Fracture Mechanics & Failure Analysis:Lecture Toughness and fracture toughness
Mixed-Mode Fracture Toughness Determination USING NON...
Transcript of Mixed-Mode Fracture Toughness Determination USING NON...
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.
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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.
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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.
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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
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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.
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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
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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.
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conventional tests [mixed-mode I+II] 10
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
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(1)
(2)
conventional tests [mixed-mode I+II] 11
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®
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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.
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non-conventional tests [mixed-mode I+II] 13
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.
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non-conventional tests [mixed-mode I+II] 14
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
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non-conventional tests [mixed-mode I+II] 15
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)
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non-conventional tests [mixed-mode I+II] 16
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)
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non-conventional tests [mixed-mode I+II] 17
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
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non-conventional tests [mixed-mode I+II] 18
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.
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non-conventional tests [mixed-mode I+II] 19
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.
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
The specimen was modelled with 7680 plane strain 8-node quadrilateral elements and 480 6-node interface elements with null thickness placed at the mid-plane of the bonded specimen.
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non-conventional tests [mixed-mode I+II] 20
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)
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non-conventional tests [mixed-mode I+II] 21
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
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non-conventional tests [mixed-mode I+II] 22
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
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non-conventional tests [mixed-mode I+II] 23
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
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non-conventional tests [mixed-mode I+II] 24
spurious effect phenomenon
Figure 21. Spurious effect phenomenon.
the curves were cut at the beginning of the inflexion caused by the referred effects
envelope
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non-conventional tests [mixed-mode I+II] 25
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
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non-conventional tests [mixed-mode I+II] 26
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
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non-conventional tests [mixed-mode I+II] 27
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
non-conventional tests [mixed-mode I+II] 28
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]
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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.
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non-conventional tests [mixed-mode I+II]
non-conventional tests [mixed-mode I+II] 30
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
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(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
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(22) (21)
(24) (23)
non-conventional tests [mixed-mode I+II]
32
numerical model spelt
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.
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
The specimen was modelled with 3992 plane strain 8-node quadrilateral elements and 382 6-node interface elements with null thickness placed at the mid-plane of the bonded specimen.
h = 12.7 mm 2L = 260 mm L2 = 35 mm b = 25 mm
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non-conventional tests [mixed-mode I+II]
33
numerical model spelt
P54
P96
Figure 31. Job manager (left) and two combinatons .
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non-conventional tests [mixed-mode I+II]
34
numerical model spelt
Linear criterion
Figure 32. Spelt numerical fracture envelope plot.
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non-conventional tests [mixed-mode I+II]
35
numerical model spelt
Figure 33. numerical envelope plot for MMB, DAL and SPELT.
Linear criterion
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non-conventional tests [mixed-mode I+II]
non-conventional tests [mixed-mode I+II] 36
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
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37
spelt experimental envelope
Figure 37. Experimental envelope (Araldite 2015)
June 28th, 2012
non-conventional tests [mixed-mode I+II]
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