a comparison of the predictive capabilities of current failure theories for composite laminates...

A comparison of the predictive capabilities of current failuretheories for composite laminates: additional contributions

A.S. Kaddoura,*, M.J. Hintonb, P.D. Sodenc

aQinetiQ FST, Farnborough, Hampshire, GV14 0LX UKbQinetiQ Fort Halstead, Kent, TN14 7BP UK

cMechanical Aerospace and Manufacturing Engineering Department, UMIST, UK

Received 1 March 2003; accepted 1 May 2003

Abstract

Following the publication of an article by Soden, Hinton and Kaddour (Compos Sci Technol, V58, pp. 1225–54, 1998), whichcompared the predictions of 14 internationally recognised failure theories for fibre reinforced polymer composite laminates, thepresent paper extends that comparative study to include five more theories, supplied by their originators. Evaluation of the pre-

dictive capabilities of the additional theories was carried out in an identical manner to the original study. The same test cases wereutilised, covering a wide range of lay-ups, materials and in-plane loading conditions. The results (initial and final failure envelopesand representative stress–strain curves) have been superimposed to show similarities and differences between the predictions of the

19 theories. Final failure predictions for the additional theories fall within the existing range obtained from the original 14 theoriesbut the initial failure predictions for the additional theories have widened the previously observed range. Comments are providedon the possible reasons for the increased spread.

# 2003 QinetiQ Ltd. Published by Elsevier Ltd. All rights reserved.

Keywords:Multiaxial failure envelopes; Failure criteria; B. Stress–strain curves

1. Introduction

The present paper expands the number of theoriesconsidered within the ‘World-Wide Failure Exercise’(WWFE), Ref. [1]. In the exercise, the originators ofvarious failure theories used their own theory to predictthe performance of specified carbon and glass fibrereinforced epoxy laminates subjected to a range ofbiaxial loads, using the same material properties, lami-nate arrangements and loading conditions, defined bythe organisers (the authors of this paper), Ref. [2]. Eachcontributor has described their theory in some detail ina separate paper, Refs. [3–14] and analysed 14 test casescontaining biaxial failure envelopes and stress–straincurves. A total of 11 groups took part in the exerciseand they provided 14 different theories. Their predic-tions were made ‘blindly’, i.e. without prior knowledgeof the experimental data. The ‘blind’ predictions of the

14 theories were compared systematically with eachother in Ref. [15] in order to identify some of the majorsimilarities and differences between them.Five more theories, Refs. [16–20], have emerged since

the publication of the first comparative study, Ref. [15].These additional theories were applied by their origina-tors, whilst maintaining the ‘blind’ approach to solvethe 14 Test Cases. This paper utilises the same metho-dology as applied in Ref. [15] to compare the predic-tions of the five theories with each other and, whereappropriate, with those of the 14 previous theories. Atthis stage of the exercise, no attempt will be made todraw conclusions about which theory is the best, theemphasis is on clarifying the similarities and the differ-ences between their predictive capabilities.

2. Description of the failure theories employed in the

exercise

The current work is an extension of the methodologyreported in Ref. [15]. An identical format is used todescribe the additional theories and, to aid the reader,

0266-3538/$ - see front matter # 2003 QinetiQ Ltd. Published by Elsevier Ltd. All rights reserved.

doi:10.1016/S0266-3538(03)00226-4

Composites Science and Technology 64 (2004) 449–476

www.elsevier.com/locate/compscitech

* Corresponding author. Tel.: +44-1252-395978; fax: +44-1252-

395077.

E-mail addresses: [email protected] (A.S. Kaddour),

[email protected] (M.J. Hinton), [email protected]

(P.D. Soden).

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http://www.elsevier.com/locate/compscitech/a4.3d

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details of both original and additional theories arepresented here to provide a comprehensive view of all 19theories examined in the exercise.

2.1. Identification of the theories

Table 1 lists the ‘original’ and ‘additional’ participantsand indicates the approaches they employed. The newcontributions are the last five entries in Table 1 and theseare Cuntze, Bogetti, Mayes, Huang and Hart-Smith (3).For identification purposes each of the theories is referredto by a single name, (see the last column in Table 1). Incertain instances, the named author may not be the origin-ator of the theory as for example the case of the Max-imum Strain theory presented by Hart-Smith, Ref. [7].It’s worth mentioning here that although the Hart-

Smith (3) theory has already been published in Part B ofthe WWFE, Ref. [16], and was compared with test datain Ref. [22], his prediction has not been compared withthe ‘blind’ prediction of other theories. Therefore, andfor the sake of consistency, it is felt appropriate tocompare his published curves with those of the addi-tional theories.

2.2. Characteristics of the theories

Each of the theories can be characterised by a numberof key features. Table 2 lists some of the features,including the following:

2.2.1. Method of analysisAll of the contributors utilised classical laminate the-

ory as the principal ‘calculation engine’ on which tobase their theory. Sun, Mayes and Chamis also usedfinite element codes.

2.2.2. Type of analysisThis refers to whether linear or nonlinear material

properties were considered. Chamis, Eckold, Hart-Smith, McCartney, Tsai, Sun (L) and Zinoviev usedlinear analysis of material properties whereas the rest(Edge, Rotem, Puck, Wolfe, Sun (NL), Bogetti, Cuntze,Huang and Mayes) used nonlinear analysis.

2.2.3. Thermal stressesA significant portion of the participants (Eckold,

Hart-Smith, Rotem, Wolfe, Zinoviev and Bogetti) didnot include residual thermal stresses while the othersdid, but not all in the same way, see Table 2. Forexample Huang was the only contributor to provide ananalysis that considers residual stresses at the micro-mechanics (ie resin and fibre) level.

2.2.4. Modes of failureAlmost all of the theories were able to discriminate

between two or more modes of failure. Various modes

of failure and failure criteria were postulated in thetheories, see Table 3. The modes of failure basicallyrange from fibre failure (tension, compression or shear)to matrix failure (due to transverse tension, transversecompression, shear or a combination of these three). Inmost cases the properties used to predict the modes offailure are lamina properties rather than constituent(fibre or matrix) properties. However, Mayes, Huangand Chamis used the constituent fibre and matrixproperties (provided to all participants) for determiningfailure.

2.2.5. MicromechanicsOverall, approximately half of the theories relied on

micromechanics in their formulation. Chamis, Hart-Smith (1), Puck, Rotem, Tsai and the new participantsCuntze, Mayes and Huang, explicitly required certainproperties of the constituents.

� The use of micro-mechanics by the original par-ticipants (Chamis, Hart-Smith, Puck, Rotem,Tsai) was described in Ref. [15].

� Cuntze’s theory, like Puck’s, requires some of theproperties of the fibres.

� Huang’s theory, like Chamis’, relies on micro-mechanics for determining the unidirectional(UD) lamina properties. Huang used the non-linear shear stress/strain curves of the matrix todetermine the nonlinear behaviour of the lamina.Using a ‘bridging’ model, Huang adjusted theconstituent properties to give lamina propertiesclose to those provided in the exercise

� Mayes’s theory is also micro-mechanics based.

2.2.6. Post-initial failure degradation modelsThe entry in the seventh column of Table 2 specifies

whether or not the theory used any degradation modelto account for post initial failure behaviour. Multi-directional laminates subjected to uniaxial or biaxialstresses may still be capable of carrying a load after firstply failure or initial failure has occurred. Modelling postfailure behaviour of a laminate requires that certainassumptions be made regarding the properties of thedegraded lamina. Table 4 summarises the different postinitial failure models adopted.The post failure methods employed do differ and, for

sake of simplicity, they can be classified into three maingroups:

(i) Model employing no post failure analysis. These
include Eckold and Hart-Smith.
(ii) Models employing sudden reduction in the
properties of the failed lamina. These are utilisedby Tsai, Wolfe, Sun (L), Chamis, Bogetti, Huangand Mayes.
450 A.S. Kaddour et al. / Composites Science and Technology 64 (2004) 449–476

Table 1

A summary of participants and approaches represented in the exercise

Contributor(s) Organisation Approach represented Theory designation

Chamis C C, P K Gotsis,

L Minnetyan

NASA Lewis, Cleveland, USA. -ICAN (micro-mechanic ) -Chamis (1)

-CODSTRAN -Chamis (2)

Hart-Smith L J Boeing, USA. Generalised Tresca theo Hart-Smith(1)

Hart-Smith L J Boeing, USA. Maximum Strain Theor Hart-Smith(2)

Eckold G C AEA Technology, UK British Standard pressur l design codes Eckold

Edge E C British Aerospace, Military Aircraft Division, Warton, UK. British Aerospace, In-ho sign method Edge

McCartney L N National Physical Laboratory, London, UK. Physically based ‘Dama hanics’ McCartney

Puck A and H Schurmann Technische Hochchule, Darmstadt, Germany. Physically based 3-D ph nological models Puck

Wolfe W E and T S Butalia Department of Civil Engineering, Ohio State University, Ohio, USA. Maximum strain energy d, due to Sandhu Wolfe

Sun C T and J X Tao Purdue University School of Aeronautics & Astronautics, Indiana, USA. Linear and non-linear a

(non-linear is FE based)

Sun(L)

Sun(NL)

Zinoviev P, S V Grigoriev,

O V Labedeva and L R Tairova

Institute of Composite Technologies, Orevo, Moskovkaya, Russia. Development of Maxim ss theory Zinoviev

Tsai S W and K-S Liu Aeronautics and Astronautics Department, Stanford University,

California, USA.

Interactive progressive q ic failure criterion Tsai

Rotem A Faculty of Mechanical Engineering, Technion-Israel Institute of

Technology, Haifa, Israel.

Interactive matrix and fi lure theory Rotem

Cuntze R and A Freund MAN Technologies, Germany. Failure mode concept (F Cuntze

Bogetti T, C Hoppel, V Harik,

J Newill and B Burns

U.S. Army Research Laboratory, AMSRL-WM-MB, Aberdeen

Proving Ground, MD 21005.

3-D Maximum strain Bogetti

Mayes S J and A C Hansen US Naval Surface Warfare Center, West Bethesda, MD,

and Alfred University.

Multi-continuum micro nics theory Mayes

Z-M Huang Department of Engineering Mechanics, Tongji University,

Shanghai, China.

Bridging model, micro-m ics Huang

Hart-Smith L J Boeing, USA. Ten-Per-Cent rule Hart-Smith(3)

A.S.Kaddouretal./Composites

Scien

ceandTechnology64(2004)449–476

451

s based

ry

y

e vesse

use de

ge Mec

enome

metho

nalysis

um stre

uadrat

bre fai

MC)

-mecha

echan

452

Table 2

A summary of key features of the theories used by various contributors

Contributor Method Type of analysis Thermal stresses Failure Modes Micro- mechanics Degradation model Failure criter Computer program used

Chamis CLT+FE Linear (a) Yes(b) Yes Yes Micromechan ed ICAN and CODSTRAN

Eckold CLT Linear No No No No BS4994 None

Edge CLT Nonlinear Yes Yes No Yes Grant-Sander Modified ESDU package

Hart-Smith CLT Linear (c) No Yes Yes No Maximum str ories,

Generalised T riteria

and 10% rule

None

McCartney CLT Linear Yes Yes No Yes Fracture mec Program developed at NPL, UK

Puck CLT Non linear Yes (d) Yes Yes Yes Puck’s theory FRACUAN developed in

Kessel, Germany

Rotem CLT Nonlinear Yes Yes Yes Yes Rotem theory In-house program

Sun CLT(e) Linear Yes Yes No Yes Rotem–Hash ry In-house program

Sun CLT+FE (f) Non-linear Yes No No Yes Plasticity mo ed

on Hill’s yield

ABAQUS program

Tsai CLT Linear Yes(g) Yes Yes Yes Tsai–Wu qua heory Mic-Mac

Wolfe CLT Nonlinear No Yes No Yes Sandhu’s stra gy model In-house program

Zinoviev CLT Linear No Yes No Yes Maximum str ory STRAN software

Bogetti CLT Nonlinear No Yes No Yes Maximum str In-house program

Mayes CLT+FE Nonlinear (a) Yes Yes Yes Multi-continu ory FE-based

Cuntze CLT Nonlinear Yes(d) Yes Yes Yes Failure mode t (FMC) In-house program

Huang CLT Nonlinear Yes(a)(h) Yes Yes Yes Bridging mod In-house program

(a) Not in all cases.

(b) The theory identifies failure modes but Chamis chose not to present them for some of the cases he analysed.

(c) Secant properties rather than initial properties are occasionally used in the analysis.

(d) Only part of the thermal residual stresses are considered.

(e) Used to generate the failure envelopes and stress–strain curves.

(f) The finite element (FE) analysis was used only to generate the stress–strain curves.

(g) Tsai introduced a certain amount of moisture to compensate for the thermal stresses.

(h) Huang attempted to consider the micro thermal stresses generated in the constituents.


Scien


ion

ics bas

s

ain the

resca c

.

hanics

in theo

del bas

dratic t

in ener

ess the

ain

um the

concep

el

Table 3

Modes of failure and failure criteria used by the participants

Mode of failure Failure criterion eory

Fibre failure �K1fI21f þ K4fI4f ¼ 1 where �K1f ¼

1�S 2

11f

, K4f ¼1

S 212f

and I1f and I4f are fibre stress invariants.

yes

Fibre failure �eq ¼ �ð1Þ, when �ð2Þ 4 0 and ½ð�ð1ÞÞqþ ð�ð2ÞÞ

q�1q, when �ð2Þ > 0, 1 < q41 ang

Fibre failure in tension1

"1T"1 þ

�f12Ef1

m�f �2

� �¼ 1 k

Longitudinal tension failure s1=XT oviev, Rotem, Sun, Edge,

rt-Smith (3) and Cuntze

Longitudinal tensile failure e1=e1T (and Eckold’s e1=0.004) rt-Smith (2), Eckold and Bogetti

Fibre tension/compression and matrix

tension and compression and shear

�1XTXC

� �2

þ�2

YTYC

� �2

þ1

XT

1

XC

� ��1 þ

1

YT

1

YC

� ��2 þ

2F12�1�2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXTXCYTYC

p

� �þ

�12S12

� �2

¼ 1 i

Fibre failure (in tension and compression)

Ð"1�1d"1Ð

"u1�1d"1

" #m1, Pi¼1;2;6

Ð"i�id"iÐ

"ui�id"i

" #mi5 0:1 lfe

Fibre failure in compressive1

"1C"1 þ

�f12Ef1

m�f �2

� � ¼ 1 1021ð Þ

2 k

Longitudinal compressive failure s1=XC (and Eckold s1=XT) oviev, Edge, Rotem, Sun, Hart-Smith (1)(3),

ntze and Eckold

Longitudinal compressive failure e1=e1C (and Eckold e1=0.004) rt-Smith (2), Eckold and Bogetti

Shear of fibres Tresca type criterion rt-Smith (1)

Transverse tensile failure s2=YT oviev, Edge, Eckold and Sun

(continued on next page)


Scien


453

Th

Ma

Hu

Puc

Zin

Ha

Ha

Tsa

Wo

Puc

Zin

Cu

Ha

Ha

Zin

454

Table 3 (continued)

Mode of failure Failure criterion ory

Transverse tensile cracking � þ k� t >

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4p

1

EAð2pÞ

1

EAðpÞ

vuuut þ ��0 where 2g is

fracture energy, see also Ref [16]

Cartney

Transversal tensile failure

(IFF1)�2

Eff ?� Rt?¼ 1 ntze

Transverse tensile failure e2=e2T ( and Eckold e2=0.001) rt-Smith (2), Eckold and Bogetti

Transverse tension YT=XT/10 or XC/10 whichever is the greatest rt-Smith (3)

Transverse compression YC=XT/10 or XC/10 whichever is the greatest rt-Smith (3)

Transverse compressive failure s2=YC oviev, Edge, Eckold, Sun and Huang

Transverse compressive failure e2=e2C (and Eckold e2=0.001) rt-Smith (2), Eckold and Bogetti

Inter-fibre failure Mode A

(for transverse tension)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�21S21

� �2

þ 1 p þð Þ

?jj

YTS21

� �2 �2YT

� �2s

þ p þð Þ

?jj

�2S21

¼ 1�1�1D

k

Inter-fibre failure Mode B

(for moderate transverse

compression)

1

S21

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�221 þ p

ð Þ

?jj�2

� �2rþ p

ð Þ

?jj�2

!¼ 1

�1�1D

k

Inter-fibre failure Mode C

(for large transverse compression)

�21

2 1þ pð Þ

??

� �S21

0@

1A

2

þ�2YC

� �224

35 YC

�2ð Þ¼ 1

�1�1D

k

Wedge failure (IFF3)b�? 1� �

�2 þ �3ð Þ

Eff ?� Rc?þb�? �2 �3ð Þ

2þb�?jj �

221

Eff ?� Rc?� �2 ¼ 1 ntze

Inter-fibre failure (IFF2)�321 þ b?jj2�2�

221

Eff ?jj R?jj

� �3 ¼ 1 ntze

In-plane shear failure S12=XT/20 or XC/20 whichever is the greatest rt-Smith (3)

In-plane shear failure t12=S12 oviev, Edge, Hart-Smith (1),

rt-Smith (2), and Sun

(continued on next page)


Scien


The

Mc

Cu

Ha

Ha

Ha

Zin

Ha

Puc

Puc

Puc

Cu

Cu

Ha

Zin

Ha

Table 3 (continued)

Mode of failure Failure criterion ory

Combined transverse tension

and shear�2

YT

� �2

þ�12S12

� �2

¼ 1e

Combined longitudinal

compression and shear

�1

H1c

� �þ

�12S12

¼ 1 e

Delamination t12�te>or e

Matrix failure �K3mI3m þ K4mI4m ¼ 1 where �K3m ¼1

�S 222m þ�22S 2

33m

, K4m ¼1

S 212m

and I3m and I4m are matrix stress invariants.

yes

Matrix failure�2

Y

� �2

þ�12S12

� �2

¼ 1 (Chamis obtains Y and S12 from micromechaincs) amis and Sun

Matrix failureEm "1ð Þ"1

Ym

� �2

þ�2

Y

� �2

þ�12S12

� �2

¼ 1 tem

Matrix failure

Ð"1�1d"1Ð

"u1�1d"1

" #m1, Pi¼1;2;6

Ð"i�id"iÐ

"ui�id"i

" #mi< 0:1 lfe

Matrix failure �eq ¼ � 1ð Þ; when � 2ð Þ 4 0; � 1ð Þ� �q

þ � 2ð Þ� �q� �1

q;when � 2ð Þ > 0; 1 < q41

nang


Scien


455

The

Edg

Edg

Edg

Ma

Ch

Ro

Wo

Hu

(iii) Models employing a gradual drop in the prop-
erties of the failed lamina. These were utilised byCuntze, Puck, Edge, Rotem, Zinoviev, McCart-ney, and Sun (NL).
Full details of all the models are given in the con-tributors papers, Refs. [3–14,16–20].

2.2.7. Failure criterionColumn 8 of Table 2 identifies the origin, nature or

the name of the failure theory used. The governingequations that constitute the various failure criteria aredescribed, by the participants, in their own papers (seealso Table 3 for a brief summary). It should be notedthat the exercise features two variants of the Maximum

Strain Theory. Hart-Smith, Ref. [7], provided his 2-dimensional interpretation as used in the aircraft indus-try, without incorporating any progressive failureanalysis. The additional paper by Bogetti, Ref. [17],applied the theory in its original 3-dimensional formtogether with a progressive failure model.

2.2.8. Title of computer program usedThe entry in the last column of Table 2 lists the name,

if any, of the computer program used by the partici-pants. The participants have either coupled their analy-sis into a commercial code (Chamis, Mayes, Sun(NL),or written their own (Tsai, Zinoviev, Puck, McCartney,Rotem,Edge, Sun(L), Wolfe, Cuntze, Huang andBogetti) or used simple calculations (Hart-Smith-1,Hart-Smith-2, Hart-Smith-3 and Eckold).

Table 4

Summary of the post initial failure degradation models used in the theories

Name
Failure mode Properties degraded
Eckold
No post failure
Hart-Smith
No post failure
Rotem
After final matrix failure E2=0.0, G12=0.0, E1=E10 exp(-k e1),
k is a large constant

McCartney
Lamina cracking Detailed mathematical analysis for reducing
stiffness

Puck
Cracking under tension Mode (A) E2=Z E20, G12=Z G12
0 , n12=Z n120 where Z is a

parameter which varies with stress

Cracking under compression

Modes (B) and (C)

G12=Z1 G120 , n12=n120

Z1 is smaller than Z
Chamis Matrix failure Em is replaced by a negligible value and E2,
G12, n12 and E1 are computed from micro-mechanics

Edge
Matrix failure E2=b1 E20, G12=b2 G12
0 , n12=b3 n120

Where b1, b2 and b3 are empirical parameters

that decrease with increasing strain

Wolfe
Matrix failure E2=0.0, G12=0.0, n12=0.0
Sun (linear)
Shear matrix failure E2=0.0, G12=0.0
Transverse matrix failure
E2=0.0
Sun (NL)
Matrix shear failure E2=E20 exp(-aE l) and G12=G12
0 exp(-aG l), aE and aGare constants, l is normalised crack density

Transverse matrix failure
E2=E20 exp(-aE l)
Tsai
Matrix failure (e2>0) Em=0.15 Em0,n12=0.15 n120 where E2 and G12 are computed from
micromechanics

Matrix failure (e240)
E2=0.01 E20, G12=0.01 G12
0 , n12=0.01n120 , E1=0.01 E10

Zinoviev
Open cracks s2 >0 For |g12|< |g*12| (a) when e2 <e*2: E2=c2 E20, G12=c3 G12
0

where c is a function of strain (b) when e2=e*2 : E2=0.0,

G12=c3 G120 For |g12|=|g*12| (a) when e2 <e*2: E2=c2

E20, G12=0.0 (b) when �e2>0: E2=0.0, G12=0.0.

Closed cracks s2 <0
For �e2<0: (a) when |g12|< |g*12|: G12=c3 G120
(b) when �|g12|>0 : G12=0.0

Bogetti
Shear failure G12=0.0
Bogetti
Transverse strain failure E2=0.0
Cuntze
IFF1, IFF2 and IFF3 Curves describing ‘softening’ behaviour
Huang
Matrix or fibre Failure Em=Ef=0.0 or E2=0.0, G12=0.0, n12=0.0, E1=0.0
Mayes
Matrix failure Em=0.01 Em0 where E2 and G12 are computed
from micromechanics


2.3. Breadth of cases analysed by each participant

The participants were set 14 Test Cases for analysisand these are summarised in Table 5. The composite

systems, laminate configurations and load combinationshave been described in detail in Ref. [2]. Table 6 showsthe Test Cases attempted by each contributor. Four ofthe additional participants (Bogetti, Cuntze, Huang andMayes) were able to analyse all the 14 Test Cases.

Table 5

Details of the laminates and loading (Test) cases, Ref [15]

Laminate lay-up
Material Loading case Description (a wide range of biaxial stress ratios
unless other wise indicated)

0�
E-glass/LY556/HT907/DY063 1 Biaxial failure stress envelope under
transverse and shear loading (sy versus txy)
T300/BSL914C 2 Biaxial failure stress envelope under
longitudinal and shear loading (sx versus txy)
E-glass/MY750/HY917/DY063 3 Biaxial failure stress envelope under longitudinal
and transverse loading (sy versus sx)

(90�/�30�/90�)
E-glass/LY556/HT907/DY063 4 Biaxial failure stress envelope (sy versus sx)
5
Biaxial failure stress envelope (sx versus txy) (0�/�45�/90�) AS4/3501-6 6 Biaxial failure stress envelope (sy versus sx)
7
Stress–strain curves under uniaxial tensile
loading in y direction (sy : sx=1 : 0)

8
Stress–strain curves for sy : sx=2 : 1
�55�
E-glass/MY750/HY917/DY063 9 Biaxial failure stress envelope (sy versus sx)
10
Stress–strain curves under uniaxial tensile
loading for sy : sx=1 : 0

11
Stress–strain curves for sy : sx=2 : 1
(0�/90�)
E-glass/MY750/HY917/DY063 12 Stress–strain curve under uniaxial tensile
loading for sy : sx=0 : 1

�45�
E-glass/MY750/HY917/DY063 13 Stress–strain curves for sy : sx=1 : 1
14
Stress–strain curves for sy : sx=1 :-1
Table 6

Summary of the cases analysed by the participantsa

Theory
Loading cases analysed (see Table 2 for details of these cases) Remarks
1
2 3 4 5 6 7 8 9 10 11 12 13 14
McCartney
X X X X X X X X X X X X X X X X X X X X X X & & X X No final failure
Sun (NL)
X X X X X X X X X X X X & & X X & & & & &&
Hart-Smith (2)
&& && && X & X & X & XX XX X & X X X X X X X X X X No stress–strain
curves

Hart-Smith (1)
&& && && X & X & X & X X X X X & X X X X X X X X X X No stress–strain
curves

Eckold
&& X X && && X X X X X X X X && & X & X & X & X & X No carbon fibre
results

Hart-Smith (3)
&& && && X & X & X & X & X & X & X & X & X & X & X &
Chamis
&& && && && && & X && && && && && && && &&
Edge
&& && && && && && && && && && && && && &&
Puck
&& && && && && && && && && && && && && &&
Rotem
&& && && && && && && && && && && && && &&
Sun (L)
&& && && && && && && && && && && && && &&
Tsai
&& && && && && && && && && && && && && &&
Wolfe
&& && && && && && && && && && && && && &&
Zinoviev
&& && && && && && && && && && && && && &&
Cuntze
&& && && && && && && && && && && && && &&
Bogetti
&& && && && && && && && && && && && && &&
Huang
&& && && && && && && && && && && && && &&
Mayes
&& && && && && && && && && && && && && &&
a X, Case not analysed. , Final failure not reached; &, Initial failure predicted only; &, Final failure predicted only.

A.S. Kaddour et al. / Composites Science and Technology 64 (2004) 449–476 457

3. Comparing the predictions

In order to facilitate the task of comparing the pre-dictions, the following steps were taken, Ref. [15]:

� The 14 Test Cases can be broadly grouped intothree classes (a) biaxial failure of unidirectionallaminae (Test Cases 1, 2 and 3), (b) biaxial failureenvelopes of multidirectional laminates (Test

Table 7

Ratios of the highest:lowest predicted lamina strengths for selected stress ratios

No
Laminate studied Stress ratio Final failure prediction Highest: lowest ratio predictions
Highest
Lowest
1
E-glass/LY556
(Test Case 1)

sy:txy=-1.58:1
Edge, Hart-Smith (2), Zinoviev Eckold 3.22
2
sy:txy=1:2.06 Hart-Smith (2), Zinoviev, Eckold Tsai 1.54
3
sy:txy=0:1 Huang Eckold 3.6
4
T300/914C
(Test Case 2)

sx:txy=-12.5:1
Zinoviev, Sun, Hart-Smith (1), Hart-Smith (2) Edge 1.64
5
sx:txy=18.75:1 Edge, Zinoviev, Hart-Smith (1) (2), Sun Chamis 1.414
6
E-glass/MY750
(Test Case 3)

sx:sy=8.83:1
Edge, Zinoviev and Sun Eckold 3.6
7
sx:sy=-32:1 Eckold Wolfe 4.77
8
sx:sy=7.8:1 Hart-Smith (2), Bogetti Rotem 4.4
9
sx:sy=0:1 Hart-Smith (3) Eckold 3.2
10
sx:sy=-1:0 Eckold Wolfe 3.8
11
sx:sy=-3.15:-1 Bogetti Eckold 6.9
12
sx:sy=-14:-1 Tsai Eckold 2.72
Table 8

Summary of theoretical results showing the range of initial and final failure predictions

No.
Test Case Stress ratio Final failure prediction Initial failure prediction Largest final/initial prediction
Highest
Lowest Ratio Highest Lowest Ratio Name Ratio
1
4 sy :sx=1:3 Hart-Smith (1) Rotem 6.2 Huang Chamis (2) 13.3 Edge 34
2
sy :sx=1:-1 Hart-Smith (1) Chamis (2) 8.42 Huang Chamis (2) 4.7 Eckold 5.7
3
sy :sx=-1:-3 Eckold Zinoviev 3.1 Huang Bogetti 3.2 Eckold 3.74
4
sy :sx=1:1 Hart-Smith (2) Wolfe 7.0 Huang Chamis (2) 9.19 Chamis (2) 19
5
sy :sx=-1:-1 Edge Tsai 1.66 Huang Eckold 4.55 Eckold 3.66
6
sy :sx=1:0 Hart-Smith (1) Wolfe 2.42 Huang Chamis (2) 5.2 Edge 7.5
7
sy :sx=-4.26:1 Zinoviev Chamis (2) 2.11 Zinoviev Chamis (2) 7.46 Eckold 5
8
sx:txy=-2.35:1 Puck Chamis (2) 3.8 Huang Chamis (2) 4.5 Puck 1.99
9
5 sx:txy=1:1 Hart-Smith (1) Wolfe 5.21 Huang Chamis (2) 11.6 Edge 12.8
10
sx:txy=0:1 Puck Chamis (2) 4.71 Huang Chamis (2) 7.8 Puck 6
11
sx:txy=-1:0 Eckold Tsai 3.58 Huang Eckold 7.3 Eckold 12
12
sx:txy=1:0 Eckold Chamis (2) 3.13 Huang Chamis (2) 15.5 Eckold 24
13
sy :sx=0:-1 Huang Wolfe 1.88 Sun Chamis (2) 9.23 Huang 1.54
14
6 sy :sx=1.5:-1 Zinoviev Wolfe 2.71 Huang Chamis (2) 22.6 Edge 15.7
15
sy :sx=-1:-1 Tsai Wolfe 1.72 Tsai Bogetti 3.05 Bogetti 2.15
16
sy :sx=2:1 Sun Huang 3.0 Bogetti Chamis (2) 17.6 Chamis (2) 51
17
sy :sx=1:0 Zinoviev Chamis (2) 2.0 Huang Chamis (2) 17.6 Chamis (2) 24
18
sy :sx=1:1 Zinoviev Huang 4.45 Bogetti Chamis (2) 21.2 Edge 28
19
sy :sx=0:-1 Zinoviev Eckold 2.53 Edge Eckold 3.64 Eckold 1.47
20
9 sy :sx=-1:0 Eckold Chamis (2) 3.61 Sun Eckold 4.95 Eckold 9.6
21
sy :sx=-2:-1 Eckold Chamis (2) 4.13 Huang Bogetti 3.8 Eckold 5
22
sy :sx=2:1 Hart-Smith (1) Wolfe 8.7 Huang Chamis (2) 5.7 Chamis (2) 19
23
sy :sx=1:0 Eckold Chamis (2) 4.47 Puck Eckold 4.88 Eckold 9.55
23a
sy :sx=2.728:1 Sun Rotem 7.05 Huang Chamis (2) 15.53
23b
sy :sx=1.33:1 Hart-Smith (2) Wolfe 13.0
24
13 sy :sx=1:1 Chamis (2) Wolfe 9.7 Huanga Cuntze 3.72 Puck 13
25
14 sy :sx=1:-1 Edge Chamis (2) 5.8 Huang Chamis (2) 2.47 Edge 4
26
12 sy :sx=0:1 Puck Wolfe 2.28 Huangb Edge 3.85 Puck 12
a Eckold terminated his curves at a slightly higher stress than those of Huang.b If Huang’s results without thermal stresses are considered then the ratio would be 5.5.


Cases 4, 5, 6 and 9) and (c) stress–strain curves oflaminates under uniaxial and biaxial loading(Test Cases 7, 8, 10–14).

� Graphs were produced containing the super-imposed predictions of the five new theories foreach Test Case. Superimposed graphs for all theother 14 theories were presented previously inRef. [15].

� In addition, bar charts of strengths (and, whereappropriate, strains) predicted by all of the the-ories were constructed at specific conditions ofloading for 14 Test Cases.

� The largest differences between the predictions of

all the theories for some of the loading combi-nations are recorded in Tables 7 and 8.

Brief observations are made on the differencesbetween the predictions before proceeding to draw con-clusions in Section 4.

3.1. Biaxial failure envelopes for unidirectional laminae(Test Cases 1, 2 and 3)

The predicted biaxial failure envelopes for the fivenew theories are shown in Figs. 1–3. The bar charts inFigs. 1b, 2b and c show the failure strengths for all 19

Fig. 1. (a) Biaxial failure stress envelope for 0� unidirectional E-glass/LY556 epoxy lamina under transverse and shear loading (�y versus �xy). (b)

Bar charts showing the biaxial failure stresses for a unidirectional E-glass/LY556 epoxy lamina under �y:�xy=1.58:1, 1:2.08 and 0:1.


theories (including the additional ones: Bogetti, Cuntze,Huang, Mayes and Hart-Smith(3)) at the selected stressratios indicated in Figs. 1a, 2a, and 3a.Due to the differences between lamina failure criteria

employed, the predicted biaxial failure envelopes descri-bed a variety of shapes. Adding to the observations madein Ref. [15], the following comments can be made regard-ing the shape and magnitude of the predicted stresses.

� In all of the theories used, except those of Eck-old, Hart-Smith (2), and three of the additionaltheories (Huang and Bogetti and Hart-Smith

(3)), the predicted failure envelopes passedthrough the values of strengths under uniaxialloading which were provided as part of the datapack Ref [2] to all participants.

� In Fig. 3, Bogetti does not give a unique pre-diction for the uniaxial compression strengthparallel to the fibres. He supplied two valuesbecause of his initial failure (through-thicknesstransverse tension) prediction One of those valuesis similar to an earlier prediction byWolfe [Ref 13]i.e. about one half of the uniaxial compressivestrengths for that lamina.

Fig. 2. (a) Biaxial failure stress envelope for a unidirectional T300/BSL914C lamina under longitudinal and shear loading (�x versus �xy). (b) Bar

charts showing the biaxial failure stresses for a unidirectional T300/914C lamina under �x:�xy=-12.5:1 and 18.75:1.


� Bogetti shows an inner envelope that differs fromthe outer envelope for Test Cases Nos. 1 and 3 inwhich the initial failure occurs due to exceedingthe ultimate strain in the through thicknessdirection.

� Some theories predicted envelopes in which theeffective strength in one direction is influenced bythe applied stress in the other direction. (i.e. they

employed ‘interactive’ failure criteria). Thesetheories include the new contributors Cuntze,Mayes, Bogetti and Huang.

� Huang predicts an unusual sharp peak in thecompression quadrant of the failure envelopeshown in Fig. 1.

� Like Hart-Smith (2), Bogetti predicts a sig-nificant enhancement in the strength under

Fig. 3. (a) Biaxial failure stress envelope for 0� unidirectional lamina made of E-glass/MY750 epoxy under longitudinal and transverse loading (�y

versus �x). (b) Bar charts showing the biaxial failure stresses of E-glass/MY750 lamina under �y:�x=7.8:1,-32:1,-14:-1,-3.75:-1 and 8.83:1.


certain ranges of biaxial tension–tension andcompression–compression stress states in �y�x

space (Fig. 3). Indeed, Test Case 3 (i.e. Fig. 3)highlights the largest differences between thevarious theoretical predictions for laminastrength. Table 7 shows that at stress ratiosaround �x:�y=-3.15:-1, the transverse compres-sive failure strength (�y) predicted by Bogetti’stheory is more than six times higher than thatpredicted by Eckold. In the tension–tensionquadrant, the largest difference between thestrength values predicted occurs at stress ratio of

�x:�y=7.8:1. At this stress ratio, the transversetensile failure strength, �y, predicted by Bogetti(and Hart-Smith (2)) is more than 4.4 fold higherthan that predicted by almost all of the othertheories.

� In complete contrast, Huang’s theory predicted atrend opposite to that of Bogetti in one portionof the tension–tension quadrant of Fig. 3. Wherestress interaction appears to increase the pre-dicted strength in the Bogetti theory it supressesthe predicted strength for Huang.

� One of the most common forms of interactionassumed that combined stresses reduced thestrength of a lamina to a value lower than itsstrength under uniaxial loading. Figs. 1–3 showmany examples of this (see the predictions ofHuang, Mayes and Cuntze).

In can be seen from Figs. 1–3 that although therewere some similarities, no two theories gave identicalshaped envelopes for the three tests cases.

3.2. Failure envelopes for multi-angled laminates (TestCases 4, 5, 6 and 9)

In order to examine the performance of the failuretheories at the laminate level, four Test Cases werechosen covering a wide range of materials, layups andbiaxial loading conditions.

3.2.1. Failure envelopes for the (90�/�30�)s E-glass/LY556 laminate under biaxial loads, (�y versus �x),(Test Case 4)Initial failure envelopes predicted by the five theories

are shown in Fig. 4a and the final failure envelopes areshown in Fig. 4b. These figures show that the new con-tributors predicted a wide range of different strengthsfor this laminate. Bar charts in Fig. 5 compare the initialand final failure stresses predicted by all 19 theories atspecific ratios of applied biaxial loads �y:�x. Table 8shows the range of predicted values at selected loadratios, including those in the bar charts.With the exception of Bogetti (and previously Eckold,

Chamis (2) and Wolfe) the initial failure envelopesshown in Fig. 4a are approximately diamond shaped,but the magnitudes of the predicted initial failure stres-ses vary considerably from theory to theory. Huang’senvelope was the largest of all, due to his high predictedlamina strengths (as indicated in Fig. 1). The initialfailure predictions were influenced, among other fac-tors, by whether or not residual thermal stresses hadbeen taken into account in a given theoretical method.For instance, the inclusion of thermal stresses by Cuntzeled to the lowest predicted initial failure stresses formost of the tensile quadrants of Fig. 4a. The largestdeviation between the theoretical predictions (more

Fig. 4. (a) Initial and (b) Final biaxial failure stress envelope for (90�/

�30�/90�) laminate made of E-Glass/LY556 epoxy under combined

loading (�y versus �x).


than a factor of 13 (Huang: Chamis)) occurred at abiaxial tensile stress ratio of �y:�x=1:3 (see Table 8).Although none of the Test Cases contained any

through thickness loading Bogetti’s predictions of lowinitial failure stresses in the biaxial compression quad-rant are due to through thickness strains generated by

Poisson’s ratio effects that exceed the allowablethrough-thickness tensile failure strain.Examination of Fig. 4b in this paper and Figs. 4b and

c in Ref. [15] shows that Huang’s final failure enveloperesembles that of Rotem’s where the prediction is verylow (i.e. conservative) in the biaxial tension quadrant,

Fig. 5. Bar charts showing the biaxial failure stresses of (90�/�30�/90�) E-glass/LY556 laminate under �y:�x=-1:-3, 1:1,-1:-1 and 1:0.


compared with that of other theories. It is a sign thatHuang’s theory may contain a similar shortfall to thatexposed in Rotem’s theory during Part B of the WWFE(see Ref. [22]). This is addressed in Ref. [21].Table 8 shows that the highest ratio of maximum:

minimum final failure stresses was 8.4:1 (Hart-Smith (1):Chamis (2)) at stress ratio �y:�x=1:-1 and hence thenew contributions have not altered the range of predic-tions reported in Ref. [15].The bar charts in Fig. 5 show that at certain stress

ratios (e.g. for biaxial compression �y:�x=-1:-1) mostof the theories, except that of Bogetti, predicted finalfailure stresses of similar magnitude to the initial failurestresses, whilst at other stress ratios (e.g. �y:�x=1:0)the final stresses are predicted to be much greater thanthe initial failure stresses. In Table 8 the largest ratioof final: initial failure loads predicted by any oneauthor for this laminate was still 34:1, by Edge at�y:�x=1:3.

3.2.2. Combined direct and shear loading (�x versus�xy) of the (90

�/�30�)s E-glass/LY556 epoxy laminate(Test Case 5)Figs. 6a and b show, respectively, the initial and final

failure envelopes predicted by the five new theories forthe (90�/�30�)s E-glass/LY556 laminate under com-bined direct �x and shear �xy loads. Failure stresses forall of the theories at selected stress ratios are shown inthe bar charts in Fig. 7.The magnitude of the predicted shear strengths of the

laminate varied greatly from one theory to the next.Huang’s initial failure envelope for this test case wasagain the largest predicted by any theory in the exercise(see Fig. 7 and Table 8)The Tsai, Rotem, Wolfe, and Chamis final failure

envelopes, Ref [15], tended to be smaller than the othersincluding those shown in Fig. 6b. The majority of

Table 9

Examples of the effect of thermal stresses on the initial failure stresses of certain laminates

Laminate
SR Mode of failure Initial failure stresses (MPa)
Without thermal stresses
With thermal stresses
0�/90� GRP
1:0 Transverse tension 78 (a) 55.4 (b)
�55� GRP
2:1 Transverse tension 112 (a) 68.6 (b)
0�/�45�/90� CFRP
1:1 Matrix failure 318 (d) 35 (b)
0�/�45�/90� CFRP
1.5:-1 Matrix failure 276.5 (e) 12.24 (c)
(a) Zinoviev

(b) Edge

(c) Chamis (2)

(d) Bogetti

(e) Huang

Fig. 6. (a) Initial and (b) Final biaxial failure stress envelope for (90�/

�30�/90�) E-glass/LY556 laminate under combined direct and shear

loading (�x versus �xy).


theories, including the additional contributions showedan enhancement in the final shear strength under theapplication of moderate tensile loads.Fig. 7 shows that the predicted final failure loads were

always larger than the initial ones for this type of load-ing. The biggest difference between initial and final fail-ure loads occurred under uniaxial tension �x:�xy=1:0.A difference of a factor of 18 was predicted by Chamis,Ref. [15]. All of the additional theories predicted asmaller difference.

3.2.3. Biaxial envelope for (0�/�45�/90�)s AS4/3501-6carbon/epoxy quasi-isotropic laminate under combined�y and �x, (Test Case 6)This family of laminates is typical in aircraft struc-

tures and is frequently known as ‘black aluminium’. Theinitial failure envelopes for this quasi-isotropic laminateare presented in Fig. 8a and the final failure envelopesare shown in Fig. 8b for the five new theories. Compar-ison between the initial and final stresses at selectedstress ratios is shown in Fig. 9, for all of the theories.

Fig. 7. Bar charts showing the biaxial failure stresses of (90�/�30�/90�) E-glass/LY556 laminate under combined direct and shear loading

�x:�xy=2.35:1, 1:1, 0:1 and 1:0.


Fig. 8. (a) Initial and (b) Final biaxial failure stress envelope for (0�/�45�/90�) AS4/3501-6 laminate under combined loading (�y versus �x).


3.2.3.1. Initial failure envelopes. As one might expect, allof the envelopes are symmetric about the 1:1 diagonal.The strength under equal biaxial compression is similarfor all of the theories except that for Tsai’s interactivetheory which predicts higher biaxial compression

strength than the other theories (see the bar charts for�y:�x=-1:-1 in Fig. 9). Bogetti predicts a lower strengththan the others due to through-thickness failure (onceagain caused by a through-thickness Poisson’s straineffect) at this stress ratio.

Fig. 9. Bar charts showing the biaxial failure stresses for (0�/�45�/90�) AS4/3501-6 laminate under �y:�x=1:0, 1.5:-1, 2:1,-1:-1 and-1:0.


There were large differences in magnitude of initialfailure strengths predicted in the other quadrants of thefailure envelope by the different theories. The biggestratio of maximum: minimum predicted initial failurestrengths shown in Table 8 was 22.6 (Huang: Chamis(2)). Some of these differences may be attributed to theeffect of thermal residual stresses, see Table 9.

3.2.3.2. Final failure envelopes. The predicted final fail-ure envelopes for the 0�/�45�/90� laminate reported inRef. [15] and those in Fig. 8b fall broadly into twogroups. The majority of theories are in the first groupand predict diamond shaped failure envelopes, similarin shape and in magnitude to those of Hart-Smith (3),

Bogetti, Cuntze and Mayes in Fig. 8b. In the secondgroup (Tsai, Wolfe, Rotem, and Huang), each theorypredicts an envelope that is unique in shape and inmagnitude. Huang and Rotem predict lower final failurestresses than the other theories in the biaxial tensionquadrant and Huang and Tsai Ref. [15] predict largerstrengths than any of the other theories in differentparts of the biaxial compression quadrant.The largest difference between the theoretical predic-

tions of final failure strength (Zinoviev: Huang=4.45:1)occurred under biaxial tensile loading (�y:�x=1:1). Thebar charts in Fig. 9 show the variation between predic-tions for other stress ratios.Almost all of the theories predicted the initial failure

strength to be the same as the final failure strength (i.e.a single, catastrophic failure) over the whole of thebiaxial compression quadrant of the failure envelope.The exceptions were Bogetti and Huang’s theories.All of the theories predicted final failures that were

different from the initial failures (i.e. the presence of aprogressive failure) when tensile loads were applied(except for Hart-Smith, who did not predict initial

Fig. 10. Stress–strain curves for (0�/�45�/90�) AS4/3501-6 laminate

under uniaxial tensile loading in y direction (�y:�x=1:0).

Fig. 11. Stress–strain curves for (0�/�45�/90�) AS4/3501-6 laminate

under biaxial tensile loading (�y:�x=2:1).

Fig. 12. Initial biaxial failure stress envelope for angle ply �55�

E-glass/MY750 epoxy laminate under combined loading (�y versus �x).

Fig. 13. Final biaxial failure stress envelope for angle ply �55�

E-glass/MY750 epoxy laminate under combined loading (�y versus �x).


failure). The greatest ratio of final: initial failurestrength (see Table 8) for this Test Case was 51:1, pre-dicted by Chamis (2), Ref. [15].

3.2.4. Biaxial envelope for (�55)s E-glass/MY750epoxy laminate under combined �y and �x (Test Case9)The �55� angle-ply laminate is commonly employed

in pipes and pressure vessels. Many thousands of tonsof such pipes are in service around the world. The initialfailure envelopes are shown in Fig. 12 and the final

envelopes in Fig. 13. The bar charts (Fig. 14) compareinitial and final failure stresses at selected loading ratios.The various theories gave a range of results for the

initial failure loads. Cuntze joined several other theoriesthat predict very low initial failure strengths in biaxialtension whereas Huang’s envelope was the largest of all(see Fig. 12). Bogetti again predicted low initial failurestresses under biaxial compression due to a through-thickness Poisson’s strain effect.The biggest difference between predicted initial failure

envelopes was at the stress ratio of 2.73:1 where the

Fig. 14. Bar charts showing the biaxial failure stresses for �55� E-glass/MY750 laminate under �y:�x=1:0, 2:1,-2:-1 and-1:0.


ratios of predicted initial failure strengths was 15.5:1(Huang and Chamis (2)), see Table 8.An equally striking range of predictions was evident

for the final failure envelopes, although the stress ratioat which the largest deviations took place are different.Fig. 13 shows a wide variety of shapes of final failureenvelopes with Mayes joining Rotem and Wolfe in giv-ing much lower final failure strengths than the othertheories in the biaxial tension quadrant (see for instancethe bar chart in Fig. 14 for �y:�x=2:1). The biggestdiscrepancy was at �y:�x=1.33:1 where the ratio ofpredicted final failure stresses was 13:1 (Hart-Smith (1):Wolfe), Ref. [15].

With the exception of Sun, Edge, Puck and Bogetti,all of the theories predicted initial and final failures asbeing coincident events in the biaxial compressionquadrant, see Fig. 14. However, many of the theoriespredict very large differences between the initial andfinal failure loads in the tension–tension quadrant. Thelargest difference between initial and final failure shownin Table 8 is at �y:�x=2:1 where most of the theoriespredict matrix tension failure at low stress and finalfailure due to tensile fracture along the lamina fibredirection at high stress. The initial and final strengthsdiffer by a factor of up to 19 (Chamis (2)) in thisinstance, Ref. [15].

Fig. 15. Stress–strain curves for �55� E-glass/MY750 under uniaxial tensile loading with �y:�x=1:0.

Fig. 16. Stress–strain curves for �55� E-glass/MY750 laminate under biaxial tensile loading with �y:�x=2:1.


3.3. Stress–strain curves (Test Cases 7, 8, 10–14)

3.3.1. Stress–strain curves for (0�/�45�/90�)s AS4/3501-6 carbon/epoxy quasi-isotropic laminate underuniaxial tension �y:�x=1:0 and biaxial tension�y:�x=2:1. (Test Cases 7 and 8)The stress–strain curves for these loading cases are

shown in Figs. 10 and 11. Under uniaxial loading(Fig. 10), all of the curves are very similar in shape

except for those of Chamis and Rotem, Ref [15]. Mostpredictions showed only a small reduction in stiffnessafter initial failure. The initial failure stress was in therange 15 (Chamis (2) to 265 MPa (Huang)) whilst thefinal failure stress was in the range of 385 (Chamis (2))to 728 MPa (Zinoviev). Huang’s final failure stress pre-diction was lower than many others.Failure was predicted to take place in one stage by all

of Hart-Smith’s theories, two stages by Edge and Puck,

Fig. 17. Bar charts showing failure strains for (a) �55� E-glass/MY750 under �y:�x=2:1 (strain in y direction), (b) �55� E-glass/MY750 under

�y:�x=2:1 (strain in x direction) and (c )�45� E-glass/MY750 under �y:�x=1:-1 (strain in y direction).


three stages by the new contributors Bogetti, Huang,Mayes, Cuntze (and previously by Wolfe, Zinoviev andTsai) and in four stages by Sun.For Test Case 8 (Fig. 11), the stress–strain curves

under biaxial tension (SR=�y:�x=2:1) predicted bynearly all of the various theories were also remarkablysimilar to one another, with a very small change in slopeafter initial failure, as can be seen from Fig. 11. Huang(and Rotem in Ref. [15]) were exceptional. They showeda step in the �y and �x curves and their stress–straincurves were truncated at very low stresses (once againgiving early indication that Huang’s theory may have asimilar limitation to that identified in Rotem’s duringPart B (Ref. [22]).The initial and final failure stresses predicted by all of

the theories are compared in Fig. 9, forSR=�y:�x=2:1. The initial failure stress ranged from15 [Chamis (2)] to 264 MPa (Bogetti) while the finalstress ranged from 280 (Huang) to 840 MPa (Sun) butthe majority of theories gave values of final failurestresses which were close to one another.Stages of failure were similar to those shown for

SR=1:0, but the number of stages predicted by sometheories increased by one over that described above.Cuntze, like Wolfe, Sun, Puck, Tsai, and Zinovievshowed four stages of failure, all predicting initial fail-ure due to transverse tension (matrix failure or Inter-fibre Mode A failure) in the 0� plies, which were per-pendicular to the loading direction in this case, and finalfailure by longitudinal tension in the 90� plies. Failure inthe second and third stages occurred in the �45� plies

and 90� plies respectively with the same mode of failureas that in the 0� plies.

3.3.2. Stress–strain curves for (�55)s E-glass/MY750epoxy laminate under uniaxial tension �y:�x=1:0 andbiaxial tension �y:�x=2:1. (Cases 10 and 11)For uniaxial tension of the �55� laminate (Test Case

10, SR=1:0), the stress–strain curves for the additionalcontributors are shown in Fig. 15, bar charts of pre-dicted failure stresses for all theories in Fig. 14 and forpredicted failure strains in Fig. 17. The initial Young’smodulus is identical in all predictions, except for Hart-Smith (3) who did not provide a prediction of the shapeof the stress–strain curve. All of the new contributionspredicted non-linear curves with failure strains in theregion of 2–3% except for Huang who predicted alarger strain at final failure (ex=5.33).The predicted final failure stresses for all of the the-

ories, shown in the bar chart in Fig. 14, differ by amaximum factor of 4.5 between Eckold (640 MPa) andChamis (2) (140 MPa), Ref. [15]. All of the new con-tributors predicted failure stresses in the range 220–320MPa. Furthermore, like nearly all of the original parti-cipants, the new contributors all predicted that initialand final failures were coincident events and that failurewas dominated by in-plane shear.Fig. 16 shows the new contributors stress–strain

curves for biaxial tension Test Case (11) (at �y:�x=2:1).The original participants had predicted a very widerange of results for this case (see Ref. [15]), with a par-ticularly large variation in the strains predicted in the xdirection. Fig. 16 shows that the new contributors alsopredicted a variety of results with Huang and Mayesproducing curves that were truncated at very low stres-ses and strains, like two of the previous contributors,Rotem and Wolfe. The initial and final failure strains inthe x and y directions predicted by all of the theories areshown as bar charts in Fig. 17.The range of predicted initial failure stresses was 45–

276 MPa with general agreement (where identified) thatthe mode of failure at the lamina level was due totransverse tension (i.e. stresses perpendicular to thefibres). Some of these differences may again be attrib-uted to the effect of thermal residual stresses, seeTable 9. The predicted final failure stress ranged from112 to 977 MPa, again quite a wide range.

3.3.3. Stress–strain curves for (0�/90�)s E-glass/MY750epoxy laminate under uniaxial tension �y:�x=0:1. (TestCase 12)The stress–strain curves for uniaxial tensile loading of

a 0�/90� laminate predicted by the new contributors areshown in Fig. 18 and a comparison between the pre-dicted failure stresses for all of the theories is shown inFig. 19c. All of the predictions in Fig. 18, presented hereand the same figure in Ref. [15] are superficially rather

Fig. 18. Stress–strain curves for (0�/90�) E-glass/MY750 laminate

under uniaxial tensile loading with �y:�x=0:1.


similar but a close examination of the results does revealsome differences. For example, in Ref. [15], the "y fail-ure strain ranged from +0.05% (Sun (L)) to 0.776%(Rotem). This implies that the original theories showedthe largest spread in the prediction of this failure strain.The final failure stress was in the range 293–714 MPa

with a significant number of failure theories includingCuntze, Bogetti, Mayes and Hart-Smith (3) predictingfinal failure by tensile fracture of the fibres at a laminatestress of around 640 MPa This equates to a failure stresslevel in the 0� laminae of 1280 MPa (i.e. exactly the

value for the uniaxial tensile strength of the unidirec-tional ply as provided by the organisers Ref. [2], therebybeing an entirely plausible figure). The final failure loadspredicted by Huang, Tsai and Wolfe were lower thanthe others (see Fig. 19c). It was noted from the work ofHuang, Ref. [20], that his final strength predictionincreased from 488 to 690 MPa if thermal stresses areneglected.In all cases where predictions of both initial and final

failure stress were provided by participants the finalstresses were much larger than the initial stresses, the

Fig. 19. Bar charts showing failure stresses for (a) �45� E-glass/MY750 under �y:�x=1:1, (b) �45� E-glass/MY750 under �y:�x=1:-1 and 0�/90�

E-glass/MY750 under �y:�x=1:0.


largest ratio of final: initial stresses being more than11:1 for Puck and Edge, Ref. [15] and for the new par-ticipant Cuntze.

3.3.4. Stress–strain curves for (�45)s E-glass/MY750epoxy laminate under equal biaxial tension �y:�x=1:1.(Test Case 13)This Test Case, where equal biaxial tension is applied

to a�45� laminate, is equivalent to a 0�/90� laminate

loaded under equi-biaxial tension. It differs from TestCase 12 in that the ex and ey strains are expected to beof the same sign and of equal magnitude, such thatcracking would be expected to occur in all of the layersat an intermediate stress level well below final failure.The stress–strain curves for the new contributors areshown in Fig. 20 and bar charts comparing the pre-dicted initial and final failure stresses for all theories arepresented in Fig. 19a.

Fig. 20. Stress–strain curves for �45� E-glass/MY750 laminate under biaxial tensile loading with �y:�x=1:1.

Fig. 21. Stress–strain curves for �45� E-glass/MY750 laminate under biaxial tensile loading with �y:�x=1:-1.


In general, the predictions indicated that the laminatewould fail initially, in all layers, by transverse tension fail-ure, at a stress of 49–175 MPa. Final failure would occureither by fibre failure (e.g. Cuntze, Bogetti, Hart-Smith(3)) at a laminate stress up to 714 MPa or by matrixfailure (Mayes, Huang, Rotem and Wolfe) at a stressbelow 175 MPa. The very low final failure stresses pre-dicted by Mayes, Huang, Rotem and Wolfe are likely tobe due to the rather limited post failure models theyemployed (more of which will be reported in Ref. [21]).

3.3.5. Stress–strain curves for (�45)s E-glass/MY750epoxy laminate under biaxial tension-compression�y:�x=1:-1. (Test Case 14)This case is equivalent to a 0�/90� laminate subjected

to pure in-plane shear, Ref. [15]. The stress–straincurves for all of the additional theories are super-imposed in Fig. 21. The predicted failure stresses andstrains are shown in the bar charts in Figs. 19b and 17c,respectively. Fig. 21 shows that all of the new theoriesexcept Hart-Smith (3), predicted non-linear stress–straincurves. Some of the previous contributors [Tsai, Sun(L), Eckold, and Chamis] had predicted linear stress–strain curves and failure at very low strains. Mayes,Cuntze, Bogetti, predicted failure in shear at about 2%strain but Huang showed an extended stress–straincurve up to 10% strain at failure. In the absence of anypost failure modelling, a simple Mohr’s circle analysiswould indicate that the laminate strains at failureshould equate in magnitude to half of the lamina ulti-mate shear-failure strain, specified as 4% in Ref. [2].Therefore, some immediate doubts are raised over thevalidity of Huang’s results but this issue will be con-sidered again when the theoretical predictions are com-pared with the experimental results (Ref. [21]).All five additional theories predicted initial and final

failure to be coincident events. This should be con-trasted with a few of the theories described in Ref [15],which showed two stage failure process.

4. Conclusions

Conclusions relating to the original 14 theories com-prising Parts A and B of the WWFE have already beenpresented in Ref [15]. In light of the five additional the-ories discussed above, the conclusions have been upda-ted to ensure that they are pertinent to all of the 19theories now studied.

1 On a lamina level, each theory generated a uniquebiaxial failure envelope for the three test cases con-sidered and differences as great as 690% wereobserved in the strength predictions. Some theories(Huang, Bogetti, Hart-Smith (2), Hart-Smith (3),Eckold and Wolfe) predicted uni-axial strength data

of the lamina that were different to those measuredand provided as input data. The largest deviationbetween measured and predicted uniaxial strengthvalue (transverse tensile strength) was a factor of 3.2.This highlights some very significant problems withincertain of these theories.2 A number of major issues emerged in predicting thelaminate responses:-

� There was little unanimity between the partici-pants in how to account for the residual thermalstresses resulting from elevated temperature cur-ing of the laminates. For example, as can be seenfrom Table 9, predictions of initial failure loadsfor the quasi-isotropic carbon/epoxy laminatevaried (maximum:minimum) by 2260% in one ofthe worst instances.

� For many of the test cases the most extremepredictions of initial failure strength came fromtwo sources. The new contribution from Huangcontained the highest predictions, whereas Cha-mis (one of the original contributors) gave thelowest (possibly because he made full allowancefor thermal stresses).

� The participants used a variety of methods topredict laminate behaviour after initial failure.The type of post failure modelling employed byRotem, Wolfe and the new contributors Mayesand Huang produced much lower final failurestrength values than the other contributors inmany cases.

� Even for very familiar cases, the spread in thefinal failure strengths predicted by the partici-pants was very large. Ratios of highest:lowestpredicted final strengths as great as 445% wereobserved for the quasi-isotropic aircraft laminate(Test Case 6), 970% for the cross ply (�45�)laminate (Test Case 13) and 1300% for the �55�

GRP piping/pressure vessel laminate (Test Case9).

� Micromechanics featured in several of the theo-retical approaches. Three participating groupsChamis, Huang and Mayes, relied on the prop-erties of the fibres and matrices to compute thelinear elastic constants, strengths and non-linearbehaviour of the composite laminae. Othersrequired properties of the fibres and/or thematrix to establish failure conditions and to carryout post-initial failure modelling.

3 Any judgements as to which theoretical approachpredicts the most realistic results should be reserveduntil the theoretical results have been compared withthe available experimental data, see Ref. [21].


Acknowledgements

We wish to thank all of the contributors of the failureexercise for their generous, active, sustained and posi-tive participation in this exercise. A portion of this workwas carried out on behalf of the UK MoD CorporateResearch Programme and we gladly acknowledge thissupport.

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a comparison of the predictive capabilities of current failure theories for composite laminates...

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predictions of the19

final failure predictions

additional contributionsa

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original study

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introductionthe present