Indentation resistance of sandwich beams

A. Petras, M.P.F. Sutcli�e *

Engineering Department, Cambridge University, Trumpington Street, Cambridge CB2 1PZ, UK

Abstract

High-order sandwich beam theory is used to model the local deformation under the central indentor for sandwich beams loaded

under three-point bending. `High-order' refers to the non-linear variations of in-plane and vertical displacements through the height

of the core which the model incorporates. The analysis is elastic, which is appropriate to describe the beam response up to peak load

for the material combination of GFRP skins and Nomex honeycomb core which is the focus of this paper. Reasonable agreement is

found between theoretical predictions of the displacement ®eld under the indentor and experimental measurements using a beam

with GFRP skins and Nomex honeycomb core. By using the model to consider the way in which di�erent wavelengths of sinusoidal

pressure loading on the top skin are transmitted to the core, a spreading length scale k is introduced. k, which is a function of the

beam material and geometric properties, characterises the length over which a load on the top surface of a beam is spread out by the

skin. Calculations of the e�ect of roller diameter on indentation behaviour illustrate the importance of this length scale. When k is

small compared with the roller radius R, corresponding to a ¯exible skin, the contact load at the roller-skin interface is transmitted

relatively unchanged to the core. Conversely, when k=R is greater than about 0.25, corresponding to a relatively rigid skin, the load

from the roller is spread out by the skin and the pressure in the core is distributed over a length of the order of k. Ó 1999 Elsevier

Science Ltd. All rights reserved.

Keywords: Sandwich beams; Indentation; Localised e�ects; High-order sandwich beam theory; Surface displacement analysis; Nomex; GFRP

Composite Structures 46 (1999) 413±424

Notation

Greek symbolsd length of contact between the

central roller and the top skinD length of non-zero stresses in the

top skin-core interfacek spreading length scalemc Poissons's ratio of coremf Poissons's ratio of the skin

materialms Poissons's ratio of core

constituent materialqc honeycomb core densityqs density of honeycomb's solid

constituent materialrcc out-of-plane compressive

strength of the honeycomb corersc compressive strength of core

constituent material

rtxx maximum in-plane stresses in theskins

rzz out-of-plane normal stresses inthe core

sx shear stresses in the core

Latin symbolsAt, Ab in-plane sti�ness of the skinsb beam widthc core thicknessCm Fourier coe�cientDt, Db ¯exural rigidity of the skinsd distance between the midplanes

of top and bottom skinEc out-of-plane Young's modulus of

the coreEf Young's modulus of the face

materialEs Young's modulus of core

constituent materialGc out-of-plane shear modulus of

the coreGs shear modulus of core

constituent materialL span of the sandwich beam

* Corresponding author. Tel.: +44-1223-332996.

E-mail address: mpfs@eng.cam.ac.uk (M.P.F. Sutcli�e).

0263-8223/99/$ - see front matter Ó 1999 Elsevier Science Ltd. All rights reserved.

PII: S 0 2 6 3 - 8 2 2 3 ( 9 9 ) 0 0 1 0 9 - 9

1. Introduction

Sandwich panels are widely used in lightweightstructures, where their excellent speci®c bending resis-tance makes them an obvious choice in transport ap-plications. Skins are typically made of metal, GFRP orCFRP, while the core material is frequently made of¯exible foams or honeycomb material. A particularlywidespread honeycomb core uses Nomex paper andphenolic resin. Indentation failure of sandwich struc-tures can arise from a number of causes including failureat the loading points or joints or due to impact damage.Indentation failure is of particular concern for Nomexcores, as the low transverse ¯exibility of these materials,as compared with metallic honeycombs or rigid foamcores, reduces their resistance to local indentation.

Fundamental analyses of sandwich beams are pre-sented by Allen [1] and Plantema [2], assuming that thecore is incompressible in the out-of-plane direction.These models assume that the skins have only bendingrigidity while the core has only shear rigidity. This ap-proach is good for sandwich structures with incom-pressible cores. However, to model the local e�ects at theload points for non-metallic honeycomb sandwich panelswith low transverse ¯exibility, compressibility of the corein the vicinity of the applied loads must be included. Acommon approach is the elastic foundation model [3±6].However these foundation models neglect interactionsbetween the top and bottom skin. Frostig and Baruch [7]

overcame this di�culty by treating the beam as an or-dinary incompressible sandwich beam connected to aspecial elastic foundation element. The non-planar de-formed cross-section of the sandwich beam, observedexperimentally, suggested the need for a model whichallows non-linear variations of in-plane and verticaldisplacement ®eld through the core. Frostig et al. [8,9]used variational principles to develop a high-ordersandwich panel theory, which includes the transverse¯exibility of the core. `High-order' refers to the non-linear way in which the in-plane and vertical displace-ments are allowed to vary through the height of the core,in contrast to simple beam theory where the core in-planedisplacements are assumed to vary in a linear waythrough the depth, and the out-of-plane displacementsare assumed to be constant. A comprehensive review andcomparison between this method and conventional beamtheories was presented recently by Frostig [10]. As theaccuracy of Frostig's formulation as applied to inden-tation of sandwich panels had already been veri®ed, thiswas used here in preference to alternative higher-orderbeam formulations such as that of Reddy [11].

In the literature there are only a few references re-lating to the contact stresses exerted between the topskin of a simply supported sandwich beam and an in-dentor which applies three-point bending on the beam.Frostig and Baruch [7] examine analytically the e�ect offour types of localised load distributions; a point load,uniform or sinusoidal loading or two concentratedloads. These represent very ¯exible, intermediate andvery sti� indentors, respectively. Johnson [12, p. 143]investigates thin plates in contact with a rigid indentorand concludes that the contact stress distributionchanges from having a maximum in the centre to one inwhich the pressure is concentrated at the edges, as thethickness of the plate increases.

Indentation failure in sandwich panels is generallypredicted, for example in handbooks on sandwich panelconstruction [13], by assuming that the load is appliedover a known contact width. This contact pressure isthen assumed to be transferred directly to the core,leading to core failure when the core compressivestrength is exceeded. Previous experimental measure-ments of indentation failure in GFRP/Nomex honey-comb beams have shown that this approach is notadequate [14]. The reason for this is understood by re-ferring to two extreme cases. For very ¯exible skins(Fig. 1a), there is large local deformation under the loadwhich can easily lead to core failure. For very rigid skinsindentation failure will be relatively hard, as the skinsspread the load (Fig. 1b).

It is the aim of this paper to understand the indenta-tion behaviour of sandwich panels by examining theroles of skin and core rigidity on the localised behaviourunder an indentor. High-order sandwich beam theory(HOSBT) is used to analyse the behaviour of sandwich

long, trans longitudinal, transversehoneycomb ribbon direction

m index giving the wavelength ofthe Fourier term

M number of terms in the Fourierseries

N number of pairs of symmetricalpressure elements covering acontact width d

Pi height of triangular pressureelement

qt contact pressure distributionR roller radiust skin thicknessut, ub in-plane deformations of the top

and bottom skinv base width of triangular pressure

elementwt, wb vertical displacements of the top

and bottom skinW contact load per unit widthx coordinate along beam lengthxi coordinate of pressure element

414 A. Petras, M.P.F. Sutcli�e / Composite Structures 46 (1999) 413±424

beams subject to three point bending and to examine thedetails at the load points. The accuracy of HOSBT inmodelling localised e�ects is veri®ed experimentally inSection 3. Section 4 introduces a spreading length scalethat characterises the ¯exibility of a sandwich beam un-der indentation loading. Finally Section 5 predicts thecontact stress distribution between a cylindrical indentorand the top skin of a sandwich beam, interpreting resultsin terms of the spreading length scale de®ned in Section 4.

2. High-order sandwich beam theory

The HOSBT introduced by Frostig and Baruch [8]is used in this paper. The basic assumptions of theirHOSBT are:· The shear stresses in the core are uniform through the

height of the core.· The core vertical displacement variation is a quadrat-

ic polynomial in z (see Eq. (11)), allowing the core todistort and its height to change.

· The core is considered as a 3D elastic medium, whichhas signi®cant out-of-plane compressive and shear ri-gidity, but negligible resistance to in-plane normalshear stresses.

This high-order analysis is based on variational princi-ples; the details of derivation of the governing equationsand associated boundary conditions are presented in [8].Consider a sandwich beam of span L and unit width,consisting of a core with thickness c, Young's and shearmodulus Ec and Gc, respectively, and two skins with thesame thickness t � tt � tb, Young's modulus Ef andPoisson's ratio mf , as depicted in Fig. 2(b).

A distributed load qt is applied to the top skin. Themodel is two-dimensional, so that variations across thewidth are neglected. The displacement and stress ®eldsof the core are expressed in terms of the followings ®veunknowns: the in-plane deformations ut and ub in the x-direction at the mid-plane of the top and bottom skin,respectively; their corresponding vertical displacementswt and wb; and the shear stresses sx in the core. Therelevant geometric parameters and the notation used forstresses and displacements are given in Fig. 2. The

governing equations given by [8], adapted to a two-di-mensional model, are:

Atut;;x � sx � 0;

Abub;;x ÿ sx � 0;

Dtwt;;;;x ÿ Ec

c�wb ÿ wt� ÿ c� t

2sx;x � qt;

Dbwb;;;;x � Ec

c�wb ÿ wt� ÿ c� t

2sx;x � 0;

ut ÿ ub ÿ c� t2

wt;x ÿ c� t2

wb;x ÿ cGc

sx ÿ c3

12Ec

sx;;x � 0;

�1�where At � Ab � Ef t=�1ÿ m2

f � and Dt � Db �Ef t3=�12�1ÿ m2

f �� are the in-plane and ¯exural rigiditiesof the skins. The notation ��;;x, for example, denotes thesecond partial derivative with respect to x. For a simplysupported sandwich beam the solution can be expressed

Fig. 2. (a) Non-linear displacements, (b) Beam geometry and stresses.

The origin of the z-coordinate is always taken at the top of the beam

element, either the skin or core, which is being considered.

Fig. 1. The behaviour of ¯exible and rigid skins.

A. Petras, M.P.F. Sutcli�e / Composite Structures 46 (1999) 413±424 415

as a Fourier series expressing the variation of the rele-vant variables in the x direction as

ut�x� �XM

m�1

Cutm cos

mpxL

; �2�

ub�x� �XM

m�1

Cubm cos

mpxL

; �3�

wt�x� �XM

m�1

Cwtm sin

mpxL

; �4�

wb�x� �XM

m�1

Cwbm sin

mpxL

; �5�

sx�x� �XM

m�1

Csxm cos

mpxL

; �6�

where m is an index for the wavelength of the Fourierterm and M the number of terms in the Fourier series.The Fourier coe�cients Cut

m ;Cubm ;C

wtm ;C

wbm and Csx

m areconstants to be determined. We assume that the externalloads are exerted only on the top skin, so in terms of theFourier series we set

qt�x� �XM

m�1

Cqtm sin

mpxL

; �7�

where Cqtm is a constant that depends on the distribution

of the external load. After substituting every term of theFourier series (2)±(6) into the governing Eqs. (1), theproblem can be expressed in matrix form as

�D� � �C� � �Q�; �8�where

D �ÿAt�mp

L �2 0 0 0 1

0 ÿ Ab�mpL �2 0 0 ÿ 1

0 0 ÿ Dt�mpL �4 � Ec

c ÿ Ec

cc�t

2mpL

0 0 ÿ Ec

c ÿ Db�mpL �4 � Ec

cc�t

2mpL

1 ÿ 1 ÿ c�t2

mpL ÿ c�t

2mpL

cGc� c3

12Ec�mp

L �2

2666666664

3777777775;

C �

Cutm

Cubm

Cwtm

Cwbm

Csxm

266664377775 and Q �

00

Cqtm

00

266664377775:

By solving (8) for the matrix C for each m � 1; 2; . . . ;M ,the Fourier coe�cients in Eqs. (2)±(6) can be deter-mined. These equations can then be used to calculate allthe in-plane and vertical displacements, the core shearstresses and the in-plane normal stresses rtxx in thetopskin. 1 The out-of-plane normal stresses rzz in the topskin-core interface are calculated indirectly as

rtxx � Efut;x; �9�rzz � c

2sx;x � Ec

c�wb ÿ wt�: �10�

3. Experimental veri®cation of the high-order sandwich

beam theory

An experimental veri®cation of HOSBT has beenundertaken by Thomsen and Frostig [15], who foundgood agreement between analytical predictions of thestress ®eld and a series of photoelastic experiments onsandwich beams with a polyurethane rubber core. Inthis section a similar comparative study is conductedwith respect to the displacement ®eld in the core. Specialemphasis is given to the local compression of the core, asthis will be important in determining core failure.

3.1. Experimental method

Sandwich beams of width 30 mm and span 60 mmwere loaded under three-point bending by rollers ofdiameter 20 mm at a constant displacement rate of0.5 mm/min. The beams were comprised of a 9.4 mmthick core of 48 kg/m3 Nomex honeycomb (with cell size3 mm) and 0.38 mm thick skins of cross-ply GFRPlaminates. Material properties for the skin and the solidconstituent of the core are summarised in Table 1.Corresponding material properties for the honeycombcore (Ec, Gc) are derived using simple mechanical modelsas detailed in [16].

The experimental setup is shown in Fig. 3. A clipgauge on one side of the beam measures the midspancore compression during loading, while a video camerarecords images of the other side of the beam. The ver-tical displacements on the side face of the core are de-termined using surface displacement analysis (SDA)software developed by Instron Corporation. A randomspeckle pattern is needed, achieved here by ®rst paintingthe surface black and then by applying a ®ne spray ofwhite paint (cf. Fig. 4). A series of images of this surfaceis recorded during loading using a video camera andframe grabber.

1 Only top skin compressive failure is considered here.

Table 1

Material properties of Nomex and laminate skin

Nomex [17] Laminate skin

[Hexcel]

Young's modulus

(GN/m2)

Es � 0:9 Ef � 20:5

Shear modulus (GN/m2) Gs � 0:32 Gf � 4:2

Compressive strength

(MN/m2)

rsc � 80 rfY � 300

Poisson's ratio ms � 0:4 mf � 0:17

Density (Mg/m3) qs � 0:724 ±

416 A. Petras, M.P.F. Sutcli�e / Composite Structures 46 (1999) 413±424

SDA calculates the displacement ®eld for a number ofanalysis cells within a designated `area of interest' bycomparing the pattern in each cell of the current frameto the corresponding pattern in a reference frame. Al-though buckling of the cell walls leads to inaccuracies inthe estimated displacement ®eld at high load, for theinitial stages of loading the measurements satisfactorilycapture the local deformation under the central load.

3.2. Theory

The equations described in Section 2 are implementedusing the Matlab programming language to calculatethe beam response. The vertical displacement ®eldwithin the core is given by [8]

wc�x; z� � ÿsx;xÿz2 � cz

2Ec

� �wb ÿ wt� zc� wt: �11�

The beam material and geometric properties detailedin Section 3.1 are used in the calculations. The total

applied load is taken as 6 kN/m, corresponding to ex-perimental measurements, and is applied uniformly overa length of 2 mm. The calculation is insensitive to thedetails of the loading arrangement, for reasons ex-plained in Section 5.

3.3. Results

The variation in load and midspan core compression(i.e., wt ÿ wb measured at the midspan) with top skinde¯ection are shown in Fig. 5. For the SDA, the refer-ence frame was taken just before loading (point A inFig. 5). Fig. 6(a) shows the measured experimentalvertical displacement ®eld in the core at a line load of6 kN/m, corresponding to point B on Fig. 5.

The ®gure includes contours of equal vertical dis-placements within the core (labelled in mm). As thecamera was ®xed relative to the central roller, thespecimen appears to move upwards relative to the roller,and the vertical displacement at the contact with theroller is equal to zero. A number of closely spacedcontours directly under the roller have been omitted forclarity. The vertical displacement at the bottom of thebeam of around 0.045 mm, taken from the bottomcontour of Fig. 6(a), agrees well with the equivalentmeasurement of total core compression from the clipgauge (see the dashed line in Fig. 5), con®rming theaccuracy of the SDA measurements.

The measured displacement ®eld of Fig. 6(a) can becompared with the theoretical predictions in Fig. 6(b).The scaling of both these plots is the same. Theory and

Fig. 4. SDA software window, showing the reference image frame of a side cross-section (painted to have a random speckle pattern) and the `area of

interest' where the analysis takes place.

Fig. 3. Experimental setup.

A. Petras, M.P.F. Sutcli�e / Composite Structures 46 (1999) 413±424 417

experiments are in good qualitative agreement, withHOSBT accurately predicting the shape and extent ofthe region of local deformation under the indentor.Although theory underestimates the relative displace-ments under the indentor by a factor of around 1.5, theagreement is close enough to give us con®dence in usingHOSBT to model indentation behaviour of these sand-wich beams. This quantitative di�erence is most likely toarise from errors in estimating the core sti�ness, whichare subject to considerable scatter depending on the testarrangement [14], rather than from approximations inthe high-order theory. A particular cause of concernarises from the way that the core material is treated as acontinuum in the theory, so that details of the behaviouron the scale of the cell size are not treated accurately. Inthis case the cell size is 3 mm, so that we might expectsome errors arising from this e�ect where, as in Fig. 6,there are signi®cant changes over this length scale. In-deed the asymmetry observed in Fig. 6 probably comes

about because the indentor is not placed symmetricallywith respect to the cell structure at this `snapshot' po-sition at the edge of the core. However, for the linecontact loading used in these experiments, such localvariations will tend to be averaged out along the lengthof the contact, so that this e�ect will not be as severe as asimple comparison of the cell size and the contact di-mensions might lead us to expect. Were a circular con-tact with diameter of the order of the cell size to be usedinstead, we might expect signi®cant variations in be-haviour depending on the exact position of the indentorwith respect to the cell structure (e.g., with the indentorplaced over a cell wall or a void). Although it is likelythat a closer agreement between theory and experimentscould have been obtained by using a di�erent core ma-terial, Nomex honeycomb cores were retained for thissection of the work as the project was particularlyconcerned with modelling failure for sandwich beamsmade with this material.

Fig. 6. Vertical displacement ®eld in the core produced by (a) SDA

and (b) HOSBT model. All contour values are in mm and the scaling in

both plots is the same.

Fig. 5. Load±de¯ection and core compression curves. A and B cor-

respond to the frames used for processing by SDA. The dashed line

indicates the total midspan core compression at a line load of 6 kN/m.

418 A. Petras, M.P.F. Sutcli�e / Composite Structures 46 (1999) 413±424

4. A spreading length scale k for indentation loading

In this section we use high-order sandwich beamtheory to investigate how the contact pressure under anindentor is transmitted to the core of a sandwich beam,where indentation failure generally occurs. An arbitrarycontact pressure is considered and elastic behaviour isassumed, so that the results will be useful to model thefailure of beams made with Nomex cores and ®brecomposite skins, where the response is essentially elasticup to maximum load. A spreading length scale k is in-troduced which describes whether the skin acts in a rigidor a ¯exible manner (cf. Fig. 2).

We follow the analysis of Section 2, in which atransverse pressure on the beam is expressed in terms ofa Fourier series (Eq. (7)). Re-writing the matrices D andQ introduced in Eq. (8) as

Q �

0

0

Cqtm

0

0

266666666664

377777777775; D �

Af 0 0 0 1

0 Af 0 0 ÿ 1

0 0 Df Ec Ct

0 0 Ec Df Ct

1 ÿ 1 ÿ Ct ÿ Ct Gc

266666666664

377777777775;

where

Af � ÿAt�mpL �2;

Df � ÿDt�mpL �4 � Ec

c ;

Ec � ÿ Ec

c ;

Ct � c�t2

mpL ;

Gc � cGc� c3

12Ec�mp

L �2;

8>>>>>>>>>>><>>>>>>>>>>>:and solving Eq. (8) for C, using the Symbolic Toolbox ofMatlab, we have

�C� � �Q��D�ÿ1 )

Cutm

Cubm

Cwtm

Cwbm

Csxm

26666666666664

37777777777775� Cqt

m

ÿ Ct

R1�R2

Ct

R1�R2

R1

�DfÿEc��R1�R2�ÿ R2

�DfÿEc��R1�R2�Af Ct

R1�R2;

266666664

377777775;

�12�where R1 � Df�AfGc ÿ 2� �Af�Ct�2 and R2 � Ec�AfGc

ÿ 2� �Af�Ct�2.Substituting Eqs. (4)±(6) and the above Fourier co-

e�cients into (10), the normal stresses in the top skin-core interface are given as

rzz �XM

m�1

�ÿ c

2

mpL

Csxm �

Ec

c�Cwb

m ÿ Cwtm ��

sinmpx

L

�XM

m�1

Cqtm

�ÿ cmp

2LAfCt

R1 � R2

� Ec

Df ÿ Ec

�sin

mpxL

:

�13�Eq. (13) shows that the Fourier coe�cients for the stressrzz at the skin-core interface can be separated into twoparts. The ®rst part consists of the coe�cient Cqt

m (unitsof stress), which depends only on the distribution of theload. The second part, which we denote as a dimen-sionless `transmission coe�cient' Crzz

m , is given by

Crzzm � ÿ

cmp2L

AfCt

R1 � R2

� Ec

Df ÿ Ec

� ÿ AfC2t

R1 � R2

� Ec

Df ÿ Ec

if t� c: �14�

This coe�cient Cqtm depends on the geometric and ma-

terial properties of the sandwich beam and on L=m, thesemi-wavelength of the mth term of the Fourier series.Crzz

m varies from zero for small L=m to one for large L=m.A value of zero implies no transmission of this Fouriercomponent of applied stress through the top skin, whileone implies total transmission, so that this parametercharacterises the transparency of the skin as a functionof the wavelength of the applied load.

Fig. 7 plots the results of a parametric study, showingthe variation in transmission coe�cient Crzz

m with thesemi-wavelength L=m of the applied sinusoidal load.Table 2 details the range of material and geometric pa-rameters considered. Each sub-plot in Fig. 7 containscurves for two values of skin thickness t and core depthc, while changing from one sub-plot to the next repre-sents a change in either skin sti�ness Ef or core densityqc. Fig. 7 shows that, at su�ciently short wavelengths,the transmission coe�cient Crzz

m falls to zero, indicatingthat the contact stresses at these short wavelengths aremodi®ed signi®cantly as they pass through the skin tothe core. This tendency of the skin to spread out the loadincreases with increasing skin sti�ness Ef and thickness tas expected. The e�ect of core depth c and density qc isless marked.

It is helpful to extract a characteristic wavelengthbelow which the contact stresses are modi®ed signi®-cantly by the skin. Examining Fig. 7, it can be seen thatthe in¯ection point in each curve (marked by a circle)can be used to identify where Crzz

m becomes small. Thesemi-wavelength at this in¯ection point, which we de-note the spreading length k, characterises the suscepti-bility of sandwich beams to localised e�ects. Contactstresses with semi-wavelengths below k are spread outby the skin, while for longer wavelengths the appliedstresses pass through the skin to the core relatively un-modi®ed. Table 3 shows the in¯uence of changes in each

A. Petras, M.P.F. Sutcli�e / Composite Structures 46 (1999) 413±424 419

of the relevant parameters on the spreading length k.The ®rst row is a baseline case, using the low valuesfrom Table 2, while subsequent rows change one of theparameters as indicated. Table 3 shows that skin thick-ness t is the most signi®cant factor as expected. Thethicker the skin, the better the skins are able to spreadthe load. A similar e�ect is shown with increasing skinsti�ness Ef although, given the big change in sti�ness of�100, this e�ect is less important. Core density alsoa�ects k while core thickness c plays a minor role.Separate calculations con®rmed that k did not dependon the beam length L.

In summary, Crzzm describes how indentation loads are

transmitted through the skin into the core. Plots of Crzzm

as a function of load semi-wavelength L=m (Fig. 7) areused to extract a characteristic wavelength k, belowwhich the contact stresses are modi®ed signi®cantly bythe skin, for a given beam geometry and material com-bination. This analysis, which is based on elastic be-haviour, allows us to understand the beam behaviourfor a given applied pressure distribution. In the follow-ing section this work is related to models of indentationby cylindrical rollers. Petras [18] extends this analysis tomodel indentation failure.

5. Indentation by a cylindrical roller

Section 4 describes the behaviour of a sandwich beamunder a general contact load represented by a Fourierseries. In this section the modelling work described inSection 2 is extended to consider indentation by a cen-tral cylindrical roller for a beam loaded under three-point bending. Illustrative case studies are presented andinterpreted using the spreading length k de®ned in Sec-tion 4.

5.1. Theory

Fig. 8 shows a rigid cylinder of radius R which ispressed into contact with a sandwich beam of unit widthand length L, such that the contact width is d. For a

Fig. 7. Variation of transmission coe�cient Crzzm with semi-wavelength L=m for various values of t;Ef ; c and qc. The boxed labels identify the values of

Ef and c for each subplot. Points of in¯ection in the curves are marked by a � symbol.

Table 2

The range of the geometric and material parameters used in the

parametric study. qc is the density of Nomex honeycomb. The density

of the honeycomb solid constituent material qs � 724 kg/m3

Parameter Low value ! High value

t (mm) 0.381 ! 4 � 0.381

Ef (GPa) 1 ! 100 � 1

qc (kg/m3) 48 ! 3 � 48

c (mm) 2.5 ! 4 � 2.5

Table 3

Change of spreading length k

Parameter changed k (mm)

Baseline case; `low values' from Table 2 1.6

t ! 4t 4.4

c! 4c 2.1

Ef ! 100Ef 5.0

qc ! 3qc 1.2

420 A. Petras, M.P.F. Sutcli�e / Composite Structures 46 (1999) 413±424

given contact pressure distribution, Eq. (4) can be usedto ®nd the beam de¯ection. Conversely, for a givenroller geometry, we can evaluate the contact width andcontact pressure by inverting this approach. Followingstandard contact methods [12], we divide the contactinto a number of overlapping elements of width v dis-tributed over the contact along the beam so that themidpoint of each element is at xi, as illustrated in Fig. 8a.A symmetrical load is assumed, with 2N elements incontact, so that only one half of the beam is considered.A triangular pressure distribution of peak height Pi isapplied to each element, so that the contact pressure ispiecewise-linear, with values Pi at the ith node points.

To use the results of Section 2, the triangular pressuredistribution qi at the ith element is represented by theFourier series

qi � Piqi � Pi

XM

m�1

Cqim sin

mpxL

; �15�

where

Cqim � ÿ

8Lm2p2v

sinmpxi

L

� ��ÿ 1� cos

mpv2L

� ��are the Fourier coe�cients for a triangular pressuredistribution qi of unit amplitude. The `hat' is subse-quently used for variables corresponding to this unitpressure distribution. The number of terms M in theFourier series is typically taken as 3000. Using theprocedure described in Section 2, the vertical displace-ments wi

t�x� of the top skin associated with each of thetriangular pressure loads are given by Eq. (4)

wit�x� � Pi

XM

m�1

�Cwtm �i sin

mpxL� Piwi

t: �16�

Summing over the N pairs of elements in the contact, thetotal top skin de¯ection is given by

wt�x� �XN

n�1

wit�x� �

XN

n�1

Piwit: �17�

We assume that the indentor is in contact with the skinover the whole contact width d, so that within the con-tact with L

2ÿ d

2< x < L

2� d

2, the top skin de¯ection is

given by

wt�x� ���������������������������������R2 ÿ xÿ L

2

� �2s

ÿ R� wt�L=2�

���������������������������������R2 ÿ xÿ L

2

� �2s

ÿ R

�XN

n�1

Pi

XM

m�1

�Cwtm �i sin

mp2

���������������������������������R2 ÿ xÿ L

2

� �2s

ÿ R�XN

n�1

Piwit�L=2�; �18�

giving

XN

n�1

Pi wit�x�

�ÿ wi

t�L=2���

��������������������������������R2 ÿ xÿ L

2

� �2s

ÿ R: �19�

The above equation, which is true for each of the N nodepoints on the half-contact considered, can be written inmatrix form as

�P� � �W� � �R�; �20�where

P � P1 P2 � � � PN� �;

W �

w1t�x1� ÿ w1

t�L=2� w1t�x2� ÿ w1

t�L=2� � � � w1t�xN � ÿ w1

t�L=2�

w2t�x1� ÿ w2

t�L=2� w2t�x2� ÿ w2

t�L=2� � � � w2t�xN � ÿ w2

t�L=2�

..

. ... . .

. ...

wNt�x1� ÿ wN

t�L=2� wNt�x2� ÿ wN

t�L=2� � � � wNt�xN � ÿ wN

t�L=2�

2666666664

3777777775;

�21�

R �

����������������������������R2 ÿ �x1 ÿ L

2�2

qÿ R����������������������������

R2 ÿ �x2 ÿ L2�2

qÿ R

..

.�����������������������������R2 ÿ �xN ÿ L

2�2

qÿ R

26666664

37777775

0

: �22�

Fig. 8. (a) Details of the contact pressure elements; (b) de®nition of the

overall geometry.

A. Petras, M.P.F. Sutcli�e / Composite Structures 46 (1999) 413±424 421

To use this result, a contact width d is chosen, the ma-trices W and R are computed from Eqs. (21) and (22),respectively, and the unknown pressure distribution Pi

found from Eq. (20). Substitution of Pi into Eq. (15) andsummation for all N triangular pressure elements givesus the contact pressure distribution

qt�x� �XN

i�1

qi: �23�

The total load is then found from integrating the pres-sure distribution over the contact. Superposition can beused to calculate the other stresses, for example rzz, fromEq. (10). By varying the assumed contact width d, thevariation in load distribution with total load W can befound. In practice a problem arises at small d. As it isnecessary to span the whole beam length with elements,the number of elements needed to deal with sharpvariations in pressure then becomes prohibitively large.

5.2. Results

This section presents theoretical calculations for in-dentation by a central cylindrical roller of radius R � 5or 10 mm. Results for beams with two extreme values ofskin ¯exural rigidity are presented to illustrate the e�ectof skin ¯exibility on the way that the indentation pres-sure is transmitted to the core. The e�ect of indentor sizeis also investigated.

Two sandwich beams `A' and `B' are considered.Both beams, which are 60 mm long and of unit depth,have a Nomex honeycomb of thickness 9.4 mm withcore density of 128 kg/m3 (Ec � 0:15 GPa,Gc � 0:03 GPa). Sandwich beam A has relatively ¯exibleskins, consisting of one 90° ply with thickness t � 0:191mm and Ef � 1 GPa, to give a spreading length k of 0:8mm. Beam B, which is typical of those used in industry[13], has relatively rigid skins, each skin consisting of a0° and a 90° ply with t � 0:381 mm and Ef � 20 GPa, togive k � 3:7 mm.

Figs. 9 and 10 show the e�ect of roller radius R forthe beams `A' and `B' with ¯exible and sti� skins re-spectively. These ®gures plot the variation with positionof the contact stress qt=rcc and normal stress in the topskin-core interface rzz=rcc across the width of the con-tact patch. Stresses are normalised by the core out-of-plane compressive strength rcc. d is adjusted manually toensure that the maximum normal stresses in the topskin-core interface are approximately equal for bothvalues of indentor radius R. For beam B the input valueof d is chosen to give the maximum value of rzz=rcc � 1.Due to the much greater ¯exibility of beam A, de¯ec-tions are excessive at this load and a reduced maximumstress of rzz=rcc � 0:4 was chosen. As the calculationsare elastic, this di�erence in peak stress between the twobeams is immaterial.

A striking feature of the contact pressure distributionfor the ¯exible beam A with R � 5 mm (Fig. 9) and forthe sti�er beam B with R � 10 mm (Fig. 10) is the sharppeaks in pressure at the edges of the contact. (It is likelythat numerical inaccuracies have obscured this featurefor the sti�er beam with R � 5 mm.) This feature hasbeen noted by Johnson [12] for similar contacts. Thesecontact peaks are not transmitted through the skins, asthey are of shorter wavelength than the spreading lengthscales of both beams. Fig. 9, shows that, aside fromthese pressure peaks, the pressure at the contact for the¯exible beam is similar to that transmitted into the core.The skin is not su�ciently sti� to spread out the load atthis wavelength. For the sti�er beam, Fig. 10, thestresses in the core are spread over a much greater widththan the contact stresses. In this case the skin is su�-ciently sti� to spread out the load.

We characterise the degree to which the skins spreadout the load by comparing the width of the contact

Fig. 9. Indentation stresses for beam A (k � 0:8 mm, k=L � 0:013).

Distribution of contact stress qt=rcc and normal stress in the top skin-

core interface rzz=rcc, normalised by the core's out-of-plane compres-

sive strength.

422 A. Petras, M.P.F. Sutcli�e / Composite Structures 46 (1999) 413±424

patch d and the width D over which there are signi®cantpressures transmitted across the interface into the core.The width D is taken as the length over which the in-terface pressures are positive, as illustrated in Figs. 9and 10. Fig. 11 shows the e�ect of beam spreadinglength scale k on the ratio D=d of the width of the loadedregion at the skin-core interface to the contact pressurewidth. This plot is calculated for the two roller radii 5and 10 mm by considering a range of skin thicknesses tbetween 0.18 and 0.3 mm and Young's moduli Ef be-tween 1 and 20 GPa. A smooth curve is drawn throughthe calculated points for each roller radius. Fig. 11shows that, for very ¯exible skins with small k, the widthof the contact equals the width of the region in whichthere are signi®cant pressures across the interface, withD=d � 1 as expected. For larger values of the spreadinglength k, the distribution of pressure in the interface isgreater than the contact width, with D=d greater thanone. This change depends on roller diameter. If we take

the transition from ¯exible to rigid behaviour when D=dequals two, then this occurs for k=R equal to 0.24 and0.25 for the 5 and 10 mm rollers, respectively.

This case study demonstrates the usefulness of thespreading length k in characterising the beam ¯exibility.It also allows us to answer some common modellingquestions. Firstly, for practical beams with large k, thedistribution of pressure over the contact does not a�ectthe core loading signi®cantly, and the assumption thatthe external load is applied uniformly across some ap-propriate contact width is reasonable. However, thecontact load is not transmitted through the skin into thecore unchanged. In fact, Fig. 10 shows that it is essentialto model beams with sti� skins accurately to make areasonable estimate of the stress distribution in the core.The commonly used assumption that the failure can bepredicted using

Failure line load � Core compressive strength ; rcc

� Contact width; d �24�will be poor unless a judicious choice of d is made basedon experimental measurements.

6. Conclusions

A HOSBT is used to analyse the indentation behav-iour of sandwich beams with properties typical of No-mex honeycomb core and composite skins. Thecalculation, which is elastic, will be appropriate for thefailure analysis of these materials. The capabilities ofHOBST are veri®ed experimentally by measurements of

Fig. 11. Variation with beam spreading length scale k of the ratio D=dof the width of the loaded region at the skin-core interface to the

contact pressure width.

Fig. 10. Indentation stresses for beam B (k � 3:7 mm, k=L � 0:062).

Distribution of contact stress qt=rcc and normal stress in the top skin-

core interface rzz=rcc, normalised by the core's out-of-plane compres-

sive strength.

A. Petras, M.P.F. Sutcli�e / Composite Structures 46 (1999) 413±424 423

the displacements at the free surface of a beam under acylindrical indentor. Although there are some quanti-tative di�erences, the results are encouraging and justifythe use of the model in an analysis of indentation be-haviour of such beams. By considering the generalloading case using a Fourier series, the e�ect of theapplied wavelength of the contact stress on the beambehaviour is investigated. Results are used to extract aspreading length k for a given beam material and geo-metric properties. This is a property of a sandwichbeam, depending mainly on the skin's ¯exural sti�nessand characterising the susceptibility of the sandwichbeam to indentation loads. Small values of k correspondto sandwich beams with very ¯exible skins whichtransmit external loads straight into the core. Largevalues of k indicate rigid skins that spread the indenta-tion load over a wider area. HOSBT is then used toexamine indentation loading by a cylindrical indentor.The way in which the contact pressure is transmittedthrough the core is examined. A case study shows thatsandwich beams used for standard aircraft ¯oor panelsin industry have skins which are rigid enough to spreadthe external loads. For such beams the maximum corestresses, which are responsible for indentation failure,cannot be predicted in a straightforward way, even if thecontact width d is known. For sandwich beams withvery ¯exible skins, the approximation of dividing thetotal line load W by d can give us a reliable estimation ofthe failure stresses in the skin-core interface.

Acknowledgements

The authors are most grateful to Hexcel Compositesfor their help in supplying materials and for the tech-nical assistance of Mr. Nigel Hookham and Mr. PeterClayton. Thanks are due to Messrs. Alan Heaver andSimon Marshall for their valuable contribution to theexperiments and to Prof. Norman Fleck for his advice.The authors acknowledge with gratitude the ®nancialsupport of the Greek State Scholarships Foundation

and the US O�ce of Naval Research (grant number0014-91-J-1916).

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