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Prof. Cuntze Prof. Becker, Dr. Broede 22.6.2012 IASB

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Strength Failure Conditions of

Transversely-Isotropic Material (UD material)

IndustrieAusschussStrukturBerechnungsunterlagen(IASB).Allrightsreserv

ed.

Confidentialandproprietarydocument

Key Words: strength failure conditions, transversely-isotropic material, UD-lamina

Summary

This HSB sheet comprises standard strength failure conditions and novel ones for UD materials.

These cover a wide range of composite types especially for high-performance applications.

Results of the World-Wide-Failure-Exercises WWFE-I and -II on UD-materials are addressed.

References

[00000-04] HSB 00000-04: Glossary. Issue A, 2012 (available also at the website of CCeV)

[12512-03] HSB 12512-03: Static design values of ARALL. Issue C, 1997

[12512-04] HSB 12512-04: Determination of Material Design Values of Fiber Metal Laminates

using the Metal Volume Fraction Approach (rule of mixtures). Issue C, 1999

[12512-05] HSB 12512-05: Material Design Values of Standard Glare (with 2024-T3 aluminium sheet).

Issue A, 2006

[37102-01] HSB 37101-02: Abbreviations and Definitions used for Composites. Issue B, 2010

[37103-01] HSB 37103-01: Classical Laminate Theory for Laminates composed of Uni-directional lami-

nas, and Analysis Flow Chart. Issue E, 2012

[51100-01] HSB 51100-01: Introduction to Basics, Generation and Use of Strength Failure Conditions forStructural Materials. Issue A, (not yet issued)

[51101-01] HSB 51101-01: Strength Failure Conditions of Isotropic Materials. Issue A (not yet dis-

tributed)

[51301-02] HSB 51301-02: Strength Failure Conditions of Orthotropic Materials. Issue A, (not yet issued)

[51100-01] HSB 51100-01: Equivalent Stresses for Different Stress States. Issue B, 1997

[51101-01] HSB 51101-01: Isotropic materials. Issue A, (not yet published)

[Awa78 ] Awaji H. and Sato S.: A Statistical Theory for the Fracture of Brittle Solids under Multi-axialStresses. Intern. Journal of Fracture 14 (1978), R13-16

[Bel1885] Beltrami E.: Sulle Condizioni di Resistenza del Corpi Elastici. Rend. ist. d. lett., Cl. mat. 18,

1885, 705-714

[Boe85] Boehler J.P.: Failure Criteria for Glass-Fiber Reinforced Composites under Confining Pres-

sure. J. Struct. Mechanics 13 (1985), 371

[Cun12] Cuntze R.G.: Failure Mode Concept-based Strength Failure Conditions and their Application

to the Wide Variety of Structural Materials with some Supplementing Investigations added.

Free, living document on the CCeV-Website of Carbon Composites Augsburg, WG Engineer-

ing, 2012


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[Cun04] Cuntze R.G.: The Predictive Capability of Failure Mode Concept-based Strength Criteria for

Multidirectional Laminates - Part B. Composites Science and Technology 63 (2004), 487-516

[Cun11] Cuntze R.G.: Facts and Effects to be considered when Validating 2D and 3D UD- composite

Strength Failure Conditions. ??? (available at the website of CCeV)

[Cun12a] Cuntze R.G.: The predictive capability of Failure Mode Concept-based Strength Conditions

for Laminates composed of UD Laminas under Static Tri-axial Stress States. - Part A of the

WWFE-II. Composites Science and Technology, (to be published 2012)

[Cun12b] Cuntze R.G.: Comparison between Experimental and Theoretical Results using Cuntzes Fail-

ure Mode Concept model for Composites under Triaxial Loadings - Part B of the WWFE-II.

Composites Science and Technology, (to be published 2012)

[Chr98] Christensen R.M.: The Numbers of Elastic Properties and Failure Parameters for Fibre Com-

posites. Transactions of the ASME, Vol. 120 (1998), 110-113

[Gol65] Goldenblat I. and Kopnov V. A. (1966). Strength of glass reinforced plastic in the complex

stress state. Polymer Mechanics, vol. 1, pp. 54-60. (Russian: Mechanika Polimerov, vol. 1, pp.

70-78. 1965)

[Has80] Hashin, Z.: Failure Criteria for Unidirectional Fibre Composites. J. of Appl. Mech. 47 (1980),

329-334

[Hin02] Hinton M.J., Kaddour A.S. and Soden P.D.: A comparison of the predictive capabilities of

current failure theories for composite laminated, judged against experimental evidence. Com-

posites Science and Technology 2002 (62), 1725-97, WWFE-I

[Huf01] Hufenbach W., Kroll L., Langkamp A. and Hpken J.: Physically Based Failure Criteria for

Long-Fibre Reinforced Ceramics. Proceedings of High Temperature Ceramic Matrix Compos-

ites, Vol. 4, Munich, 1. - 3. October 2001, pp. 564-569

[Kai04a] Kaiser C., Kuhnel E. and Obst A.: Failure Criteria for Non-metallic Materials - Part I: Fibre

Reinforced Plastics. ECCM-11, May 31-June 2004, Rhodes

[Kai04b] Kaiser C., Weihs H., Wittke, H. and Obst, A.: Failure Criteria for Non-Metallic Materials

Part II: Ceramic Matrix Composites. Proceedings of the ECCM11, 31. May - 04. June

2004, Rhodes, Greece

[Puc69] Puck A.: Calculating the strength of glass fibre/plastic laminates under combined load.

Kunststoffe. German Plastics 1969, 55, 18-19 (in German, pp. 780-787)

[Puc96] Puck, A.: Festigkeitsanalyse von Faser-Matrix-Laminaten -Modelle fr die Praxis.Carl Hanser Verlag, Mnchen, 1996 (available at the website of CCeV)

[Puc02] Puck A., Kopp J. and Knops M.: Guidelines for the determination of the parameters in

Pucks action plane strength criterion. Composites Science and Technology 62 (2002),

371-378

[Puc02] Puck A. and Schuermann H.: Failure Analysis of FRP Laminates by Means of

Physically based Phenomenological Models. Composites Science and Technology 62(2002), 1633-1662


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[Sch07] Schrmann, H.: Konstruieren mit Faser-Kunststoff-Verbunden. Springer Verlag Berlin, Hei-delberg, New York, 2007

[Thi97] Thielicke B.: Determination of the Interlaminar Shear Strength of Carbon Fibre reinforced

Carbon using the Compression-Shear Test within the Temperature range of Room Temperature

and 2000 C. Ph.D. Thesis, University of Karlsruhe, 1997, in German.

[Tsa71] Tsai, S.W. and Wu, E.M.: A General Theory of Strength for An-isotropic Materials. Journal

Comp. Materials 5 (1971), 58-80

[Tsa80] Tsai S.W. and Hahn H.Th.: Introduction to Composite Material. ISBN 0-87762-288-4, Tech-

nomic Publ. Co., Westport, CT, 1980

[Tsa11] Tsai, S.W. (editor): Strength & Life of Composites. ISBN 987-0-9819143-0-5, Stanford Uni-

versity, 2011

[VDI2014] VDI 2014: Development of Fibre-Reinforced Plastic Components; Sheet 3, Analysis. Beuth-

Verlag, Berlin, 2006

[VDI97] Cuntze R.G., Deska R., Szelinski B., Jeltsch-Fricker R., Meckbach S., Huybrechts D., Kopp

J., Kroll,L., Gollwitzer S., and Rackwitz R.: Neue Bruchkriterien und Festigkeitsnachweise fr

unidirektio-nalen Faserkunststoffverbund unter mehrachsiger Beanspruchung Mod-

ellbildung und Experimente. VDI-Fortschrittbericht, Reihe 5, Nr. 506 (1997), 250

pages. In German. (New fracture criteria (Pucks criteria) and Strength Proof of Design

for Uni-directional FRPs subjected to Multi-axial States of Stress Model development and

experiments. Available at the website of CCeV)

[ZTL79] Berechnung und Versagensvorhersage von Faserverbundstrukturen. Zukunft Technik

Luft (ZTL) FAG 2, Do.2.02/11A, MBB 2.24/5, VFW 2.12/5, DFVLR 2.2.16, 1979


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Contents

1 General 5

2 List of Symbols 7

3 Analysis 9

3.1 Introduction and survey on UD strength failure conditions . . . . . . 9

3.1.1 Definitions and assumptions . . . . . . . . . . . . . . . . . . 9

3.1.2 Technical terms used . . . . . . . . . . . . . . . . . . . . . . 9

3.1.3 Significa of considered materials . . . . . . . . . . . . . . . . 11

3.1.4 Strength failure behaviour of UD lamina material . . . . . . . 11

3.2 Some standard and novel strength failure conditions for UD materials 14

3.2.1 Maximum stress . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.2 Tsai-Wu (1971) . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.3 ZTL (1979) . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.4 Hashin (1980) . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.5 Puck (action-plane formulation, 1991) . . . . . . . . . . . . . 17

3.2.6 Cuntze (application of the FMC to UD-material, 1995) . . . . 20

3.3 Validation of strength failure conditions by experiments . . . . . . . . 22

4 Examples 26

4.1 UD lamina failure curves for validation . . . . . . . . . . . . . . . . 27

4.2 UD laminas composed laminate failure curves . . . . . . . . . . . . 30

4.3 CFRP UD-material, Failure curve 21(2) and Failure Body . . . . . . 32

5 Application Hints and Remarks 33

6 Annex 34

6.1 Automatic use of the UD-failure conditions in 3D applications . . . . 34

6.2 Measurement of material friction value . . . . . . . . . . . . . . 35

6.3 Test data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Bitte an die IASB-Mitglieder: mir erst einmal bergeordnete Kommentare, Wnsche etc. fr diesen ersten Blattentwurf

zu senden.


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1 General

This sheet describes a variety of classical and novel strength failure conditions (SFC) of transversely-isotropic structural materials. Effort is put on the objective: provision of simple, plausible formu-

lations that map the courses of test data reliably well. The given formulas are failure conditions

F = 1. Nevertheless, these are most often termed criteria (F >=< 1).In chapter 3 and 4, for reasons of simplification, a bar - considering a typical average behaviour

- is not always put over the strength letter R despites of the fact that the formulations are usedhere as mapping functions for model validation (average behaviour addressed) and not for strength

verification (minimum strength values applied).

To perform a strength analysis of laminates composed of transversely-isotropic UD laminas (layers,

plies) formulations are necessary which treat 2D and 3D states of stresses as well. For the 2D task

the stresses to be inserted come from Classical Laminate Theory (CLT) analysis and from FEA.The formulations are termed macro-mechanical. The denotations follow [VDI2014] and Fig. 1.

Transversely-isotropic means that the material (homogenized UD lamina) possesses identical prop-

erties in all transverse directions but not in the longitudinal principal direction, where superior

properties are usually encountered.

Tri-axial states of stresses are encountered in many structural parts: in submarines, bolted and

screwed joints, bearings such as sealed polymer bearing cartridges pressurized up to 600 MPa, in

cases of impact and ballistics, and other applications like high pressure vessels, anchor points of

tension cables in civil engineering, load carrying UD hangers of helicopter blades, load introduction

points, CFRP tubes for deepwater umbilicals, underwater blast.

The contents of this work sheet can be applied to laminates composed of a stack of UD laminas.

Non-crimp fabrics (NCF) which are built up from many UD-plies in different directions and stitched

together by z-threads are dealt with in the orthotropic sheet [51302-01]. The reason for that is

that such a NCF laminate is often treated as a macroscopically orthotropic material.

Also fiber-metal laminates (FML) are not addressed here, because they are usually not modelled

ply-by-ply but as a complete laminate stack or as a sublaminate. Such a FML laminate consists of

a lay-up built up from sublaminates consisting of several FRP plies with outer metal plies.

Note: This sheet supersedes HSB 51301-01 Failure Criteria and Reserve Factors of Uni-directional Layersbecause the contents of which is covered by the new sheet. The failure condition that is pointed out there is

just a 2D formulation and stems from the so-called ZTL-program (1979). It is a global condition which can

not correctly map the hump of the (21, c2) failure curve.


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Figure 1: 2D UD lamina stresses and definition of lamina angle


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Abbreviations Description

A, B, C Pucks IFF modes

CMC ceramic matrix compositeCFRC ceramic fiber reinforced ceramic (matrix)

COS coordinate system

DYL, DUL design yield load, design ultimate load

FF, IFF fiber failure (FF1, FF2), inter fiber failure (IFF1, IFF2, IFF3)

FEA finite element analysis

FI failure index

FRP fiber reinforced plastic (polymer)

NF, SF normal fracture, shear fracture

NY, SY normal yielding, shear yielding

SFC strength failure conditionUD uni-directional

WWFE world-wide-failure-exercise

1D, 2D, 3D one-dimensional (uni-axial), bi-axial, tri-axial

Indices Descriptiont, c tensile, compressive

1, 2, 3 related to the coordinate axes of the UD lamina

x, y, z related to the coordinate axes of the UD-laminas composed laminate

, symbolic denotation of lamina quantities

,r

superscript for laminate COS, reference planeT, 1 transposed, inverse


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3 Analysis

3.1 Introduction and survey on UD strength failure conditions

3.1.1 Definitions and assumptions

The following definitions and assumptions are made for the lamina:

The UD-lamina is macroscopically homogeneous. It can be treated as a homogenized (smeared)

material

The stress-strain relationship may be linear and non-linear

The UD-lamina is transversely-isotropic. On planes, parallel with the fibre direction x1

x||,

it behaves orthotropically and on planes transverse to the fibre direction it behaves isotropi-cally. A UD material cube is considered in the figure

Uniform stress state in the critical stress area

The embedded lamina with its stress states is described as in Fig. 1 and Fig. 2 depicted

Using experimental data: Pore-free material, specimen surfaces polished and well sealed,

fiber volume content Vf is constant, tube specimens show no warping and do not bulge,perfect bonding, no layer waviness, edge effects do not exist (all this is not always true with

provided

Mind: As with [Tsai71], the used 12 is the larger Poissons ratio in the HSB! In the [VDI2014,||] and now also in [Tsai11, 21] the application of the suffixes follows the old convention

location,cause as it is always used for load affected quantities.

3.1.2 Technical terms used

Failure theory : theory composed of a strength failure condition, a non-linearity capturing stress-

strain description, and a non-linear analysis method

Failure index = numerical value of the failure condition : many failure conditions are not mathe-

matically homogeneous. This means that linear (L), quadratic (Q) and possibly higher stress terms

may build up a SFC such as F = L/R + Q/R2

. In this non-homogeneous case, the failure indexF I= |F| does not represent the material stressing effort Ef f or Pucks stress exposure fE. Oneexample is the used ZTL-SFC (chapter 3.2.3) for the determination of the three IFFs. In this case it

becomes Ef f = fE = 0.5 (L +

L2 + 4 Q)/R = F I(material) Reserve factor : reserve factor (fRF), calculated by linear analysis, where the stress isproportional to load. Therefore often, the material reserve factor does not represent the - in fact -

load-defined reserve factor RF

Strength failure condition (SFC) : stress interacting condition where more than one stress is used

for assessment

Interaction of failure modes : interaction of both the failure mode effects in their common transitiondomain


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Material failure : failure induced by failure of the material

Structural failure : failure induced by failure (e.g. insatibility) of the structural part which may be

triggered by material failure

Figure 2: 3D UD lamina stresses with in-plane (intralaminar) stresses and interlaminar stresses

(transversal to the lamina plane x1, x2)

For transversely-isotropic UD material the associated stresses and invariants are depicted in Fig.

3. There is also to be seen that three kinds of stresses may be applied: lamina COSbased

stresses, Mohrs fracture plane stresses, and as there exists a quasi-isotropic plane - quasi-

principal stresses. The invariants of transversely-isotropic material are added in the figure.

Figure 3: Transversely-isotropic material, 3D stress states and invariants,


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As performed by Hashin/Puck, formulations of UD-SCFs may also follow Mohrs postulate: Frac-

ture is determined by the stresses in the fracture plane! This has an advantage in a simpler formu-

lation of an SFC but requires the determination of the angle of the inclined fracture plane on theother side. The SFC is not scalar any more.

3.1.3 Significa of considered materials

Fiber-reinforced plastic (FRP):

Composite material made of a polymer matrix reinforced with fibers. Significa are:

Fibers are usually glass, carbon, or aramid

Polymer is usually an epoxy, vinylester or polyester thermosetting plastic

Moduli of the fibers may differ by one magnitude. The fiber is at least one magnitude stiffer than

the plastic matrix

Matrix is well bonded to the fiber.

Ceramic matrix composite (CMC):

Composite material where matrix and fiber may consist of any ceramic material. The interest-

ing sub-group is ceramic fiber-reinforced ceramic (CFRC). CFRC has three significant differences

compared with FRP:

Modulus of the ceramic matrix remains high even at high temperatures. No much difference to

fiber modulus

Matrix is initially cracked

Bonding betwen fiber and matrix is (usually intentially) rather low to get some quasi-plasticity of

the macroscopic CMC material. Fiber pull-out is the key mechanism for CMC properties.

3.1.4 Strength failure behaviour of UD lamina material

Yielding and quasi-yielding

Yielding and fracture of brittle behaving composites looks different compared to ductile behaving

materials.

In the usually faced loading situations failure means fracture failure and yielding failure does prac-

tically not exist, according to the inherent brittle behaviour of the UD material. Only the constituent

matrix experiences some real yielding. At the most, a so-called quasi-yielding from diffuse damag-

ing or micro-cracking at the stress concentration locations within the lamina will occur. Then in the

failure process, single filament breaks will follow as well as a growing of localized micro-cracks

that initiate micro-delamination. All this might be termed quasi-yielding before real delamination

takes place. Delamination is not discussed here.

Fig. 4 indicates how onset of yielding is confined by IFF.


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Figure 4: Guess for onset of yielding (Bhm etc., Bilder?)

Fracture

Fracture morphology of transversely-isotropic UD material witnesses (Fig. 5) that 5 fracture failure

modes exist and that each single fracture mode is governed by one strength, only. Hence 5 strengths

are observed and may be collected in a vector {R} = (Rt||, R

c||, R

t, R

c, R||)

T

.

Figure 5: UD lamina fracture failure modes and strengths

Stress-strain behaviour in hardening and softening (degradation) domain

Significant for the non-linear behaviour are the shear stresses and the lateral stressses. The upper

part of Fig. 6 depicts some 2(2)-curves and 21(21)-curves. These curves are characteristic forthe so-called (strain-)hardening domain which is terminated at the IFF fracture loading. The lamina,

however, is not at its end at IFF.

The embedded, laminate deformation-controlled lamina has still some decreasing load-carrying

capacity, Fig. 6 lower part. This effect is considered in the strain-linked decreasing stiffness and

material stressing effort.


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21(21)-curves fehlen noch

Figure 6: (a) Measured isolated 2(2)- curves, provided by the WWFE-II; (b) stress-strain curvewhen another stress is occurring too; (c) Example of an effective embedded IFF-related

stress-strain curves of a UD-lamina with measured strain-hardening branch (valid for

isolated and embedded lamina) and an assumed phenomenologically-based strain-

softening branch (embedded lamina)


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3.2 Some standard and novel strength failure conditions for UD materials

From the variety of SFCs just a few are depicted here. Decision making for its choice was thesuccess (at least partly) obtained in the two WWFEs.

In the introduction sheet is discriminated between global and modal conditions. A global one

includes all occurring failure modes. Such a global formulation has on the one hand numerical

advantages because one has to apply only one condition, but on the other hand, it may lead to

erroneous results due to its physical shortcoming because it tries to map several failure modes at

once. Material internal friction is not explicitely included in most of the SFCs.

The formulation of a SFC may be invariant-based. Such failure conditions are advantageously and

have been formulated for a large number of isotropic materials.

3.2.1 Maximum stressHypothesis: Failure occurs if the respective stress reaches the associated strength depicted in the

following strength vector {R} = (Rt||, Rc||, Rt, Rc, R||)T.There are 5 modes, two FF and three IFF. They refer to NF under tension and to SF (kinking) under

compression loading. The single strength failure conditions for plane states of stresses read:

F F1 :1Rt||

= 1, F F2 :1Rc||

= 1, (3-1)

IF F1 :2Rt

= 1, IF F 2 :2Rc

= 1, IF F 3 :|21|R||

= 1 (3-2)

Comments:

This approach uses fully independent conditions and exhibits the most simple UD SFCs available.

However, they neither consider failure mode interaction nor internal material friction. Material

internal friction is not explicitely included.

The conditions fit to the design rule for multi-directional laminates applying a design limit strain of

about = 0.3% in order to have no problems with fatigue. Investigating this nowhere documenteddesign rule shows that the strain causing FF or IFF is practically the same for all 3 directions of a

fiber-dominated designed, multi-directional laminate such as a [0/90/45/ 45]slaminate.

3.2.2 Tsai-Wu (1971)

Hypothesis: Tensor polynomial.approach with many model parameters.

A general anisotropic expression of Zakharov and Goldenblat-Kopnov, [..] with the parameters

Fi, Fij as material model parameters

6i=1

Fi i +

6i=1

Fij i j

= 1 (3-3)

is the background of the Tsai-Wu SFC.

3D formulation


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This so-called tensor polynomial is derived from the more general tensor polynomial above by

using linear and quadratic terms

Fij i j + Fi i = 1, (i, j = 1, 2, ..6) or (3-4)

F11 21 + F1 1 + 2 F12 1 2 + 2 F13 1 3 + F22

22 + F2 2+ (3-5)

+ 2 F23 2 3 + +F33 23 + F3 3 + F44

223 + F55

213 + F66

212 = 1 with

F1 = 1/Rt|| 1/Rc||, F11 = 1/(Rt|| Rc||), F2 = 1/Rt 1/Rc, (3-6)

F22 = 1/(Rt R

c) = F33, F13 = F12, F55 = F66 = 1/R

2||,

2 F23 = 2 F22 1/R2, F44 = 2 (F22 F23) .

For the interaction term F12 is recommended to apply

F12 = F12 F11 F22 with 1 F12 1 (3-7)in order to avoid an open failure surface. This bi-axial materia property is principally obtained by

(c1, c2) bi-axial tests. In the WWFE-I Tsai used

F12 = 0.5.

2D formulation

The 3D formulation reduces to

F11 21 + F1 1 + 2 F12 1 2 + F22

22 + F2 2 + F66

212 = 1 (3-8)

Comments:

Pure interpolative global condition. Material internal friction is not explicitely included.

The formulation is mathematically elegant but suffers from the following drawbacks: (1) predicts

for F12 = 0 bi-axial failure stress values higher than the strengths Rt||, Rc|| in the (c1, c2)-domain ;(2) treats (2, 21) like (2, 31); (3) cannot map the (

c

2, 21) humb ; (4) difficult determination ofthe model parameters in the 3D formulation, and in 2D especially forF12.

Tsai-Wu opened the originally transversely-isotropic approach for orthotropic material, indicated

by 9 strengths {R} = (Rt||, Rc||, Rt, Rc, R||, Rt3, Rc3, R13, R23)T used. This is not a consequentmodelling, of course, but may be practical in some work cases. For brittle behaving materials reads

R23 = R = Rt, and in the transversely isotropic case it becomes R

t3 = R

t, R

c3 = R

c.

Note: It turns out - after skillfully collecting the terms - that the tensor polynomial is a stress

invariant-based formulation.


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3.2.3 ZTL (1979)

This 2D formulation is a compromise of a common research work in the German aerospace indus-

try. The single SFCs for plane states of stresses read:

FF1 :1Rt||

= 1, FF2 :1Rc||

= 1, (3-9)

IFF1 :2Rt

= 1, IFF2 :2Rc

= 1, IFF3 :22

Rt Rc

+ 2 (1

Rt 1

Rc) + (

21R||

)2 = 1

Comments:

Pure interpolative global condition. Cannot map the (c2, 21)humb. Material internal friction is not

explicitely included.

3.2.4 Hashin (1980)

Hypothesis: For UD-material the strength failure conditions should be invariant under any rotation

around the fiber direction, [Has 80].

Based on this, he used 5 stress invariants to set up the following 4 SFCs, wherein the strength Requals the failure shear stress of23.

3D formulation

FF1, 1 0 : 2

1Rt2||

+

2

31 +

2

21

R2

||

= 1

FF2, 1 < 0 :21

Rc2||= 1

IFF1, (2 + 3) > 0 :(2 + 3)

2

Rt 2+

223 2 3R2

+231 +

221

R2||= 1 (3-10)

IFF2, (2 + 3) < 0 : C2 + 3

Rc

+(2 + 3)

2

4 R2+

223 2 3R2

+231 +

221

R2||= 1

with C = ( Rc

2 R)2 1 and therefrom for interlaminar failure (orthotropic approach)

3 > 0 :

3Rt3

2= 1 and 3 > 0 :

3Rc3

2= 1 . (3-11)

2D formulation

FF1 1 0 : 21

Rt2||+

221R2||

= 1

FF2 1 < 0 :

2

1Rc2||

1Rc|| = 1


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IFF1 2 > 0 :(2)

2

Rt 2+

221R2||

= 1 (3-12)

IFF2 2 < 0 : C2Rc

+22

4 R2+

221R2||

= 1

Note: For pre-dimensioning -

Comments:

Based on physical considerations. Decision about failure mode possible. Does not model the in-

teraction of fiber failure and lateral bi-axial compression. Material internal friction is not included.

The SFCs may be transferred to confirm that the approach is invariant.

3.2.5 Puck (action-plane formulation, 1991)

Hypothesis: (1) Interfiber failure in a plane parallel to the fibers is governed by the three compo-

nents of the stress vector associated with such a plane. These are the (Mohr) normal stress acting

on that plane and the two tangential stresses, one parallel and the other perpendicular to the fiber

direction. (2) The two shear stresses always promote fracture, whereas normal stress promotes

fracture if it is a traction and impedes it if it is a compression.

Pucks approach uses as proposed by Hashin a modified Mohr/Coulomb theory for brittle IFF of

UD laminas. Due to basing IFF just on the three so-called action plane stresses, these stresses have

a common action plane. Puck discriminates two fundamental regimes: n 0 and n < 0. Theunknown IFF fracture angle f p is determined when the action plane of maximum stress exposureis found, Fig. 7 .

The well-known conventional global criteria apply all six stresses of the UD lamina and do not take

into consideration whether they might act on the same or on different action planes, like 21(2),31(2).

Hashins hypothesis in 1980 was If a failure plane can be observed under a fracture angle f r thenfailure (IFF) is generated by the action plane stresses (= Mohr stresses, if it is the fracture plane

angle) nand n =

2nt + 2n1 upon a fiber-parallel plane

FIF F(

A

n ,

A

n , R, R, ) = 1. (3-13)Instead of the 6 structural stresses he uses the 3 Mohr stresses and the fracture angle f r.

Action plane stresses An , An and basic (Mohr) strengths R, R depend on the angle . The un-

known fracture angle f r is iteratively searched. Hashins quadratic approach reads

FIF F =

An ()

Rt

2+

Ant()

R

2+

An1()

R||

2= 1 for An 0 (3-14)

with the fracture resistances Rt, R, R|| of the action plane versus NF and SF.

Note: For the envisaged brittle behaving UD materials Hashins R (

= Pucks RA) corresponds

to Rt.


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As early as 1968 Puck began to separate the FF modes from the IFF modes. In 1991 he further

developed Hashins idea If a failure plane can be observed under a fracture angle f r then failure

(IFF) is generated by the action plane stresses (Mohr stresses) nand n = 2nt + 2n1 upon afiber-parallel plane

F(An , An , R

A , R

A , ) = 1. (3-15)

The Mohr stresses above are derived from the lamina stresses by the transformation (responsible

for shear fracture are not 12, 13but 21, 31)

nntn1

= c2 s2 2 c s 0 0

c s c s (c2 s2) 0 00 0 0 s c

23233121

with c = cos, s = sin.That angle where the vector (n, nt, n1) touches the failure surfaces,termed here master fracture body (see Fig. 6), is the searched fracture angle f p.

3D formulation

FIF F1 = An ()

RtA

2

+ Ant()

RA

2

+An1()

RA

|| 2

= 1 for An

0. (3-16)

FIF F2 =

Ant()

RA An ()2

+

An1()

RA|| || An ()

2= 1 for An < 0. (3-17)

Action plane stresses An , An and basic (Mohr) strengths R, R depend on the angle . If the angle

is known then RtA = Rt, R

A|| = R||is valid. The model parameter R

A can be not measured via

a test (always tensile fracture), however, can be determined via a lateral compression testwith measurement of the fracture angle.

The unknown fracture angle f r is iteratively searched setting the differentiated equation abovezero. This model was numerically optimized and the model-caused kink at 21 = 0 investigated.

For numerical analysis is recommended:........Bitte an Herrn Schrmann zu prfen

.........


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Figure 7: (a) Pucks action plane (Mohr) stresses, (b) IFF modes, and (c) master fracture body

For the two fiber failures is assumed [Puc02] that the filament stress f1at failure is the same for a1D and 2D state of stress.

2D formulation

For plane stress states the 3D formulations above yield the forms:

FF1 (1 12 2 + f12 E||Ef

mf 2)/Rt|| = 1

FF2 (1 12 2 + f12 E||Ef

mf 2)/Rc|| = 1

Mode A:

21

R||

2+

1 pt||

RtR||

2

2Rt

+ pt||

2R||

= fw = fw

Mode B:

21

2 (1 + pc) R||2

+

2Rc

2

Rc(

2)= fw

Mode C:1

R||

221 + (pc|| 2)

2 + pc|| 2

= fw

applying mf 1.3 for glass fibers, mf 1.1 for carbon fibers andconsidering for mode A 2 > 0; for mode B 2 < 0, 0 21

2 21c

RA; and for mode C 2 0 : pt|| = d21d2 2=0 , 2 0 : pc|| = d21d2 2=0 , (3-19)

RA =R||

2 pc||

1 + 2 pc||

RcR||

1

, pc = pc||

RAR||

, 21c = R||

1 + 2 pc

Statistics states that IFF and FF are mutually weakening (notching of neighbour layers and first

breakages of filaments). Therefore, Puck introduces a weakening factor fw to capture this fact (seeAnnex in VDI 2014)

fw = 1 1Rt1wn , fw = 1 1

Rc1wn

with Rt1w 1.1 Rt

|| , Rc1w 1.1 R

c

|| and an exponent n = 8 (high fracture strain e||), n = 6 (lowfracture strain).

Notes: The value of the angle indicates which failure mode is the driving one. In other words,

whether modus A or B or C dominates, Fig. 7. A Mohr stress is that action plane stress for

which the action plane becomes fracture plane. For pre-dimensioning the following parameters are

recommended:

GFRP : pt|| = 0.3; pc|| = 0.25, CF RP : pt|| = 0.35; pc|| = 0.3

Comments: Is a modal formulation despite of the fact that a Mohr-Coulomb approach was

used to combine the three IFFs. Material internal friction is thereby included. The approach is

vectorial and based on physical considerations. Models all the interactions between the failure

modes.

3.2.6 Cuntze (application of the FMC to UD-material, 1995)

Hypothesis: Similar to v. Mises an invariant-based failure mode-linked SFC can be formulated.

Some key features of the FMC shall be recalled here: (1) Driven from the shortcomings of theusual global fitting SFCs the FMC-originator Cuntze looked for a failure mode-related fitting

on basis of material symmetry and for the use of invariants, related to physical mechanisms. (2) The

knowledge of the basic strengths is sufficient for pre-dimensioning. (3) A remaining unknown curve

parameter b in the authors modal failure conditions concerns the friction of the UD material (the

real crystal experiences friction). Its value can be approximately estimated. The more brittle the

material behaves the larger the friction value will be. Friction parameters are physically required.

Not physically-based global failure conditions do not use such a parameter and this leads to a

shortfall. (4) Whether a fracture type is termed NF or SF depends on the addressed model level:

For IFF3 in Fig. 4 holds that the fracture type may be a NF (constituent micro-level) or a SF (lamina

macro-level).

The final development of the UD SFCs by applying the FMC to transversely-isotropic material


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finally ended in a relatively simple set of 5 conditions (comparing the originally more complex

formulations in WWFE-I), 2 fiber failure (FF) conditions and 3 inter-fiber-failure (IFF) conditions.

In contrast to Puck the 3 IFF modes are not combined in a Mohr-Coulomb equation.

3D formulation

FF1 : Ef f|| = 1/Rt|| = ||eq /Rt||FF2 : Ef f|| = 1/Rc|| = ||eq /R

c

||

IFF1 : Ef f =

(2 + 3) +

(2 3)2 + 4 223

/2 R

t

= eq /R

t

IFF2 : Ef f =

(b 1) (2 + 3) + b

(2 3)2 + 4 223

/Rc

= eq /R

c

IFF3 : Ef f|| =

b|| I235 +

D

/(2 R2

||) = ||eq /R||, (3-20)

with 1 = t|| E||, 1 = c|| E||, (3-21)I235 = 2 2

221 + 2 3

231 + 4 23 31 21, D = (b|| I235)

2 + 4 R2

|| (231 +

221)

2

which consider that the filament might break under bi-axial Possons ratio effect-related lateral

compression.

Interaction of modes:Ef f|| m + Ef f||m + + (Ef f)m + (Ef f)m + Ef f||m = 1 (3-22)The interaction exponent 2.5 < m < 3 takes the role of Pucks weakening factor. For reasonsof simplicity the same m is taken for all interaction (mode transition) domains. The interactionequation includes all mode material stressing efforts and each of them represents a portion of load-

carrying capacity of the material. In thin laminas at maximum 3 modes of the 5 modes will physi-

cally interact. Considering 3D-loaded thick laminas, there, all 3 IFF modes might interact together

with one FF.

Typical material (internal) friction value data range:

The friction values are linked to the model parameters, [19],

b|| = ||, b = 1/(1 ). (3-23)

They are derived from bi-axial compression tests (c2, c3) or in a uni-axial compression test with

measurement of the fracture angle f p, see Annex and [Cun12a].

From investigation of some course of test data can be concluded:

0.05 < || < 0.3, 0.05 < < 0.2 . (3-24)

In consequence, there is a strong need to further validate failure conditions in the multi-axial com-

pression domain. Proving the capability of the tri-axial FMC theory requires realistic, well evalu-ated, and well understood experimental data.


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Note: For pre-dimensioning the following parameters are recommended - due to practical experi-

ence : m =2.5 (smaller is safe side), || = = 0.1.

2D formulation

When only 2D stress states are to be assessed then above equations reduce to

1Rt||

= 1,1Rc||

= 1,

1Rt

= 1,2Rc

= 1,|21|

R|| || 2= 1 . (3-25)

Note: In the 2D formulation the axial fiber stresses can be directly taken. In the 3D formulation it is consid-

ered that under bi-axial lateral compression the filament may break due to the effect of the larger Poissonsratio || which strains the filament without any applied stress 1. Under bi-axial tension or compression thefailure mode IFF1 is activated in two planes and thereby doubles fracture danger.

For m = 2.5, || = 0.3, (Ef f)

m + (Ef f)m +

Ef f||m

= 1 the interaction failure curve inFig. 8 is obtained.

Figure 8: Visualization of the interaction formula for the section (2, 21) of the 2D stress statefailure body. FMC

Comments: Based on physical considerations. Very strict modal formulation with a scalarapproach. Material internal friction is inherent. Models simply and engineeringlike all the interac-

tions between the 5 failure modes.

3.3 Validation of strength failure conditions by experiments

A formal use of test data is dangerous if no good test information is additionally provided. The

World-Wide-Failure-Exercises I and II delivers 3 paradigms for such a shortcoming. The contribu-

tions were performed in the following way: Part A in the WWFEs was a blind prediction on basis

of strength data only. Part B was a comparison of the error-freed prediction with the test results.

First paradigm: Torsionally-loaded axially (0) wound tubes, WWFE-I


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Here, in contrast to the standard 90-hoop wound tube this test specimen does not only show

bulging (barrelling) but also experiences twisting (Fig. 9). The test results are pointed out in Fig.8b

in the right diagram. Twisting under a torsional moment during shear test turns the lamina COS.Therefore, such 0 data cannot be used like 90data. The failure stress state in the laminate COS

has to be transformed into the finally achieved turned lamina COS. On top, the provided test results

show an unbelievable decay at the shear axis at the questionmark?.Further, in the WWFE-II a 0-tube data set was provided together with a 90-tube data set in one

diagram. Annex 5 in Cuntzes WWFE-II, Part B numerically proves that this means a mixture of

apples and oranges.

Lessons Learned: Discrepancies in Fig.6b, 0- tube data cannot be used together with 90 data.

Figure 9: (a) Axially wound tube test specimen [Knops]; (b) Failure curve 21(1), WWFE-I, FMC-based SFCs used

Second paradigm: 90 wound tubes, WWFE-I

In the following example the situation was: The Part A curve was the prediction with the dataR

= (1140, 570, 35, 114, 72)T and an assumed low (on the safe side) friction value || = 0.13.

For Part B, validation of the failure model, another data set was provided:

R

= (1140, 570, 38, 135, 62)T

but still no friction value. In order to map the course of test points an assumed friction value of

|| = 0.56 had to be employed. Such a high material friction related steep slope was never mea-sured anywhere the last 40 years. Hence in Fig. 10 two failure points are very doubtful. These are

the shear strength value (is not the required mean value as meanwhile was sorted out) and the peak

value.

Lesson Learned: Too few test data may lead to a physically unrealistic property input for real world

applications.


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Figure 10: Failure curve 21(2), WWFE-I, FMC-based SFCs used

Third paradigm: Laminate wound tube, WWFE-I

Fig. 11 shows a laminate example which may be used as benchmark (verification for a UD-SFC).

Here, material failure - described by a SFC as one part of a failure theory - is discriminated from

structural failure of a crushing tube. The two test curves provided for Part B should theoretically

lie upon another. In practice there is no full symmetry given from manufacturing reasons. Also

bulgening is activated and was reported for Part B. This means that the given experimental failure

stress value (was based on the nominal tube geometry) had to be increased due to the estimated

hoop strain-based larger diameter. As more test information was available for Part B, the fiber

fracture strain had to be reduced also. The inclusion of both the effects leads to a good approach.Lesson Learned: Too few test information hinders model validation.

Figure 11: Stress-strain curve y : x = 1 : 1 for a GFRP laminate tube test specimen, [45/ 45/45/ 45], WWFE-I, FMC-based SFCs used

Very essential in the application of SFCs is a physical understanding of failure and failure test data:


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The first FF in case of mass-optimized laminates (fiber-dominated laminates) means final fail-

ure, if the laminate is not multi-directionally over-dimensioned and possessing some redun-

dancy. So, usually FF in at least one lamina of a laminate means final failure of the laminate.Therefore, the biaxial failure envelopes for final failure of laminates predicted by various

theories do not differ that much, as long as the laminates have three or more fibre directions.

Also, the predicted stress-strain curves of such laminates look very similar because the fibers

which are much stiffer than the matrix carry the main portion of the loadings. Different

degradation procedures after the onset of inter-fibre failure (IFF) do therefore not influence

the predicted final failure strains very much. This is especially true for CFRP laminates.

More effect is obtained below IFF because the laminate acts as a continuum which beyond

IFF turns into a fiber net where net theory is approximately applicable.

Initial failure or onset of failure by IFF is marked by a loss in stiffness which is indicated

by a kink and a drop in the applied load-deformation curve. Failure by IFF can be always

indicated in the failure curves. Marking final failure, however, is a problem whenever strong

non-linearity occurs, such as large deformation or large strain.

For instance, a solid compression-loaded thick plate (WWFE-II, test case 12) will not fall

apart under compression. The squeezed laminate carries more and more loading, however,

just for compression. As a generally loaded structural part it cannot be used anymore after

this situation. Based on the task the structural part has to fulfil, engineers must define: What

is physical final failure and what is technical failure?

Final failure occurs after the structure has degraded to a level where it is no longer capable

of carrying additional load. This is most often caused by FF, however in specific cases by an

IFF, too. For instance, the wedge-shaped inter-fibre crack IFF2 can lead to final failure if itdamages (termed notching) the neighbouring layers

There are several sources of error besides the errors that directly happen to occur in the

always challenging tests. These are the evaluation of raw test data, placing of test data in the

graph, quality of the test data. So, enhancing the quality of test data is a permanent task Only

well understood testing and material behaviour, coupled with a careful evaluation of the raw

test data can verify the assumptions made and let validate a model.


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4 Examples

This chapter collects various UD failure curves to prove the mapping quality of the UD-failureconditions used. If a good mapping is obtained, then validation is achieved under the pre-requisite

of reliable test data.

As non-UD examples, some laminate failure curves are added in order to have some benchmarks.

These serve as verifications of the full failure theory model where the SFC is just one part of..

For a general FRP material Fig. 12 visualizes how the failure surface or failure body looks like in

the case of a 2D stress state.

Figure 12: 2D stress state failure surface (body), [Becker, TU-D]

Of interest for the designing engineer is the difference of 21(2) and 21(3). These curves do notlie upon another Fig. 13a because 3 does not have the same action plane as 21. The stress states21(2) and 31(2) do therefore show different failure curves.The shear stress23has a tensile and a compressive component. Both components are active in twodifferent failure modes, IFF1 and IFF2. Fig. 13b depicts both the single mode curves 23(2) andthe interaction curve.

Comment:

Figure 13: (a) Difference bi-axial failure stress envelope 21(2) and (dashed) 21(3) = 31(2),(b)23(2)


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4.1 UD lamina failure curves for validation

CFRP, Failure curve 21(2)In [VDI97] several sets of strength data are presented. In Fig. 14 FMC-based UD SCFs have been

applied.

Comment: Mapping of the course of test data is fine.

Figure 14: In-plane shear-transversal normal stress failure curve of a UD-based C/C. Failure

curve 21(2) of a ceramic.fiber-reinforced ceramic UD material, [Thi77],

R

=

(,, 45, 260, 59)T, m = 2.8

Failure curve 2(1)

The provided failure stresses were for the fourth quadrant ofFig. 15.

Comment: Due to the provided strength data it makes no sense to better map the course of test

data. It looks as if the test data stem from results with different tests. The gap in quadrant III is

closed by Fig. 16. It also proves that Tsai/Wu should not applied in this quadrant.

Figure 15: Failure curve 2(1) , WWFE-I, TC ?, FMC. {R} = (, , , , )T

, m =??


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Figure 16: Bi-axial failure stress envelope c2(c1), [test data IKV Aachen, Knops], FMC. {R} =(, , , , )T, m =?? neu mappen

CFRP, Failure curve 1(2 = 3):Comment:

Figure 17: Failure curve 1(2 = 3), WWFE-II, TC 7, FMC-model. E-glass/MY750epoxy cubeR

= (1990, 1500, 38, 150, 70)T, m = 2.8, || = 0.3

GFRP, Failure curve 2(1 = 3):The lower curve maps the course of test data of the lower branch very well. The upper branch is

validated by test results from [....]. Hence, this test case delivered a very good validation, after are-evaluation of the provided test data set by the WWFE organizers.


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Comment:.

Figure 18: Failure curve 2(1 = 3), WWFE-II, TC 5.

R

= (1280, 800, 40, 132, 73)T, m =2.8, b = 1.16, || = 0.28

Failure curve 21(1 = 2 = 3 = phyd)Text noch dazu setzen

Comment:

Figure 19: Shear faiure curve 21(1 = 2 = 3) WWFE-II, TC 2, FMC. {R} = (, , , , )T, m =??

CFRC, Failure curve 21(2)

From [Thi97] the following set of strength data has been provided from a slant shear test for C/Cfibre-reinforced ceramics. This is a brittle porous ceramics laminate, based on a UD tape.


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The FMC-based UD SCFs have been applied. As invariants were utilized: For friction I3 and I2,

for shear I4. This reduces for a plane stressing to the interaction equation including the three IFF

failure modes, indicated with the respective failure condition in Fig. 13.

Comment: In the frame of the small data set the mapping of the course of test data is fine.

Figure 20: In-plane shear-transversal normal stress failure curve of a UD-based C/C. Failure

curve 21(2) of a ceramic.fiber-reinforced ceramic UD material, [Thi77],

R

=

(,, 3, 99, 7)T, m = 2.5,|| = 0.3

Some basic conclusions:

* Generally, it is physically not accurate to predict a failure surface with the knowledge of strength

data only! Friction is inherent with brittle behaving materials and to be considered if applicable.

* In contrast to a dense isotropic material a dense UD material might fracture under a very high

hydrostatic compression stress (due to the Poisson effect which makes the filament strain to reach

the fracture strain ec|| = phyd (1 2 ||)/E||, that means under the pre-requisite 2 || < 1.Thisfracture strain occurs at values far beyond phyd = 1000 MP a = 10000 bar.

* Higher load carrying capacity or resistance, obtained under multi-axial compression, is the result

of the favourably affected, decreasing equivalent stress eq. Its not from an increasing technicalstrength R.

4.2 UD laminas composed laminate failure curves

For verification of the SFC and for benchmarking of FE results the following examoples may serve.

CFRP, Failure curve 21(2)..Text

Comment:


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Figure 21: WWFE-II, TC12

CFRP, Failure curve xy(z)

Fig.20 a shows a 90 lamina of the stack of a thick-walled tube milled from a laminate block. For

comparison, Fig. 20b shows a traditionally hoop wound or tape-layered tube. The lower figure

outlines that the course of test data could be mapped despites of the non-homogeneous stress field.

For analysis details, see [Cun12b].

Comment: Mapping could be achieved but the results cannot be generalized for design purpose

because of the non-homogeneous stress field.


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Figure 22: Carbon/EP-IM7/8551-7, WWFE-II, TC19

GFRP, Failure curve ty(tx) of a mat laminate

A mat is approximately quasi-isotropic in its plane and transversely-isotropic at all. The 1-axis has

to be turned upward compared to a UD lamina where it is inplane.

Comment:

Figure 23: 2D failure curves of two mat-laminates LM1, LM2 [Kothe]

4.3 CFRP UD-material, Failure curve 21(2) and Failure Body

A further example shall depict when a lamina will fail in its plane under a stress state including

shear stress and normal stress.


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Figure 24: 2D failure curve of UD material CFRP

5 Application Hints and Remarks

- The presented strength failure conditions are valid for UD laminas under 2D and partly for 3D

stress states, too. Onset of delamination can be predicted only by the given 3D SFCs. Curing

stresses can be inserted.

- Prediction is not possible if physically necessary internal material friction values must be con-sidered in a SFC of a distinct failure mode. Global UD strength failure conditions do not consider

friction and therefore have shortcomings.

- Validation of failure conditions requires a uni-form (homogeneous) stress field in the critical

domain where the test specimen should break. In the WWFE-II this was only partly given.

- Comment on modelling and assessment of composites made of technical textiles: Some textiles

can be modelled as plane or planar 2D laminas (good properties in the plane) and therefore as-

sessed by 2D SFCs. Whether a 3D lamina strength (tri-axial, has improved properties in thickness

direction and degraded properties in the plane) can be assessed or not depends on the specific task.

- A fracture surface, generally composed of various SFCs, confines the global yield surface (quasi-

yield capacity exhausted) and therefore possesses a different shape.

- Effects from imperfections on failure come from boundary conditions, support, fiber ondulation,

lay-up.

- UD lamina SFCs, applied in laminate analysis: designing a laminate without having clarified the

conditions under which its laminas fail is questionable.

- Transferability of static failure criteria to fatigue (cyclic loading): Cyclic fatigue life consists of

three phases: (1) Growth of diffuse damage up to discrete damage. This is the main phase for

the accumulation of damage portions (Schdigungen), (2) Stable local discrete (macro-)damage

growth, and (3) Final instable fracture due to delamination criticality.

Remark:

- Brittle behaving materials experience several failure modes or mechanisms. Consequence: Sev-

eral SFCs are to be employed. These are for UD material 5 SFCs.

-There is not a lack of theories but a lack of reliable test results to benchmark the available theories.

So, the WWFEs helped to clarify the situation by outlining which SFC well mapped the reliable


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test data cases in WWFE-I and WWFE-II.

AcknowlegementThe author thanks for contributions and checks of the messieurs Prof. Dr. H. Schrmann, J. Bold

(DLR-Braunschweig), T. Weber (CADCON), Dr. R. Boehm ? (ILK-Dresden), ....

.

6 Annex

6.1 Automatic use of the UD-failure conditions in 3D applications

In order to make the necessary automatic SFC use possible in a 3D state of stresses and to avoidcomplicate queries in the computer program some specific by-passes were introduced (see [Cun12b])

for the FMC-based formulations - other SFCs have similar problems:

(1) FF1, IFF1, IFF2: Caused by the automatic insertion of a state of stress physically incorrect

negative efforts or negative equivalent stresses may occur. This can be avoided by Fppl-Macauley

brackets which represent for instance the equation (eq +| eq |) /2 and make negative values zero.

Such a negativeeq may occur in the case of a combination of a high friction parameter with acertain state of bi-axial stressing.

In the case of certain compressive stress ratios the equivalent stress or the respective material stress-

ing effort can become negative. Such a negative value physically means that IFF2 will not occur

due to strong friction. For numerical reasons such a problem is by-passed by specific settings. Thesecond part in IFF2 and in IFF3 is the effort reducing friction part. The magnitude of this part

should remain with respect to the 3D validity of the model - below b

(2 3)2 + 4 223(2) IFF3: When deriving above efforts no proportional stressing (all stresses are changed by the

same factor) was applied to all stresses but just to the failure driving shear stresses. This practically

affects the interaction domain, only, which is independently dominated by m.

(3) IFF1, IFF2: Shear affects the two different failure modes, IFF1 and IFF2, according to the fact

that the shear stress can be replaced by a normal tensile stress together with a compressive stress.

Of the two failure modes just one is significant in the case of a brittle behaving UD material. Due

to this, naturally a tensile Ef f (driving material stressing effort in case of a brittle behavingmaterial) is caused and the compressiveEf fas well. As the compressive portion incorporatesa small additional failure danger this is considered via the use of principal stresses in the quasi-

isotropic 2-3 domain instead of the lamina stresses.

4) IFF2: The solution of the IFF2 equation delivers two roots and therefore two branches as can be

seen in case of WWFE-II TC 5, for instance.


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6.2 Measurement of material friction value

Figure 25: Fracture angle of the laterally compressed UD test specimen

Figure 26: Determinatin of internal material friction value


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6.3 Test data

For application of the SFCs some UD data tables are provided. Table 1 contains the propertiesprovided in the WWFE-II.


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Table 1: Lamina properties provided in WWFE-II

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