lecture 6 2d interaction and trajectory
TRANSCRIPT
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CEE 770 Meeting 6
Objectives of This Meeting
Learn theories for predicting mixed mode interaction,
crack trajectory in 2D, and stability of such trajectory:
1st Order, LEFM theories, isotropic material
Crack kinking vs crack turning: trajectory stability
2nd Order, LEFM theory, isotropic and orthotropicmaterials
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Observation and Your Thoughts?
Cracks do not usually propagate as straight lines, or flat surfaces, or perfect
ellipses.
What controls the shape (sometimes called trajectory when 2D idealization is
reasonable) of a propagating crack?
Why do some cracks in a symmetric structure with symmetric BCs notpropagate symmetrically?
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Examples of These Observations
Very non-simple crack shapes!
Symmetric structure, BCs:
Unsymmetric crack growth??
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One of My Favorite Quotes.
... NOTHING AT ALL HAPPENS IN THE UNIVERSE IN
WHICH THERE DOES NOT SHINE OUT SOME PRINCIPLE
OF MAXIMUM OR MINIMUM, WHEREFORE THERE IS
ABSOLUTELY NO DOUBT BUT THAT ALL HAPPENINGS IN
THE UNIVERSE MAY BE DETERMINED FROM FINAL
EFFECTS BY A METHOD OF MAXIMA OR MINIMA QUITE
AS SUCCESSFULLY AS FROM ACTUAL CAUSES
THEMSELVES.
L. EULER, 1744
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Recall Continuum Fracture Modes
y,v
x,u
z,wz,w
x,u
y,v y,v
x,u
z,w
Mode II Mode IIIMode I
Basic modes of crack loading. Positive sense shown for each:Mode I = crack opening
Mode II = in-plane sliding
Mode III = anti-plane tearing
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And, the four shell fracture modes
x
y
z
rh 2
membrane
KI KII
bending
K2,K3 Reissner theory
k2 Kirchhoff theory
K1 Reissner theory
k1 Kirchhoff theory
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1st Order, LEFM Crack Kinking Theories
1st Order LEFM theories are based on only the singular terms ofthe local asymptotic LEFM crack front fields.
Many such theories have been proposed and tested, and mostof these are variants of these 3:
Maximum Hoop Stress Theory
Maximum Energy Release Rate Theory
Minimum Strain Energy Density Theory
max Erdogan and Sih (1963)
G()max Hussain et al. (1974)
S()min Sih (1974)
We will study only the theory, here, but will return to the concept
of maximum energy release rate theory later. Why, and why?
max
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1st Order, LEFM, Isotropic Crack Kinking Theories:
max Theory
Recall equation 16, p.36:
y
( )1cos3sin
=
c
c
I
II
K
K
This theory asserts that, for an isotropic material, a crack will kink into the directionnormal to the maximum circumferential (hoop) stress. So maximize (16) wrt ,ignoring T-stress, set =
c, and rearrange,
( )
2cos1
2sin
2
3
2cos
2cos
2
1 2 +
=T
KKr
III(16)
c = 2tan1 1 1 + 8(KII KI)
2
4(KII KI)
Then solve for c
(65)c max
max
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Mixed-Mode Interaction According to
max Theory
Rewrite (16), again neglecting T-stress, and recognize that the new left-handside represents a measure of fracture toughness that, in the limit of Mode Ionly, must beKIc
=
sin
2
3
2cos
2cos
2
1 2III KK
r
IcIIIc KKKr =
=
sin2
3
2cos2cos2
(66)
Equations 65 and 66 comprise a parametric set in c, KI, andKII.These can be
solved to produce an interaction diagram that is analogous to a multi-axialyield interaction diagram, or a biaxial bending yield-crushing diagram.
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Mixed-Mode Interaction According to
max and Other Theories
111
= tan-1(KII/KI)
Note that each theory has itsown interaction surface, and
its own Mode II toughnessprediction
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
Gmax
max
Smin
Keff
/ c
c
K
/K
( = 0.25)
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Another Way To View Effective,
Mixed-Mode Toughness
112
= tan-1(KII/KI)
0
0.2
0.4
0.6
0.8
1
1.2
0 30 60 900 90, Load Angle,
K
/
K
(
)
Ic
maxmax
max
Gmax
Smin ( = 0.25)
(b)
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Comparison of 1st Order, Linear Elastic, Isotropic
Crack KinkingTheories: Kink Angle
Mode II Mode I
Me = 2
tan
1 KI
KII
Mode mixity parameter:
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Demonstration of the Maximum Hoop Stress Crack
Turning Criterion SEN(B) polymethylmethacrylate (PMMA) beams.
Initial crack location and length were varied among thespecimens.
10.0 10.0
9.09.0
a
b
8.0
4.02.75
2.0
2.0 0.5 dia.
typ.
P
Note: all dimensionsin inches
thickness: 0.5
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Comparisons between observations and predictions
for two different initial crack configurations
2.5 inches(from bottom
of plate)
6.0 inches(from centerline)
pre-cut slot
Analysis crack-increment lengths:
a = 0.3 inch
a = 0.2 inch
a = 0.05 inch
This is VG predicting!
How big is processzone in PMMA?
analysis
experiment
1.0 inch 6.0 inches(from centerline)
pre-cut slot
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Crack Kinkingversus Crack Turning
Crack path problems encountered in most real structural applications are notreally crack kinking problems. In an average macroscopic sense, cracks typicallypropagate in a rather smoothly turning fashion as the crack negotiates its wayamong the structural features of the part.
Since the first-order isotropic theories predict crack kinking for non-zeroKII ,the only way for a crack to propagate smoothly is for the crack to follow a pathalong whichKII=0.
Since all the first-order isotropic theories agree exactly for this condition, thecrack path is apparently independent of any first-order theory.
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Then Why Do Some Mode I Cracks Turn?
There appears to be some trajectory instability phenomenon at work.
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Then Why Do Some Mode I Cracks Turn?
Crack Turning TheoryConsider a Mode I crack subjected to a small trajectory perturbation at x= 0, i.e.the crack propagates very slightly out of its self-similar direction and feels somesmall, correspondingKII. Also, lets include the first 2
nd order field term, the
T-stress. Cotterell and Rice (1980) then asked:
What happens to continuing trajectory if we enforce thecondition that subsequentKII=0?
0
T
I
II
K
K20 =
IK
T22=Strength of the T-stress:Strength of the perturbation:
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Cotterell and Rice Crack Turning TheoryCotterell and Rice found that subsequent trajectory is
Normalized Plot of the Perturbed Crack Path of Cotterell and Rice (1980).
I
II
K
K20 =
IK
T22=
( ) ( )
=
x21xerfcxexp(x) 2
2
o
Trajectory unstable forPositive T-stress !
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M M i l E hibi M C li d B h i S h
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Many Materials Exhibit More Complicated Behavior Such as
Toughness Orthotropy and Crack Path Sensitivity to Load Level
Objective: develop a theory for crack turning in real materialsbased on LEFM concepts
120
M t i l O i t ti D fi iti f F ti
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Material Orientation Definitions for Fatigue
and Fracture
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2nd Order Theories: Role of T-Stress
Cotterell and Rice (1980) crack perturbation theory highlighted the importanceof the T-stress in trajectory predictions.
Their work inspired the creation of 2nd order theories for prediction of crackshape.
We will investigate one of these 2nd order theories, and extend our thinking
about crack shapes to the more general case of materials with anisotropictoughness.
122
Recall T Is Second Term of the Crack tip Stress Expansion
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Recall T Is Second Term of the Crack-tip Stress Expansion
( )
+
+
++=
)2sin(
)2cos(1
)2cos(1
2)1)cos(3()sin(
)sin()2(cos
)2tan(2)sin()2(sin1
2cos
2
1232
2
32
T
KK
KK
KKK
rIII
III
IIIII
r
rr
Include the T-term and the maximum hoop stress expression then becomes:
KIIKI
=2sin( 2)
3cos1cos
2
8
3
T
KI2rc cos
(67)
r
x
y
T
rc is the distance from the crack tip at which
the stresses are computed.
rc for plastic tearing is theorized to be a
material constant.
rc scales with the plastic zone size.
Kosai, Kobayashi, and Ramulu, Tear straps in aircraft fuselage,Durability of metal aircraft structures: Proc. of Int. Workshop
Structural Integrity of Aging Airplanes, Atlanta, GA, 443-457, 1992123
2nd Order Linear Elastic
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2nd Order Linear Elastic,
Crack TurningTheory, Isotropic Case
Normalized Crack Turning Plot for Isotropic Material Based on the Formulation ofKosai et al. (1992).
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125
-80
-60
-40
-20
0
20
40
60
80
-4 -3 -2 -1 0 1 2 3 48T
3KI2rc
tan1(KII KI) = 90o
tan
1(KII KI) = 0
o
90o
67.5o
67.5o
45o
22.5o
22.5o
5o
5o
1o
1o
tan1
(KII KI)
criterion
45o
Crack turning
interaction diagram
max
Pettit, Wang, and Toh,Integral airframe structures (IAS) - validated feasibility study of integrally stiffened metallic fuselage panels for
reducing manufacturing cost, Boeing Report CRAD-9306-TR-4542, NASA contract NAS1-20014, Task 34, November, 1998.
Conceptual Model of a Crack Propagation
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Conceptual Model of a Crack Propagation
Criterion for a Toughness Orthotropic Structure
rc
)( evaluated at rc
c
Predicted direction ofcrack propagation
Material toughness function
criticalcc
c21III
K)(K
),r,T,E,E,K,K(Maximum
=
(68)
Boone, Wawrzynek, and Ingraffea, Analysis of fracture propagation in orthotropic materials,Engng Fracture Mech,
Vol. 35 (1990) pp. 159-170126
i l i f h
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A Simple Representation for Toughness
Anisotropy: The Toughness Ellipse
Kp()n cos
2
Kp(0)n +
sin2
Kp(90)n
= 1
Kp is the stress intensity at which the crackpropagates, in the relevant regime of crackgrowth. Thus, for fatigue crack growth, Kpis the stress intensity at which the crackpropagates at a given rate; for stable
tearing, Kp represents the fracturetoughness.
Km Kp(90)
Kp(0)
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Typical 2nd Order
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1.0
1.2
crack
ellipse for anisotropic
fracture resistance
Kc LT
KcTL=1.2
Typical 2nd Order
Interaction Diagrams for
Orthotropic Toughness
-100
-75
-50
-25
0
2550
75
100
-4 -3 -2 -1 0 1 2 3 4
8T3KI
2rc
crack oriented at = 0o
-100
-75
-50
-25
0
25
50
75
100
-4 -3 -2 -1 0 1 2 3 4
8T3KI
2rc
crack oriented at = 45o
-100
-75
-50
-25
0
25
50
75
100
-4 -3 -2 -1 0 1 2 3 4
8T3KI
2rc
crack oriented at = 90o
128
Normalized Crack Turning Plots for an Elastically Isotropic Material
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129
with Fracture Orthotropy =1.6, n = -1,
Various Crack Orientations.
(a) Crack Oriented at =0, (b) CrackOriented at =45, (c) Crack Oriented at
=90.
Km
.inksi70KI =lli di i (L)
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1.1)L(K
)T(K
in.0.05r
ksi50T
0K
c
c
c
II
I
=
=
=
=rolling direction (L)
130
(T)
No T-Stress T-Stress T-Stress and Orthotropy
o0c =o
5.23c = o6.45c =
Propagation direction
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Observed and predicted crack
paths for 7050-T7451 DCBspecimens, Static Loading
131
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
0 1 2 3 4 5 6 7 8 9 10
Horizontal Crack Growth (in)
rc=0 , Km = 1.3
rc=.05 inches, Km = 1.3
rc=.1 inches, Km = 1.3rc-LT-15-5
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
0 1 2 3 4 5 6 7 8 9 10Horizontal Crack Growth (in)
VerticalCrackGrowth
(in)
rc-TL-15-5
rc=.05 inches, Km =1.3
rc=0, Km =1.3
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Observed and predicted crack paths for 7050-T7451
DCB specimens, Fatigue Loading
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
0 1 2 3 4 5 6 7 8 9 10
Horizontal Crack Growth (in)
VerticalCrackGrowth(in)
FRANC2D, Km=1.1, rc=0
rc-LT-15-2
rc-TL-15-2
132
What About These Data?
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What is Going on Here?
-40
-20
0
20
40
60
80
100
0 10 20 30 40 50 60 70 80 90
Mode mixity, -
c
LEFM Max stress
Pure mode virtual kink, Eq. (5.30)
Curve fit to 2024-T3 Test Data [40]
SSY CTOD Analyses [40]
2024-T3 Arcan Test Data [40]
Mode I
Dominated
Mode IIDominated
Look only at the test data
and the LEFM Max stress,information.max ,
There is an obvious, abruptchange in trajectory behavior.Why?
133
Predicted Effect of T-Stress on Kink Angle for
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g
Mode II Crack According to Maximum Shear Stress
Theory, Isotropic Case.
How would you formulatesuch a theory?
134
Predicted Effect of T-Stress on Kink Angle for
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g
Mode II Crack According to Maximum Shear
Stress Theory, =1.6, n=-1KII m
(a) Crack Oriented at = 0 (b) Crack Oriented at = 90
135