Mixed-Mode Fracture Toughness Determination USING NON …filipechaves.com/DOCs/presentation_IJSI_2012.pdf · 2018. 2. 15. · Consequently, the fracture characterization of bonded

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Mixed-Mode Fracture Toughness Determination

USING NON-CONVENTIONAL TECHNIQUES

IDMEC- Pólo FEUP

DEMec - FEUP

ESM – Virginia Tech

motivation

outline

fracture modes

conventional tests [mode I]

conventional tests [mode II]

conventional tests [mixed-mode I + II]

non-conventional tests [mixed-mode I + II]

conclusions

acknowledgments

June 28th, 2012

motivation 3

Bonded joints in service are usually subjected to mixed-mode conditions due to geometric and loading complexities.

Consequently, the fracture characterization of bonded joints under mixed-mode loading is a fundamental task.

There are some conventional tests proposed in the literature concerning this subject, as is the case of the asymmetric double cantilever beam (ADCB), the single leg bending (SLB) and the cracked lap shear (CLS).

Nevertheless, these tests are limited in which concerns the variation of the mode-mixity, which means that different tests are necessary to cover the fracture envelope in the GI-GII space.

This work consists on the analysis of the different mixed mode tests already in use, allowing to design an optimized test protocol to obtain the fracture envelope for an adhesive, using a Double Cantilever Beam (DCB) specimen.

Aloha Flight 243

Boeing 737

Photos from NTSB

June 28th, 2012

fracture modes 4

Mode I – opening mode (a tensile stress normal to the plane of the crack); Mode II – Sliding mode (a shear stress acting parallel to the plane of the crack and perpendicular to the crack front); Mode III – tearing mode (a shear stress acting parallel to the plane of the crack and parallel to the crack front)

mode I

mode II

mode III

fracture modes for adhesive joints

Figure 1. Fracture modes.

June 28th, 2012

conventional tests [mode I] 5

Mode I release rate energy GI is well known and well characterized.

DCB – Double Cantilever Beam TDCB – Tapered Double Cantilever Beam &

AST

M D

34

33

- 9

9

Aa

Figure2. DCB specimen and test. Figure 3. TDCB specimen and test.

June 28th, 2012

conventional tests [mode II] 6

Mode II release rate energy GIIC .

ENF – End Notch Flexure 4ENF - 4 Points End Notch Flexure

Aa

ELS – End Load Split

Figure 4. ENF specimen and test. Figure 5. 4 ENF scheme Figure 6. ELS scheme

June 28th, 2012

CLS - Crack Lap Shear EDT - Edge Delamination Tension

Arcan MMF - Mixed Mode Flexure

Mixed Mode Bending [MMB] ASTM D6671

Asymmetrical Double Cantilever Beam [ADCB]

Asymmetrical Tapered Double Cantilever Beam [ATDCB]

conventional tests [mixed-mode I+II] 7

SLB – Single Leg Bending

Mixed-Mode I + II release rates energies GT= GIC + GIIC .

Figure 7. Conventional test schemes for mixed-mode I + II.

June 28th, 2012

conventional tests 8

Bondline thickness = 0.2 mm

specimens [DCB,ATDCB,SLB, ENF]

Steel

Young modulus, E [Gpa] 205

Yield strength, sy [MPa] ~900

Shear strength, sy [MPa] ~1000

Strain, ef [%] ~15 Figure 8. DCB, ATDCB, SLB and ENF specimen geometries.

Table1. Adhesive shear properties using the thick adherend shear test method ISO 11003-2

Table2. Steel adherend properties

June 28th, 2012

Araldite

conventional tests [envelope] 9

Table 3. Fracture toughness obtained with the conventional testing methods (average and standard deviation).

Figure 9. Fracture envelope for conventional tests.

June 28th, 2012


Figure 10. Modified Mixed Mode Bending by Reeder.

Standard Test Method for Mixed Mode I-Mode II Interlaminar Fracture Toughness of Unidirectional Fiber Reinforced Polymer Matrix Composites

Compliance Based Beam Method applied to MMB

𝐺𝐼 =

𝐺𝐼𝐼 =

An equivalent crack length (𝑎𝑒𝑞,𝐼 and 𝑎𝑒𝑞,𝐼𝐼 ) can be

obtained from the previous equation as a function of the measured current compliance 𝑎𝑒𝑞,𝐼= 𝑓(𝐶𝐼) and

𝑎𝑒𝑞,𝐼𝐼= 𝑓(𝐶𝐼𝐼)

J.M.Q. Oliveira et al. / Composites Science and Technology 67 (2007) 1764–1771

June 28th, 2012

(1)

(2)


Figure 11. Numerical Fracture Envelope for MMB.

The specimen was modelled with 11598 plane strain 8-node quadrilateral elements and 257 6-node interface elements with null thickness placed at the mid-plane of the bonded specimen.

Elastic properties (Steel) Cohesive properties (Adhesive)

E (GPa) G (MPa) su,I (MPa) su,II (MPa) GIc (N/mm) GIIc (N/mm)

210 80.77 23 23 0.6 1.2

Table 4. Elastic and cohesive properties.

Linear Criterion I II

Ic IIc

1G G

G G

ABAQUS®

June 28th, 2012

non-conventional tests [mixed-mode I+II] 12

Figure 13. DAL frame

The Dual Actuator Load Frame (DAL) test is based on a DCB specimen loaded asymmetrically by means of two independent hydraulic actuators

Different combinations of applied displacement rates provide

different levels of mode ratios, thus allowing an easy definition of

the fracture envelope in the GI versus GII space.

Figure 12. DAL loading a DCB specimen.

June 28th, 2012


Figure 14. Loading schemes.

DAL loading schemes for this study

Classical data reduction schemes based on compliance calibration and beam theories require crack length monitoring during its growth, which in addition to FPZ ahead of the crack tip can be considered important limitations.

June 28th, 2012


2 2 2

R L T

0 0R L2 2 2

a a L

a

M M MU dx dx dx

EI EI EI

2 2 22 2R L T

0 02 22 2 2

h ha a L h

h h a hB dydx B dydx B dydx

G G G

Using Timoshenko beam theory, the strain energy of the specimen due to bending and including shear effects is:

Figure 15. Schematic representation of loading in the DAL test.

(3)

M is the bending moment

subscripts R and L stand for right and left adherends

T refers to the total bonded beam (of thickness 2h)

E is the longitudinal modulus

G is the shear modulus

B is the specimen and bond width

I is the second moment of area of the indicated section

For adherends with same thickness, considered in this analysis, I = 8IR = 8IL

June 28th, 2012


The shear stresses induced by bending are given by:

2

2

31

2

V y

Bh c

c - beam half-thickness V - transverse load on each arm for 0 ≤ x ≤ a, and on total bonded beam for a ≤ x ≤ L

PU /From Castigliano’s theorem

P is the applied load

is the resulting displacement at the same point

the displacements of the specimen arms can be written as

3 3 3 3

L L R L RRL 3 3

7 3 ( ) ( )( )

2 2 5

a L F L F F a F FL a F

Bh E Bh E BhG

3 3 3 3

R L R R LLR 3 3

7 3 ( ) ( )( )

2 2 5

a L F L F F a F FL a F

Bh E Bh E BhG

(4)

(5)

June 28th, 2012


Figure 16. Schematic representation of loading in the DAL test.

The DAL test can be viewed as a combination of the DCB and ELS tests

2

LRI

FFP

II R L P F F (6) pure mode loading

pure mode displacements I R L R LII

2

(7)

June 28th, 2012


Combining equations (5-7), the pure mode compliances become

3

II 3

8 12

5I

a aC

P Bh E BhG

3 3

IIII 3

II

3 3

2 5

a L LC

P Bh E BhG

and (8) (9)

However, stress concentrations, root rotation effects, the presence of the adhesive, load frame flexibility, and the existence of a non-negligible fracture process zone ahead of crack tip during propagation are not included in these equations

To overcome these drawbacks, equivalent crack lengths can be calculated from the current compliances CI and CII (eq. 6 and 7)

eI

1 2

6a A

A

1/33 3

eII II

3 2

5 3 3

L Bh E La C

BhG

(10)

(11)

1

33 224 27

108 12 3A

I3

8 12; ;

5C

Bh E BhG

3

eI eI 0a a

June 28th, 2012


The strain energy release rate components can be determined using the Irwin-Kies equation:

2

2

P dCG

B da(12)

(13) combined with equation 6

combined with equation 7

3

II 3

8 12

5I

a aC

P Bh E BhG

3 3

IIII 3

II

3 3

2 5

a L LC

P Bh E BhG

22

eIII 2 2

26 1

5

aPG

B h h E G

2 2

II eIIII 2 3

9

4

P aG

B h E

(14)

The method only requires the data given in the load-displacement (P-) curves of the two specimen arms registered during the experimental test.

Accounts for the Fracture Process Zone (FPZ) effects, since it is based on current specimen compliance which is influenced by the presence of the FPZ.

June 28th, 2012




210 80.77 23 23 0.6 1.2


Numerical analysis including a cohesive damage model was carried out to verify the performance of the test and the adequacy of the proposed data reduction scheme. Figure 16. Specimen geometry used in the simulations of the DAL test


June 28th, 2012


su,i

sum,i

si

om,i o,i um,i u,i

i

Pure mode

model

Mixed-mode

model

Gic i = I, II

Gi i = I, II

Figure 18. The linear softening law for pure and mixed-mode cohesive damage model.

quadratic stress criterion to simulate damage initiation

2 2

I II

u,I u,II

1s s

s s

a b

c the linear energetic criterion to deal with damage growth

I II

Ic IIc

1G G

G G

(15)

(16)

Figure 17. ABAQUS simulation mixed-mode (left) mode I (right)

June 28th, 2012


l = L/R It is useful to define the displacement ratio

l = -1

l = 1

pure mode I

pure mode II

0

0.25

0.5

0.75

1

1.25

70 100 130 160 190 220a eI (mm)

GI/G

Ic

0

0.25

0.5

0.75

1

1.25

70 100 130 160 190 220a eII (mm)

GII/G

IIc

Figure 19. Normalized R-curves for the pure modes loading: a) Mode I; b) Mode II.

a b

June 28th, 2012


imposed displacem.

Simul. # beam 1 beam 2

1 10 -9

2 10 -8

3 10 -7

4 10 -5

5 10 -3

6 10 -1

7 10 0

8 10 1

9 10 3

10 10 5

11 10 7

12 10 7.5

13 10 8

14 10 8.5

15 10 9

Table 5. Imposed displacements for each simulation

six different cases were considered in the range -0.9 ≤ l ≤ -0.1

nine combinations were analysed for 0.1 ≤ l≤ 0.9

in this case mode I loading clearly predominates

a large range of mode ratios is covered

June 28th, 2012


0

0.1

0.2

0.3

0.4

0.5

75 100 125 150 175 200 225

a eI (mm)

GI (N

/mm

)

for l= 0.7, the R-curves vary as a function of crack length variation of mode-mixity as the crack grows

no plateau

0

0.2

0.4

0.6

0.8

1

1.2

75 100 125 150 175 200 225

a eI (mm)

GII (

N/m

m)

Figure 20. R-curves for l = 0.7 (both curves were plotted as function of aeI for better comparison).

no plateau

June 28th, 2012


spurious effect phenomenon

Figure 21. Spurious effect phenomenon.

the curves were cut at the beginning of the inflexion caused by the referred effects

envelope

June 28th, 2012


Figure 22. Plot of the GI versus GII strain energy components for -0.9 ≤ l ≤ -0.1.

mode I predominant loading conditions

combinations -0.9 ≤l≤ -0.7, are nearly pure mode I loading conditions

envelope

June 28th, 2012


Figure 23. Plot of the GI versus GII strain energies for 0.1 ≤ l≤ 0.9.

combinations using the positive values oflinduce quite a large range of mode ratios during crack propagation

excellent reproduction of the inputted linear criterion in the vicinity of pure modes, presenting a slight difference where mixed-mode loading prevail

explained by the non self-similar crack growth, which is more pronounced in these cases

envelope

June 28th, 2012


Figure 24. Plot of the GI versus GII strain energies for l = 0.1 and l = 0.75.

practically the entire fracture envelope can be obtained using only two combinations ( l = 0.1 and l = 0.75)

important advantage of the DAL test

envelope

June 28th, 2012


Figure 26. Envelope for Araldite 2015 with 0.2 mm bondline and l= 0.75.

experimental

0.75 mm/min 1 mm/min

Figure 25. Schematic representation of loading scheme 2 with l= 0.75 .

From table 3. GII = 2.1 ± 0.21 [N/mm]

GI = 0.44 ± 0.05 [N/mm]

June 28th, 2012

envelope

29

spelt

G. F

ern

lun

d &

J. K

. Sp

elt,

Co

mp

osi

tes

Scie

nce

an

d T

ech

no

log

y 50

(19

94)

441

-449

Figure 27. Load jig specimen geometry.

Mixed-mode testing is being implemented with a specimen load jig similar to the one that Spelt proposed, using DCB specimens used for the pure mode I (DCB) and pure mode II (ENF) and also for mixed-mode DAL.

Figure 28. Specimen tested with the Spelt load jig.

June 28th, 2012

non-conventional tests [mixed-mode I+II]


Figure 29. Schematic representation of loading in the SPELT test.

The SPELT test can be viewed as a combination of the DCB and EENF tests

(17) pure mode loading

pure mode displacements (18)

𝑅𝐺 =𝐹1+𝐹2 2 𝐿

2𝐿−𝐿1 and 𝑅𝐻 =

𝐹1+𝐹2 𝐿1

2𝐿−𝐿1 𝑅𝐴 =

2 𝐿 𝑃𝐼𝐼

2𝐿 − 𝐿1 𝑎𝑛𝑑 𝑅𝐵 =

𝑃𝐼𝐼𝐿𝐼

2𝐿 − 𝐿1

𝑃𝐼 =𝐹1 − 𝐹2

2 𝑃𝐼𝐼 = 𝐹1 + 𝐹2

𝛿𝐼 = 𝛿1 − 𝛿2 𝛿𝐼 =𝛿1 + 𝛿2

2

June 28th, 2012

(20) (19)

31

data reduction scheme spelt

𝐺𝐼 =12 𝑃𝐼

2

𝐸𝑓,𝐼𝑏2ℎ

𝑎𝑒,𝐼2

ℎ2 +1 + u

5 𝐺𝐼𝐼 =

3 𝑃𝐼𝐼2 𝑎𝑒,𝐼𝐼

2

2 𝐸𝑓,𝐼𝐼 ∙ 𝐼

Assuming 𝐺 =𝐸

2 1+u , the pure compliances become:

𝐶𝐼 =8𝑎3

𝐸𝑏ℎ3 +24 ∙ 𝑎 1 + u

5𝐸𝑏ℎ 𝐶𝐼𝐼 =

1

𝐸 𝐼 𝑎3 +

2

3 𝐿 𝐿1

2 +12 𝐿1 𝐿 1+u

5 𝐸 𝑏 ℎ 2𝐿−𝐿1

June 28th, 2012

(22) (21)

(24) (23)


32

numerical model spelt



210 80.77 23 23 0.6 1.2


Numerical analysis including a cohesive damage model was carried out to verify the performance of the test and the adequacy of the proposed data reduction scheme. Figure 30. Specimen geometry used in the simulations of the SPELT test


h = 12.7 mm 2L = 260 mm L2 = 35 mm b = 25 mm

June 28th, 2012


33


P54

P96

Figure 31. Job manager (left) and two combinatons .

June 28th, 2012


34


Linear criterion

Figure 32. Spelt numerical fracture envelope plot.

June 28th, 2012


35


Figure 33. numerical envelope plot for MMB, DAL and SPELT.

Linear criterion

June 28th, 2012



spelt experimental Y = 56º

Figure 34. P- curve for Y = 56º Figure35. P-Da curve for Y = 56º

Figure 36. R curve for Y = 56º

June 28th, 2012

37

spelt experimental envelope

Figure 37. Experimental envelope (Araldite 2015)

June 28th, 2012


conclusions 38

a new data reduction scheme based on specimen compliance, beam theory and crack equivalent concept was proposed to overcome some problems intrinsic to the DAL and SPELT tests

the model provides a simple mode partitioning method and does not require crack length monitoring during the test, which can lead to incorrect estimation of fracture energy due to measurements errors

since the current compliance is used to estimate the equivalent crack length, the method is able to account indirectly for the presence of a non-negligible fracture process zone (very important for ductile adhesives)

for pure modes I and II, excellent agreement was achieved with the fracture values inputted in the cohesive model

for DAL tests a slight difference relative to the inputted linear energetic criterion was observed in the central region of the GI versus GII plot, corresponding to mixed-mode loading, which is attributed to the non self-similar crack propagation conditions that are more pronounced in these cases. The SPELT test has a nearly constant mixed-mode, providing better results for this central region of the fracture envelope.

with the DAL test only two combinations of the displacement ratio are sufficient to cover almost all the fracture envelope

June 28th, 2012

acknowledgments 39

The authors would like to thank the contribution of Edoardo Nicoli, and Youliang Guan for their work in experimental testing at Virginia Tech. The authors also acknowledge the financial support of Fundação Luso Americana para o Desenvolvimento (FLAD) through project 314/06, 2007 , IDMEC and FEUP.

Thank you.

June 28th, 2012


February 28, 2012

conventional techniques [data reduction scheme]

41

Data reduction schemes

Documents

Mixed-Mode Fracture Toughness Determination USING NON …filipechaves.com/DOCs/presentation_IJSI_2012.pdf · 2018. 2. 15. · Consequently, the fracture characterization of bonded