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Gregory W. Kooistra1

Department of Materials Science andEngineering,

University of Virginia,116 Engineer’s Way,

Charlottesville, VA 22904e-mail: gkooistra@watsonfurniture.com

Vikram DeshpandeEngineering Department,

Cambridge University,Trumpington Street,

Cambridge CB2 1PZ, UK

Haydn N. G. WadleyDepartment of Materials Science and

Engineering,University of Virginia,

116 Engineer’s Way,Charlottesville, VA 22904

Hierarchical Corrugated CoreSandwich Panel ConceptsThe transverse compression and shear collapse mechanisms of a second order, hierar-chical corrugated truss structure have been analyzed. The two competing collapse modesof a first order corrugated truss are elastic buckling or plastic yielding of the trussmembers. In second order trusses, elastic buckling and yielding of the larger and smallerstruts, shear buckling of the larger struts, and wrinkling of the face sheets of the largerstruts have been identified as the six competing modes of failure. Analytical expressionsfor the compressive and shear collapse strengths in each of these modes are derived andused to construct collapse mechanism maps for second order trusses. The maps are usefulfor selecting the geometries of second order trusses that maximize the collapse strengthfor a given mass. The optimization reveals that second order trusses made from structuralalloys have significantly higher compressive and shear collapse strengths than theirequivalent mass first order counterparts for relative densities less than about 5%. Asimple sheet metal folding and dip brazing method of fabrication has been used tomanufacture a prototype second order truss with a relative density of about 2%. Theexperimental investigation confirmed the analytical strength predictions of the secondorder truss, and demonstrate that its strength is about ten times greater than that of a firstorder truss of the same relative density. �DOI: 10.1115/1.2198243�

IntroductionMaterials with structural hierarchy can have significantly higher

tiffness or strength to weight ratios than their single-length scaleicrostructure counterparts. For instance, the maximum stiffness

o weight ratio of an isotropic two-phase material is set by theashin-Shtrikman �1� �HS� upper bound. A number of two-phase

omposites are known to attain the HS bounds on the bulk andhear moduli. Most of these include structural hierarchy. For ex-mple, Norris �2� and Milton �3� proposed differential schemes foronstructing composite structures with the extremal HS bulk andhear moduli. While Milton �3� used a laminate microstructure,orris �2� employed coated sphere architectures. However, therocedures suggested by both authors are iterative and require annfinite number of mixing processes. On the other hand, Francfortnd Murat �4� suggested a “rank” laminate approach which at-ained both the bulk and shear HS bounds with a finite number ofayering directions. Rank laminates are obtained by a sequentialrocess where at each stage the previous laminate is laminatedgain with a single lamina �always the same� in a new direction.hus, a rank-n laminate is produced by n such successive lamina-

ions. Francfort and Murat �4� showed that while in the two-imensional case isotropic rank-3 laminates have the extremalulk and shear moduli, in the three-dimensional �3D� case rank-6aminates are the optimal microstructures.

Hierarchical cellular structures consisting of self-similar struc-ural units can also exhibit significant strength to weight improve-

ents over comparable non-hierarchical structures. For example,hat et al. �5� manufactured sandwich panels with honeycombores; the webs of these honeycombs were in-turn made fromoneycomb sandwiches resulting in a second order structure. Thexperimental investigation of Bhat et al. �5� indicated the secondrder panel had a compressive strength about six times greater

1Author to whom correspondence should be addressed.Contributed by the Applied Mechanics Division of ASME for publication in the

OURNAL OF APPLIED MECHANICS. Manuscript received May 12, 2005; final manuscripteceived September 20, 2005. Review conducted by R. M. McMeeking. Discussionn the paper should be addressed to the Editor, Prof. Robert M. McMeeking, Journalf Applied Mechanics, Department of Mechanical and Environmental Engineering,niversity of California – Santa Barbara, Santa Barbara, CA 93106-5070, and will be

ccepted until four months after final publication of the paper itself in the ASME

OURNAL OF APPLIED MECHANICS.

ournal of Applied Mechanics Copyright © 20

than an equal mass first order honeycomb sandwich panel. A simi-lar experimental study of hierarchical hexagonal honeycombs byLakes �6� demonstrated that the compressive strength of a secondorder honeycomb was three to four times greater than a first orderhoneycomb of equal mass.

Lakes �6� and Murphey and Hinkle �7� presented models for thestiffness and strength of hierarchical cellular materials and truss-like structures. They assumed a “continuum” model for the mate-rial at each length scale that led to simple recursive expressionsfor the stiffness and strength of the hierarchical structures. In do-ing so, they assumed macroscopic elastic or plastic buckling ofthe struts to be the only operative failure modes at each lengthscale. Short wavelength failure modes at the higher length scalesassociated with the discrete nature of the hierarchical structurewere neglected in these initial analyses. Nevertheless, optimiza-tions performed using these models predict substantial strengthincreases with increasing structural hierarchy.

The widespread adoption of hierarchical materials has been im-peded by difficulties in their manufacture and the high costs andavailability of very thin sheets of material. This constraint arisesbecause of the very large differences in the structural length scalesrequired to achieve optimal second and higher order systems. Inall but the largest macro-scale structures �e.g. the Eiffel tower� thisrequires the use of very thin gauge sheet materials for the webs ofthe structures at the smallest length scales.

Recent work in large-scale, ultralight structures that can resistdynamic loading has stimulated a renewed interest in hierarchicalcellular structures as cores for sandwich panels. Prismatic sand-wich core topologies �e.g., the corrugated or folded plate core� areideal for application in sandwich beams as they provide a highin-plane stretching strength. However, low relative density two-dimensional prismatic cores collapse by elastic buckling of thewebs and thus have a low strength to weight ratio. Increasing thecompressive strength of such prismatic corrugated cores offerspotential benefits in the design of shock resistant sandwich beams.Hierarchical construction is expected to delay the elastic bucklingof the webs of these prismatic sandwich cores and is thus attrac-tive for application in large sandwich structures �readers are re-ferred to Deshpande and Fleck �8� for details on the relation be-tween the bending strength of sandwich beams and the shear and

compressive strengths of the core�.

MARCH 2007, Vol. 74 / 25907 by ASME

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Here, we investigate the “effective” mechanical properties ofierarchical corrugated cores. Analytical models are developed forhe compressive and shear stiffness and strengths of first and sec-nd order corrugated sandwich panel cores. Expressions for theollapse strengths for six competing collapse modes are employedo generate collapse mechanism maps for a second order corru-ated sandwich core and to determine optimal designs that maxi-ize the collapse strength for a given core mass. A preliminary

xperimental investigation to validate the model predictions islso reported.

Analysis of a Corrugated Sandwich CoreTwo types of prismatic corrugated sandwich cores are consid-

red here. The first is a simple �first order� corrugated core com-rising struts of thickness t, and length l, with a corrugation angle, as illustrated in Fig. 1�a�. This type of corrugated core has no

tructural hierarchy and will be subsequently referred to as therst order corrugated core. A second order corrugated core thenas one level of structural hierarchy, i.e., the monolithic struts ofhe first order core are themselves replaced by sandwich panelsomprising face sheets of thickness t and a corrugated truss coreomprising struts of thickness t1 and length l1 corrugated at anngle �1. These truss core sandwich panels �columns� are thenorrugated at an angle � to form the second order corrugated core,ee Fig. 1�b�. The sandwich columns will be referred to subse-uently as the large struts of the second order core while the strutshat form the core of these sandwich columns will be referred tos the smaller struts of the second order core.

Here we derive analytical expressions for the effective trans-erse stiffness and strength of the first and second order cores,andwiched between two rigid face sheets. In all the subsequentnalyses, we assume that the corrugated cores are made from anlastic-ideally plastic material with a Young’s modulus Es, Pois-on’s ratio �, and yield strength �Y. The width b of the corrugatedore �into the plane of the paper� is assumed to be sufficientlymall for plane stress conditions to prevail in the struts.

2.1 First Order Corrugated Core. The relative density �theatio of the volume of corrugated core material to the volume ofanel� �̄, of the first order corrugated structure sketched in Fig.

ig. 1 Sketches of „a… the first and „b… the second order cor-ugated cores sandwiched between two rigid face sheets

�a� is, to first order in t / l, given by

60 / Vol. 74, MARCH 2007

�̄ =2

sin 2�� t

l� �1�

where � is the angle of the corrugation. For small t / l, the contri-bution to the overall stiffness from the bending of the constituentstruts is negligible compared to that from stretching. Thus, thestruts can be assumed to be pin jointed to the face sheets andanalytical expressions for the effective transverse Young’s E andshear G moduli are

E = Es�̄ sin4 � �2�and

G = Es�̄

4sin2 2� �3�

respectively, where Es is the Young’s modulus of the solid mate-rial from which the corrugated core is made. As discussed above,for small t / l, it is acceptable to assume that the struts are pinjointed at the face sheets. Thus, an equilibrium analysis dictatesthat the effective peak transverse compressive �P and shear �Pstrengths are specified by

�p = �c�̄ sin2 � �4�and

�p = �c�̄

2sin 2� �5�

respectively, where �c is the maximum compressive strength ofthe struts of the first order corrugated core. For struts made froman elastic-ideally plastic material, �c is given by

�c = � k2�2Es

12� t

l�2

ift

l�� 12�Y

�2k2Es

�Y otherwise �6�

The factor k depends on the end constraints of the struts with k=1 or k=2 corresponding to pin-jointed or built-in end conditions,respectively.

2.2 Second Order Corrugated Core. The relative density ofthe second order corrugated core, sketched in Fig. 1�b�, �to firstorder in t / l and t1 / l1� is given by

�̄ = 4� t

l� 1

sin 2�+ 4� t1

l� sin �1

sin 2�1 sin 2��7�

where the strut dimensions t, t1, l, and l1 along with the corruga-tion angles � and �1 are defined in Fig. 1�b�. Again, for small t / land t1 / l1, the contribution to the stiffness from the bending of thestruts is negligible. Thus, equilibrium dictates that the effectivetransverse Young’s and shear moduli of the second order corru-gated core are

E = Es

sin3 �

cos ��2t

l� = Es�̄ sin4 � − �2t1

l� sin �1 sin3 �

sin 2�1 cos �� �8�

and

G = Es sin 2�� t

l� = Es �̄

4sin2 2� − � t1

l� sin �1 sin 2�

sin 2�1� �9�

respectively. Since the second level of corrugations �the miniaturecorrugations in Fig. 1�b�� do not contribute to the stiffness, thesecond order corrugated core is less efficient than the first ordercore from a stiffness perspective; compare Eqs. �8� and �9� withthe corresponding expressions for the moduli of the first ordercore. This is consistent with the prediction of Lakes �6�, whosuggested that the stiffness to weight ratio of framework-typestructures decreases with increasing structural hierarchy.

2.2.1 Failure Modes of the Second Order Corrugated Core.

The second order corrugated core made from an elastic-ideally

Transactions of the ASME

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lastic material can fail by six competing collapse modes. Weonsider each of these modes in turn and derive analytical expres-ions for the effective transverse compressive and shear strengths,p and �p, respectively.

2.2.1.1 Plastic yielding of the larger struts. The larger strutsith face sheets of thickness t are subjected to compressive and/or

ensile stresses. Thus, these struts may fail by plastic yielding ofhe face sheets, Fig. 2�a�. Upon assuming all struts to be pinointed, upper bound estimates of the compressive and shear col-apse strengths in this mode are given by

�p

�Y= 2� t

l�tan � �10�

nd

�p

�Y= 2� t

l� �11�

espectively.

2.2.1.2 Euler buckling of the larger struts. Under compressiveoads, the larger struts can fail by Euler buckling as shown sche-

atically in Fig. 2�b�. Here we model these large struts as built-inuler columns with a second moment of area

I = 2t� l1 sin �1

2�2

�12�

pon assuming this failure mode, the compressive and shear col-apse strengths of the second order corrugated core are specifiedy

�p

�Y=

2�2

�Y� t

l�� l1

l�2

sin2 �1 tan � �13�

nd

�p

�Y=

2�2

�Y� t

l�� l1

l�2

sin2 �1 �14�

espectively, where �Y ��Y /Es is the yield strain of the solid ma-erial.

2.2.1.3 Shear buckling of the larger struts. The larger struts

ig. 2 Schematic drawings of the failure modes in the secondrder corrugated core „a… plastic yielding of the larger struts,b… Euler buckling of the larger struts, „c… shear buckling of thearger struts, „d… elastic wrinkling of the larger strut faceheets, „e… yielding of the smaller struts, and „f… Euler bucklingf the smaller struts

re sandwich columns comprising face sheets of thickness t and a

ournal of Applied Mechanics

corrugated core. Shear buckling �Fig. 2�c�� of these sandwich col-umns is set by the shear stiffness of the core, as discussed inZenkert �9�, and occurs at a load Ps

Ps = �AG�eq �15�

The equivalent shear rigidity of the core �AG�eq is given by

�AG�eq = bl1 sin �1E

2� t1

l1�sin 2�1 �16�

where b is the width of the second order corrugated core �into theplane of the paper in Fig. 1�b��. Thus, the compressive and shearcollapse strengths in this mode are

�p

�Y=

1

�Y� t1

l�sin2 �1 cos �1 tan � �17�

and

�p

�Y=

1

�Y� t1

l�sin2 �1 cos �1 �18�

respectively.

2.2.1.4 Elastic wrinkling of the larger strut face sheets. Asmentioned above, the larger struts of the second order corrugatedcore are sandwich columns comprising face sheets and a corru-gated core. Wrinkling is short wavelength elastic buckling �Fig.2�d�� of the face sheets of these sandwich columns. Sheets ofthickness t can buckle as pin-ended Euler columns between thepoints of attachment to the smaller corrugated core. The compres-sive and shear collapse strengths of the second order corrugatedcore are then given by

�p

�Y=

�2

24�Y� t

l�3� l

l1�2 tan �

cos2 �1�19�

and

�p

�Y=

�2

24�Y� t

l�3� l

l1�2 1

cos2 �1�20�

respectively. It is worth emphasizing here that the above analysisneglects the stresses introduced into the face sheets by the coreand thus may overestimate the wrinkling strength of the hierarchi-cal core. A full finite element analysis may be required to inves-tigate this effect.

2.2.1.5 Yielding of the smaller struts. The sandwich columnsof length l that form the large struts of the second order corrugatedcore are built in at the rigid faces, see Fig. 1�b�. Hence, shearforces develop in these sandwich columns which in turn can yieldthe smaller struts that form the core of the sandwich columns, seeFig. 2�e�.

Elementary elastic beam theory dictates that the ratio of theaxial force Fa to the shear force Fs in the sandwich columns oflength l is

Fa

Fs=

2

3� l

l1�2 1

sin 2��21�

Note that yielding of the smaller struts implies that the maximumshear force in the larger struts is

Fs max = 2b�Yt1 cos �1 �22�

An equilibrium analysis then gives the compressive and shearcollapse strengths of the second order corrugated core as

�p

�Y= 2� t1

l� cos �1

cos � 1

3 cos �� l

l1�2

+ cos �� �23�

and

MARCH 2007, Vol. 74 / 261

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2

�p

�Y= 2� t1

l� cos �1

cos � cos2 �

3 sin3 �� l

l1�2

+ sin �� �24�

espectively.

2.2.1.6 Euler buckling of the smaller struts. Elastic bucklingf the smaller struts can also result in collapse of the second orderorrugated core, Fig. 2�f�. The analysis is similar to that outlinedor mode �e� but with the maximum value of the shear force nowet by elastic buckling of the smaller struts as built-in Euler col-mns. Thus, the maximum value of Fs is given by

Fs�max =2

3b�2Es cos �1� t1

l1�2

t1 �25�

nd the corresponding compressive and shear collapse strengths ofhe second order corrugated core are

�p

�Y=

2�2

3�Y� t1

l�3� l

l1�2cos �1

cos � 1

3 cos �� l

l1�2

+ cos �� �26�

nd

�p

�Y=

2�2

3�Y� t1

l�3� l

l1�2cos �1

cos � cos2 �

3 sin3 �� l

l1�2

+ sin �� �27�

espectively.The regimes of dominance of the failure modes described

bove can be illustrated in a collapse mechanism map. In con-tructing such a map it is assumed that the operative collapseode in compression or shear is the one associated with the low-

st collapse strength. An example of such a collapse map for theompressive failure modes of a second order corrugated core with1 / l=0.04, �1=45 deg and �=60 deg, made from a solid materialith a yield strain �Y =0.004 is shown in Fig. 3 with the non-imensional axes t / l and t1 / l1. The regimes of dominance of theollapse modes are marked along with contours of the non-imensional collapse strength, �̄��p /�Y, and relative density, �̄:rinkling of the faces of the large struts and elastic buckling of

he smaller struts are the dominant failure modes for this choice of

ig. 3 Failure mechanism map for Al6061-T6 „�Y=0.004… sec-nd order corrugated core with l1 / l=0.04, �1=45 deg. The solidircle marks the geometry tested in this study. The arrows tracehe path of the optimum designs that maximize the compres-ive strength �p for a given relative density �̄.

aterial properties.

62 / Vol. 74, MARCH 2007

3 Experimental Investigation

3.1 Sample Fabrication. Test specimens were fabricatedfrom 6061-T6 aluminum and designed using the failure mecha-nism map shown in Fig. 3. The analytical predictions detailedabove suggest that the second order corrugated core may outper-form the first order core at relative densities where elastic buck-ling of the monolithic struts is the operative failure mode in thefirst order core. Thus, we expect the 6061-T6 aluminum secondorder corrugated core to only outperform the first order core atrelative densities �̄�0.05 �the yield strain of 6061-T6 aluminum�Y �0.004�. We thus restricted the current experimental study tothis range of densities. Further, the choice of sample dimensionswas restricted by �i� the minimum thickness of 6061-T6 sheet thatwas readily available �0.51 mm in this case� and to a lesser extent�ii� by the maximum sample size that would fit within the avail-able testing equipment. These constraints dictated a second ordercorrugated core marked by the filled circle in Fig. 3. We describethe manufacture and measurement of the compressive response ofa first and second order corrugated core each with a relative den-sity �̄�0.02.

First and second order corrugated core specimens of width b=152 mm �into the plane of the paper in Fig. 1� were manufac-tured. These specimens comprised only a single unit cell as de-scribed subsequently. The first order core was manufactured from2.3-mm-thick 6061-T6 aluminum sheets. These sheets were in-clined at an angle �=60 deg with respect to the horizontal planeand then slotted into mounting plates on the test machine, see Fig.4�a�. The second order corrugated core was also manufacturedfrom 6061-T6 aluminum sheets. First 0.51-mm-thick sheets werefolded to create a corrugated core with �1=45 deg and strutlength l1=12.7 mm. This corrugated core was then sandwichedbetween two 1.02-mm-thick aluminum sheets which were coatedwith a 0.13-mm-thick layer of a braze alloy with composition

Fig. 4 Photographs of the as-manufactured „a… first and „b…second order corrugated cores. In this study the manufacturedcores are comprised of a single unit cell.

Al-12Si wt%. The assembly was then bonded by dip brazing

Transactions of the ASME

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Coleman Microwave Co., Edinburg, VA�, and heat treated to the6 temper. These sandwich beams of length l=310 mm were then

nclined at an angle �=60 deg with respect to the horizontal andlotted into mounting plates on the test machine to form a singlenit cell of the second order corrugated core; a photograph of thes-manufactured second order corrugated unit cell is shown inig. 4�b�. Table 1 gives the dimensions, and measured relativeensities of the manufactured first and second order corrugatedore specimens.

3.2 Measurements. Tensile specimens of dog-bone geometryere cut from a solid bar of AA6061-T6 and used to determine

he tensile mechanical response of the alloy at a nominal appliedtrain rate of 10−3 s−1. The measured true tensile stress versusogarithmic strain response is shown in Fig. 5 and can be ad-quately approximated as elastic-ideally plastic with Young’sodulus Es=69 GPa and a yield strength �Y �250 MPa.The first and second order samples were tested in compression

t nominal applied strain rate of 10−3 s−1. The measured load cellorce was used to define the nominal applied stress while theominal strain was obtained from a laser extensometer. The mea-ured compressive responses of the first and second order samplere plotted in Fig. 6 using axes of nominal stress � and nominaltrain �. The second order corrugated core achieved an initial peaktress of 1.0 MPa at an applied strain ��0.005. Face wrinkling ofhe face sheets of the large struts then set in which resulted in aharp drop in the stress. Subsequently, the stress increased until aecond wrinkle formed at approximately 0.8 MPa. The tworinkles are clearly visible in the photograph of the deformed

econd order specimen ���0.007� shown in Fig. 7�a�. This oscil-atory behavior continued until complete failure of the speciment an applied strain ��0.012. In contrast, the peak strength of therst order core was approximately 0.08 MPa and was controlledy elastic buckling of the struts �Fig. 7�b��.

In line with the collapse mechanism map in Fig. 3, the second

able 1 Measured dimensions and relative densities of com-ression test samples. Both the specimens have a width b152 mm.

pecimenype

Corrugation angle Strut dimensions �mm� Relativedensity

�̄� �1 t l t1 l1

st order 60 deg ¯2.3 311.2

¯ ¯0.017

nd order 60 deg 45 deg 1.02 310 0.051 12.7 0.018

Fig. 5 Uniaxial tensile response of the Al-6061-T6

ournal of Applied Mechanics

order core collapses by face wrinkling. This switch in the collapsemode from elastic buckling of the struts in the first order core toface wrinkling in the second order core results in a measured peakstrength of the second order core that is about 12.5 times greaterthan that of a first order core of equal mass. The analytical pre-dictions of the peak strengths of the first and second order corru-gated cores are included in Fig. 6. Both the pin-ended �k=1� andbuilt-in �k=2� elastic buckling predictions for the first order coreare plotted in Fig. 6. The observed deformation mode �cf. Fig.

Fig. 6 Measured compressive nominal stress versus nominalstrain responses of the �̄É0.02 first and second order corru-gated cores

Fig. 7 Photographs showing the failure modes of „a… second

„�É0.007… and „b… first „�É0.20… order corrugated cores

MARCH 2007, Vol. 74 / 263

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�b�� and the measured peak strength of the first order core agreeell with the pin-ended elastic buckling predictions. The analyti-

al model overestimates the �elastic wrinkling controlled� strengthf the second order core. However, the buckling strength of strutss highly sensitive to imperfections at the transition between thelastic buckling and plastic yielding modes �10�. Figure 3 clearlyhows that the design of the second order core lies at the boundaryetween face-sheet wrinkling and the plastic yielding collapseode. Thus, the measured strength is anticipated to be sensitive toanufacturing imperfections and be over predicted by the analyti-

al bifurcation calculations.

Collapse Mechanism Maps and Optimization of theecond Order Corrugated CoreThe geometry of the second order corrugated core can be opti-ized to maximize the shear and/or compressive collapse

trengths at a given relative density. To simplify the optimizationroblem, we restrict attention to the case of both the smaller andarger corrugations having equal angles, i.e., �=�1 �it will be seenubsequently that the choice �=�1=45 deg is optimal from aractical perspective�. Thus, the optimization problem under con-ideration can be stated as follows. Given

�i� the solid material �i.e., fixed value of yield strain, �Y�,�ii� the corrugation angle �=�1, and�iii� the effective relative density �̄,

hat are values of the non-dimensional strut aspect ratios t / l,1 / l1, and l1 / l that maximize the compressive or shear collapsetrengths of the second order corrugated core? Unless otherwisepecified, all results presented subsequently are for the choice oforrugation angles �=�1=45 deg and a solid material yield strainY =0.002; this yield strain is representative of many structuraletallic alloys.For a prescribed length scale separation l1 / l, the optimal design

s obtained by selecting a geometry �t / l , t1 / l1� that maximizes thetrength for a given value of �̄. To help with this optimization,ollapse mechanism maps for the compressive failure of the sec-nd order corrugated core are shown in Figs. 8�a� and 8�b�, for thehoices l1 / l=0.01 and 0.03, respectively. Euler buckling of themaller and larger struts are the dominant failure modes for the1 / l=0.01 corrugated core while Euler buckling of the smallertruts and face wrinkling and plastic yielding of the larger strutsominate the collapse mechanism map of the l1 / l=0.03 corru-ated core. To help select the optimum geometries, contours of theormalized strength �̄��p /�Y and relative density �̄ have beendded to the maps. We note that the optimal designs that maxi-ize �p /�Y for any given value of �̄ lie along the boundaries of

he collapse regimes. The arrows sketched in Fig. 8 designate theath of optimum designs with increasing �̄.

The optimized compressive strengths of the second order cor-ugated cores are plotted in Fig. 9�a� as a function of �̄, for thehoices l1 / l=0.01, 0.03 and 0.05. The results reveal that while thehoice l1 / l=0.01 maximizes the strength at low relative densities�̄�0.004� the choice l1 / l=0.03 is preferable at the higher valuesf �̄. This suggests that the performance of the second order coreay be improved by including l1 / l as an optimization variable.e shall refer to the optimal designs with l1 / l included in the

ptimizations as fully optimized while optimal designs obtained byxing the value of l1 / l as suboptimal.The fully optimized compressive strength of the second order

ore is plotted in Fig. 9�a� along with the strength versus relativeensity relation of the first order corrugated core �Eqns. �4� and6��. For the choice �Y =0.002, the first order corrugated core col-apses by elastic buckling of the struts for �̄�0.05 and by plasticielding of the struts at higher values of �̄. Thus, the strength ofhe first order core scales linearly with relative density for �̄

¯3

0.05 and is proportional to � at lower relative densities. Ap-

64 / Vol. 74, MARCH 2007

propriate designs of the second order corrugated cores increasethe strength of the struts at low values of �̄: the strength of thefully optimized second order core scales linearly with relativedensity for all �̄0.005. Thus, the fully optimized second ordercore substantially outperforms the first order core for 0.001��̄�0.05. In fact, even the suboptimal second order designs outper-form the first order core. For example, the suboptimal secondorder cores with l1 / l=0.03 and 0.05 have a performance approxi-mately equal to the fully optimized design for �̄0.008 and �̄0.015, respectively, and outperform the first order core over thewhole range of relative densities investigated here. However, it isworth noting that an inappropriately designed second order corecan have a lower strength to weight ratio than the first order core;e.g., the strength of the l1 / l=0.01 suboptimal second order core isbelow that of the first order core for �̄0.03. In this range thesuboptimal designs of the l1 / l=0.01 second order core collapse bya combination of elastic buckling of the larger and smaller struts

Fig. 8 Collapse mechanism maps for the compressive failureof the �=�1=45 deg second order core with „a… l1 / l=0.01 and„b… l1 / l=0.03. The arrows trace the path of the optimum designsthat maximize the compressive strength �p for a given relativedensity �̄. The yield strain of the solid material is taken to be�Y=0.002.

or elastic buckling and shear buckling of the larger struts. On the

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ther hand, for �̄0.05, the first order core collapses by plasticielding of the struts and thus outperforms this suboptimal designf the second order core. The geometrical parameters t / l, t1 / l1,nd l1 / l associated with the fully optimized design are plotted inig. 9�b� and clearly show that the optimal geometry is not self-imilar, i.e., the relative density at each level of hierarchy is notqual as t / l� t1 / l1.

It is worth emphasizing that the second order corrugated corean only outperform the first order corrugated core at low relativeensities when elastic buckling is the collapse mechanism for therst order core. In fact, at relative densities where plastic yielding

s the collapse mechanism of the first order core, the second orderore will be structurally less efficient than the first order core ashe smaller scale core contributes little to the strength but doesncrease the overall mass. In the optimal designs discussed above,he mass of the smaller scale core is very small at the higher

ig. 9 „a… Comparison between the normalized compressivetrengths of the optimized �=�1=45 deg second order corru-ated cores and the �=45 deg first order core. The yield strainf the solid material is taken to be �Y=0.002. „b… The corre-ponding geometries of the fully optimized second order core.

elative densities and thus the second and first order cores have a

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comparable performance at the higher densities. We do not expectsecond order corrugated cores to find application at these highrelative densities.

The corrugation angle significantly effects the competition be-tween collapse mechanisms. Compressive and shear collapsemechanism maps for second order corrugated cores with �=�1=70 deg and l1 / l=0.03 are plotted in Figs. 10�a� and 10�b�, re-spectively. The compressive and shear collapse mechanism mapsare quite similar with the main difference being that the elasticbuckling of the smaller struts occupies a larger fraction of theshear collapse mechanism map. Contours of the normalized com-pressive and shear strengths are included in Figs. 10�a� and 10�b�,respectively along with contours of the relative density. The ar-rows in these figures again trace the path of the suboptimal secondorder corrugated core designs that maximize the compressive andshear strengths with l1 / l=0.03 and �=�1=70 deg.

The effect of the corrugation angle on the fully optimized com-pressive and shear strengths of the second order corrugated cores

Fig. 10 „a… Compressive and „b… shear failure mechanismmaps for the �=�1=70 deg second order corrugated core withl1 / l=0.03. The arrows trace the path of the optimum designsthat maximize the strengths for a given relative density. Theyield strain of the solid material is taken to be �Y=0.002.

is illustrated in Figs. 11�a� and 11�b�, respectively. Results are

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resented for �=�1=30, 45, and 70 deg. The compressivetrength increases with increasing corrugation angle, while thehear strength is a maximum for �=�1=45 deg. It is worth notinghat the analytical formulas for the compressive and shear-trengths are identical in all six competing collapse modes for thease �=�1=45 deg. Thus, the values of the geometrical param-ters plotted in Fig. 9�b� optimize both the compressive and sheartrengths of the second order corrugated core. On the other hand,he geometrical parameters that maximize both the compressivend shear strengths of the �=�1=30 and 70 deg are not identical.his is illustrated in Fig. 12 where the minimum weight designs of

he �=�1=70 deg second order corrugated core for both com-ressive �red lines� and shear �black lines� loading are plotted.hile the values of l1 / l and t / l are approximately equal in the

ptimized compression and shear designs, t1 / l1 is higher in theptimized shear design compared to the corresponding value inhe optimized compression design. If in practice the through thick-ess compression and the transverse shear strengths are equallymportant parameters, the choice �=�1=45 deg is desirable ashe same design maximizes both the shear and compressive

ig. 11 Comparison between the „a… compressive and „b…hear strengths of the fully optimized second order and firstrder corrugated cores. Results are shown for three selectedalues of the corrugation angle �=�1 and the yield strain of theolid material is taken to be �Y=0.002.

trengths.

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5 Comparison With Competing Core TopologiesIt is instructive to compare the peak compressive strength �p of

the fully optimized second order core with competing sandwichcores. In particular, we compare the performance of the secondorder corrugated core with �i� a square-honeycomb core, Côté etal. �11�, �ii� prismatic-diamond core, Côté et al. �12�, �iii� a three-dimensional pyramidal core, Wadley et al. �13� and �iv� the firstorder corrugated core. In all cases we assume that the sandwichcores are made from an elastic-ideally plastic solid with yieldstrength �Y and yield strain �Y. All these cores collapse by elasticbuckling of the cell walls at low relative densities and plasticyielding at higher values of the �̄. Thus, piecewise expressions forthe strengths of these cores are specified below with the transitionbetween the elastic buckling and plastic yielding modes set byboth the geometry of the core and the yield strain of the solidmaterial from which the core is made.

The normalized peak strength �̄��p /�Y of the square-honeycomb core is given by �11�

�̄ = � �2

12�1 − �2��Y�̄3 if �̄ ��12�1 − �2��Y

�2

�̄ otherwise� �28�

while the strength of the prismatic-diamond core with square cellsis specified as �12�

�̄ = ��2

96�Y�̄3 if �̄ ��48�Y

�2

�̄

2otherwise � �29�

The three-dimensional pyramidal core with struts inclined at45 deg to the horizontal plane is more resistant to elastic bucklingcompared to its two-dimensional or prismatic counterparts �for agiven relative density, the struts of the 3D pyramidal core aremore stocky than those in a prismatic core�. An equilibrium analy-

Fig. 12 The geometries of the fully optimized �=�1=70 degsecond order corrugated core. The geometries that maximizethe compressive „red lines… and shear „black lines… strengthsare included.

sis dictates that the strength of this pyramidal core is

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�̄ =��2

24�2�Y

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comparison between the normalized peak compressive strengthsf the competing cores is shown in Figs. 13�a� and 13�b� for thehoices of solid material yield strains �Y =0.002 and 0.02, respec-ively. The strengths of the fully optimized �=�1=45 deg secondrder corrugated core and the �=45 deg first order corrugatedore have also been included in Fig. 13. With the choice �Y0.002, the fully optimized second order core outperforms allther two-dimensional cores �i.e., square-honeycomb, prismatic-iamond and the first order corrugated core� for relative densities�0.03. However, the high elastic buckling strength of the three-imensional pyramidal core ensures collapse by plastic yieldingor �̄0.003. Thus, the second order corrugated core has no per-ormance gain over the pyramidal core and in fact has a slightlyower collapse strength than the pyramidal core for �̄�0.005. Inrder to make a prismatic core such as the corrugated core out-

ig. 13 Comparison between the compressive strengths ofompeting sandwich core topologies for the solid materialield strains „a… �Y=0.002 and „b… �Y=0.02

erform the pyramidal core, at least three levels of structural hi-

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erarchy would be needed. It is also worth mentioning that with thechoice �Y =0.002, the three-dimensional pyramidal core collapsesby elastic buckling only for �̄�0.003. Thus, three-dimensionalhierarchical construction will only provide benefits for very lowrelative densities ��̄�0.003�. Structural hierarchy is more effec-tive in two-dimensional or prismatic cores where such construc-tion provides performance enhancements for relative densities �̄�0.05.

The effect of increasing the yield strain to �Y =0.02 on the per-formance of the competing sandwich cores is illustrated in Fig.13�b�. Elastic buckling becomes the operative collapse mode athigher relative densities and thus hierarchical construction givesperformance enhancements at even higher values of relative den-sity. For example, with �Y =0.02, the fully optimized second ordercorrugated core outperforms all the competing two-dimensionalcores for all �̄�0.10.

The high out-of-plane strength of 3D cores such as the pyrami-dal truss means that 3D cores have definite advantages over theirprismatic counterparts. However, unlike the pyramidal truss, pris-matic topologies like a corrugated core have a high in-planestretching resistance that is critical in applications such asclamped sandwich beams. The use of structural hierarchy to im-prove the out-of-plane compressive strength of such prismaticcores thus offers the attractive possibility of designing a low rela-tive density core with high out-of-plane as well as in-planestrengths.

6 Concluding RemarksAnalytical models for the transverse stiffness and strength of

second order corrugated cores have been presented. The collapseof the second order corrugated core occurs by six competing col-lapse modes. Analytical expressions for the collapse strengths ineach of these modes have been employed to determine the optimaldesign of second order corrugated cores that maximize the com-pressive and shear strengths for a prescribed effective relativedensity of the core. Second order corrugated cores made fromstructural alloys have significantly higher collapse strengths com-pared to their equivalent mass first order counterparts for relativedensities �̄�0.05. The fully optimized second order core has astrength that scales linearly with relative density for �̄0.005 andthus it has a collapse strength about two orders of magnitudegreater than its first order counterpart at �̄=0.005. Increasing theorder of the hierarchy to greater than two can increase the collapsestrength only at relative densities �̄�0.005. In contrast, increasingthe level of structural hierarchy yields no enhancements in thestiffness of the corrugated core. In fact, the stiffness to weightratio of the first order core is slightly greater than its second ordercounterpart suggesting that the hierarchical corrugated construc-tion has applications in strength limited applications.

Representative first and second order corrugated core structureshave been fabricated from a high strength 6061-T6 aluminum al-loy using a simple sheet folding and dip brazing process. Thenon-optimal second order panel with a relative density of 1.7% isfound to have a measured compressive strength about 12.5 timeshigher than that of a first order structure with a similar relativedensity. These results are consistent with analytical predictionsand suggest that hierarchical corrugated cores have a high poten-tial in strength limited sandwich panel applications.

AcknowledgmentWe are grateful to the Office of Naval Research for support of

the work reported here under Grant No. N00014-01-1-1051 �pro-gram manager, Steve Fishman�. The authors would also like tothank one of the reviewers for insightful comments that have

helped to improve the paper.

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eferences�1� Hashin, Z., and Shtrikman, S., 1963, “A Variational Approach to the Theory of

the Elastic Behavior of Multi-Phase Materials,” J. Mech. Phys. Solids, 11, p.127.

�2� Norris, A. N., 1985, “A Differential Scheme for the Effective Moduli of Com-posites,” Mech. Mater., 4, p. 1.

�3� Milton, G. W., 1986, “Modeling the Properties of Composites by Laminates,”in Homogenization and Effective Moduli of Materials and Media, J. L. Erick-sen, D. Kinderlehrer, R. Kohn, and J.-L. Lions, eds., Springer, New York, p.150.

�4� Francfort, G., and Murat, F., 1986, “Homogenization and Optimal Bounds inLinear Elasticity,” Arch. Ration. Mech. Anal., 94, p. 307.

�5� Bhat, T., Wang, T. G., and Gibson, L. J., 1989, “Micro-Sandwich Honey-comb,” SAMPE J., 25�3�, p. 43.

�6� Lakes, R., 1993, “Materials With Structural Hierarchy,” Nature �London�,361�11�, p. 511.

�7� Murphey, T., and Hinkle, J., 2003, “Some Performance Trends in Hierarchical

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Truss Structures,” 44th AIAA/ASME/ASCE/AHS/ASC Conference, Norfolk,VA, Apr. 7–10, 1903.

�8� Deshpande, V. S., and Fleck, N. A., 2001, “Collapse of Truss-Core SandwichBeams in 3-Point Bending,” Int. J. Solids Struct., 38, p. 6275.

�9� Zenkert, D., 1995, An Introduction to Sandwich Construction, EngineeringMaterials Advisory Service, Sheffield, UK, pp. 30–60.

�10� Hutchinson, J. W., 1974, “Plastic Buckling,” in Advances in Applied Mechan-ics, C.-S. Yih, ed., Academic, New York, vol. 14, p. 67.

�11� Côté, F., Deshpande, V. S., Fleck, N. A., and Evans, A. G., 2004, “The Out-of-Plane Compressive Behavior of Metallic Honeycombs, Mater. Sci. Eng., A,380, p. 272.

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�13� Wadley, H. N. G., Fleck, N. A., and Evans, A. G., 2003, “Fabrication andStructural Performance of Periodic Cellular Metal Sandwich Structures,”Compos. Sci. Technol., 63, p. 2331.

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