Design of Sandwich Structures

Achilles Petras

Robinson College Cambridge

Sup ervisor Dr MPF Sutclie

A dissertation submitted to

Cambridge University Engineering Department

for the degree of Do ctor of Philosophy

Decemb er

To my parents and b ebita

Declaration

This dissertation presents the results of research carried out in the Engineering

Department of the UniversityofCambridge b etween Octob er and Octob er

Except where sp ecic reference is made to the work of others this dissertation is the

original result of myown work and includes nothing which is the outcome of work done

in collab oration No part of this dissertation has been submitted for a degree at any

other University This dissertation is approximately words long and contains

gures

Achilles Petras

March Robinson College Cambridge ii

Acknowledgements

Iwould like to express my gratitude to my sup ervisor Dr Michael Sutclie for all his

kind help encouragement and dedicated guidance without which this pro ject would

not have been p ossible He gave me exibility to follow my research interests but at

the same time he kept me going in the right direction His patience with pro ofreading

my thesis is highly appreciated

I am also grateful to Professor Norman Fleck for his invaluable advice and fruitful

discussions Iwould like to thank him for supp orting nancially my pro ject US Oce

of Naval Researchgrant J and my attendance at the th International

Conference on Sandwich Construction in Sto ckholm

I acknowledge with gratitude the generous nancial supp ort of the Greek State

Scholarship Foundation IKY I would like to thank Professors Christos Panagop oulos

and Constantinos Lascarides at the National Technical University of Athens and Dr

Dimitris Niarchos at the National Centre for Scientic Research Demokritos in Greece

for supp orting and urging me to pursue my PhD studies at Cambridge

I would like to thank Nigel Ho okham and Peter Clayton from Hexcel Comp osites

in Duxford for providing materials and valuable technical information I wish to ac

knowledge the Cambridge University Engineering Department CUED and esp ecially

the Cambridge Centre for Micromechanics CCM for use of their facilities I am also

grateful to Alan Heav er and Simon Marshall for building the test rigs and helping

me during my exp eriments Sp ecial thanks to the CUEDs librarians for their kindest

assistance I wish to thank my friend Antonios Zervos for helping me out on compu

tational mo delling topics Many thanks to Roxy Pia Ingo Sinisa Guy Dongquan

Kieran Jide at the CCM for those stimulating discussions and entertaining moments

when there was achance to see them due to my night shifts

Sta students and fellows at Robinson College have help ed makemy time at Cam

bridge really enjoyable Thanks to all my friends here in Cambridge for the enjo yable

moments wehave shared during these three years Also to my friends in Athens Kostas

and Stefanos for not forgetting me and for making my holidays in Greece a real fun

I am extremely grateful to my parents Violetta and Alexandros for their constant

understanding caring love and supp ort o o

Finally but most imp ortantlyI would like to thank Georgina for her love patience

and the happy life we share together Te amo iii

Abstract

Failure mo des for sandwich b eams of GFRP laminate skins and Nomex honeycomb

core are investigated Theoretical mo dels using honeycomb mechanics and classical

beam theory are describ ed A failure mo de map for loading under p oint b ending

is constructed showing the dep endence of failure mo de and load on the ratio of skin

thickness to span length and honeycomb relative density Beam sp ecimens are tested

in p oint b ending The eect of honeycomb direction is also examined The exp eri

mental data agree satisfactorily with the theoretical predictions The results reveal the

imp ortant role of core shear in a sandwich b eams b ending b ehaviour and the need for

a b etter understanding of indentation failure mechanism

Highorder sandwich b eam theory HOSBT is implemented to extract useful infor

ab out the way that sandwich beams resp ond to lo calised loads under p oint mation

b ending Highorder or lo calised eects relate to the nonlinear patterns of the in

plane and vertical displacements elds of the core through its height resulting from

the unequal deformations in the loaded and unloaded skins The lo calised eects are

examined exp erimentally by Surface Displacement Analysis of video images recorded

during p oint b ending tests A new parameter based on the intrinsic material and

geometric prop erties of a sandwich beamisintro duced to characterise its susceptibility

to lo calised eects Skin exural rigidityisshown to playakey role in determining the

way that the top skin allo ws the external load to pass over the core Furthermore the

contact stress distribution in the interface b etween the central roller and the top skin

and its imp ortance to an indentation stress analysis are investigated

To b etter mo del the failure in the core under the vicinity of lo calised loads an Arcan

typ e test rig is used to test honeycomb cores under simultaneous compression and shear

loading The exp erimental measurements show a linear relationship between the out

ofplane compression and shear in honeycomb cores This is used to derive a failure

criterion for applied shear and compression which is combined with the highorder

sandwich beam theory to predict failure caused by lo calised loads in sandwich b eams

made of GFRP laminate skins and Nomex honeycomb under p oint b ending loading

indenters size are p erformed on appropriately Short beam tests with three dierent

prepared sp ecimens Exp eriments validate the theoretical approach and reveal the iv

Abstract

nature of pre and p ostfailure b ehaviour of these sandwich b eams HOSBT is used as

a compact computational to ol to reconstruct failure mo de maps for sandwich panels

Sup erp osition of weight and stiness contours on these failure maps provide carp et

plots for design optimisation pro cedures

Keywords

comp osite structures sandwich structures Nomex honeycomb sandwich b eams failure

mo de maps honeycomb anisotropy indentation lo calised eects highorder sandwich

beam theory mixed failure criterion for honeycombs biaxial testing of honeycombs v

Contents

Preface and Declaration ii

Acknowledgements iii

iv Abstract

Contents vi

List of Figures viii

List of Tables xi

Nomenclature xii

Intro duction and Overview

Intro duction to Comp osite Structures

olymer Comp osites Fibre Reinforced P

Structural Optimisation

Sandwich Structures

Previous Work on Indentation Analyses

Previous Work on Indentation Failure Prediction

Scop e and Outline of the Thesis

Failure Mo de Maps for Honeycomb Sandwich Beams

Intro duction

Beam Theory for Sandwich Panels

Skin Failure

Core Failure

Honeycomb Mechanics

Construction of a Failure Mo de Map

A Failure Mo de Map for Beams with a GFRP Skin and Nomex

Core

Exp eriments

Exp erimental Results vi

Contents

Discussion

Skin Failure

Core Failure

Eect of Ribb on Direction

Intracell Buckling

Conclusions

Indentation Resistance of Sandwic h Beams

Intro duction

HighOrder Sandwich Beam Theory

Surface Displacement Analysis

Eect of Spreading Stresses

Contact Pressure Distribution

Case Study

Concluding Remarks

Indentation Failure Analysis

Intro duction

Failure Envelop e for Nomex Honeycombs

Failure Analysis with HOSBT

Exp erimental Work

Concluding Remarks

Failure Maps using HOSBT

Intro duction

Reconstruction of Failure Maps

Optimisation Carp et Plots

Concluding Remarks

Conclusions and Future Work

Conclusions

Future Work

App endix A Exp erimental Results

Bibliography vii

List of Figures

Sandwich construction with honeycomb core

Sandwich panels with a corrugated b foam and c honeycomb core

Typical hexagonal honeycomb with a set of doubled walls t is the single

wall thickness and is the honeycomb cell size

Test rig for indentation fatigue tests on aircraft o or panels

a Simply supp orted b eam b Cross section on AA

Failure mo des in the skin

Failure mo des in the core

Comparison of theoretical and exp erimentally measured outofplane a

compressive mo dulus b compressive strength c shear mo dulus and

d shear strength of Nomex honeycombs

a Failure mo de map Solid lines refer to beams in which the honey

comb ribb on lies along the b eam axis dashed lines for when the ribb on

lies transverse to the beam axis The symbol identies exp erimen

tal measurements describ ed in section b Failure load surface for

ribb on lying along the b eam axis

Failure mo de map for three typical values of tL and Rt

Layup details

Test setup

Photographs of the dierent failure mo des for transverse ribb on direc

tion and corresp onding load deection curves for b oth ribb on directions

Variation of failure load W with skin thickness to span ratio tL

Photographs and results of sp ecimens with mm cell size

The behaviourofexibleand rigid skins

a Nonlinear displacement patterns b Beam geometry and stresses

The origin of the zco ordinate is always taken at the top of the b eam

element either skin or core which is b eing considered viii

List of Figures

Surface Displacement Analysis software window showing the reference

image frame of a side cross section painted to have a random sp eckle

pattern and the area of interest where the analysis takes place

Exp erimental setup

Loaddeection and core compression curves A and B corresp ond to

the frames used for pro cessing by SDA The dashed line indicates the

total midspan core compression at a line load of kNm

Vertical displacement eld in the core pro duced by a SDA and b

HOSBT mo del All contour values are in mm and the scaling in b oth

plots is the same

with wa Parametric Study change of transmission co ecient C ve

length Lm for variations in the parameters t E c and The b oxed

f c

lab els showeach time the parameters and the arrows indicate the direc

tion of each parameters increase The p osition of inection is marked

by a symbol

Discrete contact pressure elements a uniform piecewise constant b

overlapping triangles piecewise linear and c Geometric denitions

Distributions of contact stresses q and the corresp onding normal

t cc

stresses in the top skincore interface normalised by the cores

zz cc

outofplane compressive strength for b eam A with mm

Distributions of contact stresses q and the corresp onding normal

t cc

stresses in the top skincore interface normalised by the cores

zz cc

outofplane compressive strength for b eam B with mm

Dep endence of spreading eect on rollers radius R and determination

on how exible or rigid are the skins with resp ect to the indentation

resistance of the sandwich beam

Combined indentation failure mechanism

The Arcantyp e rig

Load paths and determination of failure envelop e

The three angle setups and the corresp onding loaddeection curves

here for sp ecimens with kgm core density

Failure envelop es for Nomex honeycombs The dashed line corresp onds

to the linear failure criterion given by equation

Combined failure criterion

Exp erimental setup

Typical exp erimental results for sandwich b eam with kgm core den

sity loaded by a roller with diameter of mm Midspan b ottom skin

deection midspan core compression and line load are plotted against

midspan top skin deection Lines legend longitudinal and

transverse honeycomb ribb on direction ix

List of Figures

The prefailure inuence of core density Video images captured just b e

fore failure during p oint b ending loading with a mm diameter central

roller

The p ostfailure inuence of core density Video images captured after

failure mm total deection during p oint b ending loading with a

mm diameter central roller

The p ostfailure inuence of core densityonexten t of damage Video im

ages captured after failure mm total deection during p oint b end

ing loading with a mm diameter central roller

Theory lines vs exp erimental results symbols for failure line load W

Improved failure map for Nomex honeycomb sandwich b eams Each

contour represents sandwich b eams of equal strength in Nm

Comparison with the exp erimental results from Chapter Lines show

the predictions of HOSBT failure analysis and the symb ols represent the

exp erimental data cf Fig Solid lines and symb ols corresp ond to

the longitudinal ribb on direction while dashed lines and hollow symb ols

corresp ond to the transverse ribb on direction

Strength contour plot corresp onding to Fig

Stiness contour plot

Optimisation carp et plot for tL with and cL as design

c s

parameters

Vertical displacements calculated by Abaqus nite element analysis

A For mm diameter roller midspan b ottom skin deection midspan core

compression and line load variation curves with resp ect to midspan top

skin deection Lines legend longitudinal and transverse

honeycomb ribb on direction

A For mm diameter roller midspan b ottom skin deection midspan

core compression and line load variation curves with resp ect to midspan

top skin deection Lines legend longitudinal and transverse

honeycomb ribb on direction

A For mm diameter roller midspan b ottom skin deection midspan

core compression and line load variation curves with resp ect to midspan

top skin deection Lines legend longitudinal and transverse

honeycomb ribb on direction

A Video snapshots during loading with a mm diameter roller

A Video snapshots during loading with a mm diameter roller x

List of Tables

An example of structural eciency of sandwich panels in terms of weight

Summary of failure criteria

Material prop erties of Nomex and laminate skin

Exp erimental results Photographs of failure given in Fig and

corresp ond to the entries in italics in the nal column of this table

The range of the geometric and material parameters used in the para

metric study

Change of spreading length

Expressions for p eak failure loads xi

Nomenclature

Greek symb ols

Honeycomb cell size

Length of contact between the central roller and the top skin or

length of top distributed load

Quantities dened in section

Outofplane honeycomb Poissons ratio

cxz

Poissonss ratio of the skin material

fxy

Honeycomb core density

Density of honeycombs constituent material

Outofplane compressive strength of the honeycomb core

Inplane compressive stress for intracell buckling of the skin

Inplane wrinkling strength of the skin

Maximum inplane stresses in the skins Chapter

Inplane yield strength of the skins

Inplane normal stresses in the skins

txx bxx

Outofplane normal stresses in the core

Outofplane shear strength of the honeycomb for transverse and

longitudinal ribb on direction resp ectively

Outofplane shear strength of the honeycomb core

Maximum shear stresses in the core Chapter

cxz

Outofplane shear stresses in the core

Setup angle in Arcan test rig

Latin symbols

A A Inplane rigidityoftop and b ottom skin

t b

b Beam width

C Fourier co ecient

c Core thickness

D Flexural stiness of the beam

D D Flexural rigidityof top and b ottom skin

t b

Nomenclature

d Distance b etween the midplanes of top and b ottom skin

E or E Outofplane Youngs mo dulus of the honeycomb core

E Inplane Youngs mo dulus of the honeycomb core in the x direction

E E Youngs mo dulus of the face material in the x direction

f fx

E Youngs mo dulus of the honeycombs solid material

G G Outofplane shear mo duli of the honeycomb for transverse and

longitudinal ribb on direction resp ectively

G G Outofplane shear mo dulus of the core in the xz direction

c cxz

G Shear mo dulus of the honeycombs solid material

I Second moment of area of the sandwich b eam

I Second moment of area of the skins with resp ect to their own cen

troidal axes

L Span of the sandwich b eam

long trans Longitudinal transverse honeycomb ribb on direction

M Maximum b eam b ending moment

P Heights of triangular pressure elements Chapter

q Discrete pressure elements Chapter

q External distributed load applied on top skin or contact pressure

distribution b etween the central indenter and top skin

R Rollers radius

t t t Skin thickness top b ottom

t b

t Single wall thickness of honeycomb

u u Inplane centroidal displacements of top or b ottom skin

t b

v Base width of triangular pressure elements Chapter

W W Load p er unit width at failure

w w Vertical displacements of top or b ottom skin

t b

w Vertical displacements in the core

x y z Co ordinates of sandwich beam

Note Other symbols not mentioned are dened in the text or gure when app earing xiii

Chapter

Intro duction and Overview

One daytowards the end of a circus proprietor called George May called to

seemeatFarnb orough After he had told me several Gerald Durrelltyp e stories ab out

the diculties of keeping monkeys in travelling circuses he pro duced something which

lo oked like a cross b etween a book and a concertina When he pulled on the ends of

this invention the whole thing op ened out like one of those colouredpap er festo ons

which p eople use for Christmas decorations It was in fact a sort of pap er honeycombof

very lightweight but of quite surprising strength and stiness Did I think that sucha

thing could b e of any use in aircraft The snag as George May mo destly admitted was

that since it was only made from brown pap er and ordinary gum it had no moisture

resistance at all and would fall to bits if it got wet

This must have b een one of the relatively few o ccasions in history when a group of

aircraft engineers have b een seriously tempted to throw their collective arms around the

neck of a circus proprietor and kiss him However we resisted the temptation and told

May that there could b e no serious dicultyinwaterpro ong the pap er honeycombby

means of a synthetic resin

JE Gordon Structures or why things dont fal l down

Intro duction to Comp osite Structures

Innovative high p erformance design of loadb earing comp onents is always sought in

hightech applications such as aircrafts spacecrafts satellites or F racing cars These

structures should be as as light as p ossible while having high stiness sucient

strength and some damage tolerance This requires structural ly ecient construc

tion Structural eciency can b e maximised by using the most ecient materials and

optimising the structures geometry To pro duce an optimum design b oth these factors

need to b e considered throughout the design pro cess

The catalogue of materials is one of forbidding length Thus a designer needs a

systematic way to be guided through the maze of material classes so as to gradu

ally narrow down the material choices and cho ose the optimum material Ashby

prop osed a material selection pro cedure using material selection charts Birming

ham et al have intro duced an integrated approach to the assessment of alternative

Chapter Intro duction and Overview

materials and structural forms at the concept stage of structural design based on the

ab ove metho dology

Fibre Reinforced Polymer Comp osites

Fibre Reinforced Polymer FRP comp osite materials are some of the most useful

materials available to the designer of high p erformance structures in the aerospace

or maritime industries b ecause their high sp ecic strength and stiness can lead to

signicant weight reductions The aerospace industry has proved the eectiveness of

comp osite materials in reducing comp onentweight and increasing fuel economy In the

marine eld the use of comp osites has been growing steadily since the early s

particularly driven by the low construction costs of Glass FRPs Initial applications

include small crafts lifeb oats and pleasure crafts Potential applications are radar

domes masts and piping ship hulls ship sup erstructures submersibles and oshore

structure mo dules Yachts participating in the Americas Cup race have allcomp osite

hull keel and mast Racing car b o dies are made of comp osites providing more safety

per unit weight to drivers In the last decade comp osites have b een used in applications

outside the aerospace industry Sp orting go o ds such as tennis and squash rackets golf

shafts bicycles and oars are a ma jor outlet for comp osites materials Future applica

tions include high sp eed trains subway cars and surface ships As well as weight

reductions other benets of comp osites include their ability to cop e with extreme en

vironment reliability maintainability life cycle cost and service life extension Despite

all their advantages comp osites are still regarded as exp ensive materials to purchase

to ol up for and work with

When using comp osites a designer has the freedom to tailor the prop erties Al

though this can improve the structural eciency this also increases the complexity

and number of design parameters Many design to ols have been develop ed to predict

their behaviour and so decrease the level of empiricism which has been used up to

now in design pro cedures Finding an ecient comp osite structural design that meets

the requirements of a given application can be achieved not only by sizing the cross

sectional areas and members thicknesses but also by global or lo cal tailoring of the

material prop erties through selective use of orientation numb er and stacking sequence

of laminae that make up the laminate comp osite A comprehensive presentation

of the equations which govern the mechanical b ehaviour of laminates can be found

in Hulls book Stress analysis for the design of comp osites laminates is commonly

implemented by the use of computer programs ie Cambridge Comp osite Designer

based on laminated plate theory LPT

Chapter Intro duction and Overview

This kind of software is convenient ecient and userfriendly but do es not help de

termine the optimum laminate conguration or cho ose the b est material system For

this purp ose there are pro cedures which are based on the use of carp et plots which

are themselves generated using LPT principles but also provide a graphic illustration

of the whole range of elastic prop erties available with chosen system Miki de

scrib es a highly practical to ol for design optimisation of laminates based on a graphical

pro cedure Tsai and Patterson have intro duced the laminate ranking metho d for

selecting the optimum ply angles Recently Quinn has published a design manual

facilitate the de for comp osites which provides practical information for engineers to

sign of GRP CFRP AramideRP comp osites Quinn has also intro duced a relevant

nomogram which allows the costs of the constituent materials bres matrix

in a comp osite to b e quickly assessed

Structural Optimisation

The design of full scale comp osite materials structures usually requires a building blo ck

approach to testing and design This approachinvolves increasing the testing com

plexity and size from coup on tests to full structural tests The chemical and material

tests in the rst stages are generally well dened tests As wemove from the laminate

level to subelement comp onent and substructure level the test become more appli

cationdep endent Neither standard test metho ds nor databases exist for these larger

tests Thus there is a need to reduce cost and increase the eciency of structural design

and structural failure prediction by a globallo cal testing and analysis approach The

globallo cal approach involves supp orting the global tests and analyses used in tradi

tional design approaches by critical lo cal subelement tests These tests are intended

to be in between a coup on and substructure test They are cheap er to manufacture

and test and most imp ortantly they are fully representative of structural congurations

undergoing the same manufacturing pro cesses and may even be cut from an actual

structure

A ma jor constraint to overcome in the design and construction of large brerein

forced plastic FRP structures eg in ship hulls or aircraft wings is the exibilityof

the panels Although this can be overcome by large increases in thickness this causes

molding diculties and is also very uneconomical A more satisfactory alternative is

the use of stieners which are commonly in the form of tophats Ib eam teejoint

or by use of sandwich constructions The stieners may be an integral part of

the shell or they may be attached by adhesive b onding the latter is the usual case

Bonded or laminated connections b etween two structural memb ers represent a zone of

Chapter Intro duction and Overview

potential weakness

Recent studies have resulted in computer co des for design of sp ecic stiened panel

congurations sub ject to simple loadings usually compression In the s a com

puter co de PASCO Panel Analysis and Sizing Co de was develop ed by NASA

which has been widely used for comp osite optimization pro cedures The co de has

been designed to have sucient generality in terms of panel conguration loading

and practical constraints so that it can be used for nal sizing of panels in a realistic

design situation The UWCODA University of Washington Comp osite Optimization

el and Design Algorithm designanalysisoptimization software to ol was originally dev

op ed to optimise the layup of at comp osite panels In the UWCODA analysis b oth

ply orientation angles and stiener geometries are treated as design variables Opti

mum designs are sought which minimise structural weight and satisfy mechanical

p erformance requirements for example maximum strain and minimum strength

Esp ecially within the aircraft industry nite element based optimisation metho ds

are used to size complex structures for minimum weight taking into account a va

riety of constraints including strength stiness and aero elasticity Howev er there is

always a need for simple lowcost direct metho ds which can assist in congurations

and weight studies at the initial design stage Bartholomew apply a simple

optimization metho d known as geometric programming GP In this metho d the

minimisation of a function representing weight or cost sub ject to nonlinear constraints

on the variables can sometimes b e reduced to the solution of linear equations

Sandwich Structures

Amongst all p ossible design concepts in comp osite structures the idea of sandwich

construction has b ecome increasingly p opular b ecause of the development of man

made cellular materials as core materials Sandwich structures consist of a pair of

thin sti strong skins faces facings or covers a thick lightweight core to separate

the skins and carry loads from one skin to the other and an adhesive attachment

which is capable of transmitting shear and axial loads to and from the core Fig

The separation of the skins by the core increases the moment of inertia of the panel

with little increase in weight pro ducing an ecient structure for resisting b ending

and buckling loads Table shows illustratively the exural stiness and strength

advantage of sandwich panels compared to solid panels using typical b eam theory with

typical values for skin and core density By splitting a solid laminate down the middle

and separating the twohalves with a core material the result is a sandwich panel The

Chapter Intro duction and Overview

Figure Sandwich construction with honeycombcore

new panel weighs little more than the laminate but its exural stiness and strength

is much greater by doubling the thickness of the core material the dierence is even

more striking

Thus sandwich panels are p opular in high p erformance applications where weight

must be kept to a minimum for example aeronautical structures highsp eed marine

craft and racing cars In the most weightcritical applications comp osite materials are

used for the skins cheap er alternatives such as aluminium alloy steel or plywood are

also commonly used Materials used for cores include p olymers aluminium wood and

comp osites To minimise weight these are used in the form of foams honeycombs or

with a corrugated construction Fig As well as mechanical requirements core

t 2 t 4 t

Relative Bending Stiffness 1 7.0 37 Relative Bending Strength 1 3.5 9.2

Relative Weight 1 1.03 1.06

Table An example of structural eciency of sandwich panels in terms of weight

Chapter Intro duction and Overview

materials may also b e selected based on their reresistance or thermal prop erties

(a)

(b)

(c)

Figure Sandwich panels with a corrugated b foam and c honeycomb core

The most common and some unortho dox techniques employed to manufacture sand

wich comp onents for structural applications as well as the recent developments and

future trends in terms of b oth materials and pro cessing routes are comprehensively

reviewed by Karlsson

Sandwich panels will have stiness and strength criteria to meet The stiness

of honeycomb sandwich panels is straightforward to predict but it remains dicult

to estimate the strength Typical mo des of failure are face yielding face wrinkling

intracell dimpling core shear or lo cal indentation where the load is applied to the

panel These are describ ed in detail in section The critical failure mo de and the

corresp onding failure load dep end on the prop erties of the face and core materials on

the geometry of the structure and on the loading arrangement

Prop er analysis of sandwich structures demands a thorough understanding of the

mechanical b ehaviour of b oth the skins and the core The skins behave in a relatively

simple manner and in case of comp osite laminates the aforementioned metho ds of

Chapter Intro duction and Overview

analysis ie laminated plate theory facilitate the mo delling pro cedure However the

mechanical mo delling of the core material particularly for foams or honeycombs is less

straightforward The resp onse of the core to shear loading from the skins or loading

normal to the plane of skins is required The b ehaviour dep ends b oth on the materials

used in the core and on the core relative densitywhich is the ratio of the core densityto

that of the solid material constituting the core Gibson and Ashby give a thorough

overview of the literature on cellular materials quoting many results for foam cores

Figure illustrates the honeycomb structure One of the most common core ma

terials is Nomex honeycomb b ecause it p ossesses an extremely high strengthto

weight ratio It is also electrically and thermally insulating chemically stable self

extinguishing and corrosion as well as sho ck and fatigue resistant As Nomex is used

extensively in this thesis the makeup of this core is describ ed in detail here Nomex is

constructed from ribb ons of aramid pap er running in the direction the longitudinal

ribb on direction These are glued together at intervals along the ribb on and the stack

of ribb ons is then expanded into a honeycombby pulling in the direction transverse

The pap er substrate is nally dipp ed into phenolic resin to build up the walls of the

honeycomb Because of this construction metho d the honeycomb is anisotropic with

resp ect to outofplane shear stiness and strength

Figure Typical hexagonal honeycomb with a set of doubled walls t is the single wall

thickness and is the honeycomb cell size

Nomex is a Du Pont trademark

Chapter Intro duction and Overview

Zhang and Ashby mo del the elastic and collapse b ehaviour for Nomex hon

eycomb materials under shear and outofplane compression Their mo dels agree well

with exp eriments that they made on a wide range of Nomex honeycombs Zhang and

Ashby have also investigated the inplane biaxial buckling b ehaviour of Nomex

honeycombs Shi et al and Grediac mo del the transverse shear mo dulus of a

honeycomb core

Considerable eort has been devoted to the analysis of b eams panels and struts of

sandwich construction and the results have b een summarised in the b o oks of Allen

and Plantema Mo delling a sandwich panel as a beam with the simplifying as

sumptions that the skins are thin relative to the core and that the core material is ho

mogeneous and much less sti than the skin material was presented by Allen and

develop ed by Gibson and Ashby Triantallou and Gibson have develop ed an

optimisation pro cedure which can determine the optimum values of skins and cores

thicknesses that satisfy the stiness constraint at minimum weight Although most

of research work in literature is concerned with b ending loading of sandwich b eams

Kwon et al or Pearce have investigated the overall buckling and wrinkling of

sandwich panels under inplane compression

A recent comprehensive review to the sub ject of sandwich construction and the

development of theoretical analyses up to now is given in the b o ok of Zenkert Holt

and Webb er summarise recent developments and analyse the elastic behaviour of

honeycomb sandwich b eams assuming linear elastic b ehaviour for the skin and the core

Mechanical thermal and hygrometric loading on a sandwich b eam with a honeycomb

core and laminated facings are included in reference Failure mo de maps have b een

derived by various authors for sandwich panels with exible cores These

authors have b een particularly concerned with b eams with ductile foam cores making

appropriate assumptions ab out the elastic and plastic b ehaviour of the core and skin

However there app ears to be little work on failure of panels with honeycomb cores

whose shear anisotropy can reveal the imp ortant role of core shear in the b ending of

sandwich b eams Furthermore resistance to indentation failure is rarely considered as

an imp ortant factor although it can be imp ortant in determining the durability and

service life of the structure

Previous Work on Indentation Analyses

The use of exible foams and nonmetallic honeycombs as core materials has intro

duced a new design diculty they are transversely exible compared to the eectively

incompressible metallic honeycombs or rigid foam cores This leads to signicant lo

Chapter Intro duction and Overview

cal deections of the loaded skin into the core material hence to indentation failure

Indentation failure of sandwich structures has been primarily investigated for aircraft

o or panels These panels need to be suciently sti to avoid passengers p erceiving

that the o or is unsafe b ecause of its deection High heel sho es or dropp ed ob jects

are typical causes for lo cal indentations on the top skin of these panels in areas like

the aisles thresholds toiletsgalley However a sandwich panel would continue to give

go o d service until a number of such indentations had been made Therefore the time

from incipient to nal failure should be sucient to allow maintenance op erators to

change a defective panel at the next convenient check

A go o d source of information on indentation analyses of sandwich b eams are the

books by Allen and Plantema They cover the development of the theoretical

analyses based on a socalled splitted rigidity mo del This approach assumes that

the beam consists of a comp onent with only b ending rigidity and a comp onent with

only shear rigidity They are connected through equilibrium assuming that the shear

resultants in the two comp onents are the same These mo dels oers an adequate level

of accuracy for sandwich structures with incompressible cores but are inadequate for

nonmetallic honeycomb sandwich panels since they neglect lo calised eects which

play an imp ortant role for these exible cores

To mo del indentation failure of transversely exible cores in sandwich structures

requires the use of mo dels that take into account the compressibility of the core in

the vicinity of the applied loads A common approach is the elastic foundation mo del

Selvadurai presents the principles of these mo dels Their application in sandwich

structures analysis is describ ed in Zenkerts b o ok The simplest of these is the

oneparameter Winkler foundation mo del which treats the core material as a set of

continuouslydistributed linear springs However its main drawback is that it neglects

shear interactions between the loaded skin and the core The elastic foundation ap

proach adopted by Thomsen was based on the use of a twoparameter elastic

foundation mo del including the shear interaction b etween the skins and the core Typ

ical twoparameter mo dels are those of Pasternak or Vlazov see Selvadurai

Nevertheless these elastic foundation mo dels neglect the interaction b etween the top

and b ottom skin A dierent approach by Frostig and Baruch consists of treating

the beam as an ordinary incompressible sandwich substructure interconnected with a

sp ecial elastic foundation substructure which provides the lo calised eects due to the

dierent displacements of the upp er and lower skins The nonplanar deformed cross

section of the sandwich beam which is observed by exp eriments suggested the need

for a mo del which allows nonlinear variations of inplane and vertical displacement

Chapter Intro duction and Overview

eld through the core Frostig et al used variational principles to develop the

highorder sandwich panel theory which includes the transverse exibility of the core

Highorder refers to the nonlinear way in which the inplane and vertical displace

ments are allowed to vary through the height of the core in contrast to simple b eam

theory where the core inplane displacements are assumed to vary in a linear way

through the depth and the outofplane displacements are assumed to be constant

The highorder theory enables the prediction of lo calised eects under conditions such

as concentrated loads delamination and diaphragms curved sandwich panels

hygothermal environmental eects and discontinuous skins A comprehensive

review and comparison b etween this metho d and conventional b eam theories was pre

sented recently by Frostig

In the literature the imp ortance of the core b ehaviour in aecting indentation failure

has been considered however the inuence of the skins exural rigidity is generally

overlo oked This omission derives from the fact that in practical applications the skins

are quite sti and the inuence of skin exural rigidity can be neglected One of the

main aims of this thesis is to fo cus on the role of skin rigidity in indentation failure

and give a b etter insightin the mechanism of indentation

Previous Work on Indentation Failure Prediction

The aircraft industry in collab oration with sandwich panel manufacturers has con

ducted research to establish design co des for indentation resistant and cost eective

sandwich panels However most of the design sp ecications have been based on

exp erimental results rather than accurate theoretical mo dels In manufacturers data

sheets and handb o oks the prediction of indentation failure load is derived as a

pro duct of the outofplane compressive strength of the core and the area over which

the load is applied However such an approach is at best approximate Sp ecial ex

p erimental setups have been intro duced dep ending on the application static blunt

indentation tests for freight o ors or dynamic fatigue tests for cabin o ors Bonded

Structures Division CIBA now Hexcel Comp osites have devised the fatigue wheel

machine Fig to provide data on the relative p erformance of various sandwich

panels

Similar indentation damage accumulation can b e observed in sandwich panels used

in this case b ecause of hail or stone as a part of control surfaces aps ailerons

impacts A strength analysis of the damaged sandwich panels is essential to predict

the inuence of the multisite damage Razi et al have intro duced an analytical

metho d to determine the stress distribution in sandwich panels with damage that is

Chapter Intro duction and Overview

Figure Test rig for indentation fatigue tests on aircraft o or panels

elliptical or circular in shap e and at an arbitrary lo cation

The lowvelo city or quasistatic impact b ehaviour of sandwich panels is also another

asp ect of the whole indentation problem that has b een investigated byseveral authors

The lack of accepted test metho ds for measuring impact damage resistance of com

p osite sandwich structures led Lagace et al to prop ose a new metho dology based

on static indentation and impact tests Mines et al have tested the dropp ed

weight impact p erformance of sandwich b eams with wovenchopp ed strand glass skins

and p olyester foam and aluminium honeycomb cores and havesimulated the upp er skin

p ostfailure energy absorption b ehaviour with an elasticplastic b eam b ending mo del

Low velo city damage mechanisms and damage mo des have been investigated in sand

wich b eams with Rohacell foam core byWu and Sun and with Nomex honeycomb

core by Herup and Palazotto

The pure static indentation resp onse of comp osite sandwich beams has been mo d

elled as linear elastic b ending of the top skin on a rigidp erfectly plastic foundation the

core by So den and Shuaeib Olsson and McManus intro duced a theory

for contact indentation of sandwich panels the mo del is based on the assumption of

axisymmetric indentation of an innite elastic face sheet b onded to an elasticideally

plastic core on a rigid foundation Despite the usefulness of such mo dels for analysing

sandwich beam indentation b ehaviour they are restricted only to the loading case

where a sandwich beam is indented on a rigid foundation However this loading case

is not normal in practice Most sandwich b eams in service are simply supp orted or

Chapter Intro duction and Overview

clamp ed and are indented whilst also suering b ending loads These loading conditions

make the inuence of the b ottom skin an imp ortant factor in the overall b ehaviour of

a sandwich beam under lo calised loads

Scop e and Outline of the Thesis

The use of laminate comp osites as skins and low density cellular materials as cores in

sandwich constructions has enabled a very go o d utilisation of the constituent materi

als providing structural comp onents with high stiness and strengthtoweight ratios

Nonmetallic honeycombs are p opular in high p erformance applications b ecause of their

good mechanical prop erties excellent resistance to hostile environments and b etter fa

tigue resistance than aluminium However because of the low stiness of the core

they are susceptible to indentation failure The need to take into consideration the

indentation resistance of sandwich panels when optimising their design and the imp or

tance of having a robust computational to ol to predict indentation failure in sandwich

beams under bending have b een the main motivations for this thesis

To address design including indentation failure we rst need to investigate the

other failure mechanisms describ ed in Chapter Failure maps for loading under

point b ending are constructed and the region where indentation failure is imp ortant

is identied This work uses sandwich panels made of GFRP laminates and Nomex

honeycomb cores of various densities Exp erimental measurements are compared with

simple theoretical predictions In Chapter analytical mo delling which can be used

for indentation failure is addressed This uses an existing beam theory develop ed by

Frostig et al and applies it to the GFRP Nomex construction considered in

the thesis The eect of material prop erties and the b eam and indenter geometry

on the beam resp onse are considered Chapter presents the metho dology to predict

more accurately indentation failure loads This is achieved byintro ducing a new failure

criterion for honeycomb core which takes into account b oth the outofplane normal

compression stresses and the shear stresses in the core Theoretical predictions are

compared with exp erimental data Rened failure maps including the new indentation

analysis are describ ed in Chapter Carp et plots based on these failure mo de maps

provide a preliminary optimisation metho dology for mimimum weight design of sand

wich b eams The thesis concludes in Chapter by summarising the main results of

this research together with a discussion of new research directions

Chapter

Failure Mo de Maps for Honeycomb

Sandwich Beams

Intro duction

In this chapter we consider loading under threep oint b ending of sandwich b eams made

with honeycomb and skins b oth of which fail in a brittle manner We apply the anal

ysis framework and failure mo de map technique used by Triantallou and Gibson

for ductile foam cores and ductile skin materials However here we investigate the

failure mo des of panels with honeycomb cores whose shear anisotropy can reveal the

imp ortant role of core shear in the b ending of sandwich b eams Furthermore the con

structed failure mo de maps include indentation failure and identify for what b eams

this mechanism will b e imp ortant

In section we review b eam theory for sandwich panels Section describ es

existing work on honeycomb mechanics By combining the analysis for sandwich b eams

with the honeycomb mechanics we derive in section failure loads for the various

failure mechanisms This information is used in section to draw up failure mo de

maps in which the mechanisms of failure and the corresp onding failure loads are plot

ted as a function of the core relative density and skin thicknessspan ratio Theoretical

results are illustrated using the commerciallyp opular combination of laminated glass

bre reinforced plastic GFRP skins and Nomex core Theory is compared in sec

tion with exp eriments on sandwich panels of various core relative densities and skin

thicknessspan ratios

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

Beam Theory for Sandwich Panels

In this section we outline the elastic analysis of sandwich b eams in three p oint b ending

This will b e used to evaluate the stresses in the core or skin and hence the failure loads

due to the various mechanisms Consider a simply supp orted sandwich beam of span

L and width b loaded in p oint b ending with a central load W per unit width as

illustrated in Fig The skins each have thickness t and are separated by a thick

layer of honeycomb core of thickness c

Figure a Simply supp orted b eam b Cross section on AA

We assume that the skins remain rmly b onded to the core that the b eam b ends in

a cylindrical manner with no curvature in the yzplane and that crosssections remains

plane and p erp endicular to the longitudinal axis of the beam The exural rigidity D

of the sandwich b eam is then given by

E bt E btd E bc

fx fx cx

where d is the distance between the midplanes of the upp er and b ottom skins E

and E are the inplane Youngs mo duli of the skin and core resp ectively for loading

in the x direction along the axis of the beam Subscripts f and c denote the face

material and the honeycomb core resp ectively Subscript s is used in later expressions

for the solid material from which the honeycomb is made The three terms on the

right hand side of corresp ond to b ending of the skins ab out their centroidal axes

b ending of the skins ab out the centroid of the whole beam and bending of the core

resp ectiv ely We can simplify this equation by assuming that b ending of the skins

ab out the centroid of the beam is the dominant term The contributions of the rst

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

and third terms amountto less than of this when

t d d E

and

t E c c

resp ectively so that b ecomes

btd

D E E I

fx fx

where I is second moment of area of the crosssection of the sandwich b eam With

p oint b ending the maximum b ending moment M is at the midspan and the corre

sp onding maximum stress in the skins is given by

ME d WL

D dt

However the ab ove theoretical mo del neglects the eect of shear deection in the core

which b ecomes signicant for low density cores Inclusion of this eect also allows

a prediction of observed dierences in b eam strength for dierent orientations of the

honeycomb ribb on see section for further details For the ab ove reasons we

follow the suggestion of Allen for the maximum axial stresses in the faces

WbL WL c t t

I I

c G d btd bt bt L

cxz

I I where

c E t t

G is the outofplane shear mo dulus of the core I is the second moment of area of

cxz

the sandwich with resp ect to its neutral axis and I is the second moment of area of

the faceplates with resp ect to their own centroidal axes Equation shows that

dep ends on the relative stiness of the skin and the core Finally gives

t t d

where

ht t t d

As the span length L or the core shear stiness G approach innity tends to

cxz

the simple beam mo del In the case study presented in section maximum

deviations from the simple b eam mo del due to the nite thickness of the skins and the

eect of nite shear stiness in the core amounttoamaximum of

Note that I is not negligible in this approach f

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

Skin Failure

Section gives an expression for the maximum stress in the skins This can be

used to predict beam failure due to the skin failure mo des of face yielding intracell

dimpling or face wrinkling as illustrated in Fig

a Face yielding b Intracell dimpling c Face wrinkling

Figure Failure mo des in the skin

Face Yielding

Failure o ccurs in the top skin due to face yielding when the axial stress in either of the

skins equation reaches the inplane strength of the face material for loading

along the b eam axis

fx fY

It is assumed that the skin b ehaves in a brittle manner With a symmetrical b eam the

stress is the same in the tension and compression faces For comp osite face materials

the compressive face is generally the critical one

Intracell Dimpling

A sandwich with a honeycombcoremayfailbybuckling of the face where it is unsup

p orted bythewalls of the honeycomb Fig b Simple elastic plate buckling theory

can be used to derive an expression for the inplane stress in the skins at which

intracell buckling o ccurs as

E t

fxy

where is the cell size ie the diameter of the inscrib ed circle of the honeycomband

E and are the elastic mo dulus and Poissons ratio for the skin for loading in the

fx fxy

axial direction A similar expression veried exp erimentally by Kuenzi has b een

given by Norris Equations and can be used to derive the value of cell

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

size ab ove which there is transition from face yielding to intracell buckling as

fxy

Face Wrinkling

Face wrinkling is a buckling mo de of the skin with a wavelength greater than the cell

width of the honeycomb Fig c Buckling may occur either in towards the core

or outwards dep ending on the stiness of the core in compression and the adhesive

strength In practice with p oint bending inward wrinkling of the top skin o ccurs

in the vicinity of the central load By mo delling the skin as a plate on an elastic

foundation Allen gives the critical compressive stress that results in wrinkling

of the top skin as

E E

cxz cxz

where is the outofplane Poissons ratio and E the outofplane Youngs mo dulus

cxz

of the honeycomb core see section

Core Failure

Honeycomb sandwich structures loaded in b ending can fail due to core failure Perti

nent failure mo des are shear failure or indentation by lo cal crushing in the vicinity of

the loads as illustrated in Fig

a Core shear b Lo cal indentation

Figure Failure mo des in the core

Core Shear

Assuming simple b eam b ehaviour the shear stress varies through the face and core in

a parab olic way under p oint b ending If the faces are much stier and thinner than

the core the shear stress can be taken as linear through the face and constant in the

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

core Neglecting the contribution from the skins the mean shear stress in the core is

given by

cxz

Assuming brittle b ehaviour failure o ccurs when the applied shear stress equals the

shear strength of the honeycomb core in this direction

cxz cs

Low density Nomex cores are particular susceptible to this failure mo de Due to the

anisotropy of the honeycomb structure section the shear strength of the core

dep ends on the loading direction

Lo cal Indentation

Failure of sandwich panels in p oint b ending can o ccur at the load p oint due to lo cal

indentation Failure is due to core crushing under the indenter The b ending stiness

of the skin and the core stiness determine the degree to which the load is spread

out at the point of application It is imp ortant here to mention the main failure

characteristic by which indentation diers from skin wrinkling In indentation the top

skin deects after failure with a wavelength of the same scale as the indentertop skin

contact length whereas in skin wrinkling the deection of the top skin after failure

exhibits wavelengths that are larger than the contact length b etween the indenter and

the top skin

Indentation failure has not b een adequately mo delled for honeycomb sandwich pan

els To include this imp ortant failure mechanism we use a simple empirical approach

used in handb o oks on sandwich panel construction In the next chapters a more

accurate mo del will be presented for indentation failure prediction Here we assume

that we know the length of contact between the central roller and the top skin It

is further assumed that the load is transferred uniformly to the core over this contact

length so that the outofplane compressive stress in the core is given by

Failure is then predicted when this compressive stress equals the outofplane compres

sive strength of the honeycomb core

zz cc

This length derives from the assumption that the skins are transparent enough to equate the

contact length with the length of initial damage in the top skincore interface

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

The ab ove approach is decient in three resp ects i the contact area must b e estimated

in some way in the exp eriments describ ed in section this is measured ii load

transfer from the roller to the core is oversimplied this will dep end on the relative

skin and core stinesses iii failure in the core will not be governed solely by the

compressive stress in the core but will also be inuenced by the lo cal shear stress A

more rigorous stress analysis of the contact region can be found in Chapter and its

implementation to predict lo cal failure in honeycomb panels is presented in Chapter

Honeycomb Mechanics

Toevaluate the failure mechanisms describ ed in section stiness and strength prop

erties for the honeycomb core are required In this section we use the results of refer

ences to express the prop erties of the honeycomb as a function of the prop erties

of the solid material from which the honeycomb is made and the relative density

c s

of the honeycomb Although the theory is applicable to any honeycomb in practice

we will fo cus on the Nomex honeycomb core used in the b eam failure exp eriments of

section Expressions are compared with published exp erimental data to evaluate

the applicability of the theoretical mo dels to the Nomex honeycomb core and to nd

the most suitable expressions for use in the b eam calculations The following notations

for honeycombs main axes refer to those illustrated in Fig

The honeycomb Poissons ratio required for the failure analysis section

cxz

is or for inplane Poisson strains due to outof plane loading in the direction

To a rst approximation its value can b e taken as that of the solid material eq

in ref ie

The Youngs mo dulus of the honeycomb in the outofplane direction is given by

the rule of mixtures expression

s s

In honeycombs failure under outofplane compressive stresses occurs due to fracture

of the cell walls or due to elastic or plastic buckling of the cell walls For Nomex

honeycombs failure is due to a crushing mechanism initiated by elastic buckling and

developing as a plastic buckling pro cess The relevant collapse strength can be

simply estimated using the rule of mixtures expression where is

cc sc c s sc

the compressive strength of the solid from which the core is made Wierzbicki gives

an alternative expression for the failure stress based on a plastic collapse mo del For a

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

honeycomb with regular hexagonal cells this approach predicts the collapse strength

cc sc

Zhang and Ashby show that the outofplane shear strength and stiness of hon

eycombs are indep endent of height and cell size Honeycomb cores exhibit slight

anisotropy in their outofplane shear strength and stiness due to the set of dou

bled walls By using simple mechanics mo dels based on an array of regular hexagons

and considering the double wall eect approximate expressions for the shear strengths

and are derived as

s s

and for the shear mo duli G and G as

s s

The core shear mo dulus G used in equation to calculate the skin stress should

cxz

be taken as either G or G dep ending on the orientation of the ribb on direction in

the honeycomb This anisotropy leads to a dep endence of skin failure loads on the hon

eycomb orientation Similarly the core shear strength dep ends on the honeycomb

orientation

Exp erimental Evaluation of Honeycomb Mechanics

The theoretical relations detailed in section are those that we use for our cal

culations in section In this section we compare the theoretical expressions with

exp erimental data from reference and manufacturers data sheets for hon

eycombs made of Nomex aramid pap er impregnated in phenolic resin Often it is

observed that there is a wide variation amongst data from dierent sources reecting

the wide manufacturing tolerances in the constituent aramid sheet particularly

and the diculties in making accurate measurements

Hexcel Comp osites formerly Ciba Comp osites

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

The measurements of outofplane compressive prop erties are made by testing hon

eycombs under stabilised compression As depicted in Fig a the prediction of

Equation for Youngs mo dulus lie b etween the two sets of data and close to that

of reference Fig b shows that Wierzbickis equation for compres

sive strength ts the exp erimental data b etter than the usual mixtures law reecting

the plastic collapse mechanism of Nomex under compression

Figures cd show plots of shear mo dulus and strength Manufacturers measure

ments are signicantly higher than the measurements of Ashby and Zhang b ecause of

the dierent test setup Ciba or Hexcel use a short b eam test where shear strength

and stiness are outofplane and measured indirectly see eq in ref Zhang

and Ashby has tested the honeycombs in inplane shear with an appropriate testing

rig We b elieve that the latter source gives a more direct estimate of shear prop erties

The ma jor dierence b etween equations a b and the exp erimental data of

for are due to deb onding of honeycomb sp ecimens from the rig We

c s

can use equations a b when since in the material systems

c s

considered here no deb onding o ccurs

Construction of a Failure Mo de Map

Sections and have describ ed various mechanisms of failure which o ccur with

honeycomb sandwich panels and the honeycomb mechanics needed to evaluate the

failure loads for each of these mechanisms In this section we describ e how a map can

be constructed detailing which failure mechanism actually o ccurs for a given material

combination and b eam geometry This follows the work of Triantallou and Ashby

for foamcore sandwich panels The failure mo de map is illustrated byway of examples

in section The failure loads dep end on the prop erties of the skin and honeycomb

solid material the relative density of the core the thicknesses t and c of the skin

c s

and the core and the beam span L Because we include indentation failure failure

also dep ends on the loading details In the exp eriments we use rollers to apply the

load so that failure dep ends on the roller radius R The failure line load W can then

be expressed as a function of the material prop erties and the beam parameters

W f tL tR To evaluate this function the expressions for skin and core

o c s

stresses equations are substituted into the various failure criteria

equations and as describ ed in section to givethe

critical line loads as summarised in Table The actual failure load and mo de are

Stabilised compression means restriction of the cell walls from slipping b etween the sp ecimen and the rig plates during the test

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

1.00 CIBA

Ashby-Zhang sc σ / cc σ

s w 0.10 mixture la /E c

eq eq 0.10 Relative Young's Modulus, Modulus, Young's Relative C IB A data 0.01 H excel data Relative Stabilised Com pressive Strength, A shby-Zhang

0.03 0.03 0.10 0.10

R elativ e D e n sity , ρc/ρs R e la tiv e D en sity , ρc/ρs

a b

0.010

eqa eqb s s /G c /E c G τ 0.10

0.001 eqa Relative Shear Strength, Strength, Shear Relative Relative Shear M odulus, G G τ

31 32 31 τ32 eqb

C IB A C IB A

H E X C E L H E X C E L

A shby-Zhan g A sh by-Z hang

0.01 0.03 0.03 0.10 0.10 ρ ρ

R e lativ e D e n sity , ρc/ρs R elativ e D en sity , c/ s

c d

Figure Comparison of theoretical and exp erimentally measured outofplane a com

pressive mo dulus b compressive strength c shear mo dulus and d shear

strength of Nomex honeycombs

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

given by the mo de with the minimum failure load Maps of the failure mo de and failure

load can then b e drawn as a function of the b eam geometry for a given material system

The Matlab programming language is used to evaluate the equations

Table Summary of failure criteria

Top skin yield W

o fY

t t

Intracell buckling W E

o f

Face wrinkling W B E E

Core Shear W AE d

o s

Indentation W

o sc

Note The quantities and A change according to the honeycomb ribb on orientation

A Failure Mo de Map for Beams with a GFRP Skin and

Nomex Core

The ab ove section describ es how a failure mo de map can be constructed for a honey

comb sandwich panel This is illustrated in this section using sandwich panels made of

GFRP laminate skins and Nomex honeycomb cores of dierent densities The core and

skin thicknesses c and t are mm and mm resp ectively and the nominal honey

comb cell size is mm Exp erimental results for this sandwich panel typ e are presented

in section Further details of the panel construction materials and material prop

erties are given in that section Fig b shows the dep endence of line load W at

thickness to span tL ratio for a value failure on core relative density and face

c s

of the radius of the roller to the skin thickness Rt of Each surface corresp onds to

a dierent failure mo de The failure mo de map Fig a is given by the pro jection

of the intersections b etween failure surfaces on the tL plane Slightly dierent

c s

failure maps are calculated dep ending on whether the honeycomb ribb on lies in the

longitudinal direction along the b eam axis or transverse to the b eam axis Dierences

arise from the anisotropy in the strength and stiness of honeycomb in shear cf equa

tions and ab As noted in section failure by indentation is estimated

using an empirical approach and relies on exp erimental measurements of the contact

area as describ ed in section Indentation is the only mechanism which dep ends on

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

the roller diameter This will b e a more widespread mechanism of failure with smaller

roller diameters For the honeycomb geometry chosen for these plots intracell dim

pling is not predicted At the map b oundaries the failure loads for the mechanisms

either side of the b oundary are equal In practice failure near a b oundary maybedue

to a combination of the two mechanisms and coupling between the two mechanisms

may reduce the load b elow that predicted for each of the mo des indep endently

The failure mo de map shown in Fig a is useful where a designer has sp ecied

an appropriate span or face and core thicknesses and wishes for example to select

relative density More commonlyhowever the span is xed a standard skin construc

tion and thickness is sp ecied and only the core thickness or density can b e relatively

easily changed In this case it is more useful to plot a map of the failure mo des and

loads as a function of core relative density and core thickness to span ratio cL at

a xed skin thickness to span ratio tL and roller radius to face thickness ratio Rt

Figure shows such maps for three typical values of tL with Rt Note that

for long spans Fig a the core crushing failure mo de vanishes while for short spans

face wrinkling is not predicted Fig c

Exp eriments

Sandwich panels were made of Nomex honeycomb core and GFRP laminate skins and

were supplied by Hexcel Comp osites All panels had the same skin crossply laminate

on either side of the core as depicted in Fig Each laminate comprised two glass

prepregs the outer with a resin content of and the inner with giving a

skin thickness t of mm The Nomex honeycomb cores used in the panels are

designated by the manufacturer as Aeroweb typ e A Panels with core densities of

and kgm were used For most of the tests the honeycomb had a

nominal cell size of mm but for tests describ ed in section a cell size of mm

was used The core thickness c was mm Mechanical prop erties of the skin and the

honeycombs constituent solid material aramid pap er resin are listed in Table

Since the compressive strength of the laminate has been inferred from b ending tests

of sandwich b eams this is not an indep endent measurement Hexcel quote a value

of MPa based on simple b eam theory equation for long b eams Using the

b eam mo del equation with a correction for shear in the core a revised estimate

for the compressive strength of the laminate of MPa is inferred from the data at

long spans

Panels were cut into b eams using a diamond wheel Awidthb of mm was chosen

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

0.4

Face Yield s c y 0.1 Core Crushing e densit

Face Wrinkling Relativ

Core Shear

0.02 −4 −3 −2

10 10 10

Face thickness span tL a

6 10 o

5 W 10

4 10

3 10

2 10 ailure line load F 10 1 0.1 −2 0.1 10 −3 10

−4

Relative density s

c 0.01 10

Face thickness span tL

Figure a Failure mo de map Solid lines refer to b eams in which the honeycomb ribb on

lies along the b eam axis dashed lines for when the ribb on lies transverse to the

beam axis The symbol identies exp erimental measurements describ ed in

section b Failure load surface for ribb on lying along the b eam axis

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

1 s c Face Yield y

0.1 e densit

Face Wrinkling Relativ

Core Shear 0.01 −3 −2 −1

10 10 10

Core thickness c m

a tL

1 s c

Face Yield y

0.1

e densit Face Wrinkling Core Crushing Relativ Core Shear

0.01 −3 −2 −1

10 10 10

Core thickness c m

b tL

1 s c Face Yield y

0.1 Core Crushing e densit

Core Shear Relativ

0.01 −3 −2 −1

10 10 10

Core thickness c m

c tL

Figure Failure mo de map for three typical values of tL and Rt

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

Nomex Laminate skin

Youngs mo dulus GNm E E

Shear mo dulus GNm G G

Compressive strength MNm

Poissons ratio

Density Mgm

Table Material prop erties of Nomex and laminate skin

so that it was greater than twice the sandwich heightand three times the cell size as

recommended in by ASTM standards Beams were cut with the ribb on direction

either in the longitudinal direction along the b eam axis or in the transverse direction

as depicted in Fig Beams of varying spans and core relative densities were made

to prob e the various regions of the failure mo de maps Fig Sp ecimens were tested

in p oint b ending applying the load through rollers of diameter of mm cf ASTM

standard C The crosshead sp eed was kept constant through the test and

was chosen so that the maximum load o ccurred between and min after the start

of the test Displacement of the central loading point relative to the end rollers was

monitored using an LVDT and logged on a computer The central section of the b eam

where failure invariably o ccurred was also recorded on a video recorder Fig

as the test progressed To mo del failure due to lo cal indentation section it is

necessary to dene the value the length of the contact area between the central roller

and the top skin This was estimated exp erimentally by putting carb on pap er b etween

the roller and the sp ecimen For all the sp ecimens tested was between and mm

Longitudinal Direction

Prepreg Fibre Direction Transverse Direction

Honeycomb Ribbon Direction

Figure Layup details

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

with a typical value of mm This information was used to estimate the failure load

due to indentation as describ ed in section

Figure Test setup

Exp erimental Results

Figures ad show photographs of the section under the central load just after fail

ure illustrating four failure mo des observed during the tests of those b eams with mm

honeycomb cell size These were taken from videos recorded during the tests Beam de

tails for each of these gures are given in Table to scale the photographs note that

the cell size is mm for all the sp ecimens of Fig except Fig ab which has

the larger cell width of mm For clarity the roller seen in the top half of the gures

has been outlined in Fig a Corresp onding loaddeection curves are included in

Fig With all the failure mechanisms except intracell buckling Fig ab and

core shear Fig d failure o ccurred in a brittle manner with little nonlinearity in

the load displacement curves b efore failure and a sharp drop in load at failure For

short span tests where indentation o ccurred Fig c the beam has a signicant

p ostfailure strength as the core progressively crushes Table presents the average

line loads W at failure and corresp onding observed failure mechanisms The mo de is

describ ed as Complex when failure app eared to o ccur simultaneously in the core and

in the face Those mechanisms denoted as mo de mo de ie Intracell dimpling

core crushing indicates the app earance of mo de as elastic instability b efore the

nal failure mo de

The measured eect of skin thickness to span ratio on the failure load is compared

with theory in Figs ad for the dierent core densities The theoretical graphs

are sections through the load surface Fig b at constant core relative density

Exp erimental mechanisms of failure are indicated on these graphs to allow a comparison

between the actual and predicted failure mechanisms

Most of tests were rep eated and the dierence b etween failure loads was less than

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

Core Honeycomb Span Measured Observed

Density ribb on length tL line failload failure mo des

kgm direction mm kNm

E Face wrinkling

E Core shearface wrink

Long E Core shearcrushing

E Core shearcrushing

E Face wrinkling

mm cell E Core shearface wrink

rans

T E Core shearcrushing

E Coreshearcrushing

E Face wrinkling

E Complex

Long E Core crushing

E Core crushing

E Face wrinkling

mm cell E Complex

rans

T E Core crushing

E Core crushing

E Face yield

E Face yieldwrinkling

E Complex

Long E Core crushing

E Core crushing

E Face yield

E Face yieldwrinkling

mm cell E Complex

rans

T E Core crushing

E Core crushing

E Face yield

Long E Face yield

E Complex

E Core crushing

E Face yield

mm cell E Face yield

E Face yield

rans

T E Face yield

E Complex

E Core crushing

E Intracell dimplface p eel

E Intracel l dimplface peel

E Intracell dimplcore crush

Long

E Intracell dimplcore crush

E Intracell dimplface p eel

mm cell E Intracell dimplcore crush rans

E Intracell dimplcore crush

Table Exp erimental results Photographs of failure given in Fig and corre

sp ond to the entries in italics in the nal column of this table

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

20 3 Core Density = 128 kg/m Span Length = 345 mm

15 (kN/m) W 10

Line Load, Load, Line 5 Honeycom b Direction

Longitudinal

T ransverse 0 0 10 20 30

M idspan D eflection (m m )

a Face yield kgm L mm c

8 3 Core Density = 48 kg/m Span Length = 380 mm

6 (kN/m) W 4

Line Load, Load, Line 2 Honeycom b Direction

Longitudinal

T ransverse 0 0 5 10 15 20

M idspan D eflection (m m )

b Face wrinkling kgm L mm c

60 Honeycomb Direction

L o n g itu d ina l

T ra n sve rse

40 (kN/m) W

20 Line Load, Load, Line 3 Core Density = 128 kg/m Span Length = 60 mm 0 0 2 4 6

M idspan D eflection (m m )

c Core crushing kgm L mm c

8 3 Core Density = 29 kg/m Span Length = 50 mm

6 (kN/m ) W 4

Line Load, Load, Line 2 H oneycom b D irection

Longitudinal

T ransverse

0 0 1 2 3

M id span D eflection (m m )

d Core crushing kgm L mm

Figure Photographs of the dierent failure mo des for transverse ribb on direction and

corresp onding load deection curves for b oth ribb on directions

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

100 3 3 Core Density = 128 kg/m Core Density = 64 kg/m

Long Long o o T rans

W 10 W 10

E xperim ental D ata E xperim en ta l D a ta Long T rans Lo n g T ran s Fa ce Y ield F ace Y ield Trans C o m p le x C o m ple x Line Load, (kN/m) Line Load, Load, Line (kN/m) T he o retic al C urve s Theoretical C urves F ace W rin klin g C o re C ru sh in g C o re C rush ing Face Y ield F a ce Y ie ld Face W rinkling C ore C ru sh ing C ore C rushing 1 0.001 0.010 0.001 0.010

F ace thickn ess / span , t/ L F ace thickn ess / span , t/ L

a b

c s c s

10 3 3 Core Density = 48 kg/m Core Density = 29 kg/m Lo ng 10 Long o o W W E xperim ental D ata T rans L on g T ran s

T ran s Face W rinkling Theoretical C urves C om plex Face W rinkling E xp e rim e n ta l D a ta Line Load, (kN/m ) Line Load, Load, Line (kN/m) C ore C rushing T heoretical C urves C ore S hear L o n g T ra n s Face W rinkling F ace W rin kling C ore C rushing C om plex 1 C ore C rushing C ore S hear + C rushing

0.001 0.010 0.001 0.010

F ace th ick ness / sp an, t/ L F ace th ick ness / sp an, t/ L

c d

c s c s

Figure Variation of failure load W with skin thickness to span ratio tL o

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

Discussion

Figure shows that in general the exp erimental failure loads agree satisfactorily

with theoretical values and that the observed failure mo des are generally the same

as the predicted mo des Fig Where a transition from one failure mo de to an

other o ccurs the mechanism of failure is mixed but the failure load is still adequately

predicted The main source of deviations arises from the errors in predicting the hon

eycomb material prop erties section

Skin Failure

The laminate face yielding strength has been chosen to t data at long spans as

discussed in section At the longest spans with the kgm honeycomb face yield

o ccurs although wrinkling is predicted but as shown in Fig a these p oints are

v ery close to the b oundary between these two mechanisms

Core Failure

Failure load predictions for core crushing are adequate for the higher density cores but

predictions are p o or for the lower density cores The photograph in Figure d illus

trates a b eam with a low density core in transverse direction near the failure load where

core shear failure is predicted Final failure is a mixture of core shear and crushing

under the indenter The dierence in the initial slop es of the loaddeection curves for

the sp ecimens with dierent ribb on directions reects the anisotropy asso ciated with

the core shear stiness Nonlinearity in the curve for the transverse direction indicates

that there maybe some ductile shear of the core prior to nal failure The signicant

underprediction of the failure load Fig d due to core shear is b ecause nal

failure do es not o ccur when the core fails in shear Predictions of failure based on the

core crushing strength would give a relatively good agreement with measurements in

this case Again this highlights the need for more elab orate mo dels of core failure near

loading p oints

Eect of Ribb on Direction

Sp ecimens with ribb on running in the longitudinal direction fail at a signicantly higher

load than those with ribb on in the transverse direction in all cases except where failure

The kinks in the load deection curves at a low load are an artefact of the measuring system

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

is by intracell dimpling section For face failure mo des a slight dierence is

predicted asso ciated with the eect of core shear stiness on the stress in the skins

although the measured dierences are somewhat greater than the prediction The the

oretical mo del describ ed for core crushing do es not include any anisotropic material

prop erties The substantial dierence between the strengths for the two ribb on ori

entations suggest that an improved mo del is necessary lo oking in more detail at the

stresses around an indenter and including the eect of core shear

Intracell Buckling

Section denes a critical honeycomb cell size ab ovewhich failure o ccurs b yintra

cell buckling instead of by top face yielding For the materials used here this transition

is predicted at a cell size of approximately mm This hyp othesis was tested by

using sp ecimens with a cell size mm a core density of kgm and a range of

skin thickness to span ratios All of the sp ecimens with the larger cell size exhibited

elastic intracell buckling Final failure did not o ccur immediately but at a higher

load either by delamination of the topface or by core crushing for long or short spans

resp ectively Fig a shows the section under the central roller at the onset of

intracell buckling while Fig b shows the section when nal failure o ccurs by

delamination Figure c compares the failure load and failure mechanisms for this

set of tests The theory equation underpredicts the nal failure load b ecause it

relates to the initial buckling instability and not the nal delamination or core failure

Failure loads are indep endent of the honeycomb ribb on direction because the single

and doublewalls pap er walls make up a smaller prop ortion of the honeycomb for a

given honeycomb density

Conclusions

Previous research on honeycomb mechanics and the b ehaviour of sandwich b eams in

p oint b ending have been combined to mo del the b ehaviour of honeycomb sandwich

panels It is assumed that the skin and core materials b ehave in a brittle manner The

failure mechanisms considered were face yield face wrinkling intracell buckling core

shear and indentation at the load p oints leading to core crushing The latter is treated

in an empirical way using measurements of the b earing area at the load points The

failure loads for each region are estimated assuming that there is no coupling b etween

failure mechanisms Following the work of Triantallou and Gibson failure mo de

maps are derived with axes as the core relative density and the ratio of the face thickness

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

to span length Each map is appropriate for a single value of the ratio of the radius

of the roller to the skin thickness Regions in which the various failure mechanisms

o ccur are identied on the maps Alternative axes for the map of core relative density

and the core thickness are suggested as a more useful set of parameters for a b eam

designer or manufacturer Although the maps are generated for threep oint b ending

the metho d can straightforwardly b e applied to other loading geometries for example

fourp oint b ending The metho d is illustrated using the widely used combination of

core crossply GFRP laminate skins with a Nomexphenolic resin honeycomb

Exp erimental tests showed that in general the maps predicted adequately failure

mo des and failure loads The transition from face yielding to intracell buckling for

long span beams was demonstrated by increasing the honeycomb cell size from to

mm Failure near the load p oints due either to core shear or core indentation was

not mo delled well To mo del this behaviour more sophisticated mo dels of the contact

region and of the failure criteria in the core are needed Chapters and show how

the implemen tation of the highorder sandwich beam theory of Frostig provides

a b etter insight into the mechanism of lo calised eects induced by concentrated loads

and helps to pro duce an improved indentation failure analysis for honeycomb sandwich beams

Chapter Failure Mo de Maps for Honeycomb Sandwich Beams

Honeycom b Direction

L ong itudina l

T ran sverse (a) 8

(kN/m ) (b) W

4 Line Load, Load, Line

3 Core Density = 64 kg/m Cell size = 13 mm Span Length = 230 mm 0 0 4 8 12

M idspa n D eflection (m m )

a Elastic intracell buckling b Final failure due to top skin delamination

100 E xperim ental D ata Long Trans

Intracell D im pl. + F ace P eeling

Intracell D im pl. + C ore C rushing

o 10 W

Theoretical C urves

Intracell Buckling

Line Load, Load, Line (kN/m) C ore C rushing 3 1 Core Density = 64 kg/m Cell Size = 13 mm

0.0 01 0.010

F ace thickness / span , t/ L

c Exp erimental results compared with theory

Figure Photographs and results of sp ecimens with mm cell size

Chapter

Indentation Resistance of Sandwich

Beams

Intro duction

The exp eriments describ ed in the previous chapter revealed the imp ortantrole of the

exural rigidity of the skins and the transverse shear stiness of the core on the overall

resistance of a sandwich beam to indentation In the literature the imp ortance of

the core b ehaviour in aecting indentation failure has b een considered however the

inuence of the skins exural rigidity is generally overlo oked This omission derives

from the fact that in practical applications the skins are quite sti and the inuence of

skin exural rigidity can b e neglected One of the main aims of this chapter is to fo cus

on the role of skin rigidity in indentation failure Before intro ducing any analytical

metho ds it is instructiveto consider two extreme cases as illustrated in Fig

For very exible skins Fig a there is large lo cal deformation under the load

which can easily lead to core failure

For very rigid skins Fig b indentation failure will b e relatively hard as the

skins spread the load

Skin exibility is a prerequisite for indentation failure and the cores transverse exibil

ity enhances this susceptibility In this chapter the highorder sandwich b eam theory

HOSBT is used to analyse the b ending b ehaviour of sandwich b eams under lo calised

loads and extract conclusions ab out their indentation resistance by examining the ex

ibility of the top skin to the external applied loads The ability of HOSBT to cop e with

the lo calised eects that app ear in sandwich b eams at the load p oints is exp erimen

tally veried by Surface Displacement Analysis in section Section intro duces a

length scale that characterises the susceptibility of a sandwich b eam to lo calised eects

Chapter Indentation Resistance of Sandwich Beams

In section highorder sandwich beam theory is used to predict the contact stress

distribution b etween a cylindrical indenter and the top skin of a sandwich b eam

low flexural rigidity (a) of skins

high flexural rigidity (b)

of skins

Figure The b ehaviour of exible and rigid skins

HighOrder Sandwich Beam Theory

The mathematical formulation of highorder sandwich b eam theory HOSBT is intro

duced byFrostig and Baruch Highorder refers to the nonlinear way in whichthe

inplane and vertical displacements are allowed to vary through the height of the core

The core vertical displacement is assumed to have a quadratic variation with height z

Other core displacements also vary in a nonlinear way with the exact variation b eing

expressed in terms of Fourier series This contrasts with simple b eam theory where the

core inplane displacements are assumed to vary in a linear way through the depth

and the outofplane displacements are assumed to b e constant These highorder vari

ations allows mo delling of the more complicated changes in core geometry whichoccur

at loading p oints Thus the basic assumptions of HOSBT are

The shear stresses in the core are uniform through the height of the core

The core vertical displacemen tvariation is a quadratic p olynomial in z see equa

tion allowing the core to distort and its height to change

The core is considered as a D elastic medium which has outofplane compres

sive and shear rigidity whereas its resistance to inplane normal shear stresses

is negligible

Outofplane refers to direction as illustrated in Fig or the direction normal to the plane

of the sandwichpanel

Inplane refers to plane as illustrated in Fig or the plane of the sandwichpanel

Chapter Indentation Resistance of Sandwich Beams

This highorder analysis is based on variational principles the details of derivation of

the governing equations and asso ciated b oundary conditions are presented in ref

In this study we examine the b ending behaviour and the lo calised eects of sandwich

beams formulating a two dimensional mo del of a sandwich b eam with unit width and

span L which consists of a core with thickness c Youngs and shear mo dulus E and

G resp ectively and two skins with the same thickness t t t Youngs mo dulus

c t b

E and Poissons ratio as depicted in Fig b An external distributed load q is

f f t

on the top skin The displacement and stress elds of the core are expressed applied

in terms of the followings ve unknowns the inplane deformations u and u in the

t b

xdirection of the centroid of the top and b ottom skin resp ectively their corresp onding

vertical displacements w and w and the shear stresses in the core The relevant

t b x

geometric parameters and the notation of stresses and displacements app ear in Fig

The governing equations see equations and in ref are given

below

A u

t tx x

A u

b bx x

E ct

D w w w q

t tx b t xx t

ct E

w w D w

b t xx b bx

ct c ct c

u u w w

t b x xx tx bx

G E

c c

where A A E t and D D E t are the inplane and

t b f t b f

f f

exural rigidity of the skins The notation for example denotes the th partial

derivative with resp ect to x For a simply supp orted sandwich beam the solution can

b e expressed as a Fourier series expressing the variation of the relevantvariables in the

x direction as

m x

C cos u x

m x

C cos u x

m x

C sin w x

m x

w x sin C

m x

x C cos

ut ub wt wb

are constants where M numb er of terms in the series and C C C C and C

m m m m m

to b e determined We assume that the external loads are exerted only on the top skin

Chapter Indentation Resistance of Sandwich Beams

ut deformed top skin d

wc deformed cross section deformed bottom skin

ub wb z

x (a)

W δ

top skin qt tt σ txx σzz(z=0)

core τ c x

σ bottom skin bxx t x b unit L width

(b)

Figure a Nonlinear displacement patterns b Beam geometry and stresses The

origin of the zco ordinate is always taken at the top of the b eam element either

skin or core which is b eing considered

Chapter Indentation Resistance of Sandwich Beams

So in terms of the Fourier series we set

m x

q x C sin

where C is a constant that dep ends on the distribution of the external load After

substituting every term of the Fourier series in the governing equations

the problem can b e expressed in matrix form as

D C Q

where

m E E ct m

c c

L c c L

E m E ct m

c c

c L c L

c m ct m ct m c

L L G E L

c c

qt wt

and Q C

C C

m m

By solving with resp ect to matrix C for every m M the Fourier co e

cients in equations can be determined These equations can then be used

to calculate all the inplane and vertical displacements and core shear stresses The

inplane normal stresses in the top skin and the outofplane normal stresses

txx zz

in the top skincore interface are calculated indirectly as

E u

txx f tx

c E

w w

zz xx b t

Surface Displacement Analysis

An exp erimental verication of highorder sandwich beam theory HOSBT has b een

undertaken by Thomsen and Frostig They conducted a series of photo elastic

Only top skin compressive failure is considered

Chapter Indentation Resistance of Sandwich Beams

exp eriments compared the results with analytical calculations and demonstrated a

go o d agreementbetween the photo elastic measurements and the predictions based on

the HOSBT In this section a similar comparative study is conducted with resp ect

to the displacement eld in the core The ob jective is to compare the analytical and

exp erimental displacement distributions in sandwich b eams with a soft core as for

example the Nomex cores used in Chapter Sp ecial emphasis is given to the lo cal

outofplane compression of the core under the lo calised loads as this will b e imp ortant

in determining failure of the core

For the analytical predictions the equations describ ed in section are implemented

in a Matlab co de to calculate the vertical displacement eld within the core by the

following equation given in ref

z cz z

w x z w w w

c xx b t t

E c

The exp erimental determination of vertical displacements in the core is achieved by

Surface Displacement Analysis SDA is a Windows based software develop ed by Instron

Corp oration which detects and maps horizontal and vertical displacement of p oints

on the surface of a sp ecimen or structure There are two main parts to the pro cedure

image capture and image analysis SDA captures images of the surface of a material

or structure using a video camera and a frame grabb er b oard The surface requires

a random sp eckle pattern either natural or applied which has to be illuminated to

ensure go o d con trast Image frames of the surface of the sp ecimen are captured at

intervals while it deforms under load

Then SDA compares and analyses pairs of these image frames using a designated area

of interest as depicted in Fig Each area of interest contains one or more analysis

cells During analysis the pattern in each cell of the current frame is compared with

the pattern in the corresp onding cell of the reference frame The software maps the

deformation of the surface as a series of vectors displayed on the image one vector

a for each cell The software also calculates the strain for each cell SDA contains

graphics mo dule that plots displacement and strain If the software is calibrated to

the dimensions of the image absolute displacementvalues can be obtained

In this study we tested sandwich b eams cut out from the panels describ ed in

section Testing was under p oint b ending at a constant displacement rate of

mmmin The b eams width was mm the span mm and the rollers diameter

mm The exp erimental setup is shown in Fig The clip gauge measures the

midspan core compression during loading while the video camera captures images of

the other cross section of the b eam T o apply a random sp eckle pattern of that cross

Chapter Indentation Resistance of Sandwich Beams

Figure Surface Displacement Analysis software window showing the reference image

frame of a side cross section painted to have a random sp eckle pattern and

the area of interest where the analysis takes place

clip gauge beam camera measured

core compression

Figure Exp erimental setup

Chapter Indentation Resistance of Sandwich Beams

section at rst it was painted black and then nesprayed with white paint to pro duce

a pattern of random white dots on a blackbackground see the image in Fig The

three dimensional surface of the cross section arising from the honeycomb structure

and the subsequent buckling of the cell walls causes errors in the image pro cessing

and mapping of displacements using SDA However in the initial stage of loading with

little buckling the results are satisfactory and illustrativeof the lo calised eects that

o ccur under the central load

For a sandwich beam with Nomex honeycomb core with density of kgm the

load deection curve is shown in Fig The same gure also shows the clip gauge

measurements of midspan core compression against top skin deection For Surface

Displacement Analysis two frames are chosen the rst reference frame just b efore

loading p oint A in Fig and the second when the applied line load reaches the

arbitrary value of kNm lab elled B in Fig

The SDA graphics mo dule after calibration plots contours of equal vertical dis

placements within the core in mm as depicted in Fig a It should be noted that

the camera was xed relative to the central roller so that the sp ecimen app ears to move

upwards with zero displacements at the roller The SDA measurements demonstrate

the o ccurrence of lo calised deections under the applied load due to the core lo cal

compression The contour of mm close to the b ottom skin indicates the total

compression of the core in the midspan of the sandwich b eam This value agrees well

with that measured by the clip gauge see the dashed line in Fig

Fig b shows a theoretical plot at the corresp onding load of kNm using

equation These calculations use the material prop erties of the skin and the

honeycomb and the geometric attributes of the sandwich beam Table The total

applied load is taken as kNm and applied uniformly over an area of mm long A

comparison between the theory and exp eriments shown in Fig show that HOSBT

provides an analytical to ol that is capable of predicting the lo calised eects due to

concentrated loads b oth qualitatively and quantitatively

Eect of Spreading Stresses

After demonstrating the applicability of highorder sandwich b eam theory in this sec

tion we use it to investigate the mechanism that controls the behaviour of a sandwich

beam under lo calised loads One of the key factors that governs the b ehaviour of a

sandwich beam under lo calised loads is the shap e of the normal stress distribution

The reason for this is explained in section

Chapter Indentation Resistance of Sandwich Beams

Line load, W (kN/m) load, Line S p eed = 0.5 m m /m in 3 C ore D en sity = 48 kg/m S p an L en gth = 60 m m A 0 0.3

0.2

0.1

B Mid-span core compression (mm) compression core Mid-span A 0.0 0.0 0.5 1.0 1.5 2.0

T op sk in d eflection (m m )

Figure Loaddeection and core compression curves A and B corresp ond to the frames

used for pro cessing by SDA The dashed line indicates the total midspan core compression at a line load of kNm

Chapter Indentation Resistance of Sandwich Beams a

7 mm

z 0.035 6 0.15 0.045

5 0.04 0.06

0.1 3 0.15

ertical co ordinate 2 V

1 0.1

24 26 28 30 32 34 36

Horizontal co ordinate x mm

Figure Vertical displacement eld in the core pro duced by a SDA and b HOSBT

mo del All contour values are in mm and the scaling in b oth plots is the same

Chapter Indentation Resistance of Sandwich Beams

along the b eam in the top skincore interface see Fig Two extreme cases can b e

considered a very rigid top skin that spreads the external stress in the interface or a

membrane skin allows the applied stresses to pass intact through the interface into

the core

In this section we extract a dimensionless quantitywhichcharacterises the b ehaviour

of a sandwich b eam under p oint b ending indep endently of the applied central load

We dothisby dening an index which measures the transparency of the top skin ie

to what extent the top skincore interface feels the stresses exerted on the external

surface of the top skin For this purp ose we rewrite the matrices D and Q intro duced

in equation as

A A

t f

m E

D D

f t

L c

E qt

Q and D where

C D E C

f c t

E D C

c f t

ct m

C C G

t t c

c c m

G E L

c c

Using the Symb olic To olb ox of Matlab equation is solved with resp ect to the

matrix C

R R

C QD C

D E R R

A C

R R

where R D A G A C and R E A G A C

f f c f t c f c f t

Substituting equations and the ab ove Fourier co ecients into the

normal stresses in the top skincore interface are given as

c m E m x

wb wt

C C C sin

m m m

L c L

m x cm A C E

f t c

sin C

L R R D E L

f c

Chapter Indentation Resistance of Sandwich Beams

Equation shows that the Fourier co ecients for the stress can be separated

into two parts one part containing the co ecient C units of stress which dep ends

on the distribution of the load and a second part represented by the dimensionless

transmission co ecient C whichisentirely dep endent on the geometric attributes

and material prop erties of the sandwich b eam where C is given by

A C E cm

f t c

L R R D E

f c

E A C

c f t

if t n c

R R D E

f c

For any geometry and material combination this transmission co ecient is a discrete

function of the parameter Lm which is the semiwavelength of every term in the

Fourier series The value of C varies from small Lm to large Lm

implies no transmission of this Fourier comp onent of applied stress while implies

total transmission through the top skin Thus we conduct a parametric study in which

we calculate the variation of C with Lm for the range of material and geometric

parameters given in Table Fig plots this variation Each subplot shows curves

for variations in the skin thickness t and core depth c Changing from one subplot to

the next represents achange in either skin stiness E or core density as indicated

f c

by the arrows to the left and top of the gure

Parameter Low Value High Value

t mm

E GPa

kgm

c mm

Note The refers to the density of Nomex honey

combwith kgm

Table The range of the geometric and material parameters used in the parametric study

To characterise the transparency of the skin it is helpful to identify a semiwave

are close to zero so that length Lm below which the values of the co ecient C

these wavelengths cannot be transmitted Examining the curves in Fig it can be

seen that the curves all have similar shap es and that the inection p ointineach curve

can b e used to characterise where C is small The semiwavelength at the inection

point characterises the susceptibility of sandwich beams to lo calised eects and this

elength is dened as the spreading length In general the larger is the semiwav

broader is the distribution of the normal outofplane stresses in the top skincore

interface and vice versa

Chapter Indentation Resistance of Sandwich Beams

GPa E GPa E

f f

1 1

c mm

zz zz

mm 0.8 0.8 c m m

t mm

C C

t mm t

t 0.6 0.6

0.4 t 0.4

t mm kgm

c 0.2 0.2

c mm

Co ecien Co ecien

c mm 0 0

0 5 10 15 20 0 5 10 15 20

Lm mm Lm mm

1 1

c mm

zz zz

c mm

mm 0.8 t 0.8 m m

C C

t mm t 0.6 t 0.6

kgm

t mm

t mm 0.4 0.4

c 0.2 0.2

c mm Co ecien Co ecien

c mm 0 0

0 5 10 15 20 0 5 10 15 20

Lm mm Lm mm

Figure Parametric Study change of transmission co ecient C with wavelength Lm

for variations in the parameters t E c and The boxed lab els show each

time the parameters and the arrows indicate the direction of each parameters

increase The p osition of inection is marked bya symbol

Chapter Indentation Resistance of Sandwich Beams

Spreading length is a function of the skin thickness t skin Youngs mo dulus E

core density and core thickness c but it is indep endent of span length L Table

shows the inuence of each parameter to spreading length This table shows the

inuence of changing the various parameters from the lowvalues given in the left hand

column of Table Table shows that the skin thickness t is the most signicant

factor as exp ected the larger the skin thickness the stronger the eect of spreading

stresses A similar eect is shown with increasing stiness E although given the big

change in stiness of this eect is less imp ortant Core density also aects

the stier is the core the less the external stresses spread Core thickness c seems to

play a small role

Table Change of spreading length

Lowvalues from Ta t t c c E E

f f c c

ble for t c E

At this point it is imp ortant to note that is not directly related to the strength

of the sandwich beam Instead it measures how the top skin is able to spread the

external load over the core is prop erty of the beam material and geometry In the

following sections we showhow it can b e normalised and provide a dimensionless index

to characterise the spreading eect in sandwich b eams under indentation loads

Contact Pressure Distribution

Having used the highorder sandwich beam theory to characterise the way in which

through the top skin in this section we investigate how contact loads are transmitted

this metho dology can be applied to the contact between an indenter and a sandwich

beam In the literature there are only afew references relating to the contact stresses

exerted between the top skin of a simply supp orted sandwich b eam and an indenter

which applies p oint b ending on the b eam Frostig Baruch examine analytically

the eect of four typ es of lo calised load distributions a p oint load uniform sinusoidal

distributions or two concentrated loads These represent very exible intermediate

plates in and very sti indenters resp ectively Johnson p investigates thin

contact with a rigid indenter and concludes that the contact stress distribution changes

from having a maximum in the centre to one in which the pressure is concentrated at

the edges when the thickness of the plate increases A dierent problem is examined

by Keer and Ballarini who investigate the problem of a rigid punch in smo oth

Chapter Indentation Resistance of Sandwich Beams

contact with an initially stressed transversely isotropic elastic b eam In this study

we investigate how sandwich b eams b ehave under contact with a rigid central roller

calculating the pressure distribution under the indenter and through the b eam

Fig shows a rigid cylinder of radius R which is pressed into contact with a sand

wich b eam of unit width and length Lsuch that the contact width is Equation

can be used to calculate the top skin displacements for a given distribution of the ex

ternal load Following standard contact metho ds we can divide the contact into a

ts resulting from a unit load on number of discrete sections calculate the displacemen

each of these sections and sup erimp ose the results to nd deections for any contact

load In particular for a given pressure distribution q x it is p ossible to nd the

corresp onding distribution of heights P Fig ab along the top skin Conversely

we shall show that for a given roller geometry we can evaluate the contact width and

contact pressure by inverting this approach

The shap e of discrete pressure elements can be step functions Fig a or o ver

lapping triangles Fig b The second choice is b etter b ecause the total distributed

load function is smo other along the x direction In addition the Fourier series converges

more quickly for a triangle shap e rather than for a rectangular one The pressure ele

ments on the top skin can be describ ed using the series

m x

q P q P C sin

i i i i

where P is the heightofthe triangular distribution and

L m x m v

C sin cos

m v L L

are the Fourier co ecients for an isosceles triangle of unit magnitude Each unit

length has co ordinate x and base width v q is the distribution of pressure q with

i i i

unit amplitude For this and subsequent symb ols the hat refers to variables relating

to this unit pressure distribution Using the pro cedure describ ed in section the

vertical displacements w x of the top skin asso ciated with each of the triangular

pressure elements are given by equation

m x

w i i i

C sin w xP P w

i i

m t t

If N is the number of pairs of symmetrical triangular pressure elements covering a

contact width then the total overall top skin deection is given by

N N

X X

i i

x P w w x w

i t

t t

n n

Chapter Indentation Resistance of Sandwich Beams

(a)

v/2 (b)

top skin surface

v Contact width δ

xi x=L/2 End of beam

R (c) δ

Figure Discrete contact pressure elements a uniform piecewise constant b over lapping triangles piecewise linear and c Geometric denitions

Chapter Indentation Resistance of Sandwich Beams

In the case where load is applied by a rigid indenter which is in p erfect contact with

the top skin the top skin deection follows the shap e of the indenter within the contact

area In this section we examine the case of a cylindrical indenter for comparison with

our exp eriments but the calculations can be applied to any indenter shap e Having a

cylinder of R radius as a central indenter and assuming p erfect contact with the top

L L

skin within contact of width then the top skin deection for all x is

R w w x R x

t L

M N

X X

m L

C sin P R R x

i i

m n

R P w R x

t L

where the sup erscript R denotes the deection at the roller We can also write the

L L

equality only for x as

w xw x

i i

P w w giving R x R

t x t L

L L

If we write the ab ove equality for N dierentvalues of x which represent

dierentvalues of w x w x along the left half of the contact width then we

t j j

get asystem of N equations with N unknowns which are the distributions of pressure

P This system can be written in a matrix form

P W R where

h i

P P P

w w w w w w

t L t x t L t x t L t x

w w w w w w

t x t L t x t L t x t L

N N N N N N

w w w w w w

t x t L t x t L t x t L

h i

q q q

L L L

R R R R x R x R x

Or right half b ecause of symmetry

Chapter Indentation Resistance of Sandwich Beams

Substitution of P into equation and summation for all N triangular pressure

elements gives us the contact pressure distribution

q q x

i t

The same concept of sup erp osition can b e used to calculate the other stresses ie

The total load is derived by integration along the x direction

Z Z

N M

L L

X X

m x

sin q x P C W

t i

M N

X X

mC for m M P W

In principle this approach can be used to determine the variation in load distribution

and the contact arc as the total load W increases In practice a problem arises at

small As it is necessary to span the whole b eam length with elements the number of

elements needed to deal with sharp variations in pressure b ecomes prohibitively large

This metho d is used in the following section to answer the following questions

How accurate is it to assume that the load is distributed uniformly over the

contact length

How transparent is the top skin to the external loads In other words howdoes

the distribution of normal stress x in the top skincore interface compare

with the contact stress distribution q x

Case Study

In this section the results from the aforementioned metho dology for two extreme cases

of skins exural rigidity are presented to illustrate the eect of indenters size on

the contact stress distribution and the top skins transparency to external loads We

consider one sandwich b eam denoted as b eam A with very exible skins ie ply

mm and GFRP laminates with t mm and E GPa which gives

another b eam B with very rigid skins ie ply laminates with t mm

and E GPa which corresp onds to mm Both b eams have unit depth

are mm long have Nomex honeycomb with core density of kgm are simply

supp orted and loaded in p oint b ending by a central roller For each beam two radii

for the central roller are considered R mm and R mm The metho dology

describ ed in the previous section requires the contact width as an input data To

Chapter Indentation Resistance of Sandwich Beams

compare results for each roller is adjusted manually in a simple numerical algorithm

so that for each b eam the maximum normal stresses in the top skincore interface are

approximately equal

Fig and Fig show the distributions of contact stresses q and the cor

t cc

resp onding normal stresses in the top skincore interface normalised by the

zz cc

cores outofplane compressive strength for beams A and B resp ectively For b eam

B the input value of is chosen to give max However for beam A this

zz cc

y of the beam means that it is not p ossible is not feasible The much greater exibilit

to generate stresses in the core greater than half the compressive strength of the core

without the b eam b ending excessively Further loading by increasing the value of

induces large skin deformations which also make HOSBT incapable to mo del the be

haviour of the sandwich b eam Thus for b eam A the input value of is chosen to give

max However this dierence in peak stress between the two b eams is

zz cc

immaterial as the calculations are elastic and the stress distributions are indep endent

of the applied load

A particular feature of the contact pressure distribution for the exible b eam A

with R mm and for the stier b eam B with R mm is the sharp p eaks in

pressure at the edges of the contact This feature is also noted by Johnson for

similar contacts These p eaks however are not transmitted through the skins

Tocharacterise the degree to which the skins are able to spread the load we need to

compare the width of the contact patch and the width over which there are signicant

pressures transmitted across the interface This latter width can b e characterised bya

length given when the pressures have p ositive nonzero values

Results show that for the beam B with a rigid skin Fig the pressure width

of the pressure distribution in the interface is considerably more than the contact

width and do es not change signicantly with roller diameter or contact width The

skin is suciently rigid to spread the lo cal pressure distribution out For b eam A with

a more exible skin Fig the width of the pressure distribution in the interface

corresp onds to that of the contact The ratio can b e used to measure howwell the

contact load has b een spread out For a very exible skin this will equal while for a

very rigid skin this will tend to a large value which will dep end on the roller radius

By considering a range of skin stinesses changing the thickness t orand Youngs

mo dulus E this metho d can b e used to nd the variation of for a range of b eam

spreading lengths for the two roller diameters and mm This is illustrated in

Fig This gure shows that for very exible skins with small the width of the

contact equals the width of the region in which there are signicant pressures across

Chapter Indentation Resistance of Sandwich Beams

1 Load distribution

0.9 mm

0.8 R

R mm 0.7 cc

0.6 t q 0.5

0.4

Normalised 0.3

0.2

0.1 δ 0

1 Normal stresses in the interface

cc 0.8 zz

0.6

0.4

0.2 ∆ Normalised 0

−0.1 −0.05 0 0.05 0.1

x LL

Figure Distributions of contact stresses q and the corresp onding normal stresses

t cc

in the top skincore interface normalised by the cores outofplane

zz cc

compressive strength for b eam A with mm

Chapter Indentation Resistance of Sandwich Beams

1 Load distribution 0.9

0.8

0.7 cc

0.6 t

R mm mm 0.5 R

0.4

Normalised 0.3

0.2 δ 0.1

1 Normal stresses in the interface

cc 0.8 zz

0.6

0.4

0.2 ∆ Normalised 0

−0.1 −0.05 0 0.05 0.1

x LL

Figure Distributions of contact stresses q and the corresp onding normal stresses

t cc

in the top skincore interface normalised by the cores outofplane

zz cc

compressivestrengthforbeamBwith mm

Chapter Indentation Resistance of Sandwich Beams

the interface with as exp ected As the spreading length increases the

distribution of pressure in the interface broadens with resp ect to the contact width

with larger This change dep ends on roller diameter If we take the transition

from exible to rigid b ehaviour when equals then this o ccurs for R equal to

and for the and mm rollers resp ectively

This case study demonstrates the usefulness of the spreading length in character

ising the b eam exibility It also allows us to answer the questions p osed in section

relating to the assumptions commonly used to mo del the contact Firstly for practical

beams with large the distribution of pressure over the contact is not imp ortant

and the assumption that the external load is applied uniformly across the b eam will

be reasonable However this contact distribution will not be transmitted to the in

terface and the core unchanged By contrast Fig shows that it is essential to

mo del b eams with sti skins accurately in order to get a reasonable estimate of the

stress distribution in the core The commonly used assumption that the failure can b e

predicted using Failure line load Cores outofplane compressive strength

Contact width will be p o or unless a judicous choice of is made based on ex

p erimental measurements In the nal section we summarise the conclusions from

this chapter including guidelines of a more accurate failure analysis

Concluding Remarks

The b enets of using the highorder sandwich b eam theory HOSBT to analyse the

behaviour of sandwich b eams under indentation are presented The capabilities of

HOBST are veried exp erimentally by Surface Displacement Analysis The symb olic

manipulation of the governing equations of HOSBT is used to extract a spreading length

This is a prop erty of a sandwich b eam dep ending mainly on the skins exural

stiness and characterising the susceptibility of the sandwich beam to indentation

loads

A further manipulation of HOSBT gives an insight into the contact mechanics for

loading of a beam by a cylindrical indenter The way in which the contact pressure is

transmitted through the core is examined Acase study shows the fact that sandwich

beams used as standard in industry have skins which are rigid enough to spread the

external loads

The most imp ortant conclusion from this section is that the maximum stresses

which are resp onsible for indentation failure cannot b e predicted in a straightforward

way for sandwich b eams with rigid skins even if the contact width is known For

Chapter Indentation Resistance of Sandwich Beams

20 R = 5 mm R = 10 mm 15

δ 10 Flexible Rigid

/ skins skins ∆

0 0 1 2 3 4

λ (mm)

Figure Dep endence of spreading eect on rollers radius R and determination on how

exible or rigid are the skins with resp ect to the indentation resistance of the

sandwichbeam

Chapter Indentation Resistance of Sandwich Beams

sandwich beams with very exible skins the approximation of dividing the total line

load W by can give us reliable estimation of the failure stresses in the skincore

interface However for commercially applied sandwich congurations the former case

is the dominant

Thus in practice one should follow the following steps for indentation failure analysis

of sandwich beams

For a particular indenters size and shap e do one test to measure the just b efore

failure since develops with the increase of W This value for can b e assumed

for other b eam geometries

Use the approximation that the external stresses are uniform within this measured

length For sandwich b eams with small this approximation is good while

for sandwich b eams with large the exact distribution of contact load do es not

aect the core stresses signicantly

Use high order sandwich b eam theory and apply the equations and

to calculate the stress eld in the core Compare them with the maximum

allowables to predict failure

This approach is adopted in the next chapter to undertake an indentation failure anal ysis

Chapter

Indentation Failure Analysis

Intro duction

If we assume that the behaviour of a sandwich b eam is elastic up to failure the high

order sandwich b eam theory can be used to calculate the stresses in the core due to

the indentation loading This can then b e compared with the cores outofplane com

pressive strength However a more accurate failure criterion for the core is determined

from biaxial tests on Nomex honeycomb cores and used in this analysis Finally short

b eam p oint b ending tests with three dierent central roller diameters verify the the

oretical predictions and show the imp ortance of the skin exural rigidity on the b eams

overall strength

Failure Envelop e for Nomex Honeycombs

A straightforward way to calculate the failure load by indentation for honeycomb cores

is by equating the maximum of the outofplane normal stresses with the allowable

stresses How ever the exp eriments in Chapter showed that the interaction of core

shear and outofplane stresses was imp ortant The concept of the indentation failure

due to these combined stress comp onents is depicted in Fig Therefore b efore p er

forming an indentation analysis we investigate b oth theoretically and exp erimentally

in this section the twodimensional failure envelop e of Nomex honeycombs under shear

and outofplane compression loading and we dene a new failure criterion

References to biaxial loading tests for cellular materials are not common in the

literature Gibson et al have p erformed a series of tests on a variety of foams

under biaxial axisymmetric and hydrostatic loading conditions Zhang and Ashby

haveinvestigated the inplane biaxial buckling b ehaviour of Nomex honeycombs More

Chapter Indentation Failure Analysis

Figure Combined indentation failure mechanism

recently Stronge and Klintworth have studied the yield surfaces for honeycombs

under biaxial macroscopic stresses

In this chapter Nomex honeycombs are tested under combined shear and compres

sive outofplane loading For this purp ose we used the Arcan test rig as illustrated in

Fig This geometry was adapted for shear testing of comp osite materials by Ar

can et al and extended to biaxial testing of butteryshap ed sp ecimens byVoloshin

and Arcan The rig consists of two pairs of plane circular Sshap ed parts with an

tisymmetric cutouts To accommo date honeycomb sp ecimens we added two loading

platens The b ottom pair of semicircular xtures are gripp ed to the frame of a servo

hydraulic testing machine The upp er pair are mounted to the load cell and actuator

transferring the load to the sp ecimen which is attached between the two rectangular

plates To ensure a uniform stress through the sp ecimen the grips are clamp ed rather

than b eing pivoted since Marlo s nite element analysis shows that stress uni

formity under biaxial conditions is not go o d The p erimetric holes in the semicircular

plates allowa change of the angle between the plane of the sp ecimen and the loading

direction and therefore the ratio tan of compression to shear applied to the sp eci

men b etween the central plates The sp ecimens are rectangular plates of no more than

mm in width or length

The success of such tests dep ends on being able to attach the sp ecimen to the

loading platens eectively In this case the at external surface of the skins can be

easily attached with standard acrylic adhesive Sp ecimens were prepared by cutting

rectangular pieces from each of the available sandwich panels of dierent densities

as describ ed in Chapter Although the aim of this work is to examine the b ehaviour

of the honeycomb core the presence of the skins do es not p ose any diculties on the

contrary they restrict movement of the honeycombwalls relative to the grips and help

attachment of the sp ecimen to the rig For each core densitytests were conducted for

Chapter Indentation Failure Analysis

Section A-A′ A Fitted to crosshead P Grip ∅200mm Pin Loading 10mm Loading platens platen 15mm Specimen 50mm Specimen ϕ

Fitted to P frame

A′

Figure The Arcantyp e rig

Chapter Indentation Failure Analysis

b oth orientations of the honeycomb ribb on longitudinal and transverse Loading was

monotonic up to failure with a constant displacement rate of mmmin In order

to cover a wide range from nearly pure compression to nearly pure shear tests were

o o o

made at three angles of and Fig shows the corresp onding

load paths Failure for each of the shear angles denes one p oint on the failure

surface with corresp onding values of the core normal and shear stresses and at

zz x

failure In Fig photos of the three angle setups and typical loaddeection curves

are presented The deection is measured from the crosshead displacement of the

machine

Pure compressive Load path strength, σcc zz

σ Failure surface σzz

τx

ϕ = 81.5°

ϕ = 51.5°

Compressive stresses, Pure shear strength, τcs ϕ = 21.5°

Shear stresses, x

Figure Load paths and determination of failure envelop e

The failure p eak loads give the combination of normal compressive load P sin and

shear load P cos which leads to failure Assuming that these loads are distributed

uniformly over the external surfaces of area b L of b oth skins the compressiveand

shear stresses in the core are given by P sin bLandP cos bL resp ectively The

calculated failure stresses and are normalised by the cores outofplane com

zz x

pressive strength and shear strength resp ectively as given by equations

cc cs

are plotted against for every a and b Normalised stresses

cc x cs zz

Chapter Indentation Failure Analysis

8 Load (kN) Load

0 0.0 0.5 1.0 D eflection (m m )

4 Load (kN) Load

0 0.0 0.5 1.0 D eflection (m m )

4 Load (kN) Load 2

0 0.0 0.5 1.0

D eflection (m m )

Figure The three angle setups and the corresp onding loaddeection curves here for

sp ecimens with kgm core density

Chapter Indentation Failure Analysis

angle core density and b oth honeycomb ribb on directions in Fig Except for the

honeycombs with density kgm the failure envelop es are well approximated by a

linear failure criterion given by

x zz

cc cs

as illustrated in Fig by the dashed line The inconsistency observed in the kgm

honeycombs is due to an inconsistency in the measured shear stresses and do es not

aect the accuracy of our calculations

Thus equation allows us to dene a failure criterion for combined loading cases

as required for an indentation failure analysis

Failure Analysis with HOSBT

In section the basics of highorder sandwich b eam theorys implementation to sand

wich b eam b ending behaviour analysis were presented Here the equations needed for

a failure analysis for honeycomb sandwich b eams loaded under p oint b ending are

assembled To simplify our calculations we assume that the external loads applied by

the central roller are uniform within a given width in the midspan of the b eam This

assumption is discussed in section So for a uniformly distributed load applied in

the midspan of the b eam the Fourier co ecient C cf eqn reads

W m m

qt qt qt

C sin sin or C W C

m m m

m L

where W is the total line load For the failure analysis we need to know the stress

eld in the coretop skin interface ie the outofplane normal stresses and

zz x

see Fig b Substitution of the ab ove co ecients C into equation gives the

wt wb

Fourier co ecients C C and C for the top and b ottom skin deections and for

m m m

core shear stresses

qt wt

C C

m m

D E R R

f c

wb qt

C C

m m

D E R R

f c

A C

f t

C C

m m

R R

Chapter Indentation Failure Analysis

1.0 1.0 cc cc σ σ / /

zz 0.8 zz 0.8 σ σ

0.6 0.6

0.4 0.4

3 3 C ore de nsity = 29 kg/m Co re density = 48 kg/m 0.2 0.2 Long Lon g

Tran s T rans Rel. Compressive Strength, Rel. Compressive Strength, 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 τ τ τ τ R el. S h e a r S tren gth , x / cs R el. S h e a r S tren gth , x / cs

1.0 1.0 cc cc 3 C ore density = 128 kg/m σ σ / / L on g zz 0.8 zz 0.8 σ σ T rans

0.6 0.6

0.4 0.4

3 C ore de nsity = 64 kg/m 0.2 0.2 Long

Trans Rel. Compressive Strength, Rel. Compressive Strength, 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 τ τ τ τ

R el. S h e a r S tren gth , x / cs R el. S h ea r S tren g th , x / cs

Figure Failure envelop es for Nomex honeycombs The dashed line corresp onds to the

linear failure criterion given by equation

Chapter Indentation Failure Analysis

Then from equations and the required stresses are given as follows

cm m x A C E

f t c

sin W x xW C

zz zz

D E L R R L

f c

A C m x

f t

xW C cos W x

x x

R R L

The variation of the stresses x and x with p osition x along the b eam and with

zz x

load W can b e represented in space by lines such as that shown dashed in Fig

For every W the lowest part of the curve corresp onds to the edge of the b eam x

where shear stresses are dominant but still there are some outofplane normal stresses

hence the curvedoesnottouchthexaxis Towards the midspan of the b eam ie the

upp er section of the curve the outofplane normal stresses increase while the shear

stresses decrease falling to zero at x L As W increases the stresses increase

until the failure envelop e shown by a solid straight line is reached at a failure load W

on pure compressive loading This failure load will dier signicantly from that based

σ cc x = L/2 (at centre of beam)

⊆ zz Location of failure, xf [0, L/2] σ

Failure surface Increasing x

Increasing W Locus of stresses in beam Normal stresses,

x = 0 (at end of beam)

τ cs τ

Shear stresses, x

Figure Combined failure criterion

Chapter Indentation Failure Analysis

W can b e calculated after substituting equations and into equation

and nding the minimum value of the following expression

x x

zz x

Exp erimental Work

Tovalidate the theoretical predictions for the indentation failure loads p oint b ending

tests were p erformed on short sandwich beams Beams mm long and mm wide

were cut in both honeycomb ribb on directions from Nomex core sandwich panels

Panels had nominal core densities and kgm with a mm cell size and

kgm with a mm cell size To observe in detail the vertical deections during

loading we painted one of the sp ecimens long cross sections so that the surface of the

honeycomb walls b ecame black and the edges of both skins and of honeycomb walls

b ecame white

The exp erimental setup is depicted in Fig It consists of a video camera and

recorder which keep track of the high contrast black and white cross sections of the

sp ecimens during loading The sp ecimens are simply supp orted bytwo rollers of mm

diameter The distance b etween them is mm p oint b ending tests were p erformed

with dierent diameters and mm of the central roller in a servohydraulic

testing machine The loading rate was mmmin As well as the midspan top skin

given by the machines transducers deection and the load measurements which are

we attached in the midspan of every sp ecimen a clip gauge to measure the relative

displacementbetween the skins

clip gauge beam camera measured

core compression

Figure Exp erimental setup

The exp erimental measurements of midspan core compression from the clip gauge

of load from the load cell and of top skin deection from the crosshead displacement

Chapter Indentation Failure Analysis

transducer are plotted for every central roller size and every core density in Figs A

A and A in App endix A A representative combined plot is shown here in Fig

The b ottom skin deection vs top skin deection curves show clearly the dierent

resp onse of the two skins the top skins exhibit larger absolute deections than the

bottom skins This is due to the compression of the core as shown clearly by the

core compression vs top skin deection curves Core compression is almost linear up

to failure For all core densities the core compression is more severe for the b eams

with longitudinal honeycomb ribb on direction due to the higher shear stiness of the

Nomex honeycomb in the longitudinal direction than in the transverse direction This

dierence conrms the imp ortance of the core shear stiness in indentation failure

The dierence between longitudinal and transverse ribb on directions is also reected

in the line loaddeection curves in the former case the beam has a higher overall

exural rigidity

The letters A B C and D in the upp er plot of Fig indicate critical stages of

the loading pro cedure ie b efore loading just b efore failure just after failure and well

after failure At these points video snapshots were captured These are presented for

every central roller size and core density in Figs A A and A in App endix A Here

we present some key photos for discussion Fig shows two video images captured

just b efore failure p osition B in Fig during p oint b ending loading with a mm

diameter central roller It is apparent how the low density core is deformed with

signicant shear causing the honeycombcellwalls to buckle elastically Fig shows

the eect of core density after failure when the total deection is approximately mm

The high outofplane stiness of the high density core do es not allow the damage to

propagate within the core and the skins fail eventually by compressive macrobuckling

For the same reason the indented top skin follows tightly the curvature of the roller

for high density cores while for low density cores the top skin is indented over a wider

area without following the indenters curvature see Fig

After comparing the corresp onding curves amongst Figs A A and A we realise

that there are no signicant dierences due to the central rollers size This is in accor

dance with the prediction made in Chapter This insensitivity to the indenters size

is conrmed by comparing the corresp onding failure p eak loads in Fig The same

gure also shows the satisfactory theoretical predictions using the highorder sandwich

beam theory in conjunction with the mixed failure criterion The values of outof

plane core compressive strength and core shear strength calculated in section

for every core density and ribb on direction are substituted in equation and the

failure loads which are minimum value of this provides the theoretical predictions of

represented by the lines in Fig The advantage of using the mixed failure crite

Chapter Indentation Failure Analysis

B C 6 D

Line load, W (kN/m) 2

A 0 0.3

0.2

0.1 Core compression (mm)

0.0 2.0

1.5

1.0

0.5 Bottom skin deflection (mm) deflection Bottom skin

0.0 0.00.51.01.52.0

T op skin deflection (m m )

Figure Typical exp erimental results for sandwich b eam with kgm core density

loaded by a roller with diameter of mm Midspan b ottom skin deection

midspan core compression and line load are plotted against midspan top skin

deection Lines legend longitudinal and transverse honeycomb ribb on direction

Chapter Indentation Failure Analysis

29 kg/m3 128 kg/m3

Figure The prefailure inuence of core density Video images captured just b efore

failure during p oint b ending loading with a mm diameter central roller

29 kg/m3 128 kg/m3

Figure The p ostfailure inuence of core density Video images captured after failure

mm total deection during p oint b ending loading with a mm diameter

central roller

29 kg/m3 64 kg/m3

Figure The p ostfailure inuence of core density on extent of damage Video images

captured after failure mm total deection during p oint b ending loading

with a mm diameter central roller

Chapter Indentation Failure Analysis

rion instead of simply checking when the maximum outofplane stresses in the top

skincore interface reach the outofplane compressive strength of the honeycomb core

is illustrated in Fig It is shown that without the correction of the mixed crite

rion the theoretical predictions are larger than the exp erimental data measurements

of the failure loads esp ecially as the core density increases Also the use of the mixed

failure criterion catches the dierent indentation resistance of sandwich b eams due

to the dierent honeycomb ribb on direction The sandwich beams with longitudinal

tly higher strengths than those with a transverse honeycomb ribb on direction havesligh

ribb on direction The dierence increases with the core density

Concluding Remarks

A systematic approach has b een develop ed to determine the failure load of sand

wich honeycomb structures under indentation loading Firstly the failure envelop e

for Nomex honeycombs under simultaneous outofplane compression and shear has

been determined by biaxial tests using an Arcan rig A linear dep endence on pure

compression and shear proves to be a go o d approximation for the failure envelop e of

Nomex honeycombs

The implementation of highorder sandwich beam theory allows a more accurate

failure analysis for sandwich b eams sub jected to lo calised loads Using the failure

envelop e determined by the biaxial tests a mixed failure criterion has b een intro duced

to predict the indentation failure caused by the simultaneous action of outofplane

compressive and shear stresses exerted in the vicinity of a lo calised load This criterion

can predict failure that ranges from pure core crushing to pure core shear

The short beam b ending tests validated the theoretical predictions of high order

beam mo del Video captures of the deformed side cross section illustrate the involve

ment of core shear in the elastic b ehaviour of sandwich b eams with low density cores

wed the dierent p ostfailure damage extent between sandwich Also these tests sho

beams with dierent core densities They showed that the mixed failure criterion of

fers a signicant improvement in predicting indentation strength as compared with

mo dels which do not include combined loading This mo del is particularly needed in

high core densities which induce more severe stress elds in the core Moreover this

approach explains observed dierences in behaviour for longitudinal and transverse ribb on directions

Chapter Indentation Failure Analysis

60 In d en ter L on g T ran s d iam eter

20 m m

10 m m

6 m m 40 (kN/m)

o L ongitudinal W

T ransverse 20 F ailure C riteria Line load, load, Line O nly com p ression

W ith m ixed shear

an d com p ression

0 0 40 80 120

C ore den sity, (kg/m 3)

Figure Theory lines vs exp erimental results symb ols for failure line load W o

Chapter

Failure Maps using HOSBT

Intro duction

In Chapter failure mo de maps for Nomex honeycomb sandwich b eams with GFRP

laminate skins are pro duced using simplied b eam mo dels and honeycombmechanics

However the simplied mo del for indentation failure prediction used in chapter can

not be used with great condence esp ecially where core crushing and shear interact

In the previous chapter a theoretical mo del based on highorder sandwich b eam theory

HOSBT and a mixed failure criterion for Nomex honeycombs has been shown to

give accurate predictions of the p eak loads when sandwich b eams made of GFRP

skins and Nomex honeycomb fail by indentation Here the use of highorder b eam

theory is extended as a compact computational to ol which determines the maximum

stresses in the skins and the core Thus by comparing the calculated maximum stresses

with the relevant allowable stresses we nd the range of design parameters where

each failure mo de is dominant Frostig and Shenhar have conducted an analogous

failure analysis for foam core sandwich panels by using the HOSBT but for failure

patterns and criteria dierent than those we present here In particular the failure

mo de of skin wrinkling is not examined Furthermore sp ecically for the failure mo de

of indentation the failure criterion for the core was based on the maximum principal

stresses Although this approach is appropriate for foams that are isotropic in case

of honeycombs which are orthotropic a more suitable failure criterion is that based

on the normal and shear stresses exerted in the top skincore interface Thus in this

chapter wemake use of the mixed failure criterion intro duced in the previous chapter

Based on the ab ove approach new failure mo de maps for Nomex honeycomb sandwich

beams are constructed and provide the basis for design optimisation carp et plots

Chapter Failure Maps using HOSBT

Reconstruction of Failure Maps

In chapter we presented the basics of high order b eam theorys implementation to

sandwich b eam b ending behaviour analysis By assuming a linear elastic resp onse for

the sandwich b eam up to failure it is feasible to use the appropriate equations derived

by high order b eam theory to calculate the maximum stresses in the skins and the core

under p oint b ending The advantages of such approach are use of a compact

computational to ol to calculate all the maximum stresses in the sandwich b eam and

ability to use mixed failure criteria for the core to mo del more accurately the

indentation failure as shown in the previous chapter

The exp erimental results presented in Chapter show that the short b eams nally

failed by core crushing under the central load but with an increased interaction of

core shear as the core density gets smaller Thus in this chapter the core shear is not

considered as a separate failure mo de The failure in the honeycomb core is treated as

one mixed failure mo de and denoted as core indentation The theoretical determination

of the p eak load for this mo de is presented thoroughly in the previous chapter and gives

W min

x x

zz x

These calculations are based on the assumption that the external loads applied by

the central roller are uniform within a measured width in the midspan of the b eam

As explained in Chapter the rigidity of the laminate skins used in the examined

sandwich b eams is high enough that the calculations are insensitivetothecontact area

between the indenter and the topskin It is assumed in this chapter that this remains

true

For the skin failure we need to know the maximum normal stresses exerted on

txx

the crosssection of the top skin Equation after substitutions gives

m C m x

C xW sin W x

txx txx

R R L L

This is a function of the x co ordinate and the maximum value is reached as exp ected

in the midspan of the b eam ie for x L

max W

txx txx xL

Now the failure loads for every failure mo de are calculated by equating these maxi

mum stresses with the allowable stresses for skin compressive yield and skin

fY fw

wrinkling resp ectively

Chapter Failure Maps using HOSBT

As in Chapter a failure map for this typ e of sandwich b eam is constructed by

nding the minimum failure loads for each of the failure mo des in Table for each

value of the design parameters ie tL and Then we plot the minimum load

c s

Table Expressions for p eak failure loads

Top skin yield W

o fY txx

Face wrinkling W

o fw txx

W min Indentation

x x

zz x

W against the skin thickness over span ratio tL and the core relative density

o c s

The pro jection of the lines where two mo des intersect provides the b oundaries for the

corresp onding failure mo de map as depicted in Fig By comparing this map with

those given in Fig one can see that the failure mo des of pure core crushing or shear

are now replaced by a core indentation region The upp er part of the indentation area

higher core densities corresp onds to core crushing under the vicinity of the lo calised

loads whereas the lower part lower densities is characterised mainly by core shear

A comparison between the failure loads calculated with HOSBT and the exp er

imental results presented in Chapter is depicted in Fig for all four examined

core densities The theoretical predictions for core crushing are more accurate for

the new failure prediction esp ecially for the lower core densities of or kgm

cf Fig Also this new approach predicts the observed dierences in b ehaviour

for longitudinal and transverse honeycomb ribb on direction Although the highorder

sandwich beam theory provides a robust and compact computational to ol for all the

mo des there are two drawbacks rstly when the skin fails the mo del calculates the

same failure load for the longitudinal and transverse direction of the honeycomb core

and secondly the predictions for the mo de of skin wrinkling are overpredicted esp e

cially closer to the transition from skin wrinkling mo de to indentation mo de The latter

is due to fact that HOSBT allows for lo cal b ending of the top skin b efore failure and

therefore the calculated inplane normal stresses in the midspan are lower than

txx

those calculated by simplied b eam mo dels Indeed the distribution of inplane normal

stresses along the b eam is triangular when predicted by simple b eam theory while

txx

HOSBT gives a curve tangential to this triangular distribution close to midspan This

discrepancy causes the dierences in prediction

Chapter Failure Maps using HOSBT

6 10

5 10 Nm o W

4 10

3 10 ailure line load F 2 10 0 10

−2 10

−1 10 −3 10

−2 −4

e density

Relativ 10 s

c 10

Face thickness span tL

newmap2.m 0 10 54600

s Skin yielding

c 26400 y

−1 10 6160

e densit 12700 relative density Skin wrinkling Relativ

2980 Core indentation 695 1440

−2 336 10 −4 −3 −2 10 10 10

skin thickness / span

Face thickness span tL

Figure Improved failure map for Nomex honeycomb sandwich b eams Each contour

represents sandwich b eams of equal strength in Nm

Chapter Failure Maps using HOSBT

100 3 3 C ore D ensity = 128 kg/m C ore D ensity = 64 kg/m (kN/m) (kN/m)

o o C ore W W Indentation 10 C ore Indentation 10

Peak Load, Load, Peak S kin Y ield

Peak Load, Load, Peak S kin Y ield or W rinkling

0.001 0.010 0.001 0.010 F ace thickness / span, t/ L F ace thickness / span, t/ L

(a) ρc /ρs = 0.160 (b) ρc /ρs = 0.080

1 0 3 C ore D ensity = 48 kg/m (kN/m) (kN/m )

10 o W W C ore Indentation

C ore Peak Load, Load, Peak Peak Load, Load, Peak Skin Indentation W rinkling Skin 3 W rinkling C ore Density = 29 kg/m 1 0 .00 1 0.010 0.00 1 0 .010 F ace th ickn ess / sp an , t/ L F ace th ick n ess / span , t/L

Figure Comparison with the exp erimental results from Chapter Lines show the pre

dictions of HOSBT failure analysis and the symb ols represent the exp erimental

data cf Fig Solid lines and symb ols corresp ond to the longitudinal rib

b on direction while dashed lines and hollow symb ols corresp ond to the trans

verse ribb on direction

Chapter Failure Maps using HOSBT

Optimisation Carp et Plots

When designing a loadb earing structure the most imp ortant functional requirements

are stiness strength and weight In general these are attributes that we want to

maximise or minimise Minimum weight design is considered by Triantallou and

Gibson for foam core sandwich b eams and extended in Gibsons b o ok In b oth

cases indentation is not examined as a p ossible core failure mo de only core shear

failure is considered They conclude that in minimum weight design of a foam core

sandwich beam of a given strength the top skin and core must fail simultaneously

In this work we use the failure maps to optimise the design of honeycomb sandwich

b eams with a graphical metho dology Here it is assumed that the failure in the core is

by indentation and is a result of the simultaneous eect of core shear and outofplane

compression Fig

Fig shows a useful practical form of a failure map In this section we consider

a typical case study seeking an optimum design for a sandwich b eam which is loaded

under p oint b ending and for which the material prop erties and skin thickness are

given The ratio cL and the relative core density are variables to b e optimised

c s

Hence it is most useful to present failure maps in terms of cL and Fig

c s

presents the data for Fig in this form showing contours of equal failure line load

By adding to this map contours of equal stiness and mass carp et plots are pro duced

which can be used to nd the optimum design conguration Such a carp et plot is

shown in Fig where for clarity the mass contours and only one strength and one

stiness contour are shown The full set of stiness contours are plotted in Fig

Two optimisation cases are considered strength limited design or stiness limited

design These constraints are represented as hatched contours in Fig In addition a

geometric constraint is included In this case we supp ose that the ratio of core thickness

to span cannot exceed Where a minimum strength constraint is applied the

potential choices are restricted to the upp er right area of the map By considering

the mass contours it is clear that the design which minimises the mass lies at p oint

A on the intersection of the skin wrinkling core indentation region on the strength

contour However if other considerations imp ose a value of cL less than

the value of cL which corresp onds to p oint A this conclusion would no longer hold

It may also b e desirable to cho ose a panel which fails by indentation b ecause this is a

safer failure mo de than the alternatives For the material combination considered the

optimum design p ointliesonthe b order between core indentation and skin failure for

a range of strength constraints This accords with the relevant conclusion of ref

Chapter Failure Maps using HOSBT

0 10 strength contours

skin yielding s

ρ / c

increasing -1 10 strength

Relative density, Relative density, skin wrinkling

core indentation

-2 10 -2 -1 10 10

Core thickness / span, c/L

Figure Strength contour plot corresp onding to Fig

0 10 stiffness contours skin yielding s

ρ / c

increasing -1 stiffness 10

Relative density, Relative density, skin wrinkling core indentation

-2 10 -2 -1 10 10

Core thickness / span, c/L

Figure Stiness contour plot

Chapter Failure Maps using HOSBT

and that the minimum design is such that the skin and core fail at the same p eak

load

The ab ove illustration assumes that stiness criteria are always satised Consider

instead a stiness limited design with a stiness constraint shown by the hatched

contour in Fig Now the contours of mass suggest that the beam with minimun

mass would b e that with the minimum available relative density of the core In practice

some other constraint for example a geometric constraint a strength constraint or a

manufacturing constraint will limit the extenttowhich the core density can b e reduced

Indeed this stinesslimited case study illustrates the need to mo del strength of these

panels as this will b e necessary to determine the optimum design

0 10 t

skin yield s

ρ / c

ρ strength constraint stiffness constraint stiffness decreasing geometric constrain mass

-1 10

A Relative density, core indentation skin wrinkling

mass contours -2 10 -2 -1 10 10

Core thickness / span, c/L

Figure Optimisation carp et plot for tL with and cL as design

c s parameters

Chapter Failure Maps using HOSBT

Concluding Remarks

The highorder sandwich beam theory can be used not only to predict indentation

failure loads but also the failure mo des of skin failure enabling the construction of

failure mo de maps This approach oers the advantage of using only one computa

tional to ol to calculate the failure loads for every failure mo de The predictions follow

satisfactorily the exp erimental data However the disadvantages of using the HOSBT

for failure maps are overprediction of the failure loads for skin wrinkling and inability

to predict the dierentbehaviour b etween the two honeycomb ribb on directions when

the top skin fails either by yield or wrinkling

Finally it is shown that failure maps can help with the preliminary design of sandwich

beams under bending by sup erimp osing contours of mass stiness and strength

Chapter

Conclusions and Future Work

This thesis addresses theoretical and exp erimental asp ects of the design of honeycomb

sandwich panels The research fo cuses on sandwich beams made of crossply GFRP

laminate skins with a Nomexphenolic resin honeycomb core a widely used combina

tion for aircraft o oring panels and freight loading pallets Indentation resistance of

such sandwich panels with transversely exible cores is an imp ortant factor in their

design An accurate mo del of the elastic deformation of such sandwich b eams under

three p oint b ending has been derived In particular this mo del includes the lo cal

behaviour under the central load allowing indentation failure to b e examined

Conclusions

Firstly the b ehaviour of sandwich b eams of diering lengths and core densities is con

sidered to examine the imp ortance of the p ossible failure mo des including failure by

skin yielding skin wrinkling intracell buckling core shear and indentation At this

initial treatmentindentation is treated in an empirical way using measurements of the

b earing area at the load p oints Previous research on honeycombmechanics and simple

b eam mo dels has b een combined to derive failure mo de maps for p oint b ending with

axes as the core relative density and the ratio of the skin thickness to span length

These maps are based on those of Triantallou and Gibson who fo cused on sand

wich b eams with ductile aluminium skin and isotropic foam cores This thesis app ears

to b e the rst attempt to construct maps for sandwich b eams with laminate skins and

honeycomb core Since commercial panels are generally provided with standard skin

thicknesses but with diering core thickness and density alternative maps with these

two core variables are presented since these will b e more useful for a b eam designer or

manufacturer Although the maps are generated for threep oint b ending the metho d

can straightforwardly be applied to other loading geometries for example fourp oint

Chapter Conclusions and Future Work

b ending

Exp erimental results for failure under three p oint b ending are summarised as fol

lows A transition from face yielding to intracell buckling for long span b eams was

observed for a honeycomb cell size ab ove a critical value The measured peak loads

were dep endent on the direction of the honeycomb ribb on This dierence is due to

the shear anisotropy of the honeycomb illustrating the imp ortant role that core shear

plays in the b ending b ehaviour of these sandwich b eams Exp erimental results veried

satisfactorily the predicted failure loads The b oundary between skin and core failure

on the failure mo de maps was also predicted with go o d accuracy However failure near

the load p oints due either to core shear or core indentation was not mo delled well

A highorder sandwich b eam theory HOSBT was implemented to provide a b etter

deformation mo del of lo calised eects under concentrated loads and so to pro duce an

improved indentation failure analysis for honeycomb sandwich b eams The b enets

of using the highorder sandwich beam theory to analyse the behaviour of sandwich

beams under indentation are presented and veried exp erimentally by measurements

of the core deformation close to an indenter HOSBT is used to extract a characteristic

spreading length This is a prop erty of a sandwich b eam dep ending mainly on the

skins exural stiness and characterising the susceptibility of the sandwich b eam to

indentation loads Small values of corresp ond to sandwich beam with very exible

skins which are transparent to the external loads Large values of indicate rigid skins

that restrict the transmission of lo calised loads to the core HOSBT is further used to

give an insightinto the contact mechanics for a b eam loaded by a cylindrical indenter

The wayinwhich the contact pressure is transmitted through the core is examined A

case study shows the fact that sandwich b eams used as standard in industry have skins

which are rigid enough to spread the external loads This spreading eect allows to

predict the indentation b ehaviour of a sandwich b eam with rigid skins without having

to mo del accurately the contact b etween the indenter and the top skin The assumption

of a uniform distribution over a roughly estimated width which dep ends on the size of

the indenter and the use of the highorder sandwich b eam theory can provide reliable

predictions of the stress eld in the core

The most imp ortant conclusion from this analysis is that the maximum normal

stresses in the top skincore interface which are mainly resp onsible for indentation

failure cannot be predicted in a straightforward way for sandwich b eams with rigid

skins even if the contact width is known For sandwich b eams with very exible skins

the approximation of dividing the total line load W by can give us reliable estimation

of the failure stresses in the skincore interface However for commercially applied

Chapter Conclusions and Future Work

sandwich congurations the skin is rather rigid and the exible skin approximation

cannot be used

A systematic approach has b een develop ed to determine the failure load of sandwich

honeycomb structures under indentation loading Biaxial tests of Nomex honeycombs

using an Arcan rig show that a linear dep endence on pure compression and shear proves

to be a go o d approximation for the failure envelop e of these honeycombs Using this

failure envelop e and the core stress eld determined by HOSBT a mixed failure crite

rion has b een intro duced to predict the indentation failure caused by the simultaneous

action of outofplane compressive and shear stresses exerted in the vicinity of a lo

calised load This criterion can predict failure that ranges from pure core crushing to

pure core shear

Short beam bending tests validated the theoretical predictions of high order b eam

mo del and showed that the mixed failure criterion oers a signicant improvement

in predicting indentation strength as compared with mo dels which do not include

combined loading This new approach is particularly needed in high core densities

which induce more severe stress elds in the core It also predicts observed dierences

in behaviour for longitudinal and transverse ribb on directions Video captures of the

deformed side cross section of the tested sandwich b eams illustrate the involvementof

core shear in the elastic behaviour of sandwich b eams with low density cores Also

they sho wed the dierent p ostfailure damage extent between sandwich b eams with

dierent core densities the higher the core density the less damage propagates into the

core and skins can suer eventually macrobuckling failure

The highorder sandwich b eam theory can also b e used to predict the failure mo des

of skin failure enabling the construction of failure mo de maps This approach oers the

advantage of using only one computational to ol to calculate the failure loads for every

failure mo de The predictions follow satisfactorily the exp erimental data However

the disadvantages of using the HOSBT for failure maps are overprediction of the failure

loads for skin wrinkling and inability to predict the dierent behaviour between the

two honeycomb ribb on directions when the top skin fails either by yield or wrinkling

Finally it is shown that failure maps can help with the preliminary design of sandwich

beams under bending by sup erimp osing contours of mass stiness and strength

Future Work

With this study a framework of failure analysis of Nomex honeycomb panels has b een

established and provides the basis for further investigation in the following areas

Chapter Conclusions and Future Work

Bending tests should be done with sandwich b eams with dierent skin thicknesses

and core densities than those tested in this research work in order to validate further

the applicability of the prop osed metho dologies for failure analysis and for failure

map construction The failure envelop e for biaxial loading of Nomex honeycombs

determined in section should also b e investigated for other core materials

In section a parametric study is conducted to dene the dep endence of the spread

ing length on the material and geometric prop erties of a sandwich beam The inu

ence of each parameter has b een determined qualitatively Further work on this area

with resp ect to Lm instead of could attempt to derive a single normalised curve C

all the curves in Fig that accurately captures the indentation resistance b ehaviour

of a wide range of sandwich b eam designs This approachwould lead to simple formu

lae relating to the material and geometric prop erties of a sandwichbeam Miller

adopts such an approach to mo del the indentation b ehaviour of foamed metals

A nite element analysis of a mo del see Fig of a rigid indenter applying loads

to a simply supp orted b eam would b e useful to determine the contact pressure b etween

the indenter and the top skin and the corresp onding normal stresses transmitted to

the core These stresses can b e compared with those calculated by the highorder b eam

theory mo del in section

In Chapter we showed the ability of the spreading length parameter to char

acterise the static indentation resistance of sandwich beams It should be p ossible

to compare values with exp erimental results on dynamic indentation resistance and

investigate the applicabilityof as a quality factor in sandwich panel manufacturing

For this purp ose the software co de used in section to calculate could b e provided

to industry for assessment of its practical usefulness

Finally the metho dology for failure analysis prop osed in this study should be used

to explore the b enets of unsymmetrical sandwich b eams with b ottom skins thinner

than the top ones This option would exploit rstly the fact that comp osite laminates

have approximately tensile strength twice as much their compressive and secondly the

higher indentation resistance that a thicker top skin can provide

Chapter Conclusions and Future Work

ABAQUS

2 DISPLACEMENT MAGNIFICATION FACTOR = 6.77 ORIGINAL MESH DISPLACED MESH RESTART FILE = indent1 STEP 1 INCREMENT 1 3 1 TIME COMPLETED IN THIS STEP 2.220E-16 TOTAL ACCUMULATED TIME .000E+00

ABAQUS VERSION: 5.6-1 DATE: 17-APR-98 TIME: 03:53:10

Figure Vertical displacements calculated by Abaqus nite element analysis

App endix A

Exp erimental Results

App endix A Exp erimental Results

Figure A For mm diameter roller midspan b ottom skin deection midspan core com

pression and line load variation curves with resp ect to midspan top skin deec

tion Lines legend longitudinal and transverse honeycomb ribb on direction

App endix A Exp erimental Results

Figure A For mm diameter roller midspan b ottom skin deection midspan core

compression and line load variation curves with resp ect to midspan top skin

deection Lines legend longitudinal and transverse honeycomb ribb on direction

App endix A Exp erimental Results

Figure A For mm diameter roller midspan b ottom skin deection midspan core

compression and line load variation curves with resp ect to midspan top skin

deection Lines legend longitudinal and transverse honeycomb ribb on direction

App endix A Exp erimental Results

Figure A Video snapshots during loading with a mm diameter roller

App endix A Exp erimental Results

Figure A Video snapshots during loading with a mm diameter roller

App endix A Exp erimental Results

Figure A Video snapshots during loading with a mm diameter roller

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