Composite Shuchms Vol. 38, No. l-4, pp. 29-36 0 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0263-8223/97/$17.00 + 0.00 PII:SO263-8223(97)00038-X
Buckling analysis of functionally graded plates subjected to uniaxial loading Esther Feldman
& Jacob Aboudi
Department of Solid Mechanics, Materials and Structures, Faculty of Engineering, Tel Aviv University,Ramat Aviv 69978, Ismel
Elastic bifurcational buckling of functionally graded plates under in-plane compressive loading is studied. It is supposed that the gradients of material properties throughout the structure are produced by a spatial distribution of the local reinforcement volume fraction vf = v,(x, y, z). To analyze the problem, a method based on a combination of micromechanical and structural approaches is employed. This establishes the effective constitutive behavior at every point of a nonhomogeneous composite plate and provides a buckling criterion. The derived criterion enables one to calculate the critical buckling load G for a given distribution v,(x, y, 2). Furthermore, with the aim to improve the buckling resistance of the functionally graded plate, the functional K(vr> is maximized. This yields an optimal spatial distribution VAX,y, z) of the reinforcement phase. Results are presented for both short- and long-fiber Sic/Al plates in which the fibers are nonuniformly distributed in the x-, y-, or z-directions. The effects of length-to-width ratio of the plate, and of different types of boundary conditions are studied. Buckling load improvements of up to lOO%, as compared to the corresponding uniformly reinforced structure, are shown. 0 1997 Elsevier Science Ltd.
properties at the macroscopic or continuum level. Such an approach offers a number of advantages over the more traditional methods of tailoring the material properties and opens up new horizons for novel applications. Grading or tailoring the internal microstructure of a composite material or a structural component allows the designer to truly integrate both material and structural considerations into the final design and final product. This brings the entire structural design process to the material level in the purest sense, thereby increasing the number of possible material configurations for specific design applications. The potential benefits that may be derived from functionally graded composites have led to increased activities in the areas of processing and materials science of these materials. However, in order to develop a component made of FGM it is necessary to model such a component and investigate its required properties. Thus, an accurate modeling of the FGM is essential to its
The traditional approach to fabricating composite materials implies that the reinforcement phase is distributed either uniformly or randomly such that the resulting mechanical, thermal, or physical properties do not vary spatially at the macroscopic level. Recently, a new concept involving tailoring or engineering the microstructure of a composite material to specific applications has taken root. This idea has given rise to the term functionally graded materials’ (FGM) to describe this newly emerging class of composites. FGMs are a new generation of composite materials in which the microstructural details are spatially varied through nonuniform distribution of the reinforcement phase, by using reinforcements with different properties, sizes and shapes, as well as by interchanging the roles of reinforcement and matrix phases in a continuous manner. The result is a microstructure that produces continuously changing thermal and mechanical 29
E. Feldman, J. Aboudi
development. There are presently two approaches for the modeling of FGM. The first is based on a homogenization of the FGM in which the microstructural effects are decoupled from the global response by calculating pointwise effective thermoelastic properties without regard as to whether the actual microstructure admits the presence of a representative volume element (RVE), and subsequently using these properties in the global analysis of the heterogeneous material. In the second approach the coupling between the microstructural and the global macrostructural effects is accounted for. Recent special issues edited by Pindera et al. [l-3], and by Needleman and Suresh  have been devoted to various topics dealing with the mechanics and material aspects of multiphased and functionally graded composites. A review paper that summarizes a higher order theory for functionally graded composites which explicitly couples the microstructural details with material’s macrostructure has been presented by Pindera et al. [S]. The concept of FGM can also be utilized for the management of a material’s microstructure so that the buckling behavior of a structure made of this material can be improved. In this investigation the idea of tailoring the microstructure of a composite material for the purpose of improving the buckling behavior of a plate is pursued. It is supposed that the gradients of material properties throughout the structure are produced by changing the local reinforcement volume fraction V~over the planform and/or through the thickness of the plate. To determine the buckling load of a plate for a given spatial distribution of reinforcement volume fraction V&K,y, z), the proposed microto-macro approach [6,7] is further extended to include the buckling analysis of functionally graded plates. The micromechanical analysis performed in the present study relies on the RVE-based version of the method of cells . As a result, a buckling criterion incorporating the effective constitutive behavior at every point of the plate, is obtained. This criterion allows one to calculate the critical buckling load RT for a given function v~(.x,y, z). Consequently, for a greater buckling resistance of a FGM plate the function RF(vf) should be maximized. This leads to an optimal spatial distribution v#,y,z) of the reinforcement phase.
To illustrate the proposed approach, results are presented for both short- and long-fiber Sic/Al unidirectional plates, with reinforcements nonuniformly distributed in the X-, y-, or z-directions. The effects of length-to-width and length-to-thickness ratios and different types of boundary conditions are studied. Substantial buckling load improvements, as compared to the corresponding uniformly reinforced plate, are shown.
THEORETICAL FORMULATION An elastic, midplane symmetric, functionally graded rectangular composite plate, subjected to an in-plane compressive loading, is considered. The structure is reinforced by either long fibers or discontinuous ones, with the reinforcements volume fraction vf being a function of the spatial coordinates X, y, z. The coordinates x and y define the plane of the plate, and z-axis is oriented in the thickness direction. The nondimensional coordinates 5 = x/u, q = y/b, c = z/h, and the aspect ratio of the plate /z = a/b are introduced, where a, b and h represent the plate’s length, width, and thickness, respectively. Governing equations to bifurcational buckling of a nonhomogeneous plate In the present study, the buckling behavior of a nonhomogeneous plate is described in terms of the stress function @ and the out-of-plane displacement w, in the framework of the classical plate theory. The corresponding system of governing equations may be obtained elsewhere
PI (0, r~,~~ +~*D,~w,~~+~~D,~w,~~),~; [email protected]
,2w.r~ +J%,Y,, +2Ws~&~,, +Why,, + A2D2,w,,,
Here A* and D are the stiffness matrices which are given, for example, by Whitney [lo]. Further, the behavior of simply supported (SS) and clamped (C) plates subjected to com-
Buckling analysis of functionally graded plates
pressive edge displacements & in the x-direction will be analyzed. Depending on the in-plane boundary conditions at the edges q = 0, 1, two cases are considered, namely: the edges q = 0, 1 are immovable in the y-direction (SSland Cl-cases); the edges q = 0, 1 are unloaded (SS2- and CZcases). In all instances, the inplane displacements at the edges are unrestricted in tangential direction. Introducing the average edge shortenings in both in-plane directions
(where u and using W, given conditions
ML 1)-v(t, O)ldt
and v are the in-plane_ displacements), the expressions for A,, 4 in terms of a, in an earlier paper , the boundary may be formulated as follows:
@6 Y, z> = cc6 Y, ZMX, y, z) where 0 and E are stress and strain tensors, and C represents the effective stiffness tensor. Determination of the critical buckling load
SSl at t=O, 1: @,Ts=~=M5t=0; -
where Mas (a, fi = 5, q) are the moment resultants. To fulfill the buckling analysis of a FGM plate, one needs to determine the constitutive behavior at every point of the structure, so that the stiffnesses involved in eqns (l)-(3) can be calculated. The required constitutive law can be obtained by a suitable micromechanical approach. In the present work, the micromechanical analysis performed relies on the RVE-based method of cells . That is, at every point (x, y, z) of the plate this micromechanical method is employed, using the values of material properties of fiber and matrix phases, as well as the reinforcement volume fraction at that point V&V,y, z). This enables one to establish the effective constitutive law of a functionally graded composite in the form
An approximate solution to the above-formulated problem is sought in the form a2
at q=O, 1: @.Tq=~=Mvv=O; ’ ss0
’ (i22A;[email protected]
SS2 at t=O, 1: Q’,rq--w=MCr=O; M
~(5, VI= E, E, W,,
at q=O, 1: Q,ST=Trl=~=Mqq=O
Cl at c=O, 1: @,,,- -wwy=o; (for C-cases)
at q=O, 1:
’ (;12A;,@,,, + A;[email protected]
C2 at l=O, -
1: a,,,- -w=w,c=o;
a ss0 0
at yI=O, 1: @‘,re--aQrl=w=w,v=o
where Xi(i = 1, 2, . . .) are beam eigenfunctions Xi(O) = Xi( 1) satisfying boundary conditions =X:(O) =X:(l) = 0. It can be readily seen that one needs to determine the coefficients of the series (4), and the values of R,, R,, (having the meanings of normal in-plane loads at the edges x=O,a andy=O,b). To calculate the coefficients Fp4, IV,,, the series (4) are substituted into the governing eqns (1) and (2) and the Galerkin procedure is employed. The compatibility equation (2) is multiplied by X,.(0X,(n) and integrated over the
E. Feldman, J. Aboudi
(1) is multiplied by plate surface. Equation sin ri[ sin rcjq for SS-cases or by Xi(oIyi(~) for C-cases and integrated over the plate surface as well. For SS-cases, to account for the boundary conditions on the bending moments, boundary integrals are included in the formulation of the Galerkin equations (see Whitney [lo] for details). As a result, the following equations are obtained
(r= 1, . . . . P, s= 1, . . . . Q)
From substitution of relationships (8) into eqn (5) it is possible to get Fp4 in the form F,, = &,y&
Finally, substituting (8) and (9) into eqn (6), one obtains
+ RySFy) W,, = 0
The coefficients S;SPy, . . . , $y, which are analogous to those given in an earlier paper  involve surface integrals from different elements of stiffness matrices A* and D, multiplied by the beam or trigonometric functions. To determine the values of Rx, RY, the approximation (4) for the stress function is substituted into the boundary conditions (2) for the edge shortenings. This yields Sl, Cl A
T,dSdvl + R?
xa +L a4
Z C F,,S,P,Y;
+ e.$$!?$.y+ e[~S!!~) -
From eqn (lo), the buckling criterion obtained det[Q.; IS3- &rJ = 0
is readily (II)
where I is an identity matrix. The buckling criterion (11) enables one to calculate the critical buckling displacement @ (as the minimal eigenvalue of the matrix QxP 'S,) and, according to (8) to obtain the critical buckling load Rz for a given reinforcement volume fraction Vf(5, VI,i).
Optimal reinforcement distributions
For greater buckling resistance of a FGM plate, the functional R~;‘(v,~) should be maximized. In the present work, this is achieved as follows. Function V~is expanded into the Legendre polynomials
-!C C F,,S;; a4 P 4
AT,dtW CZF,,S;:;R>,=O a4 p 4
=-- %+l a
For all the boundary conditions solution to (7) may be written as R, = elx&+
C Z e$‘ZFpy; P
with Vi,, being unknown coefficients to be determined. This allows one to represent the buckling load as ~~=~~(Vijk). It is further required that the total amount of reinforcements (namely the plate weight) remains a given constant
Buckling analysis of functional+ graded plates
Thereafter the objective function RF&) is maximized, with l& being a set of design variables subject to the above constraint (13). Two additional constraints stem from the following requirements on the volume fraction vf: v,-> 0, v+ 1. Once an optimal set of design variables has been obtained, the corresponding optimal distribution VT’ of the reinforcement phase is easily calculated from (14). The described procedure should be repeated for different numbers of terms retained in the series (4) and (14) to ensure the convergence of the proposed procedure. Considering in more detail the affect of the plate thickness on the buckling behavior, it may be shown that, for symmetrically laminated nonhomogeneous plates with the same aspect ratio A, the same reinforcement distribution VA<, q, [), and subjected to the same boundary conditions, the following relationship holds R’1 k’[email protected]
= h: Jh;
sidered; for each of them one of the coordinate directions is chosen as a functionally graded one. Short-fiber plate with reinforcements nonuniformly distributed in the x-direction Consider a Sic/Al plate reinforced by short Sic fibers oriented in the n-direction, with the fibers’ aspect ratio equal to 7. The fibers’ distribution is nonuniform only in the x-direction, such that vf = VAX). In Fig. l(a), optimal distributions of a Sic phase along the x-axis are exhibited for a rectangular plate subjected to different types of boundary conditions. Due to the symmetry of the problem, here and further only a half of a functionally graded coordinate axis is shown. It may be seen that the distributions v~~~(x/u) are dissimilar for simply supported and clamped
w heie R(’ w is the critical buckling load for a plate of ;hickness h,, and RF)” is the critical buckling load for a plate of thickness h2. From (15) follows, in particular, that the above-mentioned plates will possess the same optimal volume fraction distribution vTp’([, v, 5).
RESULTS AND DISCUSSION To illustrate the proposed approach, consider Sic/Al unidirectional plates with reinforcements oriented in the x-direction. Results are presented for the temperature T = lOO”C, at which the material properties are taken as follows: for Sic fibers, Young’s modulus E = 414 GPa, and Poisson’s ratio v = 0.3; for the aluminum matrix (2024-T4 alloy), E = 70.5 GPa, and v = 0.33. It is assumed that Sic reinforcements comprise 30% of the plate volume, i.e. vf* = 0.3. As an initial guess (which is needed to start an optimization procedure) assume that a plate is uniformly reinforced, that is v#, tl, [) = v; = 0.3. For the purpose of estimation the effect of a functionally graded plate, introduce a ratio cPtI R,ho”, where ep’ and Rp” stand for the critical buckling loads for a functionally graded plate with optimal distribution vTp’(& q, [) and for its homogeneous counterpart, respectively. Furthree examples of tailoring the thermore, distribution of reinforcement phase are con-
(4 1 “f
opt 0.8 0.6
Fig. 1. Optimal reinforcement volume fraction distributions along the x-axis for short-fiber functionally graded plates (the fibers are oriented in the x-direction, fiber aspect ratio = 7), (a) for various types of boundary conditions, (b) for several values of aspect ratio A.
E. Feldman, J. Aboudi 1.5r
Fig. 2. Buckling
bound. cond. SSl
bound. cond. Cl load improvements for short-fiber functionally graded plates (the fibers are oriented fiber aspect ratio = 7); the fibers are non-uniformly distributed in the x-direction.
plates. Moreover, a beneficial effect of a nonhomogeneous reinforcement is quite different as well: for the boundary conditions SSl and SS2 R”pflRh”” = 1.11, while for Cl- and CZcases R~pt/R>m r 1.35 and 1.42, respectively. The effect of the plate aspect ratio A is illustrated in Fig. l(b) for a clamped structure (boundary condition Cl). As is apparent from the graphs, the optimal distribution of a reinforcement phase is strongly affected by the value of the length-to-width ratio. Notice that the improvements in the buckling load also
in the x-direction,
depend essentially on u/b: Rzp’lR,h”” = 1.24, 1.35, 1.23, 1.17, and 1.16 for A = 0.5, 1.0, 1.5, 2.0, and 3.0, respectively. These results indicate that for the cases considered the maximum advantage over a uniform reinforcement may be achieved for a clamped rectangular plate. An additional insight into the buckling load improvements that may be attained for the case considered is provided by Fig. 2 where the histograms of RzPtlRpm are presented for two types of the out-of-plane boundary conditions and for several values of A. Long-fiber plate with fibers nonuniformly distributed in the y-direction
Fig. 3. Optimal reinforcement volume fraction distributions along the y-axis for long-fiber functionally graded plates (the fibers are oriented in the x-direction), for several values of aspect ratio 1.
Consider a long-fiber Sic/Al plate such that the fibers are oriented along the x-axis and the spacings between them (in the y-direction) are not equal; i.e. vf= v~(Y). Regarding the influence of different boundary conditions on the buckling behavior of functionally graded plates, similar trends are observed, as compared to the above considered case of short-fiber plates with v~= VAX).That is, the optimal volume fraction distributions and the corresponding buckling loads for simply supported and clamped plates differ significantly. For rectangular plates, R~PtlR~“m= 1.09, 1.13, 1.23, and 1.27 for SSl-, SS2-, Cl-, and C2boundary conditions, respectively. Examining
Buckling analysis of functional& graded plates
the effect of plate’s aspect ratio, it should be pointed out, that, as indicated in Fig. 3, the optimal distributions v~P’(~) may differ greatly from each other for various values of u/b. The improvements in the buckling loads, corresponding to the graphs shown in the figure, are as follows. * Pp’lRho” = 1.19, 1.23, 1.12 and 1.09 x for A= 0.5, 1.0, 1.; and 3.0, respectively.
Long-fiber plate with fibers nonuniformly distributed through the thickness
(b) Fig. 4. Optimal reinforcement volume fraction distributions through the thickness for long-fiber functionally graded plates (the fibers are oriented in the x-direction), (a) for two types of simply supported boundary conditions, (b) for a clamped plate and several values of aspect ratio 1.
Consider a long-fiber Sic/Al plate such that the fibers are oriented along the x-axis and nonuniformly distributed in the direction normal to the midsurface, so that vf = vAz). It turns out that in this case, for all the boundary conditions and aspect ratios convolume sidered, the optimal fraction distributions vTpt([) are quite similar and, moreover, particularly high values of ~ptlR~m may be obtained. This is illustrated in Fig. 4(a), where the optimal reinforcement distributions through the thickness are shown for a rectangular plate subjected to the two types of simply supported boundary conditions, and in Fig. 4(b), where results are presented ‘for clamped plates with different aspect ratios 1. As viewed in Fig. 4, matrix-rich regions next to the middleplane are observed, which occupy about 50% of the plate thickness. In the vicinity of the plate surfaces, fiber clustering takes place. The buckling load improvements corresponding to the gra’ hs presented in Fig. 4(a) are as r follows. -xROPfIR xOrn= 1.95 and 2.02 for rectangu-
Fig. 5. Buckling load improvements for long-fiber functionally graded plates (the fibers are oriented fibers are nonuniformly distributed in the thickness direction.
in the x-direction);
lar plates subjected to SSl- and SS2-boundary conditions, respectively. The beneficial effect of a nonhomogeneous reinforcement for various values of the aspect ratio II is shown in Fig. 5, both for simply supported and clamped functionally graded structures.
tionally graded materials.
Phys Solids 1996,44
REFERENCES Pindera, M.-J., Arnold, Aboudi, J. of composites in functionally composites Engng 1994, (1) l-145. Pindera, M.-J., Aboudi, Arnold, S.M. W.F., of composites in multi-phased
M.-J., Aboudi, in the In:
Materials for a
The first author is grateful for the support of this research by the Ministry of Absorption of the State of Israel. The second author gratefully acknowledges the support of the Diane and Arthur Belfer Chair of Mechanics and Biomechanics.
3. Pindera, M.-J., Aboudi, Glaeser, A. Arnold, S. Use of composites multi-phased and tionally graded materials. Composites B: Engineeting), March in press. Needleman, and Suresh, Mechanics and physics layered and J. 5. Pindera,
and Arnold, M., Recent of functionally graded
Thermal Structures Era (Progress Astronautics and
Vol. 168, 181-203, Thornton, A. (ed.). Institute of Aeronautics and nautics, Washington, D.C., 6. Feldman, E. Aboudi, J., Postbuckling analysis of metal-matrix laminated plates. Composites 1994,4, 151-167. Feldman, E., Postbuckling analysis of plates of discontinuous metal composites. Strut. 32, 89-96. 8. Aboudi, Mechanics of Composite Unified Micromechanical Approach. Elsevier, Amster1991. 9. Feldman, E., effect of