Lecture 6 2D Interaction and Trajectory

7/28/2019 Lecture 6 2D Interaction and Trajectory

1/34

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

102


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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?

103


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Examples of These Observations

Very non-simple crack shapes!

Symmetric structure, BCs:

Unsymmetric crack growth??

104


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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

105


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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

106


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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

107


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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.

110


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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

114


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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 !

119

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

121


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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


22/34

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)

127

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


-100

-75

-50

-25

0

25

50

75

100

-4 -3 -2 -1 0 1 2 3 4

8T3KI

2rc


128

Normalized Crack Turning Plots for an Elastically Isotropic Material


28/34

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

Lecture 6 2D Interaction and Trajectory

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Transcript of Lecture 6 2D Interaction and Trajectory