22-2001

7/28/2019 22-2001

1/8

1144 / JOURNAL OF ENGINEERING MECHANICS / NOVEMBER 2001

T RANSVERSE SHEAR STIFFNESS OF C OMPOSITE H ONEYCOMB C OREWITH G ENERAL C ONFIGURATION

By X. Frank Xu, 1 Pizhong Qiao, 2 and Julio F. Davalos 3

ABSTRACT : Based on the zeroth-order approximation of a two-scale asymptotic expansion, equivalent elasticshear coefcients of periodic structures can be evaluated via the solution of a local function and thekl( y),ijhomogenization process reduces to solving the local function by invoking local periodic boundary con-kl( y)ij

ditions. Then, effective transverse shear stiffness properties can be analytically predicted by reducing a localproblem of a given unit cell into a 2D problem. In this paper, an analytical approach with a two-scale asymptotichomogenization technique is developed for evaluation of effective transverse shear stiffness of thin-walled hon-eycomb core structures with general congurations, and the governing 3D partial differential equations are solvedwith the assumptions of free warping constraints and constant variables through the core wall thickness. Theexplicit formulas for the effective transverse shear stiffness are presented for a general conguration of hon-eycomb core. A detailed study is given for three typical honeycomb cores consisting of sinusoidal, tubular, andhexagonal congurations, and their solutions are validated with existing equations and numerical analyses. Thedeveloped approach with certain modications can be extended to other sandwich structures, and a summary of explicit solutions for the transverse shear stiffness of common honeycomb core congurations is provided. Thelower bound solution provided in this study is a reliable approximation for engineering design and can beefciently used for quick evaluation and optimization of general core congurations. The upper bound formula,based on the assumption of uniform shear deformation, is also given for comparison. Further, it is expected thatwith appropriate construction in the displacement eld, the more accurate transverse stiffness can be analyticallyattained by taking into account the effect due to the face-sheet constraints.

INTRODUCTION

Sandwich structures with honeycomb core geometries havebeen widely applied in aerospace and other cost-weight sen-sitive applications. Conventional honeycomb structures takethe advantage of structural geometrical efciency, however,mainly at global or macroscopic level due to the restriction of isotropic metallic materials used. To fully upgrade structuralcapacity by means of geometry and material design, the con-siderations based on microstructure become more important(i.e., optimization at both micro- and macro-levels). The intro-duction of composites into sandwich structures has permittedthe optimization of structural performances by concurrentlyselecting effective geometric shapes and micro-reinforcementarchitectures. Over the last decade, a large number of honey-comb sandwich structures have been manufactured with com-posite face sheets, and all-composite sandwich structures arebeing increasingly used with extensive applications in marineand aerospace structures, such as high strength-to-weight ratiopanels for boat hulls and aircraft oors. In the eld of infra-structure composites, sandwich panels have been successfullyimplemented in highway bridge decks (Davalos et al. 2001),and buildings.

Certainly, the versatility of composite materials is revolu-tionizing the architecture of sandwich structures, and designswith multiple choices of periodic spatial core arrangements arebecoming a reality. Based on demands of mechanical behavior,cost, weight, environment, and use, the design can be ratio-

nally optimized when certain theoretical guidance is available.However, in contrast to the encouraging prospects for appli-

1Grad. Res. Asst., Dept. of Civ. Engrg., The Univ. of Akron, Akron,OH 44325-3905.

2Asst. Prof., Dept. of Civ. Engrg., The Univ. of Akron, Akron, OH44325-3905 (corresponding author). E-mail: [email protected]

3C. W. Benedum Distinguished Teaching Prof., Dept. of Civ. and Envir.Engrg., West Virginia Univ., Morgantown, WV 26506-6103.

Note. Associate Editor: Arup Maji. Discussion open until April 1,2002. To extend the closing date one month, a written request must beled with the ASCE Manager of Journals. The manuscript for this paperwas submitted for review and possible publication on July 20, 2000;revised April 30, 2001. This paper is part of the Journal of Engineering Mechanics , Vol. 127, No. 11, November, 2001. ASCE, ISSN 0733-9399/01/0011-11441151/$8.00 $.50 per page. Paper No. 22411.

cations of composite honeycomb sandwich panels, the avail-ability of analytical solutions appears to be lagging. Althougha lot of work has been accomplished on sandwich structures,a wide gap still exists between new and innovative applica-tions and available sandwich theories. It must also be realizedthat even with current high-speed computers, it is still difcultto tackle periodic sandwich structures governed by differentialequations with oscillating coefcients. Thus, this paper isaimed at narrowing the gap between theory and applicationsfor composite sandwich core geometries.

By observing the repetitive pattern of sandwich structures,it is practicable to approach the solution as a boundary valueproblem with effective stiffness properties by treating a spa-

tially heterogeneous structure as an equivalent homogeneousbulk material. The effective stiffness may be analytically pre-dicted when the local problem on a characterized 3D unit cellis reduced to a 2D case, and in general, thin-walled honey-comb core congurations with warping-free assumption maysatisfy this condition. Since previous research was mainly fo-cused on the traditional hexagonal honeycomb core made of metallic foil, there is little literature available in general spatialcongurations. With regard to computational models on sand-wich structures, Noor et al. (1996) presented a review listingmost of the relevant literature. The early research in hexagonalhoneycomb includes the work of Kelsey et al. (1958), whereupper and lower bounds of transverse shear stiffness were ob-tained with the classical energy method. Among others, Gib-son and Ashby (1988) presented the predictions for transverseshear stiffness of hexagonal honeycomb using mechanics of materials and energy method. Recently, Hohe et al. (1999)studied general hexagonal and quadrilateral grid structures byusing an energy approach. All these classical mechanics ap-proaches are effective for specic problems of isotropic gridhoneycomb, but they become somewhat limited when appliedto honeycomb problems with general shapes and anisotropicmaterials. In general, this limitation is partially related to con-ventional methods themselves, which lack rigor of mathemat-ical theory.

The mathematical description of micro-periodical compositematerials was well developed during the 1970s. The methodof homogenization is believed capable of approximating theequivalent stiffness properties effectively and unrivalled by

7/28/2019 22-2001

2/8

JOURNAL OF ENGINEERING MECHANICS / NOVEMBER 2001 / 1145

FIG. 1. Body of Honeycomb Sandwich Structure

any other known methods in terms of accuracy and closeness.When applying homogenization techniques at micro- andmacro-levels, there are basically no distinctions between prob-lems of inhomogeneous materials and discrete network struc-tures. The homogenization method for composite materials(Meguid and Kalamkarov 1994) can thus be analogically usedin structural problems of thin-walled honeycomb cores. How-ever, the application of the homogenization method to engi-neering problems is not easy, and little work has been done inthis area. A crucial issue is the solution of a special local prob-lem assisted by physical interpretation of the localized varia-bles, by means of which the homogenization process can thusbe less cumbersome and more expedient. In the eld of sand-wich structures, a good attempt was made by Shi and Tong(1995) in presenting an analytical solution for hexagonal hon-eycomb core using homogenization theory.

Of all the effective stiffness properties of honeycomb sand-wich core, the transverse shear properties are most difcult topredict. Also, due to the relatively low shear moduli of poly-mer resins in composite materials, an accurate prediction of effective transverse shear stiffness of composite honeycombcore becomes more important. In this paper, the transverseshear stiffness of honeycomb core with general congurationis investigated, and a two-scale homogenization method isused to obtain explicit formulas for general congurations of

thin-walled honeycomb structures. The analytical lower boundformula of effective transverse shear stiffness is formulated bymeans of the homogenization method. The approach adoptsbasic mathematical concepts of homogenization theory, andthe 3D partial differential equations are solved with the as-sumption of free warping and constant variables through corewall thickness. This approach can be further extended to otherhoneycomb core structures by implementing certain modi-cations when required; e.g., the effect of core wall thicknesscan be added for thick core wall structures. The basic me-chanics concepts of the homogenization method are used inthis paper, and details can be found in a number of relevantreferences (Kalamkarov 1992; Parton and Kudryavtsev 1993).Several practical examples of honeycomb cores with differentcongurations are solved with the derived formula, which isvalidated by existing approaches(solutions given by Kelseyet al. (1958) and Shi and Tong (1995) for hexagonal honey-comb core)or is veried by nite element analyses pre-sented in this paper.

APPLICATION OF HOMOGENIZATION THEORY

Honeycomb sandwich structures usually consist of twoouter face sheets and a honeycomb core with double periodic-ity in the plane normal to the thickness direction. In the ho-mogenization process, a parallelepiped unit cell is rst denedand selected to characterize the spatial periodicity of a sand-wich structure. We consider a body of honeycomb sandwich(Fig. 1) occupying a bound region in R 3 space, dened bycoordinates x1 , x2 , and x3 , with a smooth boundary = 1

2 ( 1 2 = 0) under body force P i. The regionconsists of a double periodic unit cell Y with in-plane dimen-sions of y1 , y2 , and thickness y3 in the same order. It shouldbe pointed out that the homogenization of 3D periodicity bodyis different from that of plates with thickness dimension muchsmaller than the other two. However, when we neglect thewarping constraints of sandwich face-sheet, the estimate of transverse shear stiffness can be considered independent of thickness dimension. Therefore, we can rescale the thicknessdimension to attain the same periodicity parameter by whichthe 3D asymptotic expansion (Parton and Kudryavtsev 1993)can be simply adopted in the following derivations, whereinthe notation is given with small Latin indices h , i, j, k , l = 1,2, 3 and small Greek indices , , = 1, 2. The equations of

equilibrium and the boundary conditions of the linear elasticityproblem may be written as

( )ij ( )P = 0, in ; u = u , on (1a ,b)i i i 1

x j( )n = T , on (1c)ij j i 2

where ( ) ( )

x 1 u uk l ( ) ( ) ( )= c e ; e = (2a ,b)ij ijkl kl kl 2 x xl k And the coefcient c ijkl ( y) should satisfy the elliptical sym-metric condition

c ( y) = c ( y) = c ( y) = c ( y) (3)ijkl jikl ijlk klij

By using the two-scale expansion method, the series are ex-pressed as

( ) (0) (1)u = u ( x) u ( x, y) (4a )i i i( ) (0) (1)e = e ( x, y) e ( x, y) (4b)ij ij ij

( ) (0) (1)= ( x, y) ( x, y) (4c)ij ij ij

where(0) (0) (1) (1)1 u u 1 u ui j i j(0)e ( x, y) = (5a )ij 2 x x 2 y y j i j i(1) (1) (2) (2)1 u u 1 u ui j i j(1)e ( x, y) = (5b)ij 2 x x 2 y y j i j i

(0) (1)u uk k (0) ( x, y) = c ( y) c ( y) (6a )ij ijkh ijkh x yh h

(1) (2)u uk k (1) ( x, y) = c ( y) c ( y) (6b)ij ijkh ijkh x yh h

It is noted that the variables with superscript (1) and (2) are

Y -periodic in y. Substituting (5) and (6) into (1) and retainingthe terms of O ( 0), we have1O ( )(0)ij = 0 (7)

y j(0) (1)ij ij

P = 0 (8)i x y j j

Substituting (6) into (7) gives(1) (0) u ( x, y) u ( x) c ( y)k k ijkl

c ( y) = (9)ijkh y y x y j h l jIn (9), we may introduce the Y -periodic functions bykl N ( y)n

7/28/2019 22-2001

3/8


FIG. 2. Unit Cell Element of General Honeycomb Core

(0)u ( x)k (1) klu ( x, y) = N ( y) (10)n n xl

Thus (9) becomeskl N ( y) c ( y)n ijkl

c ( y) = (11)ijnh y y y j h jLet

kl N ( y)nkl( y) = c ( y) (12)ij ijnh yh

Then, (11) can be written askl( y) c ( y)ij ijkl= (13) y y j j

Because the function y) is Y -periodic in y, (8) will have(1) ( x,ija unique solution if the following condition is satised

(1)ij (1)dy = n dS = 0 (14)ij j Y y jY Y

where Y represents a parallelepiped unit cell; and n j is the unitvector in the outward normal direction to the surface Y .

Applying volume averaging procedure to (8) over Y andconsidering the conditions given in (5), (6), and (14), we have

(0) ijP = 0 (15)i

x j

where(0)1 u ( x)k (0) (0) = dy = c (16)ij ij ijkl

Y xlY

and

1 klc = [c ( y) ( y)] dy (17)ijkl ijkl ijY Y

The effective elastic constants of a unit cell are dened byc ijkl . Based on the so-called zeroth-order approximation given

in (1) to (17), the equivalent elastic coefcients can be com-puted with the solution of the local function kl( y).ij

DERIVATION OF EFFECTIVE TRANSVERSESHEAR STIFFNESS

Description of 2D Periodic Thin-WalledHoneycomb Core

A unit cell of a general honeycomb core structure (in acoordinate system of y1 , y2 , and y3) is shown in Fig. 2, wherethe whole domain and the region of composite laminates insideare designated as Y and Y s, respectively, and the thickness of honeycomb core is denoted as . Within Y s, a segment AB witharbitrary in-plane curve function is selected for analysis. Due

to periodicity, the ends A and B are located at the oppositeside of the unit cell. Let s and denote two local in-planecoordinates, one tangential along the curve segment and theother in the normal direction. For simplicity, the compositecore material is assumed to be orthotropic and dened by nineelastic constants, where G L and G T denote the in-plane andtransverse shear stiffness properties, respectively. Obviouslythe material shear stiffness c 1313 , c2323 , and c1323 in the y-coor-dinate system (Fig. 2) can be calculated by transformations of G L and G T .

For thin-walled structures, the thickness of core wall t (s) isassumed small compared with the size of the unit cell. There-fore, it is reasonable to assume constant variables through thewall thickness. When we apply (17) to a honeycomb core con-

sisting of discrete structures, the averaging integration can bemade by a summation of all segments

1 3c = [c ( y) ] dy 3 3 3 3 3Y Y

B1 3= (c ( y) )t (s) ds dy 3 3 3 3 Y K A K (18)

where K itemizes the segments.To calculate the effective shear stiffness by (18), we must

rst obtain analytical solutions for individual segments. It isobserved that the transverse shear stiffness G L and G T of the

unit cell material are zero in Y \ Y s, and constant in Y s. Thissituation by nature is similar to that of periodically distributedholes in a bulk material. The local boundary condition can bewritten as

(0) n = 0 (19 a )

Applying (6) into (10), (19 a ) can be expressed as 3[c ( y) ( y)] n = 0, at Y (19 b)ij 3 ij j s

where n is the unit vector normal to the core vertical wall Y s.Physically, the above boundary condition represents tractionfree on vertical surface of the core wall.

The condition of periodicity [i.e., duplication of local func-tion on A and B] is expressed as 3 N ( y)i

3 3[ N ( y)] = [ N ( y)] (20)i A i B

As given in (18) (20), the homogenization process actuallyreduces to solving the unknown local function with thekl( y)ijapplication of local periodic boundary conditions.

Determination of Local Function

Based on the warping-free assumption that the local varia-bles are independent of thickness dimension y3 , (13) may bespecically expressed as

3 3 c c13 23 13 3 23 3= , in Y (21 a )s y y y y1 2 1 2 3 312 11 = 0, in Y (21 b)s

y y2 1

where 3 3 N ( y) N ( y)3 3 3 = c c (22 a )13 1313 1323 y y1 2

3 3 N ( y) N ( y)3 3 3 = c c (22 b)23 2313 2323 y y1 2

Note in (22), the term is taken as zero by the 3 N ( y)/ y3warping-free assumption. Further (21 a ) can be rewritted in lo-cal coordinates ( s, ) as

7/28/2019 22-2001

4/8


ds d 3 3( c ) ( c ) = 0, in Y (23 a )13 13 3 23 23 3 s

s dy dy1 2

ds d 3 3( c ) ( c ) = 0, in Y (23 b)23 23 3 13 13 3 s

s dy dy2 1

As in the case for thin-walled structures, the gradientthrough the thickness is approximately zero, thus the secondterms of (23) disappear and (23) reduce to

3 1(s) c (s) = cos const (24 a )13 13 3 3 2(s) c (s) = sin const (24 b)23 23 3

where denotes the anticlockwise angle from y1 to s; andconst 1 and const 2 are constants.

The consideration of local boundary conditions in (19) re-sults in the condition of constant shear ow and const 1 =const 2 = const . Then, (24) becomes

3 (s) c (s) = cos const (25 a )13 13 3 3 (s) c (s) = sin const (25 b)23 23 3

Physically, the above equations can be interpreted as constantshear ows along a segment. To further take into account thewall thickness function t (s) along the segment, (25) may be

modied asconst

3(s) c (s) = cos (26 a )13 13 3t (s)

const 3(s) c (s) = sin (26 b)23 23 3

t (s)

To determine const in the above expressions, we use therelations given in (22)

3 3 N N 3 3 3(s) = G cos G sin (27 a )13 L T s

3 3 N N 3 3 3( y) = G sin G cos (27 b)23 L T s

with coordinate transformations2 2c = G cos G sin (28 a )1313 L T

2 2c = G sin G cos (28 b)2323 L T

c = (G G )sin cos (28 c)1323 L T

Substituting (27) and (28) into (26), and after several steps of transformation, we nally have

13 1 13 N const N 3 3= cos ; = sin (29 a ,b)s t (s)G L

23 2 23 N const N 3 3= sin ; = cos (29c ,d )s t (s)G L

For a simple curve AB without ramication, the substitutionof (29 a ) and (29 c) into (20) results in

B B13 1 N const3ds = cos ds = 0 (30 a ) s t (s)G L A A

B B23 2 N const3ds = sin ds = 0 (30 b) s t (s)G L A A

If a segment is ramied, const becomes piecewise constant,and the continuity of shear ow should be considered at joints.For a segment with ramication, (30) can thus be modied as

B B13 1 m N const3ds = cos ds = 0 (31 a ) s t (s)G L A A

B B23 2 m N const3ds = sin ds = 0 (31 b) s t (s)G L A A

where m is the number of ramication joints along the segmentAB, and const 1 m and const 2 m denote the piecewise constantshear ow along the segment.

Since the plane arrangement of segments in a unit cell isgenerally doubly symmetric, the problem of ramied segments

can be simplied and is shown for hexagonal honeycomb coreof the succeeding section. For the cases of nonramication,(30) is directly solved as

d const = G (32) LT

where B B B

1d = cos ds ; d = sin ds ; T = ds (33)1 2 t (s) A A A

Substituting the solution (32) into (26), we can nally obtainthe local functions as

d 3(s)

c(s) = cos

G(34

a)13 13 3 LTt (s)

d 3(s) c (s) = sin G (34 b)23 23 3 LTt (s)

Results and Applications

By substituting the local functions (34) derived above into(18), the explicit formula of effective transverse shear stiffnesscan be expressed as

B1 3c = [c ( y) ]t (s) ds dy 3 3 3 3 3 3 Y K A K

G d d L = , , = 1, 2

T K K (35)

where B B B

1d = cos ds ; d = sin ds ; T = ds1 2 t (s) A A A

and denotes the area of unit cell in the plane of y1 and y2 ,and ds represents the innitesimal length of segment K .

As described in (35), the contribution of each segment tothe effective shear stiffness depends on the term, d d / T . If asegment is straight with constant thickness t and aligned withthe coordinate y , then d = 0, d = Tt ; thus its contributionsto c

3 3 , c 3 3 , c 3 3 are d t , 0, 0, respectively, which areexactly identical to those familiar forms used for I-sectionbeams in beam theory.

To demonstrate the generality of formula (35), three typicalhoneycomb core structures with different core congurationsare studied and discussed next.

Reinforced Sinusoidal Honeycomb Core

The conguration of a sinusoidal corrugation between hor-izontal at sections is shown in Figs. 1 and 3. The thicknessof the core wall segments is assumed to be constant, and theyare given as t 1 and t 2 for at and curve segments, respectively.The function of OB as shown in Fig. 3(b), is written as

b y a1 y = 1 cos , 0 y (36)2 1 2 a 2

7/28/2019 22-2001

5/8


FIG. 3. Sinusoidal Core: (a) Honeycomb Structure; (b) Unit Cell

FIG. 5. Hexagonal Core

TABLE 1. Constants for Tubular Honeycomb Core

AOB EOF A(CD)B E(CD)F

d 1 3 R 3 R (2 3) R (2 3) R d 2 R R R R

T 2 R3t

2 R3t

R3t

R3t

FIG. 4. Tubular Core

For segment AOB, it follows from (33) that d 1 = a , d 2 = b , T = S / t 2 , where S denotes the length of segment AOB. Similarly,we have d 1 = a , d 2 = b , T = S / t 2 for segment EOF, and d 1 =2a , d 2 = 0, T = 2a / t 1 for segment COD. By substituting theseresults and the area = 2ab into the summation of (35), theeffective shear stiffness properties are easily calculated as

t at bt 1 2 2c = G ; c = G ; c = 0 (37)1313 L 2323 L 1323 b bS aS where S is the length for the segment, S = ds . B ATubular Honeycomb Core

The conguration of a tubular honeycomb core is shown inFig. 4. The radius of curvature is equal to R. As noticed,segment AE and BF are not continuous within the unit cell;we imagine a virtual segment CD connecting them. Thususing (33), we can have the calculated values of d 1 , d 2 , and T (Table 1).

Using the area = and (35), the effective shear22 3 R stiffness properties for tubular honeycomb core are calculated

as(51 24 3) t 9t

c = G ; c = G (38 a ,b)1313 L 2323 L2 3 R 2 3 R

c = 0 (38 c)1323

Hexagonal Honeycomb Core

Hexagonal honeycomb geometry is commonly used forsandwich core, for which the effective transverse shear stiff-ness properties were evaluated early by Kelsey et al. (1958).In this study, we take this conguration as an example to il-lustrate the simplication procedure for ramication problems.As shown in Fig. 5, there are two joints C and D where ram-

ications begin. The thickness of horizontal and inclined wallsis given as t 1 and t 2 , respectively. For computation of c 1313 ,segments ACDB and ECDF are simulated as two separate onesdue to geometrical symmetry. Therefore, by direct applicationof (33), we have d 1 and T for the two segments as

2a bd = a b cos ; T = (39)1

t t 1 2

Using the summation given in (35), we obtain2

G 2(a b cos ) Lc = (40)1313 A 2a b

t t 1 2

For the computation of c 2323 , we can directly apply (31) toobtain piecewise constant shear ow along ACDB/ECDF;however, with the consideration of double symmetry, the ap-proach can be further simplied as follows. Since segment CDis geometrically neutral, segments ACE and BDF can be in-stead taken as periodic segments. Then for either of these seg-ments, the d 2 and T can be expressed as

bd = b sin ; T = (41)2

t 2

7/28/2019 22-2001

6/8


TABLE 5. Comparison between Analytical and Numerical Predictions for Tubular Core

Predictions Parameters c 1313 c 2323Numerical a A = 1 = 2 = 0.0002; F 1 = 3.0020; F 2 = 2.8649

22 3 R = 2 3; = 173324F 1 A 1

= 165414F 2 A 2

Analytical a G L = 106

G L = 17332(51 24 3) t

2 3 R G L = 16540

9t

2 3 R a

v 1 = v 2 = 0.0001; h = 0.5; R = 1; t = 0.02.

TABLE 4. Comparison between Analytical and NumericalPredictions for Sinusoidal Core

Predictions Parameters c 1313

Numerical a A = 2ab = 8; 1 = = 0.0002;

u 12h

F 1 = 1.3664

= 34164F 1 A 1

Analytical a S = 2.9274; G L = 106

G L = 3416at 2bS

Note: Computation is based on the curve portion shown in Fig. 6.a

v 1 = 0.0001; h = 0.5; a = b = 2; t = 0.01.

TABLE 3. Boundary Conditions in FE Model of Quarter Unit Cellof Tubular Core

Specied displacementon boundary v 1 v 2 v 3

EE /OO /CC Linear 0 0EO/EC 0.0001 0 FreeEO /EC 0 0 Free

Note: For computation of c 1313 .

TABLE 2. Boundary Conditions in FE Mode of Quarter Unit Cell of Sinsoidal Core

Specied displacementon boundary v 1 v 2 v 3

EE /OO Linear 0 0EO 0.0001 0 FreeEO 0 0 Free

Note: For computation of c 1313 .

FIG. 7. FE Model of Quarter of Unit Cell for Tubular Core

FIG. 6. FE Model of Quarter of Unit Cell for Sinusoidal Core

Then, using (35) we can simply write

G L 2c = 2t b sin (42)2323 2 A

VERIFICATION USING FINITE ELEMENT ANALYSIS

The formulas given in (40) and (42) for hexagonal honey-comb core conguration lead to identical results given by Kel-sey et al. (1958), Gibson and Ashby (1988), and Shi and Tong(1995). Thus, we only performed nite element (FE) analyses

to conrm the results for the honeycomb core congurationswith sinusoidal and tubular geometries.The commercial nite element program ANSYS 5.5 is used,

and eight-node isoparametric layered shell elements (SHELL99) are selected to model the thin-walled cores. To minimizethe computational effort, the FE model is developed with spe-cial consideration of periodicity. Based on periodic nature of honeycomb core structures, one quarter of a unit cell (Figs. 6and 7) is modeled with boundary conditions specied in Tables2 and 3 for sinusoidal and tubular cores, respectively. Theboundary conditions are specied as prescribed displacementsv i corresponding to coordinates yi (i = 1, 2, 3). In the FE modelof the sinusoidal core (Fig. 6), a uniform displacement of v 1is applied to all the nodes at the top surface dened by thecurve line EO, and the resulting shear force F 1 along E O can

be correspondingly obtained. To keep periodic boundary con-dition and obviate bending effect, the displacement componentalong the vertical boundary EE and OO must remain linearfrom top to bottom by imposing v 2 = 0. In the numerical sim-ulation, the parameters assumed for the sinusoidal core curveare listed in Table 4, and their physical meanings are given inthe preceding section.

Similarly for the FE model of tubular core (Fig. 7), eitherv 1 or v 2 is applied to all the nodes at the top surface (for thecurve lines EO and EC), and the resulting shear force, F 1 orF 2 , along the E O and E C is correspondingly computed.Again, to prevent bending effect, the displacement componentat vertical boundaries EE , OO, and CC must remain linearfrom top to bottom by imposing v 2 = 0. The parameters usedin the FE model for tubular core are specied in Table 5.

The elastic properties of core wall of the shell elements(SHELL 99) can be modeled as orthotropic, and a simple par-ametric study of elastic stiffness shows that the resulting shearforce F i (i = 1, 2) is only dependent on the variable G L . Thisis consistent with the derivation of (26)(29).

By obtaining the resulting shear force from the FE models,the effective shear stiffness can be calculated, and comparisonsare made with the analytical solution. The equations and cal-culations for sinusoidal and tubular cores are shown in Tables4 and 5, respectively, which illustrate the accuracy of (35).

From a structural mechanics point of view, the exactness of results is well-expected once the appropriate boundary con-

7/28/2019 22-2001

7/8

7/28/2019 22-2001

8/8


strains are applied; i.e., the condition of constant shear ow isimposed so that identical solutions result from both the ana-lytical and numerical approaches.

SUMMARY AND DISCUSSION

With the application of (35), the formulas of transverseshear stiffness for arbitrary periodic cellular honeycomb corescan be easily obtained, and they would be valuable for thedesign and optimization of honeycomb shape congurations.As a summary, the transverse shear stiffness formulas for sev-eral common core congurations are given in Fig. 8, and theyare expressed in term of apparent core density c.

It must be noted that the formulation based on the abovehomogenization process actually corresponds to the lowerbound of transverse shear stiffness of sandwich structures, forwhich the face sheet warping constraints have not been takeninto account. As indicated by other researches on hexagonalhoneycombs (Grediac 1993; Shi and Tong 1995), the facesheet effect is quite localized, especially for sandwich coreswith small to moderate ratio of unit-cell size to core height.The upper bound of transverse shear stiffness can be obtainedby the assumption of innitely stiff face sheets and the prin-ciple of minimum energy theorem (Kelsey et al. 1958; Gibsonand Ashby 1988). The upper bound of transverse shear stiff-

ness for general core conguration can be expressed as B

G L 2c = cos t ds (43 a )1313 k A

BG L 2c = sin t ds (43 b)2323

k A

As an illustration, the upper bounds of transverse shear stiff-ness are also given in Fig. 8 for common core congurations.Note in Fig. 8, the exact solutions are attained for several corecongurations (for triangle and rectangular grid cores) due tothe identical lower and upper bounds of transverse shear stiff-ness, as one would intuitively expect.

To further precisely evaluate the face-sheet effect or obtaina narrow bound solution range, the energy method can be usedas an effective approach whenever the additional local energyis rationally weighted. Based on the principle of minimumenergy theorem, the consideration of complex interactions be-tween face sheets and core can be simplied by assuming anappropriate displacement eld, which can result in improvedupper-bound solution estimation. The Raleigh-Ritz method ba-sically species two approaches, one for the face-sheet dis-placement eld, and the other for the core internal displace-ment eld. The latter, successfully implemented by Penzienand Didriksson (1964) for the hexagonal core, seems more

effective and is suggested to apply for the present formulationfor all general core congurations.

CONCLUSIONS

In this paper, an analytical approach using two-scale as-ymptotic homogenization technique is presented for effectivetransverse shear stiffness evaluations of honeycomb structures,and an explicit formula is provided for general shapes of thin-walled honeycomb cores. Three typical core congurations aresubsequently solved with the developed formula, which is val-idated by existing or numerical solutions. The derived formula(35) can be efciently used to predict the effective transverseshear stiffness of honeycomb cores with any general core con-gurations and can be applied to the optimization of honey-comb core structures. Further, this approach with certain mod-ications can be extended to other sandwich structures,including the consideration of wall thickness effect for thick wall cores.

ACKNOWLEDGMENTSPartial nancial support for this study was received from the Faculty

Research Committee of The University of Akron and the National ScienceFoundationPartnerships For Innovation program (NSF-PFI: EHR-0090472). We thank Dr. Jerry Plunkett, President and CEO of KSCI, forhis technical advice with the sinusoidal cores geometry produced byKSCI, and we appreciate the support and encouragement of Dr. S. Gra-ham Kelly, Dean of Engineering at The University of Akron.

REFERENCESDavalos, J. F., Qiao, P. Z., Xu, X. F., Robinson, J., and Barth, K. E.

(2001). Modeling and characterization of ber-reinforced plastic hon-eycomb sandwich panels for highway bridge applications. Compos.Struct. , 52(3-4), 441452.

Gibson, L., and Ashby, M. (1988). Cellular solids structure and prop-erties , Pergamon, Tarrytown, New York.

Grediac, M. (1993). A nite element study of the transverse shear inhoneycomb cores. Int. J. Solids and Struct. , 30(13), 17771788.

Hohe, J., Beschorner, C., and Becker, W. (1999). Effective elastic prop-erties of hexagonal and quadrilateral grid structures. Compos. Struct. ,46, 7389.

Kalamkarov, A. L. (1992). Composite and reinforced elements of con-struction , Wiley, New York.

Kelsey, S., Gellatly, R. A., and Clark, B. W. (1958). The shear modulusof foil honeycomb cores. Aircraft Engrg. , 30, 294302.Meguid, S. A., and Kalamkarov, A. L. (1994). Asymptotic homogeni-

zation of elastic composite materials with a regular structure. Int. J.Solids and Struct. , 31(3), 303316.

Noor, A., Burton, W. S., and Bert, C. W. (1996). Computational modelsfor sandwich panels and shells. Appl. Mech. Rev. , 49(3), 155199.

Parton, V. Z., and Kudryavtsev, B. A. (1993). Engineering mechanics of composite structures , CRC Press, Boca Raton, Fla.

Penzien, J., and Didriksson, T. (1964). Effective shear modulus of hon-eycomb cellular structure. AIAA J. , 2(3), 531535.

Shi, G., and Tong, P. (1995). Equivalent transverse shear stiffness of honeycomb cores. Int. J. Solids and Struct. , 32(10), 13831393.

22-2001

Documents