International Journal of Non-Linear Mechanics 81 (2016) 177–196

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International Journal of Non-Linear Mechanics

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Non-linear response of curved sandwich panels – extendedhigh-order approach

Y. Frostig a,n, G.A. Kardomateas b, N. Rodcheuy b

a Ashtrom Engineering Company Chair Professor in Civil Engineering, Faculty of Civil and Environmental Engineering, Technion-Israel Institute of Technology,Haifa 32000, Israelb School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, USA

a r t i c l e i n f o

Article history:Received 22 October 2015Received in revised form4 January 2016Accepted 19 January 2016Available online 27 January 2016

Keywords:Sandwich constructionCurved panelCore in-plane rigidityHigh-orderNon-linear response

x.doi.org/10.1016/j.ijnonlinmec.2016.01.01162/& 2016 Elsevier Ltd. All rights reserved.

esponding author.

a b s t r a c t

The geometrical non-linear behavior a curved sandwich panel with a stiff or compliant core whensubjected to a pressure load using the Extended High-Order Sandwich Panel theory (EHSAPT), is pre-sented. The formulation follows the EHSAPT procedure where the in-plane. i.e circumferential rigidity ofthe core is considered and the distribution of the displacements through the depth of the core arepresumed. These displacement distributions are the closed-form solutions of the 2D governing equationsof the curved core without circumferential rigidity that appear in the HSAPT curved sandwich panelmodel. The mathematical formulation includes the field equations along with the appropriate boundaryand continuity conditions that take into account the high-order stress resultants in the core due to thepresumed distributions. Finally a numerical study is conducted for a panel loaded by a distributedpressure at the upper face sheet. It reveals that the post-buckling response of a curved sandwich panels isassociated with shallow to deep wrinkling deformations of the upper face sheet in the case of a simply-supported panel or a general non-linear pattern without wrinkles in the case of pinned supports with ashort span. In both cases a stable post-buckling response is observed similar to that of a plate one.

& 2016 Elsevier Ltd. All rights reserved.

1. Introduction

In aerospace, naval and transportations industries, whereweight savings combined with high strength and stiffness prop-erties are always required, the use of curved sandwich structuresis increasing. In general, sandwich panels are made of two thinface sheets, metallic or laminated composites, bonded to a corethat is often made of honeycomb, polymer foam or balsa wood.The core usually provides the shear resistance/stiffness to thesandwich structure in the radial direction, and a kind of elasticradial support to the face sheets that yields radial normal stresses.Polymer foams of low density or low strength honeycomb cores,which are flexible in the radial direction and very flexible relativeto the rigidities of the face sheets, may affect the global responseas well as the local one through changes of the core height and thecore cross section plane which may take a nonlinear deformedpattern. In general, when using low strength foams or honeycombof any material the in-plane rigidity of the core is neglected.However, when high density foam, solid foams or balsa wood areused, the circumferential (in-plane) rigidity of the core must beconsidered in addition to the shear and the radial rigidities. Also

with heavy foams or solid materials the core undergoes dis-placements pattern that change the height of the core and distortsits cross section plane, denoted as high-order effects. The inclusionof this in-plane rigidity, based on presumed displacement patternsthat are non-polynomial which yields non-conventional high-order stress resultants, for the non-linear response of curvedsandwich panel is one of the major goals of the paper.

Analyses of flat sandwich panels that appear in the literaturecan be cataloged by two major categories. The first one assumesthat the cores are an anti-plane type, i.e. very stiff in the verticaldirections (incompressible) and with negligible in-plane rigidity inthe longitudinal direction (metallic honeycomb), see for examplethe textbooks by Allen [1], Plantema [27] and Vinson [35]. Suchpanels may be modeled by computational models such as First orHigh order shear deformable approaches that replace the layeredpanel with an equivalent single layer, ESL, and assume that thecore is incompressible. The second approach is denoted as layeredapproach and it is described through high-order models, see forexample Frostig et al. [13] that solved the core fields in a closed-form or Carrera and Brischetto [7] that presumed the displace-ments fields of the core. In such models the overall response is acombination of the responses of the face sheets and the corethrough equilibrium and compatibility.

Nomenclature

ESL Equivalent Single LayerFOSDT First-Order Shear Deformable TheoryHSAPT High-Order Sandwich Panel TheoryEHSAPT Extended High-Order Sandwich Panel TheoryODEs Ordinary Differential Equationsα Angle of curved panelβj (j¼t,b)section plane slope of face sheetsδ Variational operatorδd Dirac functionεoj, χj (j¼t,b) mid-plane strain and displacement curvature of

face sheetsθ*ej (j¼t,b) angle location from left side of concentrated load at

one of face sheetsηj (j¼t,b)mathematical quantityλjk (j¼t,b,k¼u,w) Lagrange Multiplier at face–core interfaces in

circumferential and radial directionsmrsc, msrc Poisson ratio of core in circumferential and radial

directions þof core

σssj, εssj (j¼t,b,c) longitudinal normal stresses and strains inface sheets and core

σrrc, εrrc radial normal stresses and strains in coreτrsc, γrsc shear stresses and shear angle in coreτcj,σcj (j¼t,b) shear and vertical normal stresses at upper and

lower face–core interfacesζ¼0,1 boundary condition parameterbw width of panelc thickness of coredj (j¼t,b) thickness of face sheetse subscript that defines external loading magnitude and

locationEAj, EIj (j¼t,b) in-plane (circumferential) and flexural rigidities

of face sheetsEcr, Ecs,Gcmodulus of elasticity in radial and circumferential

directions and shear modulus of core

fk¼∂f/∂k (k¼x,z) function derivative with respect to variousvariables

GAc shear rigidity of panelj¼t,b,c face sheets and core subscript indicesD(wj)(ϕ) or Dwj (j¼t,b) slope of radial displacement of

face sheetnj; qj (j¼t,b) External distributed loads in the circumferential

and radial directions, respectively,Nej; Pej;Mej (j¼t,b) External concentrated loads in the cir-

cumferential and radial directions applied at theface sheets

Nssj, Mssj, Vsrj (j¼t,b) in-plane (circumferential), bendingmoment and radial shear stress resultants of theface sheets

Nssc, Mssc, Qsrc in-plane (circumferential), bending moment andradial shear stress resultants of the core

Mss2c,Mss3c,MQsr1c,MQsr2c,MRrc, Rrc High-Order stress resultantsquantities in core

Nssej, Mssej,Vsrej (j¼t,b,c) Equivalent stress resultants: in-plane(circumferntial), bending moment, and radial shear offace sheets and core

r,θ radial and circumferential coordinates of panelrk (k¼t,b,ce,ct,cb) radius of centroid of upper face sheet, lower

face sheet and core and radii of face–core interfacesrespectively

U,Uλ,V potential energy of strain, constraints energy andexternal potential energy of loads

uojðθ; rjÞ;wjðθ; rjÞ;βjðθ; rjÞ (j¼t,b) Mid-plane circumferential,radial and section plane slope the face sheets

uc (θ,rc), wc (θ,rc) circumferential and radial displacements ofthe core

ukðθ; rcÞðk¼ o;1::3Þ;wkðθ; rcÞðk¼ o;1::2Þ Displacement patternsin core of circumferential and radial displacements

y Coordinate in width directionzj (j¼t,b) radial coordinate of each face sheetsComment Letters without indices refer to overall quantities

Y. Frostig et al. / International Journal of Non-Linear Mechanics 81 (2016) 177–196178

Research on curved or shell sandwich panel includes ESLcomputational models that consider an incompressible core fol-lowing the First-Order Shear Deformable (FOSDT) Theory due toReissner or Reissner–Mindlin works. This approach has becomethe basis for a large number of research works and to mention afew: Whitney and Pagano [36] and Noor et al. [24,25] for flat andcurved panels respectively; Kollar [19] investigated buckling ofgenerally anisotropic shallow sandwich shells and Vaswani et al.[34] performed vibration and damping analysis of curved sand-wich beams. The last two models use Flügge shell theory whileassuming that the face sheets are membranes only and the core isincompressible. Shallow cylindrical sandwich panels with ortho-tropic surfaces adopting the FOSDT models has been investigatedby Ying-Jiang [37], but different from the others the face sheethere have membrane and flexural rigidities. Similarly using theprinciple of virtual work along with the FOSD model along withSanders non-linear stress-displacement relations, Di Sciuva [9]and Di Sciuva and Carrera [10] have developed a model that takeinto account the shear deformation but assuming that the core isincompressible and linear. A stability analysis for cylindricalsandwich panels with laminated composite faces based on theReissner hypothesis has been derived by Rao and Meyer-Piening[28,29] again with an incompressible core and membrane facesheets…

A class of high-order models based on the assumption of cubicand quadratic or trigonometric through-thickness distributions for

the displacements have been suggested and summed in a reviewby Librescu and Hause [21], mainly on incompressible core. Forcylindrical shells, with a compressible core, Hohe and Librescu[22] used a cubic and quadratic polynomial tangential and radialdisplacements distributions respectively while in Hohe andLibrescu [23] a quadratic distribution has been used for the dis-placement distributions. All the referenced high-order models useintegration through the thickness along with variational principle.

A different approach that includes the effect of the transverse(radial) normal stresses on the overall behavior of sandwich shellshas been considered by Kuhhorn and Schoop [20]. They usedgeometrically non-linear kinematic relations along with pre-assumed polynomial deformation patterns for plates and thesame for shells. The effect of the compressibility of the core usingthe HSAPT approach has been implemented in a number ofresearch works on curved sandwich panel and to mention a few:Bozhevolnaya and Frostig [3] for non-linear behavior, Bozhe-volnaya's Ph.D. thesis [4] on shallow sandwich panels, Karayadi'sPh.D. thesis [17] on cylindrical shells, Frostig [14] on the linearbehavior of curved sandwich panels, Bozhevolnaya and Frostig[5] on the free vibration of curved panels, Thomsen and Vinson[33] on composite sandwich aircraft fuselage structures. Recently,thermal effects that include induced deformation as well asdegradation of properties have been implemented in the analysesof curved sandwich panel, see Frostig and Thomsen [15] on fullybonded panels and Frostig and Thomsen [16] for panels with a

Y. Frostig et al. / International Journal of Non-Linear Mechanics 81 (2016) 177–196 179

debond at the upper face–core interface. The use of the HSAPT forthe analysis of strengthened masonry or reinforced concretearches by composite laminated strips appears in Dvir thesis [11]and in Dvir and Rabinovitch [12] using special finite elements.Recently the effect of the in-plane core rigidity of flat sandwichpanels has been implemented in a series of papers through the useof the EHSAPT model, see for example Phan et al. [26] and Rod-cheuy et al. [30] that include also comparison of the EHSAPTmodel with an elasticity closed-form solution. The use of a poly-nomial distribution through the thickness of the core is quiteaccurate for a flat panel but is inaccurate for a curved panel, seecomparison of these polynomial distributions with elasticityclosed-form solutions in see Rodcheuy et al. [30]. Hence, a for-mulation that modifies the EHSAPT computational model to bevalid for curved sandwich panels is required and it ispresented ahead.

The mathematical formulation presented in this paper uses theextended high-order sandwich panel theory (EHSAPT) to modelthe non-linear response of a curved sandwich panel. The sandwichpanel is modeled as two curved faces, with membrane and flexuralrigidities following the Euler–Bernoulli hypothesis, that are inter-connected through compatibility and equilibrium with a 2-Dcompressible or extensible elastic core with shear, radial and cir-cumferential resistance. The EHSAPT model for the curved paneladopts the following assumptions: The face sheets have in-plane(circumferential) and bending rigidities with kinematic relationsof moderate deformations (see Brush and Almroth [6] and Simitses[31]) and negligible shear deformations; The core is considered as:a 2D linear elastic continuum obeying kinematic relations of smalldeformation and where the core height may change and the sec-tion plane does not remain plane after deformations; The core isassumed to possess shear, radial and circumferential normal stressrigidities; Full bond between the face sheets and the core isassumed and the interfacial layers can resist shear as well as radialnormal stresses; The loads are applied to the face sheets only.

The next part of the manuscript consists of: the mathematicalformulation that describes the field and governing equations, theappropriate boundary conditions and the a numerical investiga-tion of the non-linear response of particular curved panels. Finallya summary is presented and conclusions are drawn.

2. Mathematical formulation

The field equations and the boundary conditions are definedfollowing the steps of the HSAPT approach for the curved sand-wich panel, see Frostig [14] and Bozhevolnaya and Frostig [3]. Thefield equations are derived using the variational principle ofextremum of the total potential energy as follows:

δðUþVÞ ¼ 0 ð1Þ

Fig. 1. Dimensions, signs conventions and loads of a curved san

where U and V are the internal potential strain energy and theexternal potential one, respectively.

The internal potential energy in polar coordinates of the panelreads:

δ U ¼Z α

0

Z dt=2

�dt=2

Z 1=2 bw

�1=2 bwσsst θ; zt� �

δ εsst θ; zt� �

rt dy dzt dθ

þZ α

0

Z db=2

�db=2

Z 1=2 bw

�1=2 bwσssb θ; zb

� �δ εssb θ; zb

� �rb dy dzb dθ

þZ α

0

Z rct

rcb

Z 1=2 bw

�1=2 bw

τrsc θ; rc� �

δ γrsc θ; rc� �þσrrc θ; rc

� �δ εrrc θ; rc

� �þσssc θ; rc

� �δ εssc θ; rc

� � !

�rc dy drc dθ ð2Þwhere σssj(θ,rj) and εssj(θ,rj) (j¼t,b) are the normal in-planestresses and strains of the face sheets, respectively; τrs(θ,rc),σrr(θ,rc), σss(θ,rc) and γrs(θ,rc), εrr(θ,rc), εss(θ,rc) are the shear, radialand circumferential stresses respectively and the correspondingshear angle strain and normal strains in both directions respec-tively in the core; rc is the radial coordinate of the core through itsdepth, r and s¼rθ refer to the radial and circumferential directionsof the curved panel; α is the total angle of the curved panel, rj (j¼t,b) are the radii of the centroidal lines of the face sheets, respec-tively; rcj (j¼t,b) refers to the radii of the upper and the lowerinterface line, respectively, where rct¼rt�dt/2 and rcb¼rbþdb/2;bw and dj(j¼t,b) are the width and the thicknesses of the facesheets, respectively, and δ is the variational operator. For geo-metry, sign conventions, coordinates, deformations and internalresultants, see Fig. 1.

The variation of the external potential energy read:

δ V ¼ �Z α

0ntδ uot θ

� �þqtδ wt θ� �� �

rt dθ�Z α

0nbδ uob θ

� ��

þqbδwb θ� ��

rb dθ�Xαφe ¼ 0

Net θe� �

δ uot θe� ��Pet θe

� �δ wt θe

� �

þMet θe� �

δ βt θe� ��Neb θe

� �δ uob θe

� ��Peb θe� �

δ wb θe� �

þMeb θe� �

δ βb θe� �

where

βj θe� �¼ uoj θe

� ��D wj� �

θe� �

rjð3Þ

where nj, qj and mj (j¼t,b) are the external distributed loads in thecircumferential and radial directions, respectively, and the dis-tributed bending moment applied at the face sheets; uoj and wj

(j¼t,b) are the circumferential and radial displacements of the facesheets, respectively; βj is the slope of the section of the face sheet;Nej, Pej and Mej (j¼t,b) are the external concentrated loads in thecircumferential and radial directions, respectively, and the bendingmoment applied at the edges (θe¼0,α) of the various face sheets.For sign conventions and definition of loads see Fig. 1.

dwich panel: (a) geometry; (b) loads applied at face sheets.

Fig. 2. Stress resultants and interfacial stresses on the deformed shape (face sheetonly) of a sandwich panel segment.

Y. Frostig et al. / International Journal of Non-Linear Mechanics 81 (2016) 177–196180

The displacements pattern of the face sheets through theirdepth follows the Euler–Bernoulli assumptions and they read,(j¼t,b):

uj θ; zj� �¼ uoj θ

� �þβjðθÞzj where βj θ� �¼ ðuoj θ

� �� ddθ

wj θ� �Þ=rj ð4Þ

where uoj(θ) and βj(θ) are the in-plane displacements of thecentroid of the face sheets section and its rotation respectively; zjis the radial coordinate measured upward from the centroid ofeach face sheet, θ is the angle measured from origin, see Fig. 1 forgeometry. Hence, the strain distributions using kinematic relationsof moderate deformations read ( j¼t,b):

εssj θ� �¼ εossj θ

� �þχ j θ� �

zj ð5Þ

where the mid-plane strain includes a non-linear term and thecurvature equal:

εss;oj ¼ddθuoj θ

� �þwj θ� �

rjþ1=2 β2

j ; χ j θ� �¼ d

dθβj θ� �rj

ð6Þ

The kinematic relations for the core adopts the approximationof small deformations and they read:

εssc rc;θ� �¼ ∂

∂θuc rc;θ� �þwc rc;θ

� �rc

; εrrc rc;θ� �¼ ∂

∂rcwc rc;θ� �

γrsc rc;θ� �¼ ∂

∂rcuc rc;θ� ��uc rc;θ

� �rc

þ∂∂θwc rc;θ

� �rc

ð7Þ

where the displacements distributions through the depth of thecore read:

uc rc;θ� �¼ uo θ

� �þu1 θ� �

rcþu2 θ� �rc

þu3 θ� �

lnrcrce

� �

wc rc;θ� �¼wo θ

� �þw1 θ� �rc

þw2 θ� �

lnrcrce

� �

where wc(θ,rc) and uc(θ,rc) are the radial and the circumfer-ential displacement of the core, respectively and rce is the radius ofthe centroid of the core section. These distributions take the formof the core displacement fields that have been determined throughclosed-formed solution of the curved HSAPT model when the corecircumferential rigidity is neglected, see Frostig [14].

The compatibility conditions corresponding to perfect bondingbetween the face sheets and the core require that (j¼t,b):

ucðr¼ rcj;θÞ ¼ uojðθÞþð�1Þkdj2βjðθÞ

wcðr¼ rcj;θÞ ¼wjðθÞ ð8Þ

where k¼1 when j¼t and 0 when j¼b and rcj (j¼t,b) are the radiiof the upper and the lower face–core interfaces, respectively,uc(r¼rcj,θ) and wc(r¼rcj,θ) are the displacements of the core, inthe circumferential and the radial directions at the face–coreinterfaces. Eq. (8), actually, describes four conditions denoted bycomju (first equation) and comjw (second equations) (j¼t,b) for theupper and the lower face–core interfaces respectively. The com-patibility equations are implemented in the variational formula-tion through Lagrange multipliers as follows:

δ Uλ ¼Z α

0ðλtu θ

� �δ comtu θ

� �rctþλbu θ

� �δ combu θ

� �rcb

þλtw θ� �

δ comtw θ� �

rctþλbw θ� �

δ combw θ� �

rcb

þδλtuðθÞ comtu θ� �

rctþδλbu θ� �

combu θ� �

rcb

þδλtwðθÞ comtw θ� �

rct

þδλbw θ� �

combw θ� �

rcbÞdθ ð9Þ

where λju and λjw (j¼t,b) are the Lagrange multipliers in the cir-cumferential and radial directions at the upper and the lowerface–core interfaces respectively.

The field equations and the boundary conditions are derivedusing the variational principle, see Eq. (1); the variationalexpression of the energies and the Lagrange multipliers, see Eqs.(2), (3) and (9); the kinematic relations of the face sheets and thecore, Eqs. (5)–(7), the compatibility requirements, see Eq. (8), andthe stress resultants along with high-order ones, see Fig. 2. Thefield equations, after integration by parts and some algebraicmanipulations, for the face sheets and the core using the ordinaryand high-order stress resultants and the compatibilityequations read:

Face sheets:

�ddθMsst θ

� �rt

� ddθ

Nsst θ� �þβt θ

� �Nsst θ

� �þmt θ� �þ1=2

dtλtu θ� �

rctrt

�λtu θ� �

rct�rtnt ¼ 0

�d2

dθ2Msst θ

� �rt

þβt θ� � d

dθNsst θ

� �þ ddθβt θ� �þ1

� �Nsst θ

� �þ ddθ

mt θ� �

þ1=2dt d

dθλtu θ� �� �

rct

rt�λtw θ

� �rct�rtqt ¼ 0

�ddθMssb θ

� �rb

� ddθ

Nsst θ� �þβb θ

� �Nsst θ

� �þmb θ� �þ1=2

rcbdbλbu θ� �

rb�rbnbþλbu θ

� �rcb ¼ 0

�d2

dθ2Mssb θ

� �rb

þβb θ� � d

dθNsst θ

� �þ ddθβb θ� �þ1

� �Nsst θ

� �þ ddθ

mb θ� �

þ1=2rcbdb d

dθλbu θ� �

rb�rbqbþλbw θ

� �rcb ¼ 0

Core:

λtu θ� �

rct�λbu θ� �

rcb�ddθ

Nssc θ� ��Qsrc θ

� �¼ 0

r2ctλtu θ� ��λbu θ

� �r2cb�

ddθ

Mssc θ� �¼ 0

λtu θ� ��λbu θ

� �� ddθ

Mss2c θ� ��2MQsr1c θ

� �¼ 0 ðaÞ

λtu θ� �

rct lnrctrce

� ��λbu θ

� �ln

rcbrce

� �rcb�

ddϕ

Mss3c θ� �

þQsrc θ� ��MQsr2c θ

� �¼ 0 ðbÞ

Y. Frostig et al. / International Journal of Non-Linear Mechanics 81 (2016) 177–196 181

λtw θ� �

rct�λbw θ� �

rcb�ddθ

Qsrc θ� �þNssc θ

� �¼ 0 ðcÞ

λtw θ� ��λbw θ

� �� ddθ

MQsr1c θ� �þMss2c θ

� ��MRrc θ� �¼ 0 ðdÞ

λtw θ� �

rct lnrctrce

� ��λbw θ

� �rcb ln

rcbrce

� �� ddθ

MQsr2c θ� �

þMss3c θ� �þRrc θ

� �¼ 0 ð10Þ

where Nssj and Mssj (j¼t,b) are the in-plane and bending momentstress resultants of each face sheet, Nssc, Mssc, Mss2c, Mss3c are thein-plane stress resultants (including high-order resultants), Qsrc,MQsrc, MQsr1c, MQsr2c are the vertical shear stress resultants(including high-order resultants) and Rrc,, MRrc, are the verticalnormal stress resultants (see Eq. (11)), λju and λjw (j¼t,b) are theLagrange multipliers in the circumferential and radial direction atthe two face–core interfaces. For details see Fig. 2.

The stress resultants used in Eq. (10) read:

Mssj θ� �¼ Z 1=2 dj

�1=2 djbwσssj θ; zj

� �zj dzj;Nssj θ

� �

¼Z 1=2 dj

�1=2 djbwσssj θ; zj

� �dzj ðj¼ t; bÞ

MQsr1c θ� �¼ Z rct

rcb

bwτsrc θ; rc� �rc

drc;MQsr2c θ� �

¼Z rct

rcbbwτsrc θ; rc

� �ln

rcrce

� �drc;

Qsrc θ� �¼ Z rct

rcbbwτsrc θ; rc

� �drc

u2 θ� �¼ rct

rtη rb

�1=2 rctdt ddφwt θ

� �� �rb�1=2 d

dφwb θ� �� �

dbrcbrt

þ1=2 rctrb dt�2 rtð Þuot θ� �þ

rb �rcb2þrct2� �

u1 θ� �

þrb �rcbþrctð Þuo θ� �

þ1=2 rcbuob θ� �

dbþ2 rð

0B@

0BBBBBB@

�η�1=2 d

dθwt θ� �� �

dtþ 1=2 dt�rt� �

uot θ� �

þrt u1 θ� �

rctþuo θ� �� �

0@

1Arb

0BBBBBBBBBBBBB@

¼ 1=2

ddθwb θ

� �� �dbrcbrtþrctdt d

dθwt θ� �� �

rb�rtrcb dbþ2 rbð Þuob θ� �

þ2 �1=2 rct dt�2 rtð Þuot θ� �þ rcb�rctð Þ

rcbþrctð Þu1 θ� �þ

uo θ� � !

r

rtη rb

w1 θ� �¼ rct

η

�rcbþrctð Þwo θ� �þwb θ

� �rcb�wt θ

� �rct

� �ln rct

rce

� ��η wo θ

� ��wt θ� �� �

0@

1Aw2 θ

� �¼ ð

λbu θ� �¼ 1

η

�2 MQsr1c θ� �þ1=2 d

dθMss2c θ� �� �

rct ln rctrce

� �þMQsr2c θ

� �þ d

dθMss3c θ� ��Qsrc θ

� �0@

1Aλtu θ

� �

¼� d

dθMQsr1c θ� �þMss2c θ

� ��MRrc θ� �� �

rct

cþ

ddθQsrc θ

� ��Nssc θ� �

cλtw θ� �

where

η¼ rct lnrctrce

� ��rcb ln

rcbrce

� �

MRrc θ� �¼ Z rct

rcb

bwσrrc θ; rc� �rc

drc;Rrc θ� �¼ Z rct

rcbbwσrrc θ; rc

� �drc

Mssc θ� �¼ Z rct

rcbbwσssc θ; rc

� �rc drc;Nssc θ

� �¼ Z rct

rcbbwσssc θ; rc

� �drc

Mss2c θ� �¼ Z rct

rcb

bwσssc θ; rc� �rc

drc;Mss3c θ� �

¼Z rct

rcbbwσssc θ; rc

� �ln

rcrce

� �drc; ð11Þ

Please notice that the high-order stress resultants are non-polynomial based resultants and in general are non-physicalquantities.

Compatibility equations:

u2 θ� ��rctuot θ

� �þrctuo θ� �þrct2u1 θ

� �þ1=2rctdtuot θ

� �rt

�1=2rctdt d

dθwt θ� �

rtþrctu3 θ

� �ln

rctrce

� �¼ 0

�u2 θ� ��rcbuo θ

� ��rcb2u1 θ� �þrcbuob θ

� ��1=2rcbdb d

dθwb θ� �

rb

þ1=2rcbdbuob θ

� �rb

�rcbu3 θ� �

lnrcbrce

� �¼ 0

lnrctrce

� �w2 θ� �

rct�wt θ� �

rctþwo θ� �

rctþw1 θ� �¼ 0

� lnrcbrce

� �w2 θ� �

rcb�wo θ� �

rcbþwb θ� �

rcb�w1 θ� �¼ 0 ð12Þ

This set of field equation is reduced from 18 equations to (9) byconsolidating the following unknowns: u2(θ),u3(θ),w1(θ) andw2(θ) using the compatibility equations, and the Lagrange

bÞ

1CArt

1CCCCCCAln rct

rce

� �1CCCCCCCCCCCCCAu3 θ� �

t

!rb

rcb�rctÞwo θ� ��wb θ

� �rcbþwt θ

� �rct

η

¼ 1η

�2 MQsr1c θ� �þ1=2 d

dθMss2c θ� �� �

rct ln rctrce

� �

þddθMss2c θ

� �þ2MQsr1c θ

� � !

ηþMQsr2c θ� �þ d

dθMss3c θ� ��Qsrc θ

� �0BBB@

1CCCAλbw θ

� �

¼� d

dθMQsr1c θ� �þMss2c θ

� ��MRrc θ� �� �

rcb

cþ

ddθQsrc θ

� ��Nssc θ� �

c

ð13Þ

Fig. 3. Geometry, dimensions, mechanical properties and supporting systems of the investigated curved panel.

Fig. 4. Deformed shapes of a uniformly loaded simply-supported (rollers) long curved panel: (a) panel layout; (b) deformed shapes.

Y. Frostig et al. / International Journal of Non-Linear Mechanics 81 (2016) 177–196182

multiplier, λju and λjw (j¼t,b) are determined through equations(a) to (d) in Eq. (10) and they read:

The field equations after consolidating the unknowns of thedisplacements and the Lagrange multipliers are quite complicated,see Appendix A. The boundary conditions, which are also a resultof the variational procedure, and after substituting of theunknown displacements functions and the Lagrange multiplier(see Eq. (13)), at θ¼0 and α, read:

Face Sheets j¼ t; bð Þ :ζ Nssej θeð Þrj �Nej θeð Þrj �Mej θeð Þþζ Mssej θeð Þ

rjor uoj θe

� ��uoej θe� �¼ 0

Mej θeð Þ�ζ Mssej θeð Þrj

¼ 0 or Dwj θe� ��Dwoej θe

� �¼ 0

�mj θe� ��Pej θe

� �þζ Vsret θe� �¼ 0 or wt θe

� ��woet θe� �¼ 0

Core :

Fig. 5. Results along the panel of a long simply-supported (roller) panel at the face sheets and the core: (a) mid-height vertical displacements; and equivalent resultants of:(b) bending moments; (c) shear stress resultants; (d) Normal hoop stress resultants. Legend: face sheets: upper (t), lower (b) and

core (c).

Y. Frostig et al. / International Journal of Non-Linear Mechanics 81 (2016) 177–196 183

Nssec θe� �¼ 0 or uo θe

� ��uoec θe� �¼ 0

Mssec θe� �¼ 0 or u1 θe

� ��u1ec θe� �¼ 0

Vsrec θe� �¼ 0 or wo θe

� ��woec θe� �¼ 0

ð14Þ

And the equivalent stress resultants used in the boundaryconditions terms equal:

Face sheets:

Nsset θ� �¼ rt

ηtrcb rctMss2c θ

� �þNsst θ� �� �

lnrcbrce

� ��rct

Nsst θ� �

ln rctrce

� �þMss3c θ

� �0@

1A

0@

1A

Msset θ� �¼ rt

2ηt

rcb dtrctMss2c θ� ��2 Msst θ

� �� �ln rcb

rce

� �þ2 Msst θ

� �ln rct

rce

� �rct

�Mss3c θ� �

dtrct

0@

1A

Vsret θ� �¼ 1=2

1rtηt

2rt 1=2dt ddθMss2c θ

� �þMQsr1c θ� �

dt�rtð Þ� �

r2ct lnrctrce

� �� d

dφMss2c θ� �� �

dtηtrct�rctdt ddθMss3c θ

� �� �rt

þ2 ddφMsst θ

� �� �ηtþ2 Nsst θ

� �ηt d

dφwt θ� �

�2 η rct dt�rtð ÞMQsr1c θ� ��rtrct dt�2 rtð ÞMQsr2c θ

� �þrctdtQsrc θ

� �rt�2Nsst θ

� �ηtuot θ

� �

0BBBBBBBBBB@

1CCCCCCCCCCA

where

ηt ¼ rt rct lnrctrce

� ��rcb ln

rcbrce

� �� �

Nsseb θ� �¼ rb

ηb�rct

rcbMss2c θ� �

þNsst θ� � !

lnrctrce

� �þrcb

Nsst θ� �

ln rcbrce

� �þMss3c θ

� �0@

1A

0@

1A

Msseb θ� �¼ rb

2ηb

rct dbrcbMss2c θ� �þ2 Mssb θ

� �� �ln rct

rce

� �

�rcbdbMss3c θ

� �þ2 Mssb θ

� �ln rcb

rce

� �0@

1A

0BBBB@

1CCCCA

VserbðθÞ ¼ 1=21

rbηb

2 rb 1=2 db ddθMss2c θ

� �þMQsr1c θ� �

dbþrbð Þ� �

rctrcb ln rctrce

� ��rcbdb d

dθMss3c θ� �� �

rbþ2 ddθMssb θ

� �� �ηb

þ2 Nsst θ� �

ηb ddθwb θ

� ��rbrcb dbþ2 rbð ÞMQsr2c θ� �

þrcbdbQsrc θ� �

rb�2 Nsst θ� �

ηbuob θ� �

0BBBBBBB@

1CCCCCCCA

where

ηb ¼ rb rct lnrctrce

� ��rcb ln

rcbrce

� �� �Core :

Fig. 6. Displacements and interfacial stresses of a long simply-supported (roller) along the panel. (a) Mid-height circumferential displacements (face sheets and core); andstresses at face–core interfaces: (b) shear stresses; (c) radial normal stresses; and (d) Normal hoop stresses in core. Legend: face sheets/interfaces: upper(ct, t), lower (cb, b) and core (c).

Y. Frostig et al. / International Journal of Non-Linear Mechanics 81 (2016) 177–196184

Nssec θ� �¼ 1

�rcb rctMss2c θ� ��Nssc θ

� �� �ln rcb

rce

� �þrct rcbMss2c θ

� ��Nssc θ� �� �

ln rctrce

� ��Mss3c θ

� �rcb�rctð Þ

0BBBB@

1CCCCA

rcb lnrcbrce

� ��rct ln rct

rce

� �0B@

1CA

�1

Mssec θ� �¼ 1

�rcb r2ctMss2c θ� ��Mssc θ

� �� �ln rcb

rce

� �þrct r2cbMss2c θ

� ��Mssc θ� �� �

ln rctrce

� �þ �r2cbþr2ct� �

Mss3c θ� �

0BBBB@

1CCCCA

rcb lnrcbrce

� ��rct ln rct

rce

� �0B@

1CA

�1

Vsre c θ� �¼ 1

�rcb rctMQsr1c θ� ��Qsrc θ

� �� �ln rcb

rce

� �þrct rcbMQsr1c θ

� ��Qsrc θ� �� �

ln rctrce

� ��MQsr2c θ

� �rcb�rctð Þ

0BBBB@

1CCCCA

rcb lnrcbrce

� ��rct ln rct

rce

� �0B@

1CA

�1

ð15Þ

The displacements and stress fields of the core assuming anorthotropic core, using Eq. (13), read:

Displacements :

uc rc;θ� �¼ 1=2

�rctþcð Þdb �ηcrcþrctηcb� �

ddφwb θ

� �rbη rc

þ1=2rctdt ηcrc�rctηcbþη

� �ddθwt θ

� �rcη rt

� rct�rcð Þη�c �ηcrcþrctηcb� �� �

uo θ� �

rcη

� �rc2þrct2� �

ηþc c�2 rctð Þ �ηcrcþrctηcb� �� �

u1 θ� �

rcη

�1=2rct dt�2 rtð Þ ηcrc�rctηcbþη

� �uot θ� �

rcη rt

�1=2�rctþcð Þ dbþ2 rbð Þ �ηcrcþrctηcb

� �uob θ� �

rbη rc

Fig. 7. Equilibrium curves of distributed load values versus extreme values for face sheets and core of a long simply-supported (roller) panel: (a) vertical displacements;(b) slope and average slope of face sheets and core section, respectively; and equivalent resultants of: (c) bending moments; (d) shear stresses; (e) in-plane normal stressesand (f) mid-height circumferential displacements. Legend: lines: solid – maximum, dash – minimum, face sheets: black – upper (t), red – lower (b) and blue __core (c).(For interpretation of the reference to color in this figure legend, the reader is reffered to the web version of this article.)

Y. Frostig et al. / International Journal of Non-Linear Mechanics 81 (2016) 177–196 185

where

η¼ rct lnrctrce

� ��rcb ln

rcbrce

� �; ηc ¼ ln

rcrce

� �; ηcb ¼ ln

rctrce

� �; ηct ¼ ln

rctrce

� �ð16Þ

And the stress fields read:

σrr rc;θ� �¼ � ϵss rc;θ

� �μsrcþϵrr rc;θ

� �� �Ecr

μrscμsrc�1

σss rc;θ� �¼ �Ecs μrscϵrr rc;θ

� �þϵss rc;θ� �� �

μrscμsrc�1; τsr rc;θ

� �¼ Gcγsr rc;θ� �

ð17Þ

where the strains are defined in Eq. (7) and Ecs, Ecr and Gc are themoduli of elasticity in the circumferential and radial directionand the shear modulus respectively and mrsc and msrc are thePoisson relations for the radial to the circumferential directionsrespectively. The stresses are determined through substitution

Fig. 8. Equilibrium curves of distributed load versus extreme stresses of core, at its interfaces, of a long simply-supported (roller) panel: (a) shear stresses; (b) radial normalstresses; (c) normal hoop stresses. Legend: lines: solid – maximum, dash – minimum, interfaces: black – upper (ct), red – lower (cb). (For interpretation of the reference tocolor in this figure legend, the reader is reffered to the web version of this article.)

Y. Frostig et al. / International Journal of Non-Linear Mechanics 81 (2016) 177–196186

of the displacements fields, see Eq. (16), in the stressdefinition above.

The numerical solution uses a set of 18 first-order differ-ential equations that are defined through the equivalentunknowns, see Eq. (15), that replaces the seven consolidatedequations, see Eqs. (A.1)–(A.7) into nine equations and uses9 constitutive relations for the stress resultants in the facesheets, assuming isotropic materials, and the high-order stressresultant of the core using orthotropic core material, see Eqs.(17) and (16). Please notice that the derivatives of the unknowndisplacements, uoj,wj, and Dwj (j¼t,b,c), are determinedthrough the simultaneous solution of the constitutive rela-tions, see Eq. (15). The governing equations are not presentedfor brevity.

The numerical solution of the non-linear governing set ofdifferential equations can be achieved using numerical schemessuch as the multiple-point shooting method, see Stoer andBulirsch [32], or the finite-difference (FD) approach using tra-pezoid or mid-point methods with Richardson extrapolation ordeferred corrections, see Ascher and Petzold [2] along withparametric or arc-length continuation methods, see Keller [18].Here, the FD approach that is implemented in Maple, see Charet al. [8], has been used efficiently without any numericalinstabilities. The numerical EHSAPT results have been compared

very well with the elasticity solution, for details see Rodcheuyet al. [30].

3. Numerical study

The numerical part presents the non-linear response of asimply-supported curved panel when subjected to a dis-tributed pressure load, see Fig. 3 for various boundary condi-tions and for two spans. The non-linear response uses a loadingcontrol scheme for the load changes and it starts with lowloads that yield a linear response and reaches high load levelsthat are deep in the non-linear regime. The sandwich panel ismade of two aluminum face sheets of a thickness of 1mmand an H160 foam core made by Divinycell withEcs¼Ecr¼Ec¼170 MPa and Gc¼66 MPa with a thickness of25 mm and a Poisson ratio of 0.3. The geometry of the curvedpanel is that of an experimental set-up, see Bozhevolnaya andFrostig [3] and Bozhevolnaya thesis [4] and see Fig. 3 fordetails. Two schemes of boundary conditions are considered: Asimply-supported panel where all edges of the face sheets andthe core are supported by radial roller, denoted by ss3 (seeFig. 3) except for the left edge of the upper face sheet that isrestrained to circumferential movements, i.e pinned support,

Fig. 9. Deformed shapes of a uniformly loaded simply-supported (rollers) short curved panel: (a) panel layout; (b) deformed shapes.

Y. Frostig et al. / International Journal of Non-Linear Mechanics 81 (2016) 177–196 187

and denoted by ss1 in Fig. 3; and the second scheme consists ofpinned supports at edges of all constituents.

The results include deformed shapes, structural variables alongthe panel at various load levels and equilibria curves of load versusextreme values of structural quantities.

3.1. Simply-supported panels

In this case two panels have been considered a long span of500.0 mm and a short one of 250.0 mmwith a constant radius. Theresponse in both cases is associated with wrinkling waves at thecompressed upper face sheets but with different quantities and atdifferent load levels. In addition, for some structural quantities theresults of the two cases are different.

The results for the long panel appear in Figs. 4–8. Thedeformed shapes at various load levels, see Fig. 4, reveal smoothcurves, in the upper face sheet, for low load level and in the postbuckling (non-linear) range, in the form of non-regular shallowdimples within the central part of the panel. Please notice thatwrinkling appears only at the upper face sheets while the curvesof the lower face sheet are smooth throughout the loadingsequence.

The values of the various structural variables along the panelappear in Figs. 5 and 6. The vertical displacements of the facesheets and the core at its mid-height, Fig. 5a, at various loadlevels, reveal small wrinkling bumps at high load level. Thesewrinkles are associated with significant large bending momentsin the face sheets, see Fig. 5b, and significant changes in theshear stress resultants, see Fig. 5c. The circumferential stressresultants of the face sheets and the core appear in Fig. 5d and isassociated with ripples in the upper and lower face sheets andvery small values for the resultant in the core. Please notice thatin terms of equivalent stress resultants the contribution of thecore is minor. The in-plane displacements, see Fig. 6a, is also

associated with wrinkling ripples at the various load levels,especially in the upper face sheet and the mid-height dis-placement of the core and smother curve for the lower facesheet. The interfacial stresses of the core appear in Fig. 6b–d andreveal significant changes due to wrinkling in the upper facesheet. The shear stresses, see Fig. 6b, usually reach extremevalues at the edges of the panel. However, the wrinkling phe-nomena yields extreme values in the vicinity of the wrinkledarea that are much larger then at the edges. A similar behavioroccurs for the radial normal stresses, see Fig. 6c, but here thestresses reach extreme values at the edges of the wrinkled areamainly. The interfacial circumferential stresses, at the upper andthe lower face-core interfaces, reach extreme values near theedge of the panel as well as in the vicinity of the wrinkled facesheet, see Fig. 6d.

The equilibria curves of load versus stress resultants appearin Fig. 7 and for the interfacial stresses of the core in Fig. 8. Thecurves of the vertical displacements of the face sheets and themid-height of the core appear in Fig. 7a. It reveals a non-linearbehavior in general and slight change in slope at around 64N/mm. A similar behavior is observed for the slope of the sec-tions of the face sheets and the slope at mid-height of core, seeFig. 7b. A totally different response is observed in Fig. 7c whichdescribes the response of the equivalent bending moment. Herea significant change is observed at around a load of 64 N/mmwith a stable post-buckling pattern. The equivalent shearcurves, see Fig. 7d, has only minor changes around the bucklingload of 64 N/mm. The equivalent circumferential stress resul-tants, see Fig. 7e, follows the equivalent shear resultantresponse. The circumferential displacements at mid-height ofthe face sheets and the core, see Fig. 7f, reveal a non-linearresponse in general, but as the load increases the values of theextreme positive displacements approaches zero and the nega-tive ones become larger.

Fig. 10. Results along the panel of a short simply-supported (roller) panel at the face sheets and the core: (a) vertical displacements of face sheets and mid-height dis-placement of core; and equivalent resultants of: (b) bending moments; (c) shear stresses; (d) Normal hoop stresses. Legend: face sheets: upper (t),

lower (b) and core (c).

Y. Frostig et al. / International Journal of Non-Linear Mechanics 81 (2016) 177–196188

The wrinkling buckling phenomena is much more pro-nounce in the equilibria curves of the interfacial stresses of thecore, see Fig. 8. Here the shear stresse curves at the upperinterface, see Fig. 8a, are smooth but with abrupt change at thebuckling load. Similarly, is observed for the tensile radial nor-mal stresses, see Fig. 8b, and the circumferential normalstresses at the upper and the lower fibers of the core, seeFig. 8c.

The results for the short panel appear in Figs. 9–13 whileonly the responses that are different are discussed. Thedeformed shapes at various load levels, see Fig. 9, revealwrinkling in the upper face sheet in the post buckling range, inthe form of non-regular dimples. Here, please notice thatbuckling occurred at a higher load level of about 140 N/mm ascompared with the long panel. The values of the variousstructural variable along the panel appears in Figs. 10 and 11.

The response of the vertical displacements, see Fig. 10a, revealagain wrinkling displacements in the upper face sheet but atmuch small overall displacements in the other face sheet andthe core as compared with the long panel. The bendingresponse of the face sheets and the core, see Fig. 10b, is totallydifferent than previous case and at smaller values. Pleasenotice that the curves for the equivalent shear resultants andthe in-plane ones, see Fig. 10c and d, are similar to those of thelong panel. The in-plane displacements and the interfacialstresses, see Fig. 11, reveal a similar response as in previouscase but at smaller values.

The equilibria curves, in this case, reveal a similar behavior asthat of the previous cases but here the abrupt changes occur at aload of 140 N/mm and the changes are more significant in term ofdisplacements, bending moments, see Fig. 12. The equilibria curvesof the interfacial stresses of the core reveal a similar behavior as in

Fig. 11. Displacements and interfacial stresses of a short simply-supported (roller) panel along the panel. (a) Mid-height circumferential displacements (face sheets andcore); and stresses at face–core interfaces: (b) shear stresses; (c) radial normal stresses; and (d) Normal hoop stresses. Legend: face sheets/interfaces:upper (ct, t), lower (cb, b) and core (c).

Y. Frostig et al. / International Journal of Non-Linear Mechanics 81 (2016) 177–196 189

previous case but with significant changes around the bucklingload level, see Fig. 13. Also here, the post-buckling response isstable.

The third case investigates the non-linear response of ashort panel, see previous case, which is pinned at its edges. Theresults for this case appears in Figs. 14–18. The deformed shapeof the panel at various load levels appear in Fig. 14. It revealsthat no buckling occur and there is a smooth transition fromcompressed face sheets to tensile ones, see Figs. 15 and 16. Thevertical displacements consists of smooth curves at the variousload levels with no wrinkling patterns, see Fig. 15a. The curvesof the section slopes are also smooth, see Fig. 15b. The localizedequivalent bending moments are quite large near therestrained edges, see Fig. 15c. The equivalent shear diagrams,see Fig. 15d, are quite smooth but with larger values as com-pared with the wrinkled short panel. The equivalent stressresultants, see Fig. 15e, reveal that as the load is raised thecompression stress resultant in the upper face sheet decreaseswhile the tensile stress resultant at the lower face sheet

increases. Hence, as the load is increased the tension in thelower face sheet stabilizes and strengthen the structure,similar to the response of an arch with a tensile cord. The in-plane circumferential stresses curves are smooth, see Fig. 15f,and differ significantly from the wrinkled panel case, seeFig. 11a.

The interfacial stresses of the core appear in Fig. 16 and theyreveal a smooth response at all load levels. Please notice that thevalues here are smaller as compared with the values of thewrinkled panel.

The equilibria curves here describe a stable post bucklingpattern for all structural variables, see Figs. 17 and 18. Thedisplacements response, see Fig. 17a, is associated with a non-linear response at the beginning, up to 50 N/mm followed but alinear region, up to 70 N/mm and significant non-linearresponse with a higher slop beyond 80 N/mm which is due tothe fact that the lower face sheets starts to behave more as atensile strut rather than a bending face sheet. Similar respon-ses are observed in the case of the slope of the sections, see

Fig. 12. Equilibrium curves of distributed load values versus extreme values for face sheets and core of a short simply-supported (roller) panel: (a) vertical displacements;and equivalent resultants of: (b) bending moments; (c) shear stresses; and (d) mid-height circumferential displacements. Legend: lines: solid – maximum, dash – minimum,face sheets: black – upper (t), red – lower (b) and blue __core (c). (For interpretation of the reference to color in this figure legend, the reader is reffered to the webversion of this article.)

Y. Frostig et al. / International Journal of Non-Linear Mechanics 81 (2016) 177–196190

Fig. 17b, the bending moments, see Fig. 17c and the shear stressresultants, see Fig. 17d. Please notice that the in-plane stressresultants in the face sheets, see Fig. 17e, reach large tensilevalues deep in the post buckling range. It means that as thepost-buckling regions deepens the stiffness of the structure isenhanced due to the tensile chord response of the face sheets.Similar behavior is observed for the interfacial stresses of thecore, see Fig. 18. Please notice that the interfacial radial stres-ses, see Fig. 18b, at lower load levels the stresses at the upperand the lower face sheets are in compression and as the loadincreases these stresses reach larger values.

4. Summary and conclusions

The paper presents the non-linear response of a curvedsandwich panels, within the framework of the EHSAPT model,i.e. the circumferential rigidity of the core is considered. Theformulation is mathematically rigorous, accurate and robust,presumed that the displacements through the thickness of thecore follows the pattern that have been derived in a closed-form for the HSAPT model for the curved sandwich panel. Ingeneral, it is a sum of a number of multiplied functions that areexplicitly described in the radial direction and unknown in thecircumferential one. The formulation uses the non-linear and

linear kinematic relations, for the face sheets and the corerespectively, along with a variational principle to derive thenon-linear field equations for the face sheets and the core. As aresult of the presumed distribution the formulation involvesordinary stress resultants for the face sheets and the core aswell as high-order (non-polynomial) stress resultants for thecore. The formulation uses Lagrange multipliers to enforcecompatibility in the radial and hoop directions at the face-coreinterfaces. It yields an 18 order set of ordinary differentialequations. After consolidation of four displacement functionsof the core along with the four Lagrange multiplier andthrough the use of equivalent stress resultants the number ofequations reduces to 9. This set of equations is solvednumerically using a Maple built-in ODE solver.

The numerical study investigates the non-linear response ofsome sandwich panel with a particular layout of an experi-mental setup with some modification. Two types of boundaryconditions have been considered to identify the type of post-buckling response of such panels. The first one is pinned atedge of one face sheet while the other edges of the componentsof the panel are only radially supported while all the edges ofsecond one are pinned support. In addition two length of spanshave been considered while keeping the radii identical. A verygood comparison in the linear state is observed when

Fig. 13. Equilibrium curves of distributed load versus extreme stresses of core, at its interfaces, of a short simply-supported (roller) panel: (a) shear stresses; (b) radial normalstresses; (c) hoop stresses. Legend: lines: solid – maximum, dash – minimum, interfaces: black – upper (ct), red – lower (cb). (For interpretation of the reference to color inthis figure legend, the reader is reffered to the web version of this article.)

Fig. 14. Deformed shapes of a uniformly loaded simply-supported (pinned) short curved panel: (a) panel layout; (b) deformed shapes.

Y. Frostig et al. / International Journal of Non-Linear Mechanics 81 (2016) 177–196 191

Fig. 15. Results along the panel of a short simply-supported (pinned) panel at the face sheets and the core: (a) vertical displacements; (b) slope of face sheets and section;and equivalent resultants of: (c) bending moments; (d) shear stresses; (e) hoop normal stresses and (f) mid-height circumferential displacements. Legend: face sheets:

upper (t), lower (b) and core (c).

Y. Frostig et al. / International Journal of Non-Linear Mechanics 81 (2016) 177–196192

Fig. 16. Displacements and interfacial stresses of a short simply-supported (pinned) panel along the panel. (a) Mid-height circumferential displacements (face sheets andcore); and stresses at face-core interfaces: (b) shear stresses; (c) radial normal stresses; and (d) Normal hoop stresses. Legend: face sheets/interfaces:upper (ct, t), lower (cb, b) and core (c).

Y. Frostig et al. / International Journal of Non-Linear Mechanics 81 (2016) 177–196 193

compared with closed-form elasticity solution, see Rodcheuyet al. [30].

The numerical study of the roller simply-supported panelreveal that the non-linear response is associated with wrinkling ofthe compressed face sheets, large circumferential and bendingmoments stress resultants in the face sheets and large stresses inthe core. Please notice that the wrinkling ripples are non-periodichalf-waves that appear only within the mid-part of the panel. In allcases a stable post-buckling pattern is observed. In both cases ofshort and long panel the response is of the same nature but withdifferent values in general.

The case with pinned supports has not reveal any loss ofstability and the behavior in general is of a stable non-linearresponse, although the upper face sheet is compressed. Here, asthe radial displacements deepens the lower face sheet behaves

as a tensile strut rather than a thin bending beam which stabi-lize and enhance the resistance of the panel. In general, theresponse of such a panel has been associated with moderatedisplacements and a non-linear stable behavior when comparedwith the first case.

The proposed model enhances the physical insight of the

response of a curved sandwich panels when the circumferential

(in-plane) rigidity of the core is considered. It has the capability

to reveal correctly the non-linear response of various panels

along with localized loss of stability that is associated with

extremely large stresses and stress resultants. Hence, the loss of

stability revealed here should be considered in order to achieve

a reliable and efficient design of curved panels for various

applications.

Fig. 17. Equilibrium curves of distributed load values versus extreme values for face sheets and core of a short simply-supported (pinned) panel: (a) vertical displacements;and equivalent resultants of: (b) bending moments; (c) shear stresses; and (d) mid-height circumferential displacements. Legend: lines: solid – maximum, dash – minimum,face sheets: black – upper (t), red – lower (b) and blue __core (c). (For interpretation of the reference to color in this figure legend, the reader is reffered to the webversion of this article.)

Y. Frostig et al. / International Journal of Non-Linear Mechanics 81 (2016) 177–196194

Fig. 18. Equilibrium curves of distributed load values versus extreme stresses of core, at its interfaces, of a short simply-supported (pinned) panel: (a) shear stresses;(b) radial normal stresses; (c) hoop stresses. Legend: lines: solid – maximum, dash – minimum, interfaces: black – upper (ct), red – lower (cb). (For interpretation of thereference to color in this figure legend, the reader is reffered to the web version of this article.)

Y. Frostig et al. / International Journal of Non-Linear Mechanics 81 (2016) 177–196 195

Acknowledgments

The work presented was conducted during the period of a visitingprofessorship of the Prof. Frostig with the Department of AerospaceEngineering at Georgia Institute of Technology. The visiting pro-fessorship and the research presented herein were sponsored by theUS Navy, Office of Naval Research (ONR) under Grant N00014-11-1-0597, by Technion- Israel Institute of Technology and by the AshtromEngineering Company that supports the professorship chair of Prof.Frostig. The financial support of the Office of Naval Research and theinterest and encouragement of the Grant Monitor, Dr. Y.D.S. Rajapakse, are both gratefully acknowledged.

Appendix A. : Field equations after consolidation

�ddθMsst θ

� �rt

þrct2 ddθMss2c θ

� �η rt

lnrctrce

� �rctrt�η

� ��rct2 d

dθMss3c θ� �

η

� ddϕ

Nsst θ� �þ2

rct2MQsr1c θ� �

η rtln

rctrce

� �rctrt�η

� ��rct2MQsr2c θ

� �η

þuot θ� �� d

dϕwt θ� �� �

Nsst θ� �

rtþrct2Qsrc θϕ

� �η

�rtnsstþmt θ� �¼ 0

ðA:1Þ

rct �rtþrctð Þ d2

dθ2Mss2c θ� �

η rtln

rctrce

� �rctrt�η

� �þrct rt�rctð Þ d2

dθ2Mss3c θ� �

η

þ ddθ

MQsr1c θ� �� �

rct 2 rt�2 rctð Þη

ηrt�rct ln

rctrce

� �� ��rctrbc

c

� �

þrct rt�rctð Þ ddθMQsr2c θ� �

η�

ddθQsrc θ

� �� �rt�rctð Þc�η� �

rct

η c

�MRrc θ� �

rctrbcc

þMss2c θ� �

rctrbcc

�Nssc θ� �

rctc

�d2

dϕ2Msst θ� �

rt

þuot θ� �� d

dθwt θ� �� �

ddθNsst θ

� �rt

þrtþ d

dθuot θ� �� d2

dθ2wt θ� �� �

Nsst θ� �

rt

�rtqtþddθ

mt θ� �¼ 0 ðA:2Þ

�ddθMssb θ

� �rb

�1=2rctrbc dbþ2 rbð Þ d

dθMss2c θ� �� �

rt

rbηln

rctrce

� �

þ1=2rbc dbþ2 rbð Þ d

dθMss3c θ� �� �

rt

rbη� ddθ

Nssb θ� �

�rctrbc dbþ2 rbð ÞMQsr1c θ� �

rtrbη

lnrctrce

� �

Y. Frostig et al. / International Journal of Non-Linear Mechanics 81 (2016) 177–196196

þ1=2rbc dbþ2 rbð ÞMQsr2c θ

� �rt

rbηþ

uob θ� �� d

dθwb θ� �� �

Nssb θ� �

rb

�1=2rbc dbþ2 rbð ÞQsrc θ

� �rt

rbη�rbnssbþmb θ

� �¼ 0 ðA:3Þ

ddθ

MQsr1c θ� �� �

rctrbcc

�rbcdbrtrctrbη

lnrctrce

� �� �

þ ddϕ

Qsrc θ� �� �

�rbcc�1=2

rbcdbrtrbη

� �

þMRrc θ� �

rctrbcc

�Mss2c θ� �

rctrbcc

þNssc θ� �

rbcc

�1=2rbcdbrt d2

dθ2Mss2c θ� �� �

rct

rbηln

rctrce

� �þ1=2

rbcdbrt d2

dθ2Mss3c θ� �

rbη

þ1=2rbcdbrt d

dθMQsr2c θ� �

rbη�

d2

dθ2Mssb θ� �

rbþ

uob θ� �� d

dθwb θ� �� �

ddθNssb θ

� �rb

þrbþ d

dθuob θ� �� d2

dθ2wb θ� �� �

Nssb θ� �

rbþ ddθ

mb θ� ��rbqb ¼ 0 ðA:4Þ

ddθMss2c θ

� �� �rct

ηrt rbc�rctð Þln rct

rce

� �þη

� ��

ddθMss3c θ

� �� �rtc

η

þ2MQsr1c θ

� �rct

ηrt rbc�rctð Þln rct

rce

� �þη

� ��MQsr2c θ

� �rtc

η

þQsrc θ� �

crt�η� �η

� ddθ

Nssc θ� �¼ 0 ðA:5Þ

� ddθ

Mssc θ� �þ d

dθMss2c θ� �� �

rct

ηrbc

2 lnrctrce

� �rt� ln

rctrce

� �rct2rtþη rct

� �

�rbc2�rct2� �

ddθMss3c θ

� �� �rt

η� rbc2�rct2� �

MQsr2c θ� �

rtη

þ rbc2�rct2� �

Qsrc θ� �

rtη

þ2MQsr1c θ

� �rct

ηrbc

2 lnrctrce

� �rt� ln

rctrce

� �rct2rtþη rct

� �¼ 0

ðA:6Þ

rct ddθMQsr1c θ

� �crt

rt rbc�rctð Þln rctrce

� �þη

� ��η d

dϕQsrc θ� �

crt

þMRrc θ� �

rctcrt

rt rbc�rctð Þln rctrce

� �þη

� �

�Mss2c θ� �

rctcrt

rt rbc�rctð Þln rctrce

� �þη

� �

þη Nssc θ� �

crt� ddθ

MQsr2c θ� �þMss3c θ

� �þRrc θ� �¼ 0 ðA:7Þ

where η¼ rt rct ln rctrce

� ��rcb ln

rcbrce

� �� �

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