Fracture processes at different scales · 2018-04-17 · Dominance of near crack tip field at all...

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




2

Fracture mechanical assessment of strength




3

Classification of fracture phenomena




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+




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




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




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




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




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




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 −




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




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




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




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




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




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




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




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




19

Stress distribution at crack Tipp for rigid-ideal-plastic material (EVZ)

RAMBERG-OSGOOD power – law hardening




20

Stress distribution at crack tip according HRR-solution for the harden-ing exponents n = 3 (above) und n = 10 (below)




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 =




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




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 /




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




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




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




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




28

Isoparametric quadrilateral elements with quadratic shape functions and 8 nodes

One-dimensional quarter-point element:

a) natural coordinates, b) local cartesian coordinates




29

Griffith-crack in the infinite domain under tension

FEM-mesh and enlarged detail at crack tip




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Collapsed and distorted isoparametric 8-noded-quadrilatendelement

Arrangement of quarter-point-elements at the crack tip/front




31

Equivalent domain integral and weight function q for plane crack problems

Γ

A

dacrack

domain I domain II domain III




32

Local energy method as crack closure integral




33

Modified crack closure integral for a) linear (above) and b) quadratic displacement functions (below)




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3D-crack closure integral for 8-noded hexahedral elements along straight crack

Fracture processes at different scales · 2018-04-17 · Dominance of near crack tip field at all...

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