analysis of fgm beams by means of a uni ed formulation

Analysis of FGM beams by means of a unified

formulation

G Giunta1, S Belouettar1 and E Carrera2

1 Centre de Recherche Public Henri Tudor, 29, avenue John F. Kennedy, L-1855,Luxembourg-Kirchberg, Luxembourg.2 Politecnico di Torino, 24, c.so Duca degli Abruzzi, 10129, Turin, Italy.

E-mail: [email protected], [email protected],

[email protected]

Abstract. This paper proposes several axiomatic refined theories for the linear staticanalysis of beams made of functionally graded materials. A bi-directional variationupon the cross-section is accounted for. Via a unified formulation, a generic N-orderapproximation is assumed for the displacement unknown variables over the beam cross-section. The governing differential equations and the boundary conditions are derivedin terms of a fundamental nucleo that does not depend upon the approximation order.A Navier type, closed form solution is adopted. Beams undergo bending and torsionalloadings. Deep beams are investigated. Comparisons with three-dimensional finiteelement models are given. The numerical investigation shows that the proposed unifiedformulation yields the complete three-dimensional displacement and stress fields as longas the appropriate approximation order is considered.

1. IntroductionFunctionally Graded Materials (FGMs) are composed of two or more materials whosevolume fraction changes gradually along desired spatial directions resulting in a smoothand continuous change in the effective properties. This allows a tailored material designthat broadens the structural design space by implementing a multi-functional responsewith a minimal weight increase. Also, the study of beam structures represents animportant topic of research since many primary and secondary structural elements,such as aircraft wings, helicopter rotor blades or robot arms can be idealised as beams.A general account of FGMs (design, fabrication and applications) can be found inSuresh and Mortensen [1], Miyamoto et al. [2] and Watanabe et al. [3]. Regardingthe modelling of FGMs-based beams, Chakraborty et al. [4] developed a finite elementbased on Timoshenko’s beam theory in which the shape functions have been derivedfrom the general exact solution of the static governing equations. Exponential andpolynomial gradation of mechanical and thermal properties along the through-the-

WCCM/APCOM 2010 IOP PublishingIOP Conf. Series: Materials Science and Engineering 10 (2010) 012073 doi:10.1088/1757-899X/10/1/012073

c© 2010 Published under licence by IOP Publishing Ltd 1

thickness direction were accounted for. Static, free vibration and wave propagationanalyses were carried out. Li [5] proposed a unified approach to the formulation ofTimoshenko’s and Euler-Bernoulli’s beam models. Young’s modulus varied along thetransverse coordinate according to a power-law function. Kadoli et al. [6] proposeda fined element based on a third-order approximation of the axial displacement andconstant transverse displacement for the static analysis of beams made of metal-ceramicFGMs. Components’ volume fraction was supposed to vary according to a power-lawfunction. A discrete layer approach was adopted to account for material gradation. Asfar as elasticity solutions are concerned, Sankar [7] solved the plane elasticity equationsexactly. An Euler-Bernoulli type theory was also derived. The shear stress componentwas obtained via integration of the indefinite equilibrium equations. Simply supported,FGM beams subjected to sinusoidal loadings were investigated. Young’s modulus wassupposed to vary exponentially along the thickness direction, while Poisson’s ratio wasconstant. Zhu and Sankar [8] adopted a combined Fourier series-Galerkin method forthe analysis of simply supported FGM-based beams for which the Young’s moduluswas a third-order polynomial function of the through-the-thickness coordinate. Thepresent paper proposes a systemic manner of formulating axiomatically refined beammodels accounting for material gradation above the cross-section. Via a concise notationfor the kinematic field, the governing differential equations and the correspondingboundary conditions are reduced to a “fundamental nucleo” in terms of the displacementcomponents. The fundamental nucleo does not depend upon the approximation order.This is, therefore, assumed as free parameter of the formulation. This formulation isnamed Carrera’s Unified Formulation and it was previously applied to the modellingof isotropic beam structures (see Carrera and Giunta [9] and Carrera et al. [10]).Displacement-based theories that account for non-classical effects, such as transverseshear and in- and out-of-plane warping of the cross-section, can be formulated. It isworth mentioning that no special warping functions need to be assumed. Navier’s,closed form solution is adopted to solve the governing differential equation. Slender anddeep beams are investigated. Isotropic FGMs are accounted for. Young’s modulus isconsidered to vary exponentially along two perpendicular directions on the cross-section.Poisson’s ration is constant, since its effect on the deformation is much less than thatof Young’s modulus (see Delale and Erdogan [11]). The proposed models are validatedtowards three-dimensional finite element models that have been appositely developed.

2. PreliminariesCross-sections, Ω, that are obtainable as union of NΩk rectangular sub-domains:

Ω =N

Ωk

∪k=1

Ωk (1)

with:Ωk =

(y, z) : yk1 ≤ y ≤ yk2 ; z

k1 ≤ z ≤ zk2

(2)

are considered. Terms(

yki , zkj

): i, j = 1, 2

are the coordinates of the corner points of

a k sub-domain. Superscript ‘k’ represents the cross-section sub-domain index, while, as


2

subscript, it stands for summation over the range (1, NΩk). A Cartesian reference systemis adopted: y- and z-axis are two orthogonal directions laying on the cross-section. Thex coordinate is coincident to the axis of the beam. It is bounded such that 0 ≤ x ≤ l,being l the axial extension of the beam. The displacement field is:

uT (x, y, z) =

ux (x, y, z) uy (x, y, z) uz (x, y, z)

(3)

in which ux, uy, and uz are the displacement components along x-, y- and z-axes.Superscript ‘T ’ represents the transposition operator. The stress, σ, and strain, ε,vectors are grouped into vectors σn, εn that act on the cross-section:

σTn =

σxx σxy σxz

,

εTn =

εxx εxy εxz (4)

and σp, εp acting on planes orthogonal to Ω:

σTp =

σyy σzz σyz

,

εTp =

εyy εzz εyz (5)

Linear strain-displacement relations are accounted for:

εn = Dnpu+Dnxu, εp = Dpu (6)

where Dnp, Dnx, and Dp are the following differential matrix operators:

Dnp =

0 0 0

∂

∂y0 0

∂

∂z0 0

, Dnx = I

∂

∂x, Dp =

0∂

∂y0

0 0∂

∂z

0∂

∂z

∂

∂y

(7)

and I is the unit matrix. Under the hypothesis of isotropic linear elastic FGMs, thegeneralised Hooke law holds. According to Eqs. (4) and (5), it reads:

σp = Cpp (y, z) εp +Cpn (y, z) εn,σn = Cnp (y, z) εp +Cnn (y, z) εn

(8)

Young’s modulus, E, is supposed to vary versus y and z according to the followingexponential law:

E (y, z) = E0e(α1y+β1)e(α2z+β2) (9)

Poisson’s ratio, ν, is considered to be constant. Matrixes Cpp, Cnn, Cpn and Cnp inEqs. (8) are:

Cpp =

C22 C23 0C23 C33 00 0 C44

e(α1y+α2z), Cnn =

C11 0 00 C66 00 0 C55

e(α1y+α2z),

Cpn = CTnp =

C12 0 0C13 0 00 0 0

e(α1y+α2z)

(10)


3

where the constant stiffness coefficients Cij are:

Cii =1− ν

(1 + ν) (1− 2ν)E0e

(β1+β2) i = 1, 2, 3

Cjj =1

2 (1 + ν)E0e

(β1+β2) j = 4, 5, 6

Cij =ν

(1 + ν) (1− 2ν)E0e

(β1+β2) i = 1, 2 j = i+ 1, 3

(11)

3. Hierarchical Beam TheoriesThe variation of the displacement field towards y- and z-axis is a-priori postulated inthe following manner:

u (x, y, z) = Fτ (y, z)uτ (x) with τ = 1, 2, . . . , Nu (12)

Nu stands for the number of unknowns and depends upon the theory approximationorder,N , that is a free parameter of the formulation. Repeated indexes mean summation.Problem’s governing differential equations and boundary conditions can be derived ina unified manner in terms of a single ‘fundamental nucleo’. The complexity related tohigher than classical approximation terms is tackled and the theoretical formulationis valid for the generic approximation order and approximating functions Fτ (y, z).In this paper, Fτ are Mac Laurin’s polynomials functions, see table 1. The actual

Table 1. Mac Laurin’s polynomials via Pascal’s triangle.

N Nu Fτ

0 1 F1 = 11 3 F2 = y F3 = z2 6 F4 = y2 F5 = yz F6 = z2

3 10 F7 = y3 F8 = y2z F9 = yz2 F10 = z3

. . . . . . . . .

N(N + 1) (N + 2)

2F (N2+N+2)

2

= yN F (N2+N+4)2

= yN−1z . . . FN(N+3)2

= yzN−1

F (N+1)(N+2)2

= zN

governing differential equations and boundary conditions due to a fixed approximationorder and polynomials’ type are obtained straightforwardly via summation of the nucleocorresponding to each term of the expansion. The generic, N -order displacement fieldis:

ux = ux1 + ux2y + ux3z + · · ·+ ux(N2+N+2)

2

yN + · · · + ux

(N+1)(N+2)2

zN

uy = uy1 + uy2y + uy3z + · · ·+ uy(N2+N+2)

2

yN + · · ·+ uy (N+1)(N+2)

2

zN

uz = uz1 + uz2y + uz3z + · · ·+ uz(N2+N+2)

2

yN + · · ·+ uz (N+1)(N+2)

2

zN(13)


4

Higher order models yield a more detailed description of the shear mechanics (no shearcorrection coefficient is required), the transverse to the section deformations, the couplingof the spatial directions due to Poisson’s effect and the torsional mechanics than classicalmodels do. Classical beam models, such as Timoshenko’s and Euler-Bernoulli’s can bestraightforwardly derived from the first-order approximation model. For more details,refer to Carrera and Giunta [9].

4. Governing EquationsThe governing differential equations and the boundary conditions are obtained in termsof the displacement components through the Principle of Virtual Displacements:

δLi = δLe (14)

Li represents the strain energy. Le stands for the external work. δ stands for a virtualvariation. By substitution of the geometrical relations, Eqs. (6), the material constitutiveequations, Eqs. (8), and the unified hierarchical approximation of the displacements,Eq. (12), and after integration by parts, the virtual strain energy becomes:

δLi =

∫

l

δuTτ Kτs us dx+

[δuT

τ Πτs us

]x=l

x=0(15)

The components of the differential matrix Kτs are:

Kτsxx =

(J66kτ,ys,y + J55k

τ,zs,z − J11kτs

∂2

∂x2

)

k

,

Kτsyy =

(J22kτ,ys,y + J44k

τ,zs,z − J66kτs

∂2

∂x2

)

k

,

Kτszz =

(J44kτ,ys,y + J33k

τ,zs,z − J55kτs

∂2

∂x2

)

k

,

Kτsxy =

(−J12k

τs,y + J66kτ,ys

)k

∂

∂x, Kτs

yx =(J12kτ,ys − J66k

τs,y

)k

∂

∂x,

Kτsxz =

(−J13k

τs,z + J55kτ,zs

)k

∂

∂x, Kτs

zx =(J13kτ,zs + J55k

τs,z

)k

∂

∂x,

Kτsyz =

(J23kτ,ys,z + J44k

τ,zs,y

)k, Kτs

zy =(J23kτ,zs,y + J44k

τ,ys,z

)k

(16)

Subscripts preceded by comma represent derivation versus the corresponding spatial

coordinates. The generic term Jghkτ,(φ)s,(ξ) with φ, ξ = y, z and g, h = 1, 2, . . . , 6 is the cross-

section inertial momentum of a k sub-domain accounting for the material gradation:

Jghkτ,(φ)s,(ξ)

=

∫

Ωk

Ckghe

(α1y+α2z)Fτ,(φ)F,(ξ) dΩ (17)


5

As far as the boundary conditions are concerned, the components of Πτs are:

Πτsxx =

(J11kτs

)k

∂

∂x, Πτs

xy =(J12kτs,y

)k, Πτs

xz =(J13kτs,z

)k,

Πτsyy =

(J66kτs

)k

∂

∂x, Πτs

yx =(J66kτs,y

)k, Πτs

yz = 0,

Πτszz =

(J55kτs

)k

∂

∂x, Πτs

zx =(J55kτs,z

)k, Πτs

zy = 0

(18)

The virtual work done by the external loadings is obtained in a similar manner. Forsake of brevity it is not reported here. For more details refer to Carrera and Giunta [9]The differential equations and related boundary conditions are solved via a Navier typesolution:

(uxτ , uyτ , uzτ ) = Fτ

(Uxτ cos

(mπ

lx), Uyτ sin

(mπ

lx), Uzτ sin

(mπ

lx))

(19)

upon assumption that the external loadings vary towards x in a similar manner. In thecase of a pressure loading pyy, for instance:

pyy = Pyy sin(mπ

lx)

(20)

where m represents the half-wave number along the beam axis. Uiτ : i = x, y, z are themaximal amplitudes of the displacement components and Pyy the maximal amplitude ofthe surface loading.

5. Numerical ResultsA square cross-section is considered, see Fig. 1. Dimension a along y and z axes equals 0.1m. Deep beams (l/a = 5) are investigated. A unit maximal amplitude (1 MPa) for thesurface loading is assumed. The material exhibits a gradation along y and z directionsboth: E0 = 1000 MPa, αi such that E (a, 0) /E0 = E (0, a) /E0 = 3 and βi = 0 with i = 1and 2, see Fig. 2. Poisson’s ratio equals 0.3. The proposed models are compared witha three-dimensional FEM solution (addressed as FEM 3D) developed via MSC.Nastrancommercial code. The eight-node brick element “HEXA8” is used (see MSC.NastranUser’s Guide [12]). Elements’ sides measure 2 · 10−3 m. Each element is considered ashomogeneous by referring to the material properties at its centre point. Fig. 3 presentsthe cross-section deformation at x/l = 0.5. Fourth-order model matches the referenceFEM solution. The shape of the deformation is due to the Young modulus variation law:points at z/a = 1 are stiffer than those at z/a = 0. Beam undergoes bending and torsion.A first-order model yields an accurate description of the bending stress component σxxas shown in Figs. 4 and 5, where results are presented in the form of colour maps. Resultsare computed at beam mid-span. Normal stress component σyy computed via FEM 3Dsolution and fifth-order model is presented in Figs. 6 and 7. A good approximationof the normal stress component σzz is obtained via a sixth-order model as shown inFigs. 8 and 9. Shear stress component σxz via FEM 3D model and fifth-order theoryis presented in Figs. 10 and 11. Results are evaluated for x/l = 0. In Figs. 12 and 13,the colour maps of the shear stress component σxy are presented. A fifth-order model


6

Figure 1. Square cross-sectiongeometry and loading.

0a/2

a 0

a/2

a 1 3 5 7 9

E/E0

yz

E/E0

Figure 2. Young modulus variationalong the cross-section.

-2.8

-2.6

-2.4

-2.2

-2

-1.8

-1.6

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

(y +

10

0 x

uy)/

a

(z + 100 x uz)/a

FEM 3D

N=1

N=2

N=4

Figure 3. In-plane warping at x/l = 0.5.

has been used. As shown via the comparison with the reference FEM 3D solution, theproposed models yield an accurate description of the three-dimensional stress state. Asfar as the computational time is concerned, the proposed analytical models require lessthan a second, regardless the approximation order. The FEM solution based on theproposed models, not reported here, is obtained in few seconds for a very fine mesh. Forthe reference FEM 3D solution, the computational time is about five minutes.

6. ConclusionsA unified formulation for modelling of beam structures made of isotropic functionallygraded materials has been proposed. Higher order models that account for sheardeformations and in- and out-of-plane warping can be formulated straightforwardly.


7

0 0.2 0.4 0.6 0.8 1

z/a

0

0.2

0.4

0.6

0.8

1

y/a

-40

-30

-20

-10

0

10

20

Figure 4. σxx (MPa) at mid-span viaFEM 3D.

0 0.2 0.4 0.6 0.8 1

z/a

0

0.2

0.4

0.6

0.8

1

y/a

-40

-30

-20

-10

0

10

20

Figure 5. σxx (MPa) at mid-span viafirst-order model.

0 0.2 0.4 0.6 0.8 1

z/a

0

0.2

0.4

0.6

0.8

1

y/a

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

Figure 6. σyy (MPa) at mid-span viaFEM 3D.

0 0.2 0.4 0.6 0.8 1

z/a

0

0.2

0.4

0.6

0.8

1

y/a

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

Figure 7. σyy (MPa) at mid-span viafifth-order model.

Closed form, Navier type solution has been adopted. Deep beams under a bending-torsion loading have been investigated. Young modulus have been assumed to varyversus the cross-section coordinates according to an exponential law function. It hasbeen shown that the proposed formulation yields results as accurate as desired through


8

0 0.2 0.4 0.6 0.8 1

z/a

0

0.2

0.4

0.6

0.8

1

y/a

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Figure 8. σzz (MPa) at mid-span viaFEM 3D.

0 0.2 0.4 0.6 0.8 1

z/a

0

0.2

0.4

0.6

0.8

1

y/a

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Figure 9. σzz (MPa) at mid-span viasixth-order model.

0 0.2 0.4 0.6 0.8 1

z/a

0

0.2

0.4

0.6

0.8

1

y/a

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Figure 10. σxz (MPa) at x/l = 0 viaFEM 3D.

0 0.2 0.4 0.6 0.8 1

z/a

0

0.2

0.4

0.6

0.8

1

y/a

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Figure 11. σxz (MPa) at x/l = 0 viafifth-order model.

an appropriate choice of the approximation order. Higher order models match thereference three-dimensional FEM solution. The efficiency of the models is very high.The computational time is of the order of 10−1 seconds, while three-dimensional FEMmodels require several minutes. The assumption of a polynomial law function for the


9

0 0.2 0.4 0.6 0.8 1

z/a

0

0.2

0.4

0.6

0.8

1

y/a

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Figure 12. σxy (MPa) at x/l = 0 viaFEM 3D.

0 0.2 0.4 0.6 0.8 1

z/a

0

0.2

0.4

0.6

0.8

1

y/a

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Figure 13. σxy (MPa) at x/l = 0 viafifth-order model.

material properties will be matter of further investigation. In such a manner, a genericgradation law can be accounted for via its polynomial approximation.

AcknowledgmentsFirst and second authors are supported by the Ministere de la Culture, de l’EnseignementSuperieuer et de la Recherche of Luxembourg via the project FNR CORE 2009C09/MS/05 FUNCTIONALLY and MATERA FNR ADYMA, respectively.

References[1] Suresh S and Mortensen A 1998 Fundamentals of Functional Graded Materials (IOM

Communications Limited, London, UK)[2] Miyamoto Y, Kaysser W A, Rabin B H, Kawasaki A and Ford R G 1999 Functionally Graded

Materials: Design, Processing and Applications (Kluwer Academic, Boston, MA)[3] Watanabe R, Nishida T and Hirai T 2003 Metals and Materials International 9 513–519[4] Chakraborty A, Gopalakrishnan S and Reddy J N 2003 International Journal of Mechanical Sciences

45 519–539[5] Li X F 2008 Journal of Sound and Vibration 318 1210–1229[6] Kadoli R, Akhtara K and Ganesanb N 2008 Applied Mathematical Modelling 32 2509–2525[7] Sankar B V 2001 Composites Science and Technology 61 689–696[8] Zhu H and Sankar B 2004 Journal of Applied Mechanics 71 421–424[9] Carrera E and Giunta G 2010 International Journal of Applied Mechanics 2

[10] Carrera E, Giunta G, Nali P and Petrolo M 2010 Computers and Structures 88 283–293[11] Delale F and Erdogan F 1983 Journal of Applied Mechanics 50 609–614[12] MSC.Nastran 2002 MSC.Nastran 2001, Getting Started with MSC.Nastran, User’s Guide


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analysis of fgm beams by means of a uni ed formulation

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