Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
1
Fracture processes at different scales
Micromechanical processes at the crack tip
,ij ijσ ε
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
2
Fracture mechanical assessment of strength
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
3
Classification of fracture phenomena
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
4
Definition of the three crack opening modes
Crack in the infinite domain under tensile loading:
top) Coordinate system, bottom) crack opening and stress distribution
Γ +
Γ −
x2z
x1a2
σ
σ
x2
x1
zr θ
a z0
σ22
σσ
a− a+x1
x2
x1a− a+
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
5
Stresses at the crack tip in Cartesian and cylindrical coordinates
Analysis of the fields near the crack tip
x2
x3x1
σ22
σ11
rθ
x3
θθσ
rrσθθσ
rrσrθτ
rθτrθτ
θ
σ11
σ22
τ12 τ21
τ12
τ21
rθτ
2x
1x
rθ
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
6
Elliptical internal crack in infinite domain with coordinate system
Coordinate system along the crack front in space
x2
x3
x1
σ
σ
x2
x1
A P
Ca
a
c c
ϕ
2x
1xrv
n
2kσ
3x
θ
t
s
crack front
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
7
Dominance of near crack tip field at all specimens and components
Energy balance at crack propagation by ∆A
t t
ct
b b
A AA∆ A∆
tS tS
uS uS
Kr
( )IIij ij
K fr
σ θπ
=2
process -zone
near field
component
specimen
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
8
Relation between force–displacement–diagram and energy release rate
Work of unloading during crack propagation
F
q
a a∆
F
F∆ G B a∆
A C
D
0
a a∆+a
F q∆ ∆12
q∆q
( )int a
k1
a s∆ −
a∆a
a s∆ −
x2
,x s1
( ), ,u r a s a a∆ θ π ∆= − = +2
( ), ,r s aσ θ= =22 0
s
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
9
Towards stability of crack propagation
Transformation of external loads into equivalent crack face loads
*F3
F2
F1
( )R a∆
a0*a a
R
G
*G
CG 0
q1
q2
q3
F F F> >3 2 1
q q q> >3 2 1
a∆
c−t
t
ct
t
)a )b )c
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
10
Definition of J as contour integral around the crack tip
Path – in dependence of J–Integral
X2
X1
a
x2
x2a∆
AΓ
ds
nn2n1
εΓ
1Γ
2Γ2 1A A A= −
rΓ +
Γ −
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
11
Example for the Superposition of crack face loads
Derivation of fracture mechanical weight functions
x2x1
P
Q
Q
P
c−
( )q x1
dQ dPc+ x1
( )1ttS
( )1u
a a∆ a∆( )1b
uS
a
tS( )2t
( )2u
( )2b
uS
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
12
Common specimen types used for fracture mechanics tests:
• Compact tension specimen (CT), • Single edge notched specimen (SENB), • Centre Cracked tension specimen (CCT) and • tension specimen with semi – elliptical surface crack M(T)
a
,F q
HwB
2F S 2F
B w
,F q
a
,F q,F q
B
H
2w
2a H
B
2w
2a
2c
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
13
Relation between cyclic loading and stress intensity factor
Evolution of stress intensity with growing crack length a as function of time t and number of load cycles N
maxσ
minσ
σ
t
∆σ
( )tσ
( )tσ
KmaxK
minK
K∆
t
min
max
R σσ
=min
max
KRK
=
K
cKmaxK
minK
a
Time t
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
14
Crack growth rate as function of cyclic stress intensity factor
Crack closure effect (left) due to plastic deformations of crack faces (right)
IIIS
IIS
IS
( )c cK K R∆ = −1
( )c cK K R∆ = −1 log K∆
R
log
(da/
dN)
K
t
maxK
minK
K∆ effK∆opK
crack closed
Time envelope ofplastic zones
aktiveplasticzone
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
15
Formation of plastic zones at fatigue cracks
Effect of loading sequence: a) on single overload, b) a low-high block and c) a high-low block sequence
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
16
Stages of plastic deformation in bodies with cracks
Model of small scale yielding SSY
LF / F
brit t le fracture,no plast icdeformat ion
brit t le fracture,small plast iczone
fracture at largescale plast icyielding
duct ile fractureafter large plast icdeformat ion
L
elast icbehaviorF F < L
small scaleyieldingF F L
plast iclimit loadF = F > L
plast iccollapseF F
1
Displacement
2a
LEFM SSY LSY PC
Kr
pr
t
u
or
-fieldK
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
17
Shape of plastic zones at small scale yielding
Slip planes of maximum shear stress for plane stress state (a) and plain strain state (b)
( )I F
xK
πσ
22
( )I F
xK
πσ
12
.0 7
.0 5
.0 3
.0 2
.0 1
0 .0 2 .0 3 .0 4 .0 5
EVZ
ESZ
σ σ
B( )a ( )b
45x2
x1x3
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
18
Estimation of plastic zone at the crack tip (IRWIN)
Slip line field at crack tip ideal-plastic material (EVZ)
C
B
A
135 45
θ
elastic
elastic - plastic
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
19
Stress distribution at crack Tipp for rigid-ideal-plastic material (EVZ)
RAMBERG-OSGOOD power – law hardening
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
20
Stress distribution at crack tip according HRR-solution for the harden-ing exponents n = 3 (above) und n = 10 (below)
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
21
Influence of stress triaxiality on the shape of plastic zones at the crack tip in different specimens. Biaxial parameter
11 IT a Kβ πΤ =
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
22
2
1
20 1 3 4 5 6
3
asymptotic solution HRRsmall deformationenlarge deformationen
θθσ
rrσ
0/ijσ σ
( )0/ /r J σ
Comparison of numerical crack tip solutions at small and large defor-mations with analytical HRR-field, EVZ, n = 10
Comparison of terms of higher order with HRR-field, EVZ, n = 10, = −0,5
3,0
2,0
1,0
0,0
1,0−0 30 60 90 120 150 180
3,0
2,0
1,0
0,0
1,0−0 30 60 90 120 150 180
rrσ θθσ10
0,52,59
nTr
== −=
100,5
2,59
nTr
== −=
FEMHRR
stressQ −three terms
FEMHRR
stressQ −three terms
Angle θ Angle θ
4,0
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
23
DUGDALE-Model for strip yielding plastic zone
Limit curve of Failure diagram FAD
σσ
σ
d a2 d c2
Fσ
Fσ
plastic Zones
=
+
1
.1 2
.0 8
.0 6
.0 4
.0 2
00 .0 2 .0 4 .0 6 .0 8 .1 0 .1 2
brittle fracture
plasticCollapse
safe
r r rln cosK S Sππ
−− =
11 2
28
2
r FS σ σ=
P
Kr =
KI /
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
24
Experimental determination of JIc by means of single and multiple
specimen method
Single specimen method for determination of JIc
F
q
a a∆
F F
q q
)a )b )c
eint
F aa∆∂
∂a
int∆−
a a∆+cF
cq
Pint
q
F
Fσ zFσ
aw
F
pintpJ
eJ
eint
pq eq q
)a )b
b
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
25
Ductil crack growth resistance curve
Crack growth resistance curves a crack initiation values for different specimen geometries for steel 22NiMoCr37
J-In
tegr
al
Crack growth
¢ a mm
0 .0 25 .0 50 .1 0 .1 50 .2 0 .2 50blunt ing
initiation
growth
blunt ingline
o l0.2 mm ffset ine
fit t ing curve
validnot valid
IcJ
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
26
Integration paths around the crack tip and weighting functions q
Contour-area-Integral in case of generalized loading
x1
x2
εΓ
inin−Γ +
Γ −
inA
Γ
r
q =1
q = 0
Γ +t( )T x
Γ −
x2
x1
b
n
ΓA
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
27
Three-dimensional crack configuration with virtual crack front exten-sion
Integration domain for three-dimensional J-Integral
x3
x2
x1
endSS +
S −
endSε
jnS
Sε
endSS
endSε
x2
x1
x3
s
a∆
s∆( )l s
kν
εΓ
crack planecrack front
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
28
Isoparametric quadrilateral elements with quadratic shape functions and 8 nodes
One-dimensional quarter-point element:
a) natural coordinates, b) local cartesian coordinates
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
29
Griffith-crack in the infinite domain under tension
FEM-mesh and enlarged detail at crack tip
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
30
Collapsed and distorted isoparametric 8-noded-quadrilatendelement
Arrangement of quarter-point-elements at the crack tip/front
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
31
Equivalent domain integral and weight function q for plane crack problems
Γ
A
dacrack
domain I domain II domain III
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
32
Local energy method as crack closure integral
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
Fracture_Mechanics_Computations_Handouts2012_2.docx
33
Modified crack closure integral for a) linear (above) and b) quadratic displacement functions (below)
Institute of Mechanics and Fluid Dynamics Applied Mechanics: Prof. Dr. M. Kuna
Fracture Mechanics Computations
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3D-crack closure integral for 8-noded hexahedral elements along straight crack