Transcript Slide 1

The Elastic Stress Field Approach
The Stress Intensity Factor
The three basic modes of crack surface displacements
Derivation of the Elastic Stress Field Equations
- Concepts of plane stress and plane
strain
- Equilibrium equations
- Compatibility equations for strains
- Airy stress function
- Biharmonic equation
- Complex stress functions:
Westergaard function for biaxially
loaded plate (Mode I)
- Mode I stress / displacement fields
- Mode I stress intensity factor
Westergard (1939), Irwin (1957)
X 
 a


3 
cos 1  sin sin 
2
2
2
2r
Y 
 a


3 
cos 1  sin sin 
2
2
2
2r
 XY 
 a


3
sin cos cos
2
2
2
2r
Linear Elastic Crack-tip Fields
(general case)
Mode I:
Mode II:
Mode III:
CHARACTERISTICS OF THE
STRESS FIELDS
- The stress and displacement formulas may
reduced to particularly simple forms:
-Details of the applied loading enters
only through K !!!
for the infinite plate: K = (*a)1/2
- But for a given Mode there is a
characteristic shape of the field !!!
- Principle of Superposition: for a given
Mode, K terms from superposed loadings
are additive
Angular distributions of crack-tip stresses for the
three modes (rectangular: left; polar:right)
We consider next some other cases apart from the cracked infinite plate
- Semi infinite edge notched specimens
- Finite width centre cracked specimens
- Finite width edge notched specimens
edge notched
-Crack-line loading
-Elliptical / Semielliptical cracks
a
K I  C * a * f  
W 
finite width
Finite-width centre-craked specimens:
f(a/W)
Irwin:
K I   a
W
 a 
tan 
a  W 
2
a
a
a
a
Brown: f    1  0.256   1.152   12.200 
W 
W 
W 
W 
approx.
Isida: 36 term power series
Feddersen:
 a 
K I   a sec 
W 
3
Semi infinite edgenotched specimens:
Free edges: crack opens
more than in the infinite
plate resulting in 12%
increase in stress
K I  1.12* a
Single edge notched
(SEN)
SEN:
Finite-width
edge-notched
specimens:
Double edge notched
(DEN)
2
3
4


a
a
a
a








K I   a * 1.122  0.231   10.550   21.710   30.382  

W 
W 
W 
 W  

0.5% accurate
for a/W < 0.6
DEN:
0.5% accurate
for any a/W
a
1.122  1.122
W
K I   a *
2
3

a
a
a
  0.820   3.768   3.040 

W 
W 
W 
2a
1
W
4
TWO IMPORTANT SOLUTIONS FOR PRACTICAL USE
* Crack-line Loading
(P: force per unit thickness)
K IA 
P
K IB 
P
ax
ax
a
a
ax
ax
for centrally located force:
KI 
P
a
KI decrease when crack
length increases !
Crack under internal pressure (force per unit thickness is
now P.dx, where P is the internal pressure)
KI 
Very useful solution:
- Riveted, bolted plates
- Internal Pressure
problems
KI 
P
a
P
a

a
a

a
0
ax
P
dx 
ax
a

a
0
 ax
ax

dx

 ax

a

x


a
2 Pa 
x
dx 
arcsin
 P a
a  0
a 
a2  x2
2a
same result by end loading with  !
* Elliptical Cracks
actual cracks often initiate at surface
discontinuities or corners in structural
components !!!
We start considering idealised
situations:
from embeded elliptical crack to
semielliptical surface cracks
Example: corner crack in a
longitudinal section of a
pipe-vessel intersection in
a pressure vessel
The embeded (infinite plate) elliptical
crack under Mode I loading
Irwin solution for Mode I:
 a 
KI 

a
2
 sin   2 cos  
c



2
1
4
2
Where : elliptic integral of the second type



2
1  sin 2  sin 2  d
0
with:
sin
2
c

a/c
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0

1.000
1.016
1.051
1.097
1.151
1.211
1.277
1.345
1.418
1.493
1.571
KI varies along
the elliptical
crack front
max. at  =/2:
min. at  = 0:
KI 
KI 
1
*  a

  a 2 / c 

2

 a2
c2
circular crack:
KI 
2

*  a
During crack growth an
elliptical crack will tend to
become circular:
important in fatigue
problems
The semi- elliptical surface crack in a plate of finite dimensions
under Mode I loading
In practice elliptical cracks will generally occur as semi-elliptical surface cracks or quarterelliptical corner cracks
Best solutions for semielliptical:
FEM calculations from Raju-Newman:
Raju I.S.,Newman J.C. Jr. Stress Intensity Factors for Two Symmetric
Corner Cracks, Fracture Mechanics, ASTM STP 677, pp. 411-430 (1979).
SUPERPOSITION OF STRESS INTENSITY FACTORS
1) Crack under
Internal Pressure:
K IA  K IB  K IC  K ID  0
K ID  K IC   a
H  P
2) Semi-elliptical
Surface Crack in a
Cylindrical
Pressure Vessel:
R
B
K I  K I H
P  
K IP  P a
K I H 
C H a CPR a


B
K IP 
CP a

 R
CP1   a
 B
 K IP 

3) Cracks growing
from both sidesof a
loaded hole where
the hole is small
with respect to the
crack
  a
P  a
f 
K I  

2 a   W 
 2
where P is the force per unit
thickness
CRACK TIP PLASTICITY
First approximation:
Better approaches
-selected shape: better size estimation
1
ry 
2
 K

  YS



2
-Irwin
-Dudgale
-Better shape but first order
approximation for the size
Irwin approach:
- stress redistribution; elastic – plastic; plane stress
rp  2 * ry
First Order Aproximations of Plastic Zone Shapes
Through-thickness plastic zone in a
plate of intermediate thickness
Plastic zone shape from Von Mises
yield criterion
Empirical Rules to estimating Plane Stress vs. Plane
Strain conditions:
-Plane Stress: 2.ry ≈ B
-Plane Strain: 2.ry < 1/10 B
Planes of maximum shear stress: location of the planes
of maximum shear stress at the tip of the crack for plane
stress (a) and plane strain (b) conditions
Deformation Modes: plane strain (a) and plane stress (b)
FRACTURE TOUGHNESS
Is K a useful parameter to characterise fracture
toughness?
Under conditions of:
- small scale plasticity
- plane strain
Kc = KIC
Variation in KC with specimen thickness in a high
strength maraging steel
KIC is a material property:
fracture toughness of
linear elastic materials
Effect of Specimen Thickness on
Mode I Fracture Toughness
Limits to the Validity of LEFM:
After considerable experimental work
the following minimum specimen size
requirements were established to be in a
condition of :
- plane strain
- small scale plasticity
K
a, B, W - a   2.5 IC
  YS
1
a, B, W - a   2.5 * 2 *
2
Remember: Empirical Rules to estimate Plane Strain conditions:
 K IC

  YS
2



2

  2.5 *  * 2ry  8 * (2ry )

2*ry < 1/10 B
LEFM Testing: ASTM E-399, committee E8 Fatigue and Fracture
Fatigue pre-cracked specimens !
KI 
LOAD* S
a
*
f
 
B *W 3 2
W 
where
12
a
f 
W 
ASTM Standard Single Edge notched Bend
(SENB) Specimen
a
3 
W 
KI 
2

a
a 


a
a 
1.99  1  2.15  3.93   2.7  
W  W 
W 
W  



32
a 
a

21  2 1  
W  W 

LOAD
a
*
f
 
B *W 1 2
W 
where
ASTM Standard Compact Tension
(CT) Specimen
a
f
W



a

2 
 W
2
3
4



a
a
a
a 
0.886 4.64   13.32   14.72   5.6  

W 
W 
W 
W  


32
a

1  
 W
Clip gauge and ist
attachment to the specimen
ANALYSIS
-Line at 5% offset (95 % of tg OA equivalent
to 2 % crack extension
- Ps: intersection 5 % offset with P-v record
- if there is a P value > Ps before Ps, then
PQ = Ps
- check if Pmax / PQ < 1.10, then
- go to K(PQ): calculate KQ (conditional KIC)
- check if for KQ the specimen size
requirements are satisfied, then
- check if crack front is symmetric, then
KQ = KIC (valid test)
Principal types of load-displacement plots obtained
during KIC testing
Material Toughness Anisotropy
To provide a common scheme for
describing material anisotropy, ASTM
standardized
the
following
six
orientations:
L-S, L-T, S-L, S-T, T-L, and T-S.
The first letter denotes the direction of the
applied load; the second letter denotes
the direction of crack growth. In designing
for fracture toughness, consideration of
anisotropy is very important, as different
orientations can result in widely differing
fracture-toughness values.
When the crack plane is parallel to the rolling direction,
segregated impurities and intermetallics that lie in these
planes represent easy fracture paths, and the toughness
is low. When the crack plane is perpendicular to these
weak planes, decohesion and crack tip blunting or stress
reduction occur, effectively toughening the material. On
the other hand, when the crack plane is parallel to the
plane of these defects, toughness is reduced because the
crack can propagate very easily.
SUBCRITICAL CRACK PROPAGATION IN COMPONENTS
WITH PREXISTING FLAWS
Fatigue
Sustained load crack growth behaviour
- stress corrosion cracking
*
- cracking due to embrittlement by internal or external gaseous
hydrogen
- liquid metal embrittlement
- creep and creep crack growth
Fatigue Crack Propagation
DKmax = KIC (1-R) !!
Fatigue crack growth rate curve da/dN - DK
How to describe crack growth rate curves: crack growth “laws“
Paris Law:
Forman:
da
 C (DK ) m
dN
only Region II, no R effects
da
C (DK ) m

dN (1  R) K IC  DK
also Region III


D
K


th
1  
 
da

DK  
 C (DK ) m  
n2 
dN
  K max  
1   K  
  IC  
n1
Complete
curve
McEvily:
n3

da
DK
2

 C (DK  DK th ) 1 
dN
 K IC  K max 
complete crack
growth rate curve
n1, n2, n3,
empirically adjusted
parameters
the three
regions
EXAMPLES
of Crack
Growth rate
Behaviour
Fatigue crack growth rate for Structural
steel (BS4360) at room temperature and
with cycling frequencies 1-10 Hz.
Fatigue crack growth rate vs. DK for
various structural materials at low R values
Influence of R on
fatigue crack
growth in Al 2024T3 Alclad sheet
Effect of R:
da
 f (DK , R )
dn
CRACK
CLOSURE
Crack closure effects
Measuring the crack opening stress by means
of a stress-displacement curve
Elber:
Actually:
DKeff  Kmax  Kop
DKeff  Kmax  Kmin,eff
da
 f (DK , R )  f DK eff 
dn
Elber obtained the empirical relationship
Schijve:
DK eff
DK
DK eff
DK
 U  0.5  0.4 R
 U  0.55  0.35R  0.1R 2
Closure Mechanisms
Sustained load crack growth behaviour
Time to Failure Tests:
Initial KI !!!
preferred
technique in
the past
Generalised sustained load crack growth behaviour
KISCC
KIC or KQ
Modern techniques: based on fracture mechanics parameter K !
SPECIMENS
Increasing or decreasing K specimens
crack- line
wedge-loaded
specimen
(CLWL)
bolt loaded
cantilever beamspecimen (DCB)
tapered double
cantilever beamspecimen
(TDCB):
constant K !!!
Difference in crack growth behaviour for increasing K (cantilever beam) and
decreasing K (modified CLWL or DCB specimens)
Decreasing K specimens:
Entire crack growth with one
specimen
Advantages:
Self stressed and portable
Claer steady state and arrest
Corrosion product wedging:
Disadvantages:
gives higher crack growth rate
at a given nominal KI.
Example:
Outdoor exposure stress corrosion cracking propagation in 7000 series Al-alloy plate