A thermo-mechanical analysis of functionally graded beams via hierarchical modelling G. Giunta a,⇑ , D. Crisafulli a,b , S. Belouettar a , E. Carrera b a Centre de Recherche Public Henri Tudor, 29, Av. John F. Kennedy, L-1855 Luxembourg-Kirchberg, Luxembourg b Politecnico di Torino, 24, c.so Duca degli Abruzzi, 10129 Turin, Italy article info Article history: Available online 17 August 2012 Keywords: Thermal loads Beam structures Functionally graded materials Unified formulation Closed form solution abstract In this work, functionally graded beams subjected to thermal loadings are analysed. To this end, several beam models are hierarchically derived by means of a unified formulation that makes the formulation independent from the displacements polynomial approximation order over the cross-section. The tem- perature profile is determined by solving Fourier’s heat conduction equation. The governing equations are, then, derived from the Principle of Virtual Displacements considering the temperature field as an external load. A Navier-type, closed form solution is used. Simply supported beams are, therefore, consid- ered. Functionally graded mono-layer and sandwich cross-section configurations are investigated. Numerical results in terms of temperature, displacement and stress distributions are provided for differ- ent beam slenderness ratios. Results are assessed towards three-dimensional finite element solutions demonstrating that accurate results can be obtained with reduced computational costs. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction Beam-like structures operating in severe temperature environ- ments are examples of typical aeronautical and space applications. High-temperature resistant materials are, therefore, required. Structures made of ceramic–metal Functionally Graded Materials (FGMs) are able to combine temperature resistance and a continu- ous stress distribution because of a smooth variation of material properties along some preferred directions. A brief overview of recent works about thermo-mechanical analysis of functionally graded structures is presented below. Sev- eral applications of the theory of thermo-elasticity can be found in the book by Hetnarski and Eslami [1]. In particular, the thermal stress analysis of beams based on Euler–Bernoulli assumptions was presented. Beams made of functionally graded materials were also investigated. The problem of thermal stresses in FGMs was ad- dressed by Noda [2]. The optimal gradation profiles to decrease the thermal stresses in FGMs were discussed. The thermoelastic behaviour of functionally graded beams was also studied by Chakr- aborty et al. [3]. A beam finite element based on Timoshenko’s the- ory was developed, accounting for an exponential and a power law through-the-thickness variation of elastic and thermal properties. A thermo-elastic stress analysis of multi-layered non-homoge- neous beams was carried out by Carpinteri and Paggi [4]. Analytical solutions were provided under Euler–Bernoulli’s kinematic hypotheses. Functionally graded materials were also investigated, considering a linear variation of material properties along the beam’s thickness. A meshless method for thermo-elastic analysis of functionally graded materials combined with radial basis func- tions was presented by Wang and Qin [5]. Mahi et al. [6] studied the free vibration of FGM beams subjected to initial thermal stress. Exact solutions based on several shear deformation theories were presented considering different boundary conditions. The temper- ature profile was computed solving a one-dimensional steady- state heat conduction equation. Thermal buckling and thermo- elastic vibration analysis of FGM beams were carried out by Watt- anasakulpong et al. [7] by means of a third-order shear deforma- tion theory. Material properties were considered to dependent on the temperature via a non-linear polynomial law [8]. The static re- sponse of functionally graded plates subjected to thermal loads was addressed by Brischetto et al. [9]. The temperature field was determined by solving Fourier’s equation. Different volume frac- tions of the material constituents were considered to evaluate the temperature, displacement and stress distributions. A thermal analysis of functionally graded beams via refined models is addressed in this paper. Models are derived via a Unified Formulation (UF) that has been previously proposed for plates and shells (see Carrera [10]) and extended to solid and composite beams (see Carrera et al. [11], Giunta et al. [12,13] and Catapano et al. [14]). In the proposed UF, the displacements’ polynomial 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.08.013 ⇑ Corresponding author. Address: Department of Advanced Materials and Struc- tures, Centre de Recherche Public Henri Tudor, 29, Av. John F. Kennedy, L-1855 Luxembourg-Kirchberg, Luxembourg. Tel.: +352 54 55 80 479; fax: +352 42 59 91 555. E-mail address: [email protected](G. Giunta). Composite Structures 95 (2013) 676–690 Contents lists available at SciVerse ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Transcript
Composite Structures 95 (2013) 676–690
Contents lists available at SciVerse ScienceDirect
A thermo-mechanical analysis of functionally graded beamsvia hierarchical modelling
G. Giunta a,⇑, D. Crisafulli a,b, S. Belouettar a, E. Carrera b
a Centre de Recherche Public Henri Tudor, 29, Av. John F. Kennedy, L-1855 Luxembourg-Kirchberg, Luxembourgb Politecnico di Torino, 24, c.so Duca degli Abruzzi, 10129 Turin, Italy
a r t i c l e i n f o
Article history:Available online 17 August 2012
Keywords:Thermal loadsBeam structuresFunctionally graded materialsUnified formulationClosed form solution
0263-8223/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.compstruct.2012.08.013
⇑ Corresponding author. Address: Department of Atures, Centre de Recherche Public Henri Tudor, 29,Luxembourg-Kirchberg, Luxembourg. Tel.: +352 54 5555.
In this work, functionally graded beams subjected to thermal loadings are analysed. To this end, severalbeam models are hierarchically derived by means of a unified formulation that makes the formulationindependent from the displacements polynomial approximation order over the cross-section. The tem-perature profile is determined by solving Fourier’s heat conduction equation. The governing equationsare, then, derived from the Principle of Virtual Displacements considering the temperature field as anexternal load. A Navier-type, closed form solution is used. Simply supported beams are, therefore, consid-ered. Functionally graded mono-layer and sandwich cross-section configurations are investigated.Numerical results in terms of temperature, displacement and stress distributions are provided for differ-ent beam slenderness ratios. Results are assessed towards three-dimensional finite element solutionsdemonstrating that accurate results can be obtained with reduced computational costs.
� 2012 Elsevier Ltd. All rights reserved.
1. Introduction
Beam-like structures operating in severe temperature environ-ments are examples of typical aeronautical and space applications.High-temperature resistant materials are, therefore, required.Structures made of ceramic–metal Functionally Graded Materials(FGMs) are able to combine temperature resistance and a continu-ous stress distribution because of a smooth variation of materialproperties along some preferred directions.
A brief overview of recent works about thermo-mechanicalanalysis of functionally graded structures is presented below. Sev-eral applications of the theory of thermo-elasticity can be found inthe book by Hetnarski and Eslami [1]. In particular, the thermalstress analysis of beams based on Euler–Bernoulli assumptionswas presented. Beams made of functionally graded materials werealso investigated. The problem of thermal stresses in FGMs was ad-dressed by Noda [2]. The optimal gradation profiles to decrease thethermal stresses in FGMs were discussed. The thermoelasticbehaviour of functionally graded beams was also studied by Chakr-aborty et al. [3]. A beam finite element based on Timoshenko’s the-ory was developed, accounting for an exponential and a power lawthrough-the-thickness variation of elastic and thermal properties.
ll rights reserved.
dvanced Materials and Struc-Av. John F. Kennedy, L-18555 80 479; fax: +352 42 59 91
a).
A thermo-elastic stress analysis of multi-layered non-homoge-neous beams was carried out by Carpinteri and Paggi [4]. Analyticalsolutions were provided under Euler–Bernoulli’s kinematichypotheses. Functionally graded materials were also investigated,considering a linear variation of material properties along thebeam’s thickness. A meshless method for thermo-elastic analysisof functionally graded materials combined with radial basis func-tions was presented by Wang and Qin [5]. Mahi et al. [6] studiedthe free vibration of FGM beams subjected to initial thermal stress.Exact solutions based on several shear deformation theories werepresented considering different boundary conditions. The temper-ature profile was computed solving a one-dimensional steady-state heat conduction equation. Thermal buckling and thermo-elastic vibration analysis of FGM beams were carried out by Watt-anasakulpong et al. [7] by means of a third-order shear deforma-tion theory. Material properties were considered to dependent onthe temperature via a non-linear polynomial law [8]. The static re-sponse of functionally graded plates subjected to thermal loadswas addressed by Brischetto et al. [9]. The temperature field wasdetermined by solving Fourier’s equation. Different volume frac-tions of the material constituents were considered to evaluatethe temperature, displacement and stress distributions.
A thermal analysis of functionally graded beams via refinedmodels is addressed in this paper. Models are derived via a UnifiedFormulation (UF) that has been previously proposed for plates andshells (see Carrera [10]) and extended to solid and compositebeams (see Carrera et al. [11], Giunta et al. [12,13] and Catapanoet al. [14]). In the proposed UF, the displacements’ polynomial
G. Giunta et al. / Composite Structures 95 (2013) 676–690 677
approximation over the beam cross-section is derived in a compactform. The governing equations variationally consistent with the as-sumed kinematic hypothesis are derived through the Principle ofVirtual Displacement (PVD) in terms of a fundamental nucleo. Thisnucleo does not depend upon the displacements order of expan-sion. As a result, a set of several kinematic models can be obtainedthat accounts for transverse shear deformations and cross-sectionin- and out-of-plane warping. The temperature field is obtainedby solving Fourier’s heat conduction equation. Governing differen-tial equations are solved via a Navier-type, closed form solution.Slender and deep beams are investigated. As far as the materialgradation along the beam cross-section is concerned, the materialgradation is considered by means of a Lagrange approximationbased upon a Newton series expansion upon a Chebyshev grid ofsampling points over the cross-section. In such a manner, the fun-damental nucleo is formulated in a general manner with respect tothe particular gradation law. The numerical investigations arecarried out considering a power law variation along the beamthickness direction of the elastic and thermal material properties.The proposed models are validated through comparison withthree-dimensional FEM solutions. Numerical results show thataccurate results can be obtained with small computational costs.
2. Preliminaries
A beam, see Fig. 1, is a structure whose axial extension (l) is pre-dominant if compared to any other dimension orthogonal to it. Thecross-section (X) is identified by intersecting the beam with planesthat are orthogonal to its axis. A Cartesian reference system isadopted: y- and z-axis are two orthogonal directions laying on X.The x coordinate is coincident to the axis of the beam. It is boundedsuch that 0 6 x 6 l. The cross-section is considered to be constantalong x. The displacement field is:
in which ux, uy and uz are the displacement components along x-, y-and z-axis. Superscript ‘T’ represents the transposition operator.Stress, r, and strain, e, vectors are grouped into vectors rn, en thatlay on the cross-section:
rTn ¼ rxx rxy rxzf g eT
n ¼ exx exy exzf g ð2Þ
and rp, ep laying on planes orthogonal to X:
rTp ¼ ryy rzz ryzf g eT
p ¼ eyy ezz eyzf g ð3Þ
Fig. 1. Beam structure and reference system.
Under the hypothesis of linear analysis, the following strain–dis-placement geometrical relations hold:
eTn ¼ ux;x ux;y þ uy;x ux;z þ uz;xf g
eTp ¼ uy;y uz;z uy;z þ uz;yf g
ð4Þ
Subscripts ‘x’, ‘y’ and ‘z’, when preceded by comma, represent deri-vation versus the corresponding spatial coordinate. A compact vec-torial notation can be adopted for Eq. (4):
en ¼ Dnpuþ Dnxu
ep ¼ Dpuð5Þ
where Dnp, Dnx and Dp are the following differential matrixoperators:
Dnp ¼
0 0 0@@y 0 0
@@z 0 0
2664
3775 Dnx ¼ I
@
@xDp ¼
0 @@y 0
0 0 @@z
0 @@z
@@y
26664
37775 ð6Þ
I is the unit matrix. In the case of thermo-mechanical problems, theconstitutive equations are:
rp ¼ rpe � rpt ¼ Cppep þ Cpnen � kpT
rn ¼ rne � rnt ¼ Cnpep þ Cnnen � knTð7Þ
where subscripts ‘e’ and ‘t’ refer to the elastic and the thermal con-tributions, respectively. For isotropic materials, the matrices Cpp,Cpn, Cnp and Cnn in Eq. (7) are:
Fig. 3. Mono-layer FGM beam temperature profile [K] at x/l = 1/2 via (a) Fourier’sequation solution and (b) FEM 3Da, l/a = 10.
a Mesh 30 � 30 � 30.b Mesh 20 � 20 � 20.c Scaling factor �105 (instead of 10�6).d Result not provided by the theory.
Fig. 4. Mono-layer FGM beam, uz [m] at x/l = 1/2 via (a) N = 4 and (b) FEM 3Da, l/a = 10.
G. Giunta et al. / Composite Structures 95 (2013) 676–690 679
A Lagrange approximation on Np Chebyshev points along y and zcross-section co-ordinates based on Newton series expansion isassumed for the material stiffness coefficients Cij and thermal coef-ficients ki:
and Cij[. . .; . . .] and ki[. . .; . . .] the divided difference of the approxi-mated function, see Philips [15]. Chebyshev’s points are definedon the domain [�1,+1] via the following equation:
fm ¼ cosmpNp
� �with m ¼ 0;1; . . . ;Np ð15Þ
These points are then mapped into the cross-section domain via avariable transformation. In this manner, the software implementa-tion of the proposed models is general and does not depend upon aspecific gradation law that, once defined, will be approximated via aNewton series expansion.
The beam models are derived considering the temperature (T)as an external loading resulting from the internal thermal stresses.This requires that the temperature profile is known over the wholebeam domain. Fourier’s heat conduction equation is solved in orderto obtain T. In order to obtain a strong form solution of the problemgoverning equations, the temperature is written as follows:
Tðx; y; zÞ ¼ HnðxÞHXðy; zÞ ð16Þ
The whole solution procedure is presented in Appendix A.
3. Hierarchical beam theories
The variation of the displacement field over the cross-section canbe postulated a priori. Several displacement-based theories can beformulated on the basis of the following generic kinematic field:
uðx; y; zÞ ¼ Fsðy; zÞusðxÞ with s ¼ 1;2; . . . ;Nu ð17Þ
where Nu stands for the number of unknowns. It depends on theapproximation order N that is a free parameter of the formulation.The compact expression is based on Einstein’s notation: a repeatedindex stands for summation. Thanks to this notation, problem’s gov-erning differential equations and boundary conditions can be de-rived in terms of a single ‘fundamental nucleo’. The complexityrelated to higher than classical approximation terms is tackled andthe theoretical formulation is valid for the generic approximation or-der and approximating functions Fs(y,z). In this paper, the functionsFs are assumed to be Mac Laurin’s polynomials. This choice is in-spired by the classical beam models. Nu and Fs as functions of Ncan be obtained via Pascal’s triangle as shown in Table 1. The actualgoverning differential equations and boundary conditions due to afixed approximation order and polynomials type are obtainedstraightforwardly via summation of the nucleo corresponding toeach term of the expansion. According to the previous choice ofthe polynomial functions, the generic, N-order displacement field is:
ux ¼ ux1 þ ux2yþ ux3zþ � � � þ uxðN
2þNþ2Þ2
yN þ � � � þ uxðNþ1ÞðNþ2Þ2
zN
uy ¼ uy1 þ uy2yþ uy3zþ � � � þ uyðN
2þNþ2Þ2
yN þ � � � þ uyðNþ1ÞðNþ2Þ2
zN
uz ¼ uz1 þ uz2yþ uz3zþ � � � þ uzðN
2þNþ2Þ2
yN þ � � � þ uzðNþ1ÞðNþ2Þ2
zN
ð18Þ
Fig. 5. Mono-layer FGM beam, ux [m] at x/l = 0 via (a) TBT, (b) N = 4 and (c) FEM 3Da,l/a = 10.
Fig. 6. Mono-layer FGM beam, uy [m] at x/l = 1/2 via (a) N = 4 and (b) FEM 3Da, l/a = 10.
680 G. Giunta et al. / Composite Structures 95 (2013) 676–690
As far as the first-order approximation order is concerned, the kine-matic field is:
ux ¼ ux1 þ ux2yþ ux3z
uy ¼ uy1 þ uy2yþ uy3z
uz ¼ uz1 þ uz2yþ uz3z
ð19Þ
Classical models, such as Timoshenko beam theory (TBT):
ux ¼ ux1 þ ux2yþ ux3z
uy ¼ uy1
uz ¼ uz1
ð20Þ
and Euler–Bernoulli beam theory (EBT):
ux ¼ ux1 � uy1;xy� uz1;xz
uy ¼ uy1
uz ¼ uz1
ð21Þ
are straightforwardly derived from the first-order approximationmodel. In TBT, no shear correction coefficient is considered, sinceit depends upon several parameters, such as the geometry of thecross-section (see, for instance, Cowper [16] and Murty [17]). High-er-order models yield a more detailed description of the shearmechanics (no shear correction coefficient is required), of the
G. Giunta et al. / Composite Structures 95 (2013) 676–690 681
in- and out-of-section deformations, of the coupling of the spatialdirections due to Poisson’s effect and of the torsional mechanicsthan classical models do. EBT theory neglects them all, since itwas formulated to describe a pure bending mechanics. TBT modelaccounts for constant shear stress and strain components. In thecase of classical models, the material stiffness coefficients shouldbe corrected in order to contrast a phenomenon known in literatureas Poisson’s locking (see Giunta et al. [18]).
4. Governing equations
The governing equations and the boundary conditions are de-rived through the PVD:
dLi ¼ 0 ð22Þ
where d stands for a virtual variation and Li represents the strain en-ergy. According to the grouping of the stress and strain componentsin Eqs. (2) and (3), the virtual variation of the strain energy for athermo-mechanical case is:
dLi ¼Z
l
ZX
d�Tnðrne � rntÞ þ d�T
pðrpe � rptÞh i
dXdx ð23Þ
By substitution of the geometrical relations, Eq. (5), the constitutiveequations, Eq. (7), and the unified hierarchical approximation of thedisplacements, Eq. (17), Eq. (23) becomes:
Fig. 7. Mono-layer FGM beam, rxx [Pa] at x/l = 1/2 via (a
) TBT, (b) N = 4, (c) N = 13 and (d) FEM 3Da, l/a = 10.
682 G. Giunta et al. / Composite Structures 95 (2013) 676–690
In a compact vectorial form:
dLi ¼Z
lduT
sKssuuus dx�
Zl
duTsKs
uhHn dxþ duTsP
ssuuus
� �x¼l
x¼0
� duTsP
suhHn
� �x¼l
x¼0 ð26Þ
The components of the differential stiffness matrix Kssuu are:
Kssuuxx¼ J66
s;ys;yþ J55
s;zs;z� J11
ss@2
@x2 Kssuuxy¼ J66
s;ys � J12ss;y
� @@x
Kssuuxz¼ J55
s;zs � J13ss;z
� @
@x
Kssuuyy¼ J22
s;ys;yþ J44
s;zs;z� J66
ss@2
@x2 Kssuuyx¼ J12
s;ys � J66ss;y
� @
@x
Kssuuyz¼ J23
s;ys;zþ J44
s;zs;y
Kssuuzz¼ J44
s;ys;yþ J33
s;zs;z� J55
ss@2
@x2 Kssuuzx¼ J13
s;zs � J55ss;z
� @
@x
Kssuuzy¼ J23
s;zs;yþ J44
s;ys;z
ð27Þ
The generic term Jghsð;/Þsð;nÞ is a cross-section moment:
Fig. 8. Mono-layer FGM beam, rxz [Pa] at x/l = 0 via (a)
Jghsð;/Þsð;nÞ ¼
ZX
CghFsð;/ÞFsð;nÞ dX ð28Þ
The components of the differential thermo-mechanical couplingmatrix Ks
uh are:
Ksuhxx¼ �J1
s@
@xKs
uhyy¼ J2
s;y Ksuhzz¼ J3
s;z ð29Þ
The generic term Jgsð;/Þ is:
Jgsð;/Þ ¼
ZX
Fsð;/ÞkgHX dX ð30Þ
As far as the boundary conditions are concerned, the components ofPss
uu are:
Pssuuxx¼ J11
ss@
@xPss
uuxy¼ J12
ss;yPss
uuxz¼ J13
ss;z
Pssuuyy¼ J66
ss@
@xPss
uuyx¼ J66
ss;yPss
uuyz¼ 0
Pssuuzz¼ J55
ss@
@xPss
uuzx¼ J55
ss;zPss
uuzy¼ 0
ð31Þ
and the components of Psuh are:
Psuhxx¼ J1
s Psuhyy¼ 0 Ps
uhzz¼ 0 ð32Þ
N = 4, (b) N = 13, (c) FEM 3Da and (d) TBT, l/a = 10.
Fig. 9. Mono-layer FGM beam, rxy [Pa] at x/l = 0 via (a) N = 4, (b) N = 13 and (c) FEM3Da, l/a = 10.
Fig. 10. Mono-layer FGM beam, ryz [Pa] at x/l = 1/2 via (a) N = 13 and (b) FEM 3Da, l/a = 10.
G. Giunta et al. / Composite Structures 95 (2013) 676–690 683
The fundamental nucleo of the governing equations in a compactvectorial form is:
duTs : Kss
uuus ¼ KsuhHn ð33Þ
In explicit form:
duxs : �J11ss uxs;xx þ J55
s;zs;zþ J66
s;ys;y
� uxs þ J66
s;ys � J12ss;y
� uys;x
þ J55s;zs � J13
ss;z
� uzs;x ¼ �J1
sHn;x
duys : J12s;ys � J66
ss;y
� uxs;x � J66
ss uys;xx þ J22s;ys;yþ J44
s;zs;z
� uys
þþ J23s;ys;zþ J44
s;zs;y
� uzs ¼ J2
s;y Hn
duzs : J13s;zs � J55
ss;z
� uxs;x þ J23
s;zs;yþ J44
s;ys;z
� uys � J55
ss uzs;xx
þþ J33s;zs;zþ J44
s;ys;y
� uzs ¼ J3
s;zHn
ð34Þ
The fundamental nucleo of the natural and mechanical boundaryconditions at x = 0 and l are:
either uxs ¼ �uxs or J11ss uxs;x þ J12
ss;yuys þ J13
ss;zuzs � J1
sHn ¼ 0
either uys ¼ �uys or J66ss;y
uxs þ J66ss uys;x ¼ 0
either uzs ¼ �uzs or J55ss;z
uxs þ J55ss uzs;x ¼ 0
ð35Þ
For a fixed approximation order, the nucleo has to be expanded ver-sus the indexes s and s in order to obtain the governing equationsand the boundary conditions of the desired model.
Fig. 11. Mono-layer FGM beam, ryy [Pa] at x/l = 1/2 via (a) N = 4, (b) N = 13 and (c)FEM 3Da, l/a = 10.
Fig. 12. Mono-layer FGM beam, rzz [Pa] at x/l = 1/2 via (a) N = 13 and (b) FEM 3Da, l/a = 10.
Fig. 13. Sandwich FGM beam.
684 G. Giunta et al. / Composite Structures 95 (2013) 676–690
5. Closed form analytical solution
The differential equations are solved via a Navier-type solution.Simply supported beams are, therefore, investigated. The following
harmonic form for the axial variation of the displacement and tem-perature fields is adopted:
a Mesh 30 � 30 � 30.b Mesh 20 � 20 � 20.c Scaling factor � 0.1 (instead of 10�6).d Result not provided by the theory.
G. Giunta et al. / Composite Structures 95 (2013) 676–690 685
where a is:
a ¼ mpl
ð37Þ
m 2 N+ represents the half-wave number along the beam axis. {Uis:i = x,y,z} are the maximal amplitudes of the displacement compo-
nents. Upon substitution of Eq. (36) into Eq. (34), the algebraic fun-damental nucleo is obtained:
dUxs : a2J11ss þ J55
s;zs;zþ J66
s;ys;y
� Uxs þ a J66
s;ys � J12ss;y
� Uys
þ a J55s;zs � J13
ss;z
� Uzs ¼ �aJ1
s
dUys : a J66ss;y� J12
s;ys
� Uxs þ a2J66
ss þ J22s;ys;yþ J44
s;zs;z
� Uys
þ J23s;ys;zþ J44
s;zs;y
� Uzs ¼ J2
s;y
dUzs : a J55ss;z� J13
s;zs
� Uxs þ J23
s;zs;yþ J44
s;ys;z
� Uys
þ a2J55ss þ J33
s;zs;zþ J44
s;ys;y
� Uzs ¼ J3
s;z
ð38Þ
The determination of the temperature profile by solving Fourier’sheat conduction equation is presented in Appendix A.
6. Numerical results and discussion
A ceramic–metallic gradation along the thickness direction isconsidered. The ceramic phase is made of Zirconia (ZrO2), whereasMonel (70Ni–30Cu), a nikel-based alloy, is considered as metallicphase. Materials properties are presented in Table 2. The genericmaterial property, f, is assumed to vary versus the thickness coor-dinate z according to the following power law distribution:
f ¼ ðf1 � f2Þðazzþ bzÞnz þ f2 ð39Þ
This law is obtained through the assumption of a power grada-tion law of the volume fraction of the two constituent materialsand the rule of mixtures, see Praveen and Reddy [19] and Chakr-aborty et al. [3]. nz is the power law exponent and it is equal tothe unit, fi is the generic material property of each constituentand az and bz are two constant coefficients that depend upon thethrough-the-thickness extension of the FGM layer and thethrough-the-thickness position of the center of the reference sys-tem, which is centred at the cross-section bottom left corner. Amono-layer and a sandwich FGM cross-section configuration areinvestigated. In the case of the mono-layer FGM cross-sectionand for the assumed reference system, az = 1/b and bz = 0. Thehalf-wave number m in Eq. (37) is equal to one. The thermalboundary conditions (see Eq. (45) in Appendix A) are: Tt = +400 Kand Tb = +300 K. Square cross-sections with sides lengtha = b = 1 m are considered. The length-to-side ratio l/a is equal to
Fig. 14. Sandwich FGM beam, displacements components [m] via N = 13 and FEM 3Da, l/a = 10.
686 G. Giunta et al. / Composite Structures 95 (2013) 676–690
100 and 10. Slender and deep beams are, therefore, investigated. Asfar as tabular results are concerned, the displacements and stressesevaluated at the following points are considered:�ux ¼ uxð0; a=2; bÞ �uy ¼ uyðl=2; a; bÞ �uz ¼ uzðl=2; a=2; b=2Þ�rxx ¼ rxxðl=2; a=2; b=2Þ �rxz ¼ rxzð0;0; b=2Þ �rzz ¼ rzzðl=2; a=2; b=2Þ
ð40Þ
Results obtained using the proposed higher-order modelsare compared with three-dimensional FEM solutions obtained viathe commercial code ANSYS�. The three-dimensional quadratic
element ‘‘Solid90’’ is used for the thermal analysis, whereas the20-node element ‘‘Solid186’’ is considered for the mechanicalproblem. For a FGM layer, each element is considered as homoge-neous by referring to the material properties at its centre. Theaccuracy of the three-dimensional FEM solution depends uponboth the FEM numerical approximation and the approximation ofthe gradation law. In order to present the convergence of thethree-dimensional reference solution, two different meshes areconsidered for each analysis. The acronym FEM 3Da stands for athree-dimensional FEM model with a 30 � 30 � 30 elements mesh,
Fig. 15. Sandwich FGM beam, rn stress components [Pa] via N = 13 and FEM 3Da, l/a = 10.
G. Giunta et al. / Composite Structures 95 (2013) 676–690 687
whereas the coarser 20 � 20 � 20 mesh solution is addressed byFEM 3Db. Although the three-dimensional FEM solution and theanalytical one are different in nature, some considerations aboutcomputational effort can be addressed. The degrees of freedom(DOFs) of the three-dimensional FEM mechanical problem over abeam cross-section as function of the number of elements for eachside, n, are 3(3n + 1)(n + 1). n is as low as 20 (DOFs = 3843) and ashigh as 30 (DOFs = 8463). For a fixed approximation order N, the
DOFs of the proposed solutions are 3(N + 1)(N + 2)/2. In the caseof the highest considered expansion order (N = 13) they are 315.
6.1. Mono-layer FGM beam
Beams made of a single FGM layer, see Fig. 2, are first investi-gated. The temperature variation over the cross-section at mid-span in presented in Fig. 3. The solution of Fourier’s equation via
Fig. 16. Sandwich FGM beam, rn stress components [Pa] at x/l = 1/2 via N = 13 and FEM 3Da, l/a = 10.
688 G. Giunta et al. / Composite Structures 95 (2013) 676–690
the procedure presented in Appendix A has been obtained consid-ering 16 fictitious layers. It matches the FEM 3Da solution. Tables3–6 present the displacements and the stresses in Eq. (40) for slen-der and thick beams. Differently from a bending mechanical load, athermal load results in axial and through-the-thickness displace-ment components as well as normal stress components of compa-rable order of magnitude. This is due to the fact that in the former
case the mechanics is mainly governed by bending (and shear, forthick beams), whereas the normal stresses along the other twodirections are mainly due to the coupling governed by the Poissoneffect. In the thermo-mechanical case, the deformations aregoverned by the thermal expansion coefficients. Classical theo-ries yield a zero through-the-width displacement since, accordingto their kinematic hypotheses (see Eqs. (20) and (21)), the
G. Giunta et al. / Composite Structures 95 (2013) 676–690 689
cross-section is rigid on its own plane and the problem issymmetric versus a plane parallel to Oxz and passing at mid-width.Nevertheless, they provide a well globally estimated displacementfield. As shown in Figs. 4–6, where the variation of the displace-ment components over the cross-section is presented, lower-ordertheories match the reference solution FEM 3Da. For instance, afourth-order model is able to predict the trough-the-width varia-tion of the axial displacement component, which it is constant inthe case of TBT. As far as the stresses are concerned, higher-ordermodels (e.g., N as low as eight) are required for an accurate solu-tion. The stress field is three-dimensional. Fig. 7 shows the axialstress component at mid-span cross-section. Results are obtainedvia, TBT, fourth- and 13th-order model and FEM 3Da. The latterpresents a relevant though-the-width variation and a highthrough-the-thickness gradient, especially at cross-section top.This is very different from a typical mechanical problem of globalbending. The solution for N = 13 compares globally well with thereference solution. The shear component rxz is presented inFig. 8. TBT yields a solution several order of magnitude smallerthan the reference solution. A different scale has been used inthe figure and the tables for this reason. A steep stress gradientis present at both cross-section sides. N = 13 solution matchesthe reference one. Fig. 9 presents rxy computed via N = 4 and 13and FEM 3Da. The fourth-order theory yields an acceptable estima-tion in the neighbourhood of cross-section’s centre. The last twosolutions compare very well. This is also true for the shear compo-nent ryz presented in Fig. 10. The normal stress component ryy isshown in Fig. 11. N = 13 and FEM 3Da solutions compare fairly well.This stress component presents a relevant thought-the-width var-iation and localised stress zones. A fourth-order approximation isobviously not sufficient to describe it. N = 13 yields also a fairlyaccurate prediction of rzz as demonstrated by Fig. 12. The resultspresented in a graphical form have been all obtained for l/a = 10.The case of slender beams is very similar and it is not presentedhere for the sake of brevity.
6.2. Sandwich FGM beam
A FGM sandwich configuration is investigated, see Fig. 13. AFGM core connects the top and bottom layers that are entirelymade of zirconia and monel. The thickness, hf, of the top and bot-tom faces is 0.1 times the cross-section side length. The solution ofFourier’s equation is accurate when compared to the FEM 3Da solu-tion. It is not presented here for the sake of brevity. The main dif-ference versus the mono-layer configuration is in a globally slightlycooler bottom part of the cross-section since the ceramic layer actsas a further thermal barrier. Displacements and stresses for slenderand thick beams are presented in Tables 7–10. Higher-order mod-els match the reference three-dimensional FEM results. Lower-or-der and classical theories yield good displacements but are notcapable of predicting the stress field properly. When comparedwith the mono-layer configuration, a higher transverse displace-ment (resulting in higher stresses) is observed. For the sake ofbrevity, only a 13th-order model is considered for the plots overthe cross-section of displacements and stresses. Fig. 14 showsthe displacement components. Results are practically identical.The stress components rn are presented in Fig. 15. The proposedresults and the reference solutions match. The presence of an innerand outer homogeneous layer changes the profile of the axial stressincreasing the maximum and minimum values and introducing astress gradient also at the cross-section bottom. Finally, Fig. 16 pre-sents the stress components rp. The normal stress components alsopresent localised stress areas that make them difficult to be cor-rectly predicted, especially in the case of rzz. Nevertheless, fairlygood results are obtained. A future work perspective consists in a
the formulation of higher-order theories by means of a layer-wiseapproach that should enhance the accuracy of the approximation.
7. Conclusions
A thermo-mechanical analysis of functionally graded beams hasbeen carried out in this paper. Several one-dimensional displace-ments-based beam models have been derived by means of a uni-fied formulation. Via this formulation, higher-order theories aswell as classical Euler–Bernoulli’s an Timoshenko’s models can beformulated straightforwardly. This is possible thanks a compactnotation for the a priori displacement field approximation. A closedform, Navier-type solution has been used. The temperature fieldhas been obtained by solving Fourier’s heat conduction equationand it has been accounted for in the mechanical analysis as anexternal load. Beams made of a single FGM layer as well as a sand-wich configurations have been studied. Slender and thick beamshave been investigated in terms of temperature, displacementsand stresses. Results have been validated through comparison withthree-dimensional FEM solutions obtained via the commercialcode ANSYS�. In has been shown that the considered thermo-mechanical problems, although presenting a global bending defor-mation, are governed by three-dimensional stress fields that callfor very accurate models. Through an appropriate choice of theapproximation order over the cross-section, the proposed formula-tion yields accurate results with reduced computational costs.
Acknowledgments
The authors would like to sincerely thank the Fonds National dela Recherche (FNR) of Luxembourg for supporting this research viathe project FNR CORE C09/MS/05 FUNCTIONALLY and the AFRGrant PHD-MARP-03.
Appendix A
A solution of Fourier’s heat conduction equation for FGM beamscan be obtained by ideally dividing the cross-section X into NXk
non-overlapping sub-domains (or layers) along the through-the-thickness direction z:
X ¼ [N
Xk
k¼1Xk ð41Þ
Each sub-domain is, then, supposed to be homogeneous, being theelastic and material properties constant and equal to the value atsub-domain’s centre. For a kth homogeneous and isotropic layer,the Fourier differential equation becomes:
@2Tk
@x2 þ@2Tk
@y2 þ@2Tk
@z2 ¼ 0 ð42Þ
In order to obtain a closed form analytical solution, it is further as-sumed that the temperature does not depend upon the through-the-width co-ordinate y. This also implies that the material grada-tion law should be independent from y. The continuity of the tem-perature and the through-the-thickness heat flux qz hold at eachinterface between two consecutive sub-domains:
Tkt ¼ Tkþ1
b
qkzt ¼ qkþ1
zb
ð43Þ
Subscript ‘t’ and ‘b’ stand for sub-domain top and bottom, respec-tively. The through-the-thickness heat flux is proportional to thetemperature derivative versus z:
qkz ¼ Kk @Tk
@zð44Þ
690 G. Giunta et al. / Composite Structures 95 (2013) 676–690
being Kk the thermal conductivity. The following temperatures areimposed at cross-section through-the-thickness top and bottom:
T ¼ Tt sinðaxÞT ¼ Tb sinðaxÞ
ð45Þ
Tt and Tt are the maximal amplitudes and a has been introduced inEq. (37). The following temperature field:
Tkðx; zÞ ¼ HkXðzÞ sinðaxÞ ¼ Tk
0 expðszÞ sinðaxÞ ð46Þ
represents a solution of the considered heat conduction problem. Tk0
is an unknown constant obtained by imposing the boundary condi-tion, whereas s is obtained by replacing Eq. (46) into Eq. (42):
s1;2 ¼ �a ð47Þ
HkXðzÞ, therefore, becomes:
HkXðzÞ ¼ Tk
01 expðþazÞ þ Tk02 expð�azÞ ð48Þ
or, equivalently:
HkXðzÞ ¼ Ck
1 coshðazÞ þ Ck2 sinhðazÞ ð49Þ
For a cross-section division into NXk sub-domains, there are 2 � NXk
unknowns Ckj . The problem is mathematically well posed since
the boundary conditions in Eqs. (43) and (45) yield a linear alge-braic system of 2 � NXk equations in Ck
j . The solution convergenceversus NXk (although not presented here) has been investigated. Ithas been found that, for the considered material gradation,NXk ¼ 16 ensures a converged temperature field.
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