Large deformation analysis of functionally graded shells · R.A. Arciniega, J.N. Reddy * Texas A&M University, College Station, TX 77843-3123, USA Received 1 March 2006; received

International Journal of Solids and Structures 44 (2007) 2036–2052

www.elsevier.com/locate/ijsolstr

Large deformation analysis of functionally graded shells

R.A. Arciniega, J.N. Reddy *

Texas A&M University, College Station, TX 77843-3123, USA

Received 1 March 2006; received in revised form 24 August 2006Available online 6 September 2006

This paper is dedicated to Professor Fong Shih on the occasion of his 60th birthday.

Abstract

A geometrically nonlinear analysis of functionally graded shells is presented. The two-constituent functionally gradedshell consists of ceramic and metal that are graded through the thickness, from one surface of the shell to the other. Atensor-based finite element formulation with curvilinear coordinates and first-order shear deformation theory are usedto develop the functionally graded shell finite element. The first-order shell theory consists of seven parameters and exactnonlinear deformations and under the framework of the Lagrangian description. High-order Lagrangian interpolationfunctions are used to approximate the field variables to avoid membrane, shear, and thickness locking. Numerical resultsobtained using the present shell element for typical benchmark problem geometries with functionally graded material com-positions are presented.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Functionally graded shells; First-order shell theory; Finite element analysis; Geometrically nonlinear shell theory; Higher-order elements

1. Introduction

Laminated composite shells have been of great interest in many engineering applications. Composite lam-inates made up of a set of fiber-reinforced lamina bonded together are particularly attractive (Reddy, 2004a).A typical lamina is often characterized as orthotropic with the principal material directions of each laminacoinciding with the fiber direction and transverse to it. By changing the material type, fiber orientation, orthickness, a designer can tailor different properties of a laminate to suit a particular application. Despite theirmultiple advantages, laminated composites exhibit serious shortcomings due to stress concentrations betweenlayers. The material mismatch due to the piece-wise variation of the material properties through the thicknessof the shell is the reason for the interlaminar stress field that often leads to delamination failures. To avoid thematerial mismatch, a special material named ‘‘functionally graded material’’ (FGM) was proposed by Koiz-umi (1997) and Yamanouchi et al. (1990), in which the material properties are varied smoothly and continu-

0020-7683/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijsolstr.2006.08.035

* Corresponding author. Tel.: +1 979 862 2417; fax: +1 979 862 3989.E-mail addresses: [email protected], [email protected] (J.N. Reddy).

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R.A. Arciniega, J.N. Reddy / International Journal of Solids and Structures 44 (2007) 2036–2052 2037

ously from one surface to the other. The gradation of the material properties through the thickness eliminatesjumps or abrupt changes in the stress and displacement distributions.

In certain applications, shell structures can experience large elastic deformations and finite rotations.Geometric nonlinearity plays a significant role in the behavior of a shell, especially when it undergoes largedeformations. Previous studies show that laminated shells exhibit drastic changes in their bending response(Bas�ar et al., 1993; Vu-Quoc and Tan, 2003; Balah and Al-Ghamedy, 2002). Even for homogeneous andisotropic shells, we observe an unpredictable behavior (Simo et al., 1990; Sansour and Kollmann, 2000).Therefore, it is of vital importance to study the nonlinear response of inhomogeneous materials such asfunctionally graded shells.

This paper is motivated by the lack of studies found in the literature that address large deformation analysisof FGM shells. A review of the literature shows that few studies have been carried out to investigate the non-linear bending response of plates and shells. Most of them use von Karman or Sanders theories, which arerestricted to moderately small deformations. We cite the papers of Na and Kim (2005), who examined the effectof thermal loading and uniform pressure on the bending response of FGM plates; and Yang and Shen (2003a,b),who analyzed nonlinear bending and postbuckling behavior for FGM plates under thermomechanical load andwith various boundary conditions. Woo and Meguid (2001) provided an analytical solution for large deflectionof FGM plates and shells under mechanical and thermal loading; while Ma and Wang (2003) examined axisym-metric large deflection bending and thermal postbuckling of FGM circular plates subjected to mechanical andthermal loads. Both articles are based on the classical von Karman plate theory (Reddy, 2004a,b).

Moreover, Reddy and Chin (1998) analyzed the dynamic thermoelastic response of functionally graded cyl-inders and plates. Praveen and Reddy (1998) carried out a nonlinear thermoelastic analysis of functionallygraded ceramic–metal plates using a finite element model based on the first-order shear deformation theory(FSDT). Thermomechanical buckling, as well as bending and free vibration analysis, of FGM plates canbe found in the articles by Reddy and Arciniega (2006a,b). Further studies of bending and vibration analysesof FGMs plates are found in the articles of Reddy (2000), and Della Croce and Venini (2004).

On the subject of computational models for shell structures, we focus our attention on tensor-based finiteelement models (Harte and Eckstein, 1986). This approach is able to determine all properties of the shell’s dif-ferential geometry exactly. Additional errors, introduced by approximating the geometry of the midsurface ofthe shell (as in continuum-based finite element models), are prevented from the beginning. Previous works ofthe authors using tensor-oriented finite element formulations for linear analysis of laminated shells can befound in Arciniega and Reddy (2005), and Reddy and Arciniega (2004).

In this paper, a large deformation analysis for functionally graded shells is presented. The formulation isbased on the first-order shear deformation theory with seven independent parameters (Sansour, 1995; Bischoffand Ramm, 1997) where no plane stress assumption is required (3D constitutive equations). A tensor-basedfinite element model is developed using high-order Lagrange elements to preclude membrane shear locking.The gradation of the material properties of the FGM shell is considered through the thickness. The materialstiffness tensor is obtained by Gauss integration. Numerical results are presented for typical benchmark prob-lems with applications to functionally graded shells.

2. Theoretical formulation

The shell theory will be briefly discussed here. For a detailed development, one can consult the paper ofArciniega and Reddy (in press) and references herein. The mathematical background utilized in the followingderivation is given in the books of Naghdi (1963, 1972), Green and Zerna (1968), and Pietraszkiewicz (1979).

Consider the motion v(X, t) of the shell body B from the reference configuration BR to the current config-uration Bt. We introduce in the region BR(Bt) a convected curvilinear coordinate system {hi}, i = 1,2,3, suchthat the surface h3 = 0 defines the midsurface MR(Mt) of the region BR(Bt). The coordinate h3 is the measureof the distance between points P 2 BR ðP 2 BtÞ and M 2MR ðM 2MtÞ, with �h/2 6 h3

6 h/2, where h is thethickness of the shell (Fig. 1).

Since a convected coordinate system {hi} has been adopted, geometric quantities of the region Bt areanalogous to those defined in BR. In the Lagrangian description, the displacement of the particle X fromthe reference configuration to the current configuration is given by the vector v(X, t), i.e.

e3

e2

e1

r

MP

a3R

R

t

t

X

PM

a3

xr

u

v

1x

3x

1θ

3θ

2θ

2x

1θ

3θ

2θ

Fig. 1. Shell continuum in the reference and current configurations.

2038 R.A. Arciniega, J.N. Reddy / International Journal of Solids and Structures 44 (2007) 2036–2052

vðX; tÞ ¼ vðX; tÞ � X ¼ x� X ¼ V igi ¼ V jgj ð1Þ

wherein the last line is in component form with respect to the region BR.We introduce the first kinematical assumption for the shell model: ‘‘the displacement field is considered as a

linear expansion of the thickness coordinate around the midsurface. The transverse displacement is parabolicthrough the thickness of the shell’’.

This assumption implies that

vðhaÞ ¼ uðhaÞ þ h3uðhaÞþðh3Þ2wðhaÞ ð2Þ

where

uðhaÞ ¼ uiai; uðhaÞ ¼ uia

i; wðhaÞ ¼ w3a3 ð3Þ

The underlined term of Eq. (2) is included to avoid Poisson locking (Bischoff and Ramm, 1997).The position vector of the deformed shell can be obtained substituting Eq. (2) into (1). Thus

x ¼ �rþ h3�a3 þ ðh3Þ2w ð4Þ

where �r ¼ rþ u and �a3 ¼ a3 þ u. The vector u is also called the difference vector (change of the director of themidsurface). The director �a3 is, in general, neither a unit vector nor orthogonal to Mt. The configuration of theshell is uniquely determined by the displacement vector u of the midsurface together with the difference vectoru and the additional variable w, or by seven independent components of these vectors (Sansour, 1995).

We now introduce the Green strain tensor E as a measure of the strain for a material description

E ¼ 1

2ðC�GÞ ð5Þ

where C = FTF is the right Cauchy–Green tensor, G = gijgi � gj is the Riemannian metric in the reference con-

figuration and F ¼ �gi � gi is the deformation gradient. We define the covariant space and surface base vectorsin the current configuration as �gi and �ai, respectively.

The shifter tensor l is a two-point tensor which relates the region BR to the reference midsurface MR and itis useful to define the tensor bE as
bE ¼ U�ðEÞ ¼ lTEl ð6Þ where U*(�) is the pull-back operator.
The tensor bE can be expanded as a function of the thickness coordinate, i.e.
bE ¼ e0 þ h3e1þðh3Þ2e2 þ ðh3Þ3e3 þ ðh3Þ4e4 ð7Þ


The second assumption for the shell model asserts that: ‘‘quadratic and higher-order terms of bE, underlinedin Eq. (7), are negligible’’. Then, we arrive at the following decomposition

e0 ¼ eð0Þab aa � ab þ eð0Þa3 ðaa � a3 þ a3 � aaÞ þ eð0Þ33 a3 � a3

e1 ¼ eð1Þab aa � ab þ eð1Þa3 ðaa � a3 þ a3 � aaÞ þ eð1Þ33 a3 � a3ð8Þ

where eðiÞab, eðiÞa3 and eðiÞ33 are functions of the triple (u,u,w). After some manipulations we can write them in termsof the seven components of the displacement field (Habip, 1965), i.e.

eð0Þab ¼1

2ðuajb þ ubja � 2babu3 þ akcukjaucjb � bk

bu3ukja � bkau3ukjb

þ cabðu3Þ2 þ u3;au3;b þ bkauku3;b þ bk

buku3;a þ bkabc

bukucÞ

eð1Þab ¼1

2ðuajb þ ubja � 2babu3 � bk

bukja � bkaukjb þ 2cabu3 þ akcukjaucjb þ akcukjbucja � bk

bu3ukja

� bkau3ukjb � bk

bu3ukja � bkau3ukjb þ 2cabu3u3 þ u3;au3;b þ u3;bu3;a

þ bkauku3;b þ bk

buku3;a þ bkauku3;b þ bk

buku3;a þ bkabc

bukuc þ bkbbc

aukucÞ

eð0Þa3 ¼1

2ðua þ u3;a þ bk

auk þ akcukjauc � bkauku3 þ u3u3;a þ bk

auku3Þ

eð1Þa3 ¼1

2ðu3;a þ akcukjauc þ u3u3;a þ 2w3u3;a þ 2w3bk

aukÞ

eð0Þ33 ¼1

2ð2u3 þ akcukuc þ ðu3Þ

2Þ

eð1Þ33 ¼ 2ðw3 þ u3w3Þ ð9Þ

where cab ¼ balblb is the covariant third fundamental form of the reference surface and (�)ja. is the surface

covariant derivative. Note that the component eð1Þ33 vanishes when w3 = 0 (6-parameter formulation).The second Piola–Kirchhoff stress tensor is used for the Lagrangian formulation and is energetically con-

jugate to the rate of Green strain tensor _E (Reddy, 2004a). Like E, the second Piola–Kirchhoff stress tensor S

is transformed to the midsurface MR by
bS ¼ l�1Sl�T ¼ U�ðSÞ ð10Þ
which is the pull-back operator of the contravariant tensor S.Let Mn denote the stress resultant tensor which is a symmetric tensor. The tensor Mn is defined as

½M0;M1� ¼Z h=2

�h=2

½1; h3�bSldh3 ð11Þ

The scalar quantity l is the determinant of the shifter tensor l. The stress resultant tensors are also ener-getically conjugate to the strain resultants ei. The stress resultant tensors may be decomposed in componentform as

M0 ¼ N abð0Þ

aa � ab þ Qa3ð0Þðaa � a3 þ a3 � aaÞ þ T 33

ð0Þa3 � a3

M1 ¼ N abð1Þ

aa � ab þ Qa3ð1Þðaa � a3 þ a3 � aaÞ þ T 33

ð1Þa3 � a3

ð12Þ

where N abðnÞ;Qa3ðnÞ

and T 33ðnÞ

are membrane, shear and stretching components of the stress resultant tensor,respectively.

3. Functionally graded shells

In this section we consider a hyperelastic and inhomogeneous shell. The shell structure can undergo largedeformations (rotations and displacements) while the material response remains in the elastic regime. We also


consider the relation between the second Piola–Kirchhoff stress tensor S and the Green strain tensor E islinear. It implies that

S ¼ C � E ð13Þ

where C is the fourth-order elasticity tensor. The tensor C is represented in convected coordinates as

C ¼ Cijklgi � gj � gk � gl ð14Þ

where the components of C satisfy the following symmetry conditions

Cijkl ¼ Cjikl ¼ Cijlk ¼ Cklij ð15Þ

As discussed earlier, functionally graded materials (FGMs) are a special kind of composites in which thematerial properties vary smoothly and continuously from one surface to the other. These materials are micro-scopically inhomogeneous and are typically made from isotropic components. One of the main advantages ofFGMs is that they mitigate severe stress concentrations and singularities typically exhibited by composites atinterfaces of lamina due to abrupt transitions in material compositions and properties. Applications of FGMsare extensive especially in high-temperature environments such as nuclear reactors, chemical plants and high-speed spacecrafts.

The materials in the bottom and top surfaces are usually metal and ceramic, respectively (Fig. 2). Materialproperties at a point X are given by a combination between metal and ceramic constituents, i.e. by theweighted average of the moduli of the constituents, namely

-ðh3Þ ¼ -cfc þ -mfm ð16Þ

where the subscripts m and c refer to the metal and ceramic constituents and f is the volume fraction of thephase. The symbol - denotes a generic material property.

The volume fractions of the ceramic, fc, and metal, fm, corresponding to the power law are expressed as(Reddy, 2000; Praveen and Reddy, 1998; Reddy and Chin, 1998)

fc ¼zhþ 1

2

� �n

fm ¼ 1� fc

ð17Þ

where n is the volume fraction exponent which takes values greater than or equal to zero. The value of n equalto zero represents a fully ceramic shell. Conversely, we have a fully metal shell as n tends to infinity (Fig. 3).

The components of the elasticity tensor Cijkl(h3) are functions of the thickness coordinate. They can bewritten in terms of the convected base vectors as

C ¼ Cijklðh3Þgi � gj � gk � gl ð18Þ

which can be arranged in a matrix ½Cijkl� 2M6�6 such that

3θFully ceramic shell

Fully metal shell

1θ

2θ

Fig. 2. Functionally graded shell.

c f

0.0

0.2

0.4

0.6

0.8

1.0

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

n = 0.1n = 0.2

n = 0.5

n = 1

n = 2n = 5

n = 10

0.01,0.02,0.05n =

n = 20,50,100

3 hθ

Fig. 3. Variation of the volume fraction function fc through the thickness for different values of power-law index n.


½Cijkl� ¼

C1111 C1122 C1133 0 0 0

C1122 C2222 C2233 0 0 0

C1133 C2233 C3333 0 0 0

0 0 0 C2323 0 0

0 0 0 0 C1313 0

0 0 0 0 0 C1212

2666666664

37777777756�6

ð19Þ

The components Cijkl at each h3 are functions of only two independent variables. then

C1111 ¼ C2222 ¼ C3333 ¼ Eðh3Þð1� mÞð1þ mÞð1� 2mÞ

C1122 ¼ C1133 ¼ C2233 ¼ Eðh3Þmð1þ mÞð1� 2mÞ

C1212 ¼ C1313 ¼ C2323 ¼ Eðh3Þ2ð1þ mÞ

ð20Þ

where E(h3) = Ecfc + Emfm. The Poisson’s ratio m is considered constant through the thickness. Hence

Cijklðh3Þ ¼ Cijklc fc þ Cijkl

m fm ¼ Cijklcm fc þ Cijkl

m ð21Þ
where Cijkl
cm ¼ Cijklc � Cijkl

m and fc, fm are given in (17).

4. Weak formulation

The finite element framework is based on the principle of virtual work. Our analysis is restricted to staticcases. The virtual work statement is nothing but the weak form of the equilibrium equations and it is valid forlinear and nonlinear stress–strain relations (Reddy, 2002).

The abstract configuration solution of the shell is denoted by the set

C ¼ fU ðu;u;wÞjU : A 2 R2 ! R3 � R3 � Rg ð22Þ


where A is the parametric space of the midsurface. Note that U 2 C contains the same amount of three-di-mensional information as Eq. (2) to locate any arbitrary point in the three-dimensional shell at any time.

We express the weak formulation as

GðU; dUÞ ¼ GintðU; dUÞ þ GextðU; dUÞ ¼ 0 ð23Þ
where
GintðU; dUÞ ¼ZMR

ðM0 � de0 þM1 � de1ÞdX ð24Þ

GextðU; dUÞ ¼ �ZMR

ðp � duþ l � duþ k � dwÞdX�Z

oMR

ðps � duþ ls � duþ ks � dwÞds ð25Þ

For hyperelastic materials, the static part of the weak form of the equilibrium equations is the first variationof an elastic potential energy function. This statement is known as the principle of minimum total potentialenergy (Reddy, 2002). We define the elastic potential function Pð�Þ : C! R as

PðUÞ ¼ZBR

q0WdV �ZMR

ðp � uþ l � uþ k � wÞdX�Z

oMR

ðps � uþ ls � uþ ks � wÞds ð26Þ

The first variation of the potential energy is given by

GðU; dUÞ ¼ dPðU; dUÞ ¼ DPðUÞ½dU� ¼ 0 ð27Þ
The nonlinear equations are solved by the incremental/iterative method of Newton–Raphson. This proce-
dure requires a linearization of the weak form to generate a recurrence update formulas. The linearization pro-cess relies on the concept of directional derivatives (Hughes and Pister, 1978; Bonet and Wood, 1997). Weassume that the external forces are conservative (independent of U). Applying that procedure to Eq. (23)we obtain

LGðU; dU; DUÞ ¼ GðU; dUÞþDGðU; dUÞ½DU� þ oðDUÞ ð28Þ

where the underlined term is called consistent tangent operator. Furthermore, we can write the tangent oper-ator as

DGðU; dUÞ½DU� ¼ rGðU; dUÞ � DU ð29Þ
since dU remains constant during the increment DU.
The iterative solution procedure goes as follows: given a configuration Uk 2 C, corresponding to iteration k,solve the linearized system

GðUk; dUÞ þ rGðUk; dUÞ � DUk ¼ 0 ð30Þ
where DUk is the incremental change in the configuration of the shell. This increment is used to update theshell configuration Uk! Uk+1 2 C. Namely
Ukþ1 ¼ Uk þ DUk ð31Þ
Notice that the use of the triple (u,u,w) preserves the additive structure of the configuration update of theshell.
The consistent tangent operator is decomposed in two parts: the material tangent operator and the geomet-ric tangent operator. Thus

DGðU; dUÞ½DU� ¼ DmGðU; dUÞ½DU� þDgGðU; dUÞ½DU� ð32Þ

The contribution of the external forces vanishes because they are conservative. The first term which is thematerial part is given by

DmGðU; dUÞ½DU� ¼ZMR

X1

n¼0

ðDMn½DU� � denÞdX ð33Þ

and the geometric part by

TableNumb

Elemen

Q4Q9Q25Q49Q81


DgGðU; dUÞ½DU� ¼ZMR

X1

n¼0

ðMn �Dden½DU�ÞdX ð34Þ

The material part of the tangent operator results from the directional derivative of the stress resultants.After some manipulations we obtain

DMiðUÞ½DU� ¼X1

j¼0

Z h=2

�h=2

lðh3Þiþj bC � Dej dh3 ð35Þ

where bC is the pull-back of the contravariant fourth-order elasticity tensor C. Substituting (35) into (33) wearrive at

DmGðU; dUÞ½DU� ¼ZMR

X1

i¼0

X1

j¼0

ðdei � BðiþjÞ � DejÞdX; ð36Þ

where Dej can be easily calculated. The components of the fourth-order tensor BðkÞ are the material stiffnesscoefficients of the shell and are defined as

BðkÞ ¼Z h=2

�h=2

lðh3Þk bC dh3; k ¼ 0; 1; 2 ð37Þ

and are computed by Gauss integration using 50 points across the thickness.The computation of the virtual internal energy Gint and the tangent operator is not a trivial task. Even for

isotropic materials these expressions have an extremely complex form when displacements and rotations arelarge.

Next, the finite element equations are obtained by interpolating the covariant components of the kinematicvariables in terms of the base vectors aa. Namely

uhpðhÞ ¼Xm

j¼1

uðjÞi N ðjÞðn; gÞ !

ai; uhpðhÞ ¼Xm

j¼1

uðjÞi N ðjÞðn; gÞ !

ai

whpðhÞ ¼Xm

j¼1

wðjÞ3 N ðjÞðn; gÞ !

a3

ð38Þ

where ðuðjÞi ;uðjÞi ;w

ðjÞ3 Þ denote the nodal values of the kinematic variables.

We then arrive at a system of highly nonlinear algebraic equations which can be written in matrix form bymeans of the stiffness and tangent matrices. The solution is carried out by subroutines written in FORTRAN.

5. Numerical examples

In this section, numerical results obtained by the model developed herein are presented for shell structures.Typical benchmark shell problems are investigated using functionally graded materials.

Regular meshes of Q25, Q49 and Q81 high-order elements with seven degrees of freedom per node wereutilized in the finite element analysis (see Table 1). By increasing the p level or refining the finite element mesh,we mitigate locking problems. A full Gauss integration rule is employed in all examples.

1er of degrees of freedom per element for different p levels

t p level FSDT (DOF)

1 282 634 1756 3438 567

h

L

b

M

1θ

3θ

2θ

Fig. 4. Cantilever FGM plate strip under end bending moment.


5.1. Roll-up of a functionally graded plate strip

We consider a cantilevered FGM plate strip subjected to a bending distributed moment on the other end(Fig. 4). The isotropic and homogeneous counterpart has been considered Simo et al. (1990) as well as Betschet al. (1998). This problem is good for testing the capability of the finite element model to simulate large rota-tions on shells.

The material properties for the ceramic and metal constituents are

Em ¼ 0:7� 109 Pa; Ec ¼ 1:51� 109 Pa; m ¼ 0:3

These values will be used in all examples. For all quantities in this paper we utilize the international unit sys-tem. The geometry of the plate is

L ¼ 12:0; b ¼ 1:0; h ¼ 0:1

MREF ¼ 65; 886:17926

Figs. 5 and 6 depict tip displacements of the cantilever strip plate vs. the end bending moment for variousvolume fraction exponents n (from fully ceramic to fully metal). We utilize a regular mesh of 1 · 8Q25 ele-ments for the finite element discretization. The Newton–Raphson method exhibits a good rate of convergenceuntil some displacement level and then it diverges (for inhomogeneous shell cases). It is not clear to the authorswhy this problem occurs. It seems that for these cases we do not have real solutions. However, before arrivingat any conclusion further studies are needed.

RE

FM

M

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-2 0 2 4 6 8 10 12 14 16Deflections at the tip

Ceramic

n = 0.2

n = 0.5

n = 1.0

n = 2.0

n = 5.0

Metal

Fig. 5. Tip-deflection �uh2i vs. end moment M for the FGM plate strip.

Y

Z

X

Fig. 7. Deformed configurations of an FGM plate (n = 1.0) under end bending moment (load values M/MREF =0.075,0.15, . . . , 0.6,0.625).

q

1θ

B

Ri

Re

h

A

3θ

2θ

Fig. 8. Annular FGM plate strip under transverse end shear force.

RE

FM

M

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12 14 16Deflections at the tip

Ceramic

n = 0.2

n = 0.5

n = 1.0

n = 2.0

n = 5.0

Metal

Fig. 6. Tip-deflection uh3i vs. end moment M for the FGM plate strip.



Fig. 7 shows the undeformed and deformed configuration of a FGM strip plate for various load stages andn = 1.0. The plate shows large rotations beyond 180� with deformed configurations similar to the homoge-neous case.

5.2. Annular FGM plate under end shear force

We analyze an annular FGM plate subjected to a distributed transverse shear force (Fig. 8). This bench-mark problem was considered for homogeneous and isotropic plates by Buchter and Ramm (1992) and

F

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10 12

Deflection at point A

Ceramic

n = 0.2

n = 0.5

n = 1.0

n = 2.0

n = 5.0

Metal

Fig. 9. Transverse displacement curves at point A vs. shear force F = 4q of the cantilever annular plate strip.

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10 12 14 16

Deflection at point B

Ceramic

n = 0.2

n = 0.5

n = 1.0

n = 2.0

n = 5.0

Metal

F

Fig. 10. Transverse displacement curves at point B vs. shear force F = 4q of the cantilever annular plate strip.


Sansour and Kollmann (2000); and for multilayered composites by Arciniega and Reddy (in press). The geo-metric quantities are given by

Fig.

Ri ¼ 6; Re ¼ 10; h ¼ 0:03

for a maximum distributed force of qmax = 20.0.The plate is modeled by polar coordinates. A regular mesh of 1 · 5Q49 elements (p level equal to 6) is used

in the present analysis. Computation is performed by the Newton–Raphson method with 80 load steps andconvergence tolerance for the residual forces of 1.0 · 10�4.

The shear load vs. displacement curves for two characteristic points are depicted in Figs. 9 and 10. Thedeformed configurations of a FGM annular plate for various load levels and n = 2.0 is shown in Fig. 11. Itis clear that the plate undergoes large displacements at the corresponding load of F = 80.

5.3. Pull-out of a functionally graded cylindrical shell

The functionally graded cylindrical shell with free ends is subjected to two opposite loads (Fig. 12). Thehomogeneous case was considered by Brank et al. (1995) and Sansour and Kollmann (2000), among others.The following geometrical data is used in the analysis

L ¼ 10:35; R ¼ 4:953; h ¼ 0:094

X Y

Z

11. Deformed configurations of a FGM plate strip (n = 2.0) under transverse end shear force (load values F = 10,20, . . . , 80).

/2

h

R

A

D

C

B

P

Free edges

P

1θ2θ

3θ

L

Fig. 12. Pull-out of a FGM cylinder with free edges.

0

50

100

150

200

250

300

350

400

450

500

Pulli

ng f

orce

at A

(x1

0000

)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Radial deflection at point A

Ceramic

n = 0.2

n = 0.5

n = 1.0

n = 2.0

n = 5.0

Metal

Fig. 13. Radial displacements at point A (uh3i) vs. pulling force of a FGM cylinder with free edges.

0

50

100

150

200

250

300

350

400

450

500

Pulli

ng f

orce

at A

(x1

0000

)

0.0 1.0 2.0 3.0 4.0 5.0

Radial deflection at point B

Ceramic

n = 0.2

n = 0.5

n = 1.0

n = 2.0

n = 5.0

Metal

Fig. 14. Radial displacements at point B (�uh3i) vs. pulling force of a FGM cylinder with free edges.


An octant of the shell is modeled using 2 · 2 Q81 elements which is enough to overcome locking problems.The Newton–Raphson method with 80 load steps is utilized with equal load steps of 60,000. The adopted errortolerance for the residual was 1.0 · 10�5.

Figs. 13–15 show the radial displacements at points A, B and C of the shell, respectively. Convergence ratesfor this example are quite good (3–5 iterations per load step). As expected, the bending response of FGM cyl-inders lies in between of the fully ceramic and fully metal shells. The deformed configurations for a FGMcylindrical shell is depicted in Fig. 16 for P = 5.1 · 106 and n = 1.0.

0

50

100

150

200

250

300

350

400

450

500

Pulli

ng f

orce

at A

(x1

0000

)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Radial deflection at point C

Ceramic

n = 0.2

n = 0.5

n = 1.0

n = 2.0

n = 5.0

Metal

Fig. 15. Radial displacements at point C (�uh3i) vs. pulling force of a FGM cylinder with free edges.

X Y

ZP

P

X Y

Z

Fig. 16. Deformed configurations of the FGM cylinder under pulling forces. Load P = 5.1 · 106 (n = 1.0).

/2

h

R A

B

C

D

Fixed

1θ2θ

3θ

q

a

Fig. 17. FGM cylindrical shell under internal pressure.


Q

0

10

20

30

40

50

60

70

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Radial deflection at point A

Ceramic

n = 0.2

n = 0.5

n = 1.0

n = 2.0

n = 5.0

Metal

Fig. 18. Radial deflection at A vs. pressure load (Q = 106q) of a FGM cylindrical shell.

X Y

Z

X Y

Z

Fig. 19. Deformed configuration of a FGM cylindrical shell. Loading q = 3.6 · 106 (n = 5.0).


5.4. FGM cylinder under internal pressure

The last example considered is a cylindrical FGM shell under internal pressure (Fig. 17). This is not a fol-lowing load (independent of the displacements). The cylinder has fixed boundary conditions on both ends. Thegeometric data is as follows:

a ¼ 20:0; R ¼ 5:0; h ¼ 0:01

A regular mesh of 2 · 2Q81 elements is used in the analysis. We take advantage of the symmetry of the shelland only an octant of the shell is considered as the computational domain. Fig. 18 shows the radial deflectionsat the central point vs. the internal pressure for FGM cylinders. We notice that FGM cylinders with low valuesof n exhibit stiffer response than those with high volume fraction exponent (more metal than ceramic). We alsoobserve good rate of convergence of the Newton–Raphson scheme below some load level. When it reachesthat level of deformation, (similar to the first example) it diverges. The final configuration of a FGM cylinderfor n = 5.0 is depicted in Fig. 19.


6. Conclusions

In this paper we presented a large deformation analysis for functionally graded shells. A through-the-thick-ness variation of the material properties of the FGM shell made of two isotropic constituents, namely ceramicand metal. The gradation of properties through the thickness was assumed to be of the power law type. Atensor-based finite element model is developed for geometric nonlinear analysis of the shell. A first-order shelltheory with seven parameters with exact nonlinear deformations is used. The model incorporates thicknesschanges and utilizes the full three-dimensional constitutive equations. A family of high-order Lagrangianinterpolations was used to avoid membrane and shear locking for shells. This approach showed to be reliableand efficient. We found that the nonlinear bending response of FGM shells under mechanical loading liesbetween that of ceramic and metal shells, as expected. The patterns of load–displacement equilibrium curvesof FGM shells are similar to those of isotropic and homogeneous counterparts. Numerical examples for platesand cylindrical shells, presented herein, illustrate the validity of the present approach and the developed for-mulation for FGM shells. Extension of the study to thermally loaded shells is awaiting.

Acknowledgement

The research results reported herein were supported by the Structural Dynamics Program of the Army Re-search Office (ARO) through Grant 45508-EG.

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Documents

Large deformation analysis of functionally graded shells · R.A. Arciniega, J.N. Reddy * Texas A&M University, College Station, TX 77843-3123, USA Received 1 March 2006; received