1 6. Elastic-Plastic Fracture Mechanics Introducti on Applies when non-linear deformation is confined to a small region surrounding the crack tip LEFM: Linear elastic fracture mechanics Elastic-Plastic fracture mechanics (EPFM) : Effects of the plastic zone negligible, linear asymptotic mechanical field (see eqs 4.36, 4.40). Generalization to materials with a non-negligible plastic zone size: elastic-plastic materials crack Plastic zone Elastic Fracture Contained yielding Full yielding Diffuse dissipation D L a ,, L aDB B L D a L D a LEFM, K IC or G IC fracture criterion EPFM, J C fracture criterion Catastrophic failure, large deformations
6. Elastic-Plastic Fracture Mechanics. Plastic zone. crack. B. L. a. D. Introduction. LEFM : Linear elastic fracture mechanics. Applies when non-linear deformation is confined to a small region surrounding the crack tip. - PowerPoint PPT Presentation
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6. Elastic-Plastic Fracture Mechanics
Introduction
Applies when non-linear deformation is confined to a small region surrounding the crack tip
LEFM: Linear elastic fracture mechanics
Elastic-Plastic fracture mechanics (EPFM) :
Effects of the plastic zone negligible, linear asymptotic mechanical field (see eqs 4.36, 4.40).
Generalization to materials with a non-negligible plastic zone size: elastic-plastic materials
crack
Plastic zone
Elastic Fracture Contained yielding Full yielding Diffuse dissipation
D
L
a
, ,L a D B
B
L D a L D a
LEFM, KIC or GIC fracture criterion
EPFM, JC fracture criterion
Catastrophic failure, large deformations
2
Surface of the specimen Midsection Halfway between surface/midsection
Plastic zones (light regions) in a steel cracked plate (B 5.0 mm):
Slip bands at 45°Plane stress dominant
Net section stress 0.9 yield stress in both cases.
Plastic zones (dark regions) in a steel cracked plate (B 5.9 mm):
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6.2 CTOD as yield criterion.
6.1 Models for small scale yielding : - Estimation of the plastic zone size using the von Mises yield criterion. - Irwin’s approach (plastic correction). - Dugdale’s model or the strip yield model.
Outline
6.5 Applications for some geometries (mode I loading).
6.3 The J contour integral as yield criterion.
6.4 Elastoplastic asymptotic field (HRR theory).
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Von Mises equation:
6.1 Models for small scale yielding :
1 22 22
1 2 1 3 2 312e
e is the effective stress and i (i=1,2,3) are the principal normal stresses.
Recall the mode I asymptotic stresses in Cartesian components, i.e.
3cos 1 sin sin ...
2 2 223
cos 1 sin sin ...2 2 22
3cos sin cos ...
2 2 22
Ixx
Iyy
Ixy
Kr
Kr
Kr
yy
xy
xx
x
y
Oθ
r
LEFM analysis prediction:
KI : mode I stress intensity factor (SIF)
Estimation of the plastic zone size
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We have the relationships (1) ,
1 222
1,2 2 2xx yy xx yy
xy
Thus, for the (mode I) asymptotic stress field:
1,2 cos 1 sin2 22
IKr
rrr
x
y
Oθ
r
and in their polar form:Expressions are given in (4.36).
1 222
2 2rr rr
r
31 2
0 plane stressplane strain
and
, ,rr r
3
0 plane stress2 cos plane strain
22IKr
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Substituting into the expression of e for plane stress
1 22 2 212cos sin cos (1 sin ) cos (1 sin )
2 2 2 2 2 22 2I
eK
r
Similarly, for plane strain
1 2
2 21 31 2 1 cos sin
22 2I
eK
r
(see expression of Y2 p 6.7 )
1 22 2 21
6cos sin 2cos2 2 22 2
IKr
1 221 3
1 cos sin22 2
IKr
1 2
2 2cos 4 1 3cos2 22
IKr
1 22 2 2 2 21
4cos sin 2cos 2cos sin2 2 2 2 22 2
IKr
1 22cos 4 3cos
2 22IKr
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Yielding occurs when: e Y Y is the uniaxial yield strength
Using the previous expressions (2) of e and solving for r,
22 2
22 2 2
1cos 4 3cos
2 2 2
1cos 4 1 3cos
2 2 2
I
Yp
I
Y
K
rK
plane stress
plane strain
Plot of the crack-tip plastic zone shapes (mode I):
2
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p
I
Y
r
K
Plane strain
Plane stress (= 0.0)
increasing 0.1, 0.2, 0.3, 0.4, 0.5
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1) 1D approximation (3) L corresponding to 0pr Remarks:
221 22
I
Y
KL
Thus, in plane strain:
2) Significant difference in the size and shape of mode I plastic zones.
For a cracked specimen with finite thickness B, effects of the boundaries:
in plane stress:2
12
I
Y
KL
- Essentially plane strain in the in the central region.Triaxial state of stress near the crack tip:
- Pure plane stress state only at the free surface.
B
Evolution of the plastic zone shape through the thickness:
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4) Solutions for rp not strictly correct, because they are based on a purely elastic:
3) Similar approach to obtain mode II and III plastic zones:
Alternatively, Irwin plasticity correction using an effective crack length …
Stress equilibrium not respected.
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The Irwin approach Mode I loading of a elastic-perfectly plastic material:
02
Iyy
Kr
Plane stress assumed (1)
crackx
r1
Y
r2
(1)
(2)
(2)
To equilibrate the two stresses distributions (cross-hatched region)
r2 ?
Elastic:
Plastic correction 2,yy Y r r
r1 : Intersection between the elastic distribution and the horizontal line yy Y
12I
YK
r
1
20 2
rI
Y YK
r dxx
yy
21 2
11222
I IY
Y
K Kr r
2 1r r
2
11
2I
Y
Kr
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Redistribution of stress due to plastic deformation:
Plastic zone length (plane stress):
Irwin’s model = simplified model for the extent of the plastic zone: - Focus only on the extent of the plastic zone along the crack axis, not on its shape.
2
112 I
Y
Kr
- Equilibrium condition along the y-axis not respected.
yy
real crack x
Y
fictitious crack
Stress Intensity Factor corresponding to the effective crack of length aeff =a+r1
1 1,I effK a r K a r
2
112
3I
Y
Kr
2eff
yyK
X
X
In plane strain, increasing of Y. : Irwin suggested in place of Y 3 Y
(effective SIF)
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Application: Through-crack in an infinite plate
Effective crack length 2 (a+ry)
eff yK a r
21
2Y
effeff
KK a
Solving, closed-form solution:2
112
Y
effaK
(Irwin, plane stress)2
12
Iy
Y
Kr
with
2a
ry ry
aeff
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More generally, an iterative process is used to obtain the effective SIF:
eff eff effK Y a a
Convergence after a few iterations…
Initial: IK Y a a
2
01
2I
Y
Ka am
1 plane stress3 plane strain
m
YesNo
1i i
Algorithm:
( )Ii
eff i iK K Y a a
Application:
Through-crack in an infinite plate (plane stress):
∞= 2 MPa, Y = 50 MPa, a = 0.1 m KI = 1.1209982Keff= 1.1214469 4 iterations
Y: dimensionless function depending on the geometry.