness) direction and and D is the fourth-order elastic rigidity tensor, whose components depend on five material constants for the case of transverse isot- two Gauss points in the longitudinal direction of the ‘‘tube element” are considered (reduced integration scheme). ropy. One can readily show that s is an objective rate tensor. Ovalization analysis is conducted using one ‘‘tube element” and restraining all out-of-plane (warping) deformation parameters. a Such an analysis is referred to as ‘‘2D analysis”. To analyze buckling and postbuckling response a three-dimensional analysis (‘‘3D analysis”) is required. In such a case, uniform wrinkling is assumed and N a periodic solution along the cylinder is sought. Therefore, only the cylinder portion corresponding to a half wavelength Lhw is analyzed, with symmetry boundary conditions at the two ends with respect to the cross-sectional plane, restraining warping, that en- force the periodicity of the solution. The Lhw value is not known a b priori and, therefore, a sequence of analyses is conducted for each case considered, so that the actual wavelength is determined, as the one that corresponds to the ‘‘earliest” bifurcation point on the primary ovalization path. The nonlinear governing equations are solved through an incre- n mental Newton–Raphson iterative numerical procedure, enhanced to enable the tracing of postbuckling ‘‘snap-back” equilibrium paths through an arc-length algorithm, which monitors the value of the so-called ‘‘arc-length parameter” (Crisfield, 1983). A very small initial imperfection of the cylinder is imposed to trace the postbuckling path. The initial imperfection is considered in the Fig. 2. Schematic representation of transverse-isotropic cylinders; the anisotropy form of the buckling mode, obtained by an eigenvalue analysis just axes in the (a) undeformed and (b) in the deformed configuration are directed along prior to bifurcation on the ovalization primary path. the cylinder generators. 14 S. Houliara, S.A. Karamanos / International Journal of Solids and Structures 47 (2010) 10–24

b Tensor D is written ﬁrst in terms of its components Dijkl in a Carte- Therefore, using Eqs. (25) and (40), one can readily show that the r sian system deﬁned by the following orthonormal base vectors: stress rates s and s are related as follows:

r 1 1 e ¼ g ð26Þ s ¼ s þsðx xaÞðx xaÞs ð41Þ 1 kg1k where e2 ¼ n ð27Þ e3 ¼ e1 e2 ð28Þ x xa ¼ðn nÞd dðn nÞð42Þ where n is the unit vector along the anisotropy axis (Fig. 2) defined Equivalently, 1 in Eq. (21), g is the first vector of the reciprocal basis in the current r 1 s ¼ Dd þ s½ðn nÞd dðn nÞ ½ðn nÞd dðn nÞs ð43Þ configuration, and kg1k is the magnitude of g , so that tensor D can be expressed as follows: Using simple tensor algebra and considering Eq. (21), one can write

b ijkl ÀÁ D ¼ D ðei ej ek elÞð29Þ 1 k l ðn nÞd ¼ dkld2 g2 g ð44Þ b g 2 These components Dpqrs are given in the following matrix form k 2k (Green and Zerna, 1992; Christensen, 1979): and ÀÁ 1 l k dðn nÞ¼ 2 dkld2 g g2 ð45Þ kg2k k where dm is Kronecker’s delta. Furthermore, using Eqs. (44) and (45), one obtains after some manipulations: ÀÁ 1 k im jl sðn nÞd ¼ 2 d2s gm2g gi gj dkl ð46Þ kg2k ÀÁ 1 mj l ik dðn nÞs ¼ 2 s d2gm2g gi gj dkl ð47Þ kg2k ÀÁ ð30Þ 1 jl k i ðn nÞds ¼ 2 s d2d2 gi gj dkl ð48Þ in terms of ﬁve constants A ; A ; k; k ; l, which designate the ﬁve kg2k 11 22 2 ÀÁ independent effective properties of the media, and are related to 1 j sdðn nÞ¼ sikdl d g g d ð49Þ the engineering constants E ; E ; ; and of the material 2 2 2 i j kl 11 22 l12 m21 m13 kg2k through the following identities (Christensen, 1979; Aravas, 1992): Therefore, Eq. (43) can be written as follows

A ¼ K þ l ð31Þ r 11 13 13 s ¼ðD þ AÞd ð50Þ 2 A22 ¼ E22 þ 4l13m21 ð32Þ where the components of the fourth-order tensor A, using Eqs. k ¼ 2K m ð33Þ 13 21 (46)–(49) are k2 ¼ K13 l13 ð34Þ no 1 l ¼ l ð35Þ ijkl k im jl l jm ik jl i k ik j l 12 A ¼ 2 g2m d2s g þ d2s g s d2d2 þ s d2d2 kg2k where ð51Þ 1 E11 From continuum mechanics, the convected rate of Kirchhoff stress, K13 ¼ ð36Þ 2 1 m13 2m12m21 defined as: 1 E 11 _ ij l13 ¼ ð37Þ s ¼ s ðgi gjÞð52Þ 2 1 þ m13 is related to the Jaumann stress rate as follows: In the above equations, E11; E22 are the uniaxial moduli and they are directly measurable from a tensile test, is a shear modulus and r l12 s ¼ s Ld ð53Þ m21; m13 are Poisson’s ratios, where the notation mij represents that the first index i refers to the coordinate of the imposed stress or where L is the geometric rigidity fourth-order tensor, with compo- strain and the second index j refers to the response direction. nents with respect to the current basis: Subsequently, tensor D is written in terms of its components 1 ÂÃ ijkl Lijkl ¼ giksjl þ gjksil þ gilsjk þ gjlsik ð54Þ D with respect to the covariant base vectors in the current 2 configuration: Combining expressions (50) and (53), one finds: ijkl D ¼ D ðg g g g Þð38Þ i j k l s ¼ðD þ A LÞd ¼ Rd ð55Þ

These components are obtained through a standard tensor where R is a fourth order tensor, with components equal to: transformation ijkl ijkl ijkl ijkl R ¼ D þ A L ð56Þ ÀÁÀÁÀÁÀÁ ijkl b pqrs i j k l D ¼ D g ep g eq g er g es ð39Þ In component form, Eq. (55) is written as follows:

ij ijkl ijkl ijkl b s_ ¼ðD þ A L Þdkl ð57Þ where components Dpqrs are given in Eq. (30). r The Jaumann rate s is defined by the following equation: In the above hypoelastic model the stress rate s is always expressed r co-rotationally with the anisotropy direction vector n at the current s ¼ s_ þ sx xs ð40Þ configuration. In other words, the constitutive equation ‘‘follows” S. Houliara, S.A. Karamanos / International Journal of Solids and Structures 47 (2010) 10–24 15 the current orientation of the anisotropy axis throughout the defor- the anisotropy axes and their orientation in the deformed (current) mation history. configuration cannot be defined through a hypoelastic model. For the special case of isotropic material response the above hypoelastic material model can obtain a simpler form, where the 4. Numerical results continuum spin x is used instead of xa. In such a case, from Eqs. (40) and (41), the co-rotational stress rate s becomes the Jaumann Results for the bending response of long transversely-isotropic stress rate and and isotropic elastic cylinders are obtained, which focus on bifur- r cation from a uniform ovalization state to a uniform wrinkling pat- s ¼ Dd ð58Þ tern and postbuckling response. The cylinders are considered where the rigidity tensor D in Eq. (58) now depends on two material infinitely long in the sense that they are free of end effects and, constants instead of five. In such a case, tensor D obtains the follow- therefore, periodic solutions are sought for both buckling and ing simple form: post-buckling configurations, imposing appropriate boundary conditions. The possibility of a secondary bifurcation on the post- D ¼ kðI IÞþ2lJ ð59Þ buckling path is not investigated. Such a secondary bifurcation where I, J are the second-order and fourth-order identity tensors, may lead to localization of deformation (as in the case of uniform respectively, and k, l are the so-called Lamé constants. axial compression (Lord et al., 1997; Hunt et al., 2003)) but its examination is out of the scope of the present study. In the case of pressurized cylinders, pressure is applied first and 3.2. Hyperelastic model then, keeping the pressure level constant, bending load is gradually increased. Negative and positive values of pressure refer to internal For comparison purposes, a hyperelastic model is also imple- and external pressure, respectively. The uniaxial moduli of the mented in the present formulation. The model considers the Helm- transversely-isotropic material in the hoop and the longitudinal holtz free-energy function W as a function of the Lagrange–Green direction are denoted as E and E , respectively, and satisfy the strain tensor Eð2E ¼ FT F IÞ and the constitutive equation is ex- 11 22 condition pressed as: m12 m21 oWðEÞ ¼ ð67Þ S ¼ q ¼ DE ð60Þ E22 E11 0 oE where m12; m21 are the corresponding Poisson ratios. The valuesÀ of where S is the second Piola–Kirchhoff stress tensor, E is the La- pressure p and moment M are normalized by p ¼ E t3 4r3 ÁÀ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cr 11 grange–Green strain tensor and q0 is the mass density in the initial 2 ð1 m12m21Þ:Þ and Me ¼ E22rt qp1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m12m21 , respectively, so that configuration. In the present study, following previous works (Ara- f ¼ p=pcr; m ¼ M=Me, where q ¼ E22=E11 is a parameter indicating vas, 1992; Spencer, 1972, 1984), WðEÞ is expressed through a qua- the level of anisotropy. Bending curvature k is expressed as the ratio dratic function of the components of tensor E so that a linear of the relative rotation between the two end sections of the cylinder constitutive equation between S(E) and E is obtained in the form segment under consideration over their initial distance, which is of Eq. (11), where tensor D depends on the material constants only. equalÁÀ to Lhwffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi. The value of curvature k is normalized by In the case of transversely-isotropic materials a Cartesian sys- 2 p kN ¼ trq 1 m12m21 , so that j ¼ k=kN, whereas the buckling half tem is defined in the undeformed (initial) configuration, through wavelengthÁÀ Lhwffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiis normalized by the cylinder’s radius r. Moreover, the following orthonormal base vectors: p re ¼ E22tqr1 m12m21 is a ‘‘reference” stress used for normali-

1 1 zation purposes only. Cross-sectional ovalization is expressed e1 ¼ G ð61Þ kG1k through the following dimensionless ovalization parameter: e ¼ N ð62Þ D2 D1 2 f ¼ ð68Þ 2D e3 ¼ e1 e2 ð63Þ where D is the diameter length normal to the plane of bending and where N is the unit vector along the anisotropy axis in the initial 1 D is the diameter length on the plane of bending, both measured at conﬁguration (Fig. 2), deﬁned in Eq. (20), so that the fourth-order 2 the current state of cylinder deformation. tensor D can be expressed as follows: A parametric study was conducted using both hypoelastic and ÀÁ b pqrs hyperelastic models for r=t ratios equal to 100 and 720 and various D ¼ D ep eq er es ð64Þ levels of anisotropy. No difference was found in terms of the calcu- b where the components Dpqrs are given in Eq. (30). lated prebuckling ovalization path, bifurcation point, buckling Subsequently, tensor D can be written in terms of its compo- wavelength, and the postbuckling response. Therefore, in the fol- nents with respect to the reference covariant base vectors, as lowing results, the hyperelastic model is employed. follows:

ijkl 4.1. Ovalization instability D ¼ D ðGi Gj Gk GlÞð65Þ through the tensor transformation The ovalization primary path obtained from the present finite element formulation denoted as ‘‘uniform ovalization” is given in ijkl b pqrs i j k l D ¼ D G ep G eq G er G es ð66Þ Fig. 3. This path is denoted as ‘‘uniform ovalization”. In particular, in Fig. 3 the numerical results are compared with the predictions of For the special case of isotropy the fourth-order tensor D obtains a the closed-form expressions proposed in Kedward (1978), Stock- more simple form, expressed by Eq. (59). well and Cooper (1992), Libai and Bert (1994) and Harursampath One should note that the hyperelastic material model refers al- and Hodges (1999), using the present normalization so that a di- ways to the initial configuration and, therefore, it does not neces- rect comparison is possible. The comparison shows that, despite sarily ‘‘follows” the local anisotropy axis throughout the its simplicity, Kedward’s expression (Kedward, 1978), which is deformation history. On the other hand, it is a quite efficient model basically the Brazier solution (Brazier, 1927), provides a better in describing general cases of anisotropy (e.g. orthotropy), where prediction of the ovalization limit point than the more elaborate 16 S. Houliara, S.A. Karamanos / International Journal of Solids and Structures 47 (2010) 10–24

1.2 (c) (d) 1.0 (a) 0.8 (b) (e) 0.6 (a) numerical (b) Kedward 0.4 (c) Harursampath and Hodges (d) Libai and Bert normalized moment (m) 0.2 (e) Stockwel and Cooper 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 normalized curvature ( )

Fig. 3. Ovalization prebuckling path; numerical results versus analytical expressions (Kedward, 1978; Stockwell and Cooper, 1992; Libai and Bert, 1994; Harursampath and Hodges, 1999).

1.0

limit point 0.8

bifurcation 0.6

0.4 3D present analysis 2D present analysis

0.2 Corona and Rodrigues normalized moment (m)

0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 normalized curvature ( )

Fig. 4. Comparison of present numerical results with those of Corona and Rodrigues (1995), arrows (;) denote bifurcation and ovalization predictions from Corona and

Rodrigues (1995) and arrows (") denote the corresponding predictions of the present formulation (E22/E11 = 15.54). formulae reported in Stockwell and Cooper (1992), Libai and Bert bifurcation instability (buckling) in a form of uniform longitudinal (1994) and Harursampath and Hodges (1999). The fact that the wavy-type wrinkles, usually before a limit moment is reached on simple expression of Brazier/Kedward provides very good predic- the ovalization primary path. This buckling problem is associated tions has also been noted in several previous publications (Karam- with a highly nonlinear prebuckling state, where the compression anos, 2002; Houliara and Karamanos, 2006; Calladine, 1983). zone of the cylindrical shell has a double curvature of opposite sign Fig. 4 shows the ovalization path obtained from the present for- in the longitudinal and in the hoop direction. mulation and the corresponding path reported in Corona and As before, a comparison of the present numerical formulation Rodrigues (1995) for a composite cylinder with AS3501 material, and numerical results reported in Corona and Rodrigues (1995) is 0° layers and diameter-to-thickness ratio ðD=tÞ equal to 100. offered in Fig. 4 in terms of the buckling moment and the corre- According to Corona and Rodrigues (1995), the ovalization limit- sponding curvature. A very good comparison is obtained; accord- point instability occurs at curvature jov ¼ 0:49 and moment ing to the results of Corona and Rodrigues (1995), buckling mov ¼ 0:98, as indicated by the symbol ‘;’, whereas the results ob- occurs at jcr ¼ 0:37 and mcr ¼ 0:92 (symbol ‘;’inFig. 4 in the tained from the present formulation indicate a limit point at bifurcation area), whereas the present analysis predicts bifurcation jov ¼ 0:480 and mov ¼ 0:954, denoted by the symbol ‘"’ in the lim- at jcr ¼ 0:354 and mcr ¼ 0:882 (symbol ‘"’inFig. 4 at the bifurca- it point area. The comparison between the two sets of results is tion point area). very good. Figs. 5 and 6 show the buckling response of transversely-isotropic cylinders with radius-to-thickness ratio between r=t ¼ 100 and 4.2. Bifurcation instability 720, for different levels of anisotropy. The results show that the location of bifurcation on the m k primary path, as well as the post- Ovalization instability was considered in the previous section buckling behavior depend on the level of anisotropy s; increasing the under the assumption that bifurcation does not occur. However, level of anisotropy, bifurcation occurs earlier on the normalized bifurcation from the ovalized prebuckling conﬁguration does prebuckling path. For the case of thin-walled cylinder ðr=t ¼ 720Þ occur. The increased axial stresses at the compression side of the the normalized value of the buckling curvature jcr ranges from cylinder, accentuated because of the ovalization mechanism, cause 0.39 for the isotropic case ðs ¼ 1Þ to 0.34 for s ¼ 10. S. Houliara, S.A. Karamanos / International Journal of Solids and Structures 47 (2010) 10–24 17

0.95 r/t=100

0.90 s=1

0.85 s=3

0.80 s=5

0.75 normalized moment (m) s=10

0.70 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 normalized curvature ( )

Fig. 5. Response of non-pressurized tube for different values of anisotropy parameter s ðr=t ¼ 100; k=l ¼ 1:5Þ.

0.94

0.92 r/t=720 0.90

0.88 s=1

0.86 s=3 0.84

normalized moment (m) s=5 0.82 s=10 0.80 0.30 0.32 0.34 0.36 0.38 0.40 0.42 normalized curvature ( )

Fig. 6. Response of non-pressurized tube for different values of anisotropy parameter s ðr=t ¼ 720; k=l ¼ 1:5Þ, arrows (;) denote the critical points obtained from the numerical procedure.

1.00 r/t=10 0.95 s=1 0.90

s=3 0.85

s=5 0.80 normalized moment (m) s=10

0.75 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 normalized curvature ( )

Fig. 7. Response of non-pressurized tube for different values of anisotropy parameter s; arrows ‘;’ stand for bifurcation points ðr=t ¼ 10; k=l ¼ 1:5Þ.

Numerical results from a thick cylinder ðr=t ¼ 10Þ are shown in bifurcation point is observed on the normalized equilibrium path, Figs. 7 and 8. The material properties of the cylinder in Fig. 8 are as the anisotropy parameter s increases. obtained from Wang et al. (2005), and correspond to the mechan- The dependence of the buckling half wavelength on the level of ical properties of single-walled carbon nanotubes, calculated from anisotropy is plotted in Fig. 9 for three r=t ratios. It can be seen molecular dynamics. A signiﬁcant shift of the location of the that the critical wavelength increases as anisotropy effects are 18 S. Houliara, S.A. Karamanos / International Journal of Solids and Structures 47 (2010) 10–24

a 1.0 (a) (b) 0.8

detail below 0.6 2D analysis

0.4 (a) E /E =1.47, λ/µ=1.27 22 11

(b) E22 /E11 =0.80, λ/µ=0.52

normalized moment (m) 0.2

0.0 0.00 0.10 0.20 0.30 0.40 0.50 normalized curvature ( )

b 0.96

0.94

(a) (b) 0.92 2D_analysis

(a) E /E =1.47, λ/µ=1.27 0.90 22 11

normalized moment (m) (b) E /E =0.80, λ/µ=0.52 22 11

0.88 0.38 0.40 0.42 0.44 0.46 0.48 normalized curvature ( )

Fig. 8. Buckling analysis of thick-walled cylinders (r/t=10) with carbon nanotube mechanical properties as described in Wang et al. (2005); arrows ‘;’ indicate bifurcation points.

0.7

0.6 numerical r/t=120 0.5

0.4 analytical r/t=120 numerical r/t=480

Lhw/r 0.3 analytical r/t=480 numerical r/t=720 0.2

0.1 analytical r/t=720 0.0 12345678910 anisotropy parameter (s)

Fig. 9. Dependence of the buckling half wavelengths on the level of anisotropy for three different r=t ratios; comparison of numerical results with the corresponding analytical predictions of Eq. (70) ðk=l ¼ 1:5Þ. accentuated and this increase can be related to the longitudinal in the hoop direction. On the other hand, the buckling shape of aniso- strengthening of the cylinder. tropic cylinders is characterized by multiple short-length waves The buckling modes associated with the bifurcation points are de- within the compression zone. These wavy patterns are more pro- picted in Figs. 10 and 11. An important observation is that the buck- nounced for higher levels of anisotropy parameter s and indicate a ling mode for the isotropic case has the form of a smooth buckle in the signiﬁcant variation of stresses and strains within the buckling zone. hoop direction that ‘‘dies out” quickly outside the buckling zone, Motivated by the local character of buckling shape in the hoop especially for thin-walled cylinders, where it covers a limited range direction, an analytical solution based on a simpliﬁed heuristic S. Houliara, S.A. Karamanos / International Journal of Solids and Structures 47 (2010) 10–24 19

( s =1, rt=720, λ/μ=1.5)

( s =3, rt=720, λ/μ=1.5)

( s =10, rt=720, λ/μ=1.5)

Fig. 10. Buckling modes for a thin cylinder r=t ¼ 720 for three values of anisotropy parameter; higher values of anisotropy parameter s result in greater variation of deformation in the hoop direction, in the form of wavy patterns.

argument can be developed, in lieu of a detailed buckling analysis. ment with more rigorous buckling calculations (Houliara and This approach was initially introduced by Axelrad (1965, 1985) for Karamanos, 2006; Kempner and Chen, 1974; Fabian, 1977). the bending response of isotropic cylinders. In subsequent years, The argument is based on the so-called ‘‘local-buckling hypoth- this argument has been widely employed to obtain an estimation esis”; buckling is determined by the state of stress and deformation of the bifurcation point on the prebuckling ovalization path of iso- inside the ‘‘buckling zone”, located around the critical point, and tropic elastic cylinders under bending, which was in good agree- that stress and strain parameters are constant within the buckling 20 S. Houliara, S.A. Karamanos / International Journal of Solids and Structures 47 (2010) 10–24

( s =1, rt=100, λ /μ =1.5)

( s =3, rt=100, λ /μ =1.5)

( s =10, rt=100, λ /μ =1.5)

Fig. 11. Buckling modes for a thin cylinder r=t ¼ 100 for three values of anisotropy parameter; higher values of anisotropy parameter s result in greater variation of deformation in the hoop direction, in the form of wavy patterns. zone with respect to the hoop direction. Therefore, axisymmetric the critical point h ¼ p=2, considering prebuckling ovalization. Fur- conditions can be considered within the buckling zone and buck- thermore, the corresponding critical half wavelength can be also ling can be estimated through the following buckling equation computed as follows (Brush and Almroth, 1975): (Brush and Almroth, 1975): pffiffiffiffiffiffiffiffiffiffiffi axi 1=4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Lhw ¼ p qrh0t=½12ð1 m12m21Þ ð70Þ E E t ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11 22 rx ¼ p p ð69Þ which is a generalization of Eq. (2) for anisotropic cylinders. 3 1 m12m21 rh0 Employing the buckling condition from Eq. (69) and considering which is a generalization of Eq. (1) for axisymmetric buckling of Kedward’s solution for the local hoop curvature 1=rh0 and the longi- anisotropic cylinders, where rh0 is the local radius of curvature at tudinal stress rx at h ¼ p=2, expressed by Eqs. (87) and (88) in S. Houliara, S.A. Karamanos / International Journal of Solids and Structures 47 (2010) 10–24 21

Appendix B, a third-order algebraic equation in terms of the critical agreement with the corresponding numerical results for cylinders curvaturejcr is obtained: with small anisotropy ðs < 3Þ. The analytical predictions deviate 1 2 1 3 2 from the numerical results as the level of anisotropy is increased. jcr 1 j pffiffiffi 1 j ¼ 0 ð71Þ 1 f cr 3 1 f cr This is also verified in Fig. 9, which depicts the variation of the critical half-wavelength value Lhw with respect to the anisotropy which has the following closed-form solution: parameter s. In this graph the analytical half-wavelength predic- qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tions are those of Eq. (70). The rather poor comparison of the ana- 1ffiffiffi 2 jcr ¼ p abs½3ð2 f Þ lytical predictions with the numerical results for large values of 3 3 "#pffiffiffi !anisotropy is attributed to the fact that those cylinders exhibit p 1 3 3 rather large buckling zone of a non-smooth shape. Therefore, in cos þ arccos ð72Þ 3 3 abs½3ð2 f Þ3=2 such a case, the concept of ‘‘local buckling hypothesis”, on which the analytical solution is based, is no longer valid. where abs[ ] is the absolute value of [ ]. Subsequently, the bifurcation moment mcr is obtained from Eq. (85) with j ¼ jcr. For zero 4.3. Postbuckling behavior and imperfection sensitivity pressure case ðf ¼ 0Þ one readily obtains from Eq. (72): Initial postbuckling behavior of shells is generally characterized jcr ¼ 0:3811 ð73Þ by a significant sensitivity with respect to initial imperfections. and from Eq. (85) the corresponding critical moment is: Herein, imperfection sensitivity is examined numerically, and the results are compared with available asymptotic solutions. The ini- mcr ¼ 0:9365 ð74Þ tial imperfection is assumed periodic in the form of the first insta- It is noted that, using the normalization values adopted in the pres- bility mode (e.g. Figs. 10 and 11), and the imperfection amplitude n ent study, the value of critical curvature jcr, predicted by Eq. (72),is is defined as the ratio of the total wave height W0 over the cylinder independent of the r=t ratio and of the level of anisotropy s. For the thickness t ðn ¼ W0=tÞ. case of isotropic cylinders ðs ¼ 1Þ a good comparison of this analyt- The ‘‘snap-back” behavior shown in Figs. 5 and 6 indicates a ical solution with the numerical results obtained from the present sensitivity of the bending response in the presence of initial imper- formulation was found. This comparison is even better for thin- fections for both isotropic and anisotropic cylinders. Numerical re- walled cylinders, also noticed by the authors elsewhere (Houliara sults of cylinders with r=t equal to 100 and 720 (Figs. 12 and 13) and Karamanos, 2006). For anisotropic cylinders it was shown verify that for increasing imperfection amplitude the value of the

(e.g. Figs. 5 and 6) that these analytical predictions are in good ultimate (maximum) bending moment mmax is reduced and the

a 1.0 r/t=100, s=3

0.8

detail below 0.6

0.4 3D analysis

0.2 ξ =9.20x10-3 normalized moment (m)

0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 normalized curvature ( )

b 0.88 perfect cylinder r/t=100, s=3 0.86 (b) perfect cylinder (a) (c) 0.84 (a) ξ =2.30x10-3

(b) ξ =4.60x10-3 0.82 (c) ξ =1.15x10-2

0.80 normalized moment (m)

0.78 0.320 0.325 0.330 0.335 0.340 0.345 0.350 0.355 normalized curvature ( )

Fig. 12. Imperfection sensitivity of a transversely-isotropic cylinder; (a) entire m–k path and (b) detail at bifurcation point (r/t = 100, s =3,k/l = 1.5, E22/E11 = 3.45). 22 S. Houliara, S.A. Karamanos / International Journal of Solids and Structures 47 (2010) 10–24

0.92

r/t=720, s=3 perfect cylinder 0.90 (b) (a)

0.88 (c) (a) ξ =1.55x10-3 0.86 (b) ξ =6.20x10-3

0.84

normalized moment (m) (c) ξ =1.55x10-2

0.82 0.34 0.35 0.36 0.37 0.38 0.39 normalized curvature (κ)

Fig. 13. Detail of the imperfection sensitivity of a transversely-isotropic cylinder (r/t = 720, s =3,k/l = 1.5, E22/E11 = 3.45).

1.00 results are compared with the results of the asymptotic analysis m of Fabian (1977). The latter are obtained from the application of max = 1−Cξ 2/3 mcr the general initial postbuckling theory on systems with nonlinear 0.99 prebuckling state (Fitch, 1968), resulting in the following asymp- external totic expression:

cr pressure

m (f=2/3) / mmax a 0.98 ¼ 1 Cn ð75Þ

max no-pressure mcr

m (f=0) where mcr is the bifurcation moment of the corresponding ‘‘perfect” 0.97 numerical cylinder, C is a positive constant that depends on the r=t ratio and internal analytical pressure the level of pressure f, and exponent a is equal to 2/3. The compar- (f=-2/3) ison between the present results and Fabian’s results is shown in 0.96 Fig. 14 and indicates an excellent agreement. The values of C are 0.000 0.002 0.004 0.006 0.008 0.010 equal to 0.272, 0.567, 0.797 for f equal to 2/3, 0, -2/3, respectively. imperfection amplitude (ξ) The results also show that the presence of internal pressure despite its beneficial effect on ultimate moment capacity (Houliara and Fig. 14. Load reduction ðmmax=mcrÞ versus imperfection amplitude, f = 2/3, 0 and 2/3 (s =1,r/t = 100, k=l ¼ 1:5). Karamanos, 2006; Limam et al., 2009) results in an increase of the imperfection sensitivity of the buckling behavior, whereas the presence of external pressure has an opposite effect. response becomes smoother. Moreover, the results show that the The maximum moment reduction in anisotropic cylinders is de- presence of very small geometric imperfections, with amplitude picted in Fig. 15 in terms of initial imperfection amplitude. In this equal to a small fraction of the cylinder wall thickness (e.g. case, no asymptotic expression is available. However, from general n ¼ 103), may have a substantial influence on the maximum post-buckling theory, the maximum moment of the imperfect cyl- bending moment. inder is expected to follow an expression of Eq. (75), with exponent The reduction of maximum moment due to initial imperfections a equal to 2/3. The numerical results verify that the maximum mo- in isotropic cylinders is depicted in Fig. 14 with respect to the ment value mcr follows an exponential expression of the form of Eq. imperfection amplitude. In the same figure the present numerical (75), where the value of constant C depends on the r=t ratio and on

1.000 s=3

0.995 numerical results

cr 0.990 exponential curve m / max

m 0.985 r/t=100 r/t=720 0.980

0.975 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 imperfection amplitude (ξ)

Fig. 15. Imperfection sensitivity versus imperfection amplitude (s =3,k/l = 1.5, E22/E11 = 3.45). S. Houliara, S.A. Karamanos / International Journal of Solids and Structures 47 (2010) 10–24 23 the level of anisotropy, with a equal to 2/3. Finally, it should be where s is a parameter that indicates the level of anisotropy. Clearly, mentioned that the imperfection sensitivity of bent elastic circular if s ¼ 1, the material is isotropic elastic, A11 ¼ A22 ¼ k þ 2l, and k, l cylinders is not as severe as in the case of uniformly compressed become the Lamé constants. Using the above moduli and assuming elastic circular cylinders. This is explained by the fact that in our uniaxial stress states in the direction of axes 1 and 2, one obtains analysis, under the assumption of uniform wrinkling pattern, the the uniaxial moduli E11; E22 and the corresponding Poisson’s ratios: buckling moment (i.e. the lowest eigenvalue) is associated with a single buckling mode, as opposed to the infinite number of buck- 4lk2 þ sðk þ 2lÞ k2 ðk þ 2lÞ2 ling modes corresponding to the bifurcation load in the case of a E11 ¼ ð79Þ k2 s k 2 2 uniformly compressed cylinder (Hunt and Lucena Neto, 1991; ð þ lÞ 2 Champneys et al., 1999). This makes the present problem less sen- sðk þ 2lÞðk þ lÞk E22 ¼ ð80Þ sitive to initial imperfections. k þ l k2 kðk þ 2lÞ m12 ¼ ð81Þ 5. Conclusions k2 sðk þ 2lÞ2 k2 skðk þ 2lÞ Buckling of isotropic and transversely-isotropic elastic cylinders 82 m13 ¼ 2 2 ð Þ has been examined, with special emphasis on the constitutive mod- k sðk þ 2lÞ eling of anisotropic material behavior, using a special-purpose finite k m ¼ ¼ m ð83Þ element formulation and solution methodology. A hypoelastic mod- 21 2ðk þ lÞ 23 el, developed for the purposes of this study, and a hyperelastic model 2 E22 ðk=lÞðs 1Þ that was also employed, were incorporated in the numerical tech- q ¼ ¼ s þ ð84Þ E11 4ð1 þðl=kÞÞ nique to examine the bending problem. The comparison of these two constitutive models resulted in the same numerical results regarding prebuckling, bifurcation and postbuckling response. Appendix B. Analytical solutions for ovalization The bending response of the cylinder is characterized by significant prebuckling cross-sectional ovalization, followed by a bifurca- Kedward (1978), adopting the formulation of Brazier (1927), tion instability in the form of uniform wrinkles at the compression and assuming a different behavior in the longitudinal and the hoop side. Considering the effects of anisotropy, this bifurcation and the direction, obtained a closed-form expression for the moment–cur- postbuckling response were examined mainly numerically, vature ovalization path, and for the ovalization–curvature relation- employing the finite element technique. The results showed that ship as follows: there is a strong dependence of the bifurcation point on the level of anisotropy and on the value of the r=t ratio. Furthermore, a sim- 3j2 m ¼ jp 1 ð85Þ plified analytical bifurcation solution, based on the ‘‘local buckling 2 hypothesis” was also presented that resulted in closed-form 2 expressions for the buckling curvature and the critical moment. f ¼ j ð86Þ The analytical predictions were compared with the corresponding The corresponding expressions for the longitudinal stress and the numerical results and a good agreement was found for the cases hoop curvature are of small anisotropy. Using an initial imperfection of the cylinder in the form of the buckling mode, the imperfection sensitivity of 3j2 j2 those cylinders were examined in detail for both isotropic and r ðhÞ¼r j 1 sin h þ sin 3h ð87Þ x e 4 4 transversely-isotropic cylinders. In all cases, the initial postbuck- 1 1 ling equilibrium path was found to be unstable, indicating verifying ¼ ð1 þ 3j2 cos 2hÞð88Þ the imperfection sensitivity, and the numerical results were found rhðhÞ r to compare very well with available asymptotic expressions. Other closed-form expressions for the m j ovalization path, sim- Acknowledgements ilar to Eq. (85) have been derived in Harursampath and Hodges (1999), Libai and Bert (1994) and Stockwell and Cooper (1992). This work was supported by a Research Fellowship from the More specifically, Alexander S. Onassis Public Benefit Foundation, Athens, Greece, awarded to S. Houliara. Stockwell and Cooper (1992) 3 3 m ¼ jp 1 j2 j4 ð89Þ 2 2 Appendix A. Parameterization of transversely-isotropic materials Libai and Bert (1994) 3 5 For the purposes of conducting the parametric study in an effi- m ¼ jp 1 j2 þ j4 ð90Þ cient manner, the above assumptions are made for the five inde- 2 8 pendent material parameters in the components Dijkl of tensor D Harursampath and Hodges (1999) (Eq. (30)) in the transversely-isotropic material: 1 3 2 15 4 ðA k Þ¼l ð76Þ m ¼ jp 1 j þ j ð91Þ 2 11 2 2 8 k ¼ k2 ð77Þ To account for the effects of pressure on the ovalization re- From Eqs. (76) and (77), one trivially obtains A11 ¼ k þ 2l. It is further assumed that the following condition applies: sponse, an approach similar to the one presented in Houliara and Karamanos (2006) for isotropic cylinders. In such a case, it is read- A22 ¼ sA11 ð78Þ ily obtained that Eqs. (85)–(88) become 24 S. Houliara, S.A. Karamanos / International Journal of Solids and Structures 47 (2010) 10–24 3 Hunt, G.W., Lord, G.J., Peletier, M.A., 2003. 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