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FAILURE CRITERIA FOR POLYMER COMPOSITES UNDER 3D STRESS

STATES: THE SECOND WORLD-WIDE FAILURE EXERCISE

A. S. Kaddour* and M J Hinton

*QinetiQ, Ively Rd, Farnborough, Hampshire, GU14 0LX, UK. E-mail: [email protected]

QinetiQ, Fort Halstead, Sevenoaks, Kent, TN14 7BP, UK. E-mail: [email protected]

Copyright QinetiQ Ltd 2009

SUMMARY

The authors (hereafter referred to as the organisers) are coordinating a Second

World-Wide Failure Exercise (WWFE-II) to establish the current status of theoretical

methods for predicting structural failure in fibre reinforced composite materials whensubjected to 3-D states of stress. The exercise runs in two parts. Part A is devoted to

providing full details of the theories together with predictions, made by their

originators, for a standard set of test cases. Part B is concerned with comparing the

theoretical predictions with experimental results.

This paper is directed at exposing some of the early lessons emerging from Part A.

Particular attention is focussed on two Test Cases, the first being an isotropic material

subjected to a range of triaxial compressive stress states and the second being a

unidirectional laminate subjected to the same conditions. Theoretical predictions are

presented and preliminary observations are drawn in regard to the degree of

applicability of the current theories.

Keywords: triaxial, failure criteria, 3D stresses, isotropic, through-thickness,

hydrostatic pressure.

1 Introduction

In order to set this paper in context, it is important to provide the reader with the

background to the work. In 1992, the organisers set out on a coordinated study (known

as the World-Wide Failure Exercise or WWFE) to provide a comprehensive

description of the foremost failure theories for fibre reinforced plastic (FRP) laminatesthat were available at the time, a comparison of their predictive capabilities directly

with each other, and a comparison of their predictive capabilities against experimental

data. In the exercise, selected workers in the area of fibre composite failure theories,

including leading academics and developers of software/numerical codes, were invited

to submit papers to a strictly controlled format.

To make traction in this field, the organisers focused WWFE on the response of

classical, continuous fibre, laminated, fibre reinforced polymer composites subjected to

in-plane biaxial loading situations, in the absence of stress concentrations. WWFE

proved to be a groundbreaking effort with many achievements :-


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- It established, for the first time, an open and objective way of working in order

to compare, contrast and challenge disparate theories from around the world.

- It exposed the strengths and weaknesses of the current theories.

- It provided a stimulus for researchers to build upon the accurate theoretical

features whilst making improvements to deal with the shortfalls that were

exposed.

- It highlighted gaps in experimental data and in theoretical understanding, and

preliminary recommendations were made in terms of prioritisation and

approach to their resolution.

- It provided design engineers (the ultimate customers for such research

knowledge) with recommendations on the preferred theories to use, together

with evidence of the level of confidence and bounds of applicability.

WWFE was completed successfully in 2004, having generated numerous publications

(but best summarised in Ref[1]).

A high priority gap, identified in WWFE, was the need to examine the fidelity of

failure theories when applied to components under 3-D (ie triaxial) states of stress.

Such stress states are commonly induced in thick composite components (rotor blades,

pressure vessels), during impact and ballistic conditions and as a result of stress

concentrations (bolted joints et al). In order to meet this need, the authors launched a

Second World-Wide Failure Exercise (WWFE-II), in 2007, building upon and

employing the principles established during WWFE, with the objective of extendingthe assessment of predictive failure criteria from 2D to 3D states of stress.

WWFE-II is being run in two parts, following the guidelines adopted in WWFE :-

- Part A is devoted to providing full details of the theoretical models and failure

criteria of the participants.

- Part B is concerned with comparing the theoretical results with experimental

results.

The WWFE-II is organised to run logically through a series of activities. A description

of these activities and the associated completion dates are shown in Table 1.

Table 1 Timeline for WWFE-II

Activity Date of completion (*)

Definition of the scope of WWFE-II (Selection of Test Cases andsupporting data)

Completed Dec 2006

Identifying suitable participants /gaining their agreement to participate Completed Dec 2006

Issuing Part A data to participants Completed March 2007

Receipt of Part A submissions Completed Dec 2008-April2009

Issuing Part B experimental data to participants July 2009

Publication of Part A in special edition of a suitable journal Sept 2009


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Receipt of Part B submissions March 2010

Publication of Part B in special edition of a suitable journal June 2010

Publication of WWFE-II text book Dec 2010

(*): These timings are indicative

The arrangement allows both a blind test and a further opportunity for participants to

offer refinements to the theories. At the time of writing, the authors are nearing the

completion of Part A and this paper is aimed at :-

- Providing an overview of WWFE-II

- Identifying the participants and the theories employed

- Defining the Test Cases that have been chosen and the supporting rationale

- Previewing an initial slice of the data from Part A

2 The Participants and their associated theories

A guiding principle employed by the organisers has been to invite the originator of a

leading theory to act as the participant in the exercise, rather than utilising unconnected

experts who might stray in their interpretation of the theory from that intended

originally. Where that has proven to be impossible, connections have been made

between the participant and the originator to minimise any variations and/or identify

the reasons for such. The organisers started WWFE-II by approaching the original

participants in WWFE, many of whom had presented 3-D theories in the first instance.

Six of the participants accepted the invitation, thereby providing valuable continuitybetween the two exercises. The six were supplemented by inviting those who were

regarded as representative of contemporary modelling tools and methodologies

currently in use by research and design institutes around the world. The participants

represent some twelve institutions /groups /individuals from seven countries. Table 2

provides a summary of the participants, their institute affiliation and references to the

theory that each has employed.

Table 2: A list of the participants for the WWFE-II

I.D. No. Participants Name Organisation

1 Bogetti, Staniszewski, Burns, Hoppel, Gillespie

and Tierney, Ref[13]

U.S. Army Research Laboratory (USA)

2 Wolfe, Butalia, Zand and Schoeppner, Ref[10] Ohio State University, AFRL, Wright-Patterson,

AFB, Ohio (USA)

3 Nelson, Hansen, Mayes, Ref[5] Firehole Technologies, Wyoming University, Alfred

University (USA)

4 Deuschle and Kroeplin, Ref[6] ISD, Stuttgart university (Germany)

5 Carrere, Laurin and Maire, Ref[9] ONERA (France)

6 Cuntze, Ref[3] Retired Scientist (Germany)

7 Pinho, Darvizeh, Robinson, Schuecker,Camanho, Ref[8]

Imperial College (UK), NASA (USA), University ofPorto ( Portugal)

8 Rotem, Ref[13] Technion University (Israel)

9 Zhou and Huang, Ref[11] Tongji University (China)


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10 Kress, Ref[4] ETZ Zurich (Switzerland)

11 Ye, Zhang and Sheng, Ref[7] Leeds University, Manchester University (UK),Hefei University (China)

12 Ha, Jin and Huang, Ref[15] Hanyang University (S Korea)

3 Description of the Test Cases

The Test Cases in WWFE-II (described in Ref[2]) have been chosen carefully to

stretch each theory to the full in order to shed light on their strengths and weaknesses.

They are focused on a range of classical, continuous fibre, laminated, reinforced

polymer composites subjected, in the absence of stress concentrations, to a variety of

triaxial loading conditions. The key issues being explored are :-

- The means by which the theories distinguish (if at all) between the effects of

anisotropy and heterogeneity.

- The types of failure mechanism employed and the way that each is

implemented within any given theory.

- The accuracy and bounds of applicability of each theory

Twelve Test Cases were identified for the purpose. They employ five lay-ups :-

(1)- a base resin with isotropic properties, (2)- a unidirectional laminate, (3)- a cross

ply laminate, (4)- an angle ply laminate and (5)- a quasi-isotropic laminate.

Six different fibre/matrix combinations were used and these are:-

(1)- an epoxy, (2)- T300/epoxy, (3)- E-glass/epoxy, (4)-S-glass/epoxy, (5)- A-S

Carbon/epoxy and (6)-IM7/8551 materials.

Full details of the Test Cases are provided in Table 3 (below).

Table 3 Details of the Test Cases used in WWFE-II

Test

Case

Laminate lay-up Material Required predictions

1 Resin MY750 epoxy2 versus 3 (1 = 3 ) envelope2 0 T300/PR319 12 versus 2 (1 =2 = 3 ) envelope

3 0 T300/PR319 12 versus 2 (1 =2 = 3 ) envelope

4 0 T300/PR319 Shear stress strain curves (12-12 ) (for1 =2 = 3 =-600MPa)

5 90 E-glass/MY750 2 versus 3 (1= 3 ) envelope

6 0 S-glass/epoxy 1 versus 3 (2= 3 ) envelope

7 0 carbon/epoxy 1 versus 3 (2= 3 ) envelope

8 35 E-glass/MY750 y versus z (x= z ) envelope

9 35 E-glass/MY750 Stress-strain curves (y -x and y -y) at z = x =-

100MPa

10 (0/90/45)s IM7/8551-7 yz versus z (y =x =0 ) envelope

11 (0/90)s IM7/8551-7 yz versus z (y =x =0 ) envelope


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12 (0/90)s IM7/8551-7 Stress-strain curves (z -z, z -x and z -y) fory =

x =0

4 A Preview of an initial Slice of the Part (A) data

The present paper deals with a slice of the assessment of the theories - The attention

here is focussed on two important Test Cases, referred to as Test Case 1 and 5 in Table

2. Test Case 1 deals with an isotropic material subjected to a range of triaxial

compressive stress states and Test Case 5 is concerned with a unidirectional laminate

subjected to the same conditions. Initial observations are drawn in regard to the degree

of applicability of the current theories to both Cases.

The Test Cases selected here are important building blocks in our understanding of

traditional, high performance, continuous fibre, laminated composites which typically

contain 60% fibre volume fraction in order to gain both stiffness and strength. It is

accepted that a number of failure models rely on micro-mechanics to model the

behaviour of the composite starting from that of the constituents, namely fibres and

matrix. For this reason, a full understanding of the response of resin matrix is crucial

for gaining an insight into how these micro-mechanics models tackle the behaviour of

one of the main constituents of a composite material.

The two Cases are interrelated insofar as the epoxy polymer material studied in Case 1

is the same resin matrix used in making the E-glass/epoxy composite laminate in Test

Case 5. Hence, in choosing Test Cases 1 and 5 these will begin to illuminate the

assumptions made in each theory regarding the treatment of materialisotropy/anisotropy and material heterogeneity. The mechanical properties are

provided in Table 4. Note that the epoxy exhibits isotropic stiffness but anisotropic

strength properties, with a uniaxial tensile strength that is lower than the uniaxial

compressive strength.

Table 4 Mechanical properties for materials in Test Cases 1 and 5, Ref[2].

Test Case No Test Case 1 Test Case 5

Material Epoxy E-Glass/epoxy

Longitudinal modulus E1 (GPa) 3.35 45.6

Transverse modulus E2 (GPa) 3.35 16.2

In-plane shear modulus G12 (GPa) 1.24 5.83

Major Poisson's ratio 12 0.35 0.278

Through-thickness Poisson's ratio 23 0.35 0.4

Longitudinal tensile strength XT (MPa) 80 1280

Longitudinal compressive strength XC (MPa) 120 800

Transverse tensile strength YT (MPa) 80 40

Transverse compressive strength YC (MPa) 120 145

In-plane shear strength S12 (MPa) 54 73

Through-thickness shear strength S23 (MPa) 54 50

The loadings in both Test Cases involve the application of three direct stresses. The

coordinate system used here is shown in Figure 1 where directions 1, 2 and 3 are those


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in the fibre direction and transverse and

through-thickness directions. For the

polymer material, the same coordinate

system is used to describe the three principal

stresses.

As there are many combinations of stresses

that can be applied under 3D loadings, Test

Cases 1 and 5 deal with a section through

the 3D space and the state of stress consists

of the following:

For Case 1: combined 2 and 1 (= 3): The

stresses applied are such that those in 1 and

3 directions are equal while that in 2 direction varies proportional to that in 1 direction.

For Case 5: Combined 2 and 1 (= 3) where 1 .is applied parallel to the fibre

direction, 2 is applied in the through thickness direction and 3 is applied in the

transverse direction (see Fig 1).

In the present analysis, the application of equal triaxial compressive stresses is

commonly referred to as hydrostatic compressive loading where : (-2)= (-1)= (-3)=

-P where P is the equivalent hydrostatic pressure. The application of equal triaxial

tension (ie (2)= (1)= (3) causes an analogous hydrostatic tensile state (ie +P).

5 Comparison between predictionsIn order to avoid prejudicing the ethos of the overall exercise, for the purposes of this

preview paper the data is presented without specific reference to the originating

author (readers will need to wait until Part A is published, in full, for this information).

Instead the theories employed are referred to as A, B, C, . and L (note that these

letters have been assigned randomly). Though this reduces the impact somewhat, a

number of very useful lessons can be drawn from the information. Predictions from

two of the laminate configurations, defined earlier, have been selected and are

discussed more fully below.

Test Case 1 - MY750 Epoxy

Figure 2 shows the failure envelopes predicted by different contributors for this

configuration. The envelopes are superimposed in order to observe the general

differences between the various predictions. The envelopes are split into groups to

facilitate visualisation of differences and similarities between the curves.

Test Case 5 - 90 E-Glass / MY750 Epoxy

Figure 3 shows the failure envelopes predicted by different contributors for this

configuration. The envelopes are superimposed in order to observe the general

differences between the various predictions. The envelopes are split into groups to

facilitate visualisation of differences and similarities between the curves.

2

1

33

1

2

Figure 1: Coordinate systems used in

the Test Cases


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The data from Test Cases 1 and 5 share similar features which raise a number of

interesting issues regarding the assumptions used in the theories.

(a) The interaction between the stresses is generally nonlinear, except in the models

presented by Model F. In some portions of the envelope, other theories (models B, D,

E) predicted linear interaction between the stresses.

(b) In the tension-tension portion of the envelope, some theories predicted

enhancement in the strength and others predicted no enhancement. For Test Case 1, it

was noted that some of the participants introduced various forms of cut-offs in the

envelopes in the tension-tension space. The models involved are B, C and I and those

were introduced to indicate the dominance of different modes of failure. No equivalent

cut-off was made in Test Case 5.

(c) Table 4 provides a summary of the general shapes of the envelopes. These shapes

are categorised as (a) open and (b) closed envelopes. In the last column, the models are

described either as seamless or switched, depending upon the types of analysis used.

The results in Figures 2 and 3 and those in Table 4 may be used to classify the models

as follows:

(I)- Theories that predict an open envelope under hydrostatic compressive

stresses (all except theories A and F in Test Case 1 and all except theories A, B,

C, E, F, G, I, K and L in Test Case 5). In other words, 2 out of 12 models

predicted closed envelopes in Test Case 1 while 9 out of 12 models predicted

closed envelopes in Test Case 5.

(II)- Theories that predict an open envelope under hydrostatic tension stresses

(just one theory (D) in Test Case 1, and no theories in Test Case 5).

(d) The theories can also be classified according to the type of analysis used :-

(1)- Analysis Type 1 (Seamless Models): These theories employ identical

equations for both the isotropic material (Test Case 1) and the anisotropic

material (Test Case 5). These will be referred to as seamless models. They

include A and F. In one of these models (Model F), the application of the

composite criteria to isotropic material resulted in two sets of predictions,depending upon the interpretation of the stresses applied. Based on these two

sets of prediction, the innermost (i.e. most conservative) envelope, resulting

from the interaction between the two curves, was considered in the present

work, as recommended by the participants.

(2)- Analysis Type 2 (Switched Models): These theories use different

equations, one for the isotropic material (Test Case 1) and a separate set for the

anisotropic material (Test Case 5). These will be referred to as switched

models. They involve 10 of the models employed (B, C, D, E, G, H, I, J, K and

L). It can be then concluded that the majority of the participants used

switched models.


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Figure 2: Failure envelopes for Test Case 1 for a pure resin epoxy material under

triaxial stresses. All of the 12 curves were predicted by the participants of WWFE-II

using their own models (A to L).

-1500.0

-1000.0

-500.0

0.0

-1500.0 -1000.0 -500.0 0.0

A

B

C

G

J

L

1= 3 (MPa)

2

(MPa)

-1500

-1000

-500

0

-1500 -1000 -500 0

D

E

F

H

I

K

1= 3 (MPa)

2

(MPa)


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Figure 3: Failure envelopes for Test Case 5, describing the behaviour of E-glass/epoxy

material under triaxial stresses. All of the 12 curves were predicted by the participants

of WWFE-II using their own models (Models A to L).

-1500

-1000

-500

0

-1500 -1000 -500 0

A

B

C

G

J

L

1=3 (MPa)

2

(MPa)

-1500

-1000

-500

0

-1500 -1000 -500 0

D

E

F

H

I

K

1=3 (MPa)

2

(MPa)

Note: Model C predicted closed envelope at large stresses


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Table 5 : States of the failure envelope under hydrostatic compressive stresses for

a polymer and a UD composite obtained from the participants for the WWFE-II.

Model used Test Case 1 Test Case 5 Analysis Type(*)

A Closed Closed Seamless

B Open Open Switched

C Open Closed Switched

D Open Open Switched

E Open Closed Switched

F Closed Closed Seamless

G Open Closed Switched

H Open Open Switched

I Open Closed Switched

J Open Closed/open Switched

K Open Closed Switched

L Open Closed Switched

(*) The word Seamless means the same equations are used for both polymer and

composite materials. The word Switched means that the strength equations used

for polymer are different to those for composite materials

6 Conclusions

(a) WWFE-II is now underway with twelve leading, and internationally

recognised, groups taking part by employing their methods to solve 12 challengingTest Cases. WWFE-II is utilising the Part A /Part B format successfully pioneered via

WWFE-I.

(b) Initial results for two of Test Cases (Nos 1 and 5) from WWFE-II are presented

in the form of predicted failure envelopes. The various models were employed to

predict the failure of an isotropic un-reinforced polymer matrix material (Test Case 1)

and an anisotropic, heterogeneous, E-glass/epoxy unidirectional lamina (Test Case 5)

under triaxial stresses.

(c) The overwhelming majority of theoreticians (9 out of 12) employed separate

equations to delineate between isotropic and heterogeneous materials. This appears notto be an opaque feature of the models employed and rather it appears to require a

conscious operator intervention to make that selection, based on an examination of

the problem to be analysed. From a designers perspective, the preference would be for

a black box modelling tool that contains sufficient resilience to provide accurate

predictions in all circumstances. It remains to be seen if the theories employed within

WWFE-II will satisfy that aspiration.

(d)- There was significant diversity between the theoretical predictions in terms of

the shapes of the failure envelopes and whether or not the envelopes should be open

under hydrostatic compressive and/or tensile loading situations. Clearly, Test Cases 1

and 5 indicate that some of the theories must contain incorrect assumptions. The


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remaining Test Cases will undoubtedly provide further evidence of the resilience of the

12 theories.

References

[1]Hinton M J, Kaddour A S and Soden P D, Failure Criteria In Fibre Reinforced Polymer

Composites: The World-Wide Failure Exercise, published by Elsevier Science Ltd,

Oxford, UK, 2004.

[2]Kaddour A S and Hinton M J, Input data for Test Cases used in benchmark triaxial failure

theories of composites, to be published.

[3]Cuntze R G, The predictive capability of failure mode concept - based strength conditions

for laminates composed of UD laminae under static tri-axial stress states, to be published.

[4]Kress G, Examination of Hashin's Failure Criteria for the Second World-Wide Failure

Exercise, to be published.

[5]Emmett E. Nelson, Andrew C. Hansen and Steven Mayes, Failure analysis of composite

laminates subjected to hydrostatic stresses: A multicontinuum approach, to be published.

[6]Deuschle H M and Kroeplin B-H, FE Implementation of Pucks Failure Theory for Fibre

Reinforced Composites under 3D-Stress, to be published.

[7]Ye J, Zhang D and Sheng H, Prediction of failure envelopes and stress strain curves of

composite laminates under triaxial loads, to be published.

[8]Pinho S T , Darvizeh R, Robinson P, Schuecker C and Camanho P P, Material and

structural response of polymer-matrix fibre-reinforced composites, to be published.

[9]Carrere N, Laurin F, and Maire J-F, Micromechanical based hybrid mesoscopic 3D

approach for non-linear progressive failure analysis of composite structures, to be

published.

[10] Zand B, Butalia T S, Wolfe W E, and Schoeppner G A, A Strain Energy Based Failure

Criterion for Nonlinear Analysis of Composite Laminates Subjected to Triaxial Loading,

to be published.

[11] Zhou Y X and Huang Z-M, A bridging model prediction of the ultimate strength of

composite laminates subjected to triaxial loads, to be published.

[12] Bogetti T A, Staniszewski J, Burns B P, Hoppel C P R, Gillespie, Jr. J W and Tierney J,

Predicting the Nonlinear Response and Progressive Failure of Composite Laminates Under

Tri-Axial Loading, to be published.

[13] Rotem A, The Rotem Failure Criterion for Fibrous Laminated Composite Materials:

Three Dimensional Loading Case, to be published.

[14] Kaddour A S and Hinton M J, Comparison between the predictive capabilities of 3D

failure criteria, to be published.

[15] Ha S K, Jin K K and Huang Y. Prediction of composite laminate failure withmicromechanics of failure, to be published.

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