iwsd -module 2-2_5 introduction to fracture mechanics

Objective: The student will be introduced to fracture modes and how the stress intensity factor can be used in modelling fatigue crack growth and brittle fracture.

Module 2.5: Introduction to fracture mechanics

1

Scope: Stress intensity factor , Brittle fracture, Ductile fracture, Fatigue crack growth , Paris equation, Plastic zone, Critical stress intensity factor Expected result: Explain mechanism and common features of ductile fracture of structures. Explain mechanism and common features of brittle fracture of materials and structures. Explain fatigue crack growth based on linear elastic fracture mechanics. Illustrate relationship between fracture type and surface appearance. Review an industrial failure case involving fracture. Explain the stress intensity factor, stress intensity factor range and critical stress intensity factor. Compute stress intensity factors for a simple geometry using geometry factors. Review experimental method related to fracture.

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Introduction to fracture mechanics

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From notch to sharp cracks: How do we describe the stress state? What effect the number of cycles to failure?

0

R Kt

Mild notch

0

R

d

Blunt notch

Kt = 1+ 2d/R

0

a

Sharp notch

N is dependent on Kt N is dependent on Kf Kf = 1 + q(Kt - 1)

N is dependent on KI (SIF ) KI 0 a

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Stress concentrations

3

2b

2a

ba21

a

max

ab2

Sa

a21amax Sa

Sa

max

kt

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Stress concentration Stress intensity factor

The theoretical stress concentration approaches infinity!

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Stress Intensity Factor, SIF

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Real materials compared to ideal (model)

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aSK

K is the stress intensity factor.

It characterizes the severity of a crack in terms of stress, S, and crack size, a.

fy= 520 MPa

Nom

inal str

ess,

MP

a

600

400

200

0 0 20 40 60

a, crack length, mm

Test data

mMPaaSKc 66

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aSK

b

aKt 21

Note:

Not a stress concentration

fy= 520 MPa

Nom

inal str

ess,

MP

a

600

400

200

0 0 20 40 60

a, crack length, mm

Test data

mMPaaSKc 66

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fy= 520 MPa

Nom

inal str

ess,

MP

a

600

400

200

0 0 20 40 60

a, crack length, mm

Test data

mMPaaSKc 66

aSK

Kc is a material parameter called the critical stress intensity factor or fracture toughness

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aSK yC

Kc is a material parameter called the critical stress intensity factor or fracture toughness

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Basic modes of crack growth and fractured surfaces formation

Three crack tip displacement modes

(Mode I) (Mode II) (Mode III)

Linear Elastic Fracture Mechanics (LEFM) Method used to predict the behaviour of cracks in solids subjected to fatigue loading

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Stress intensity factor, SIF

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General stress state

Plane stress state

2

3

21

2

2

3

22

2

3

22

2

2

sinsincos

sincossin

coscossin

r

K II

xy

yy

xx

2

3

2

2

3

21

2

3

21

22

cossin

sinsin

sinsin

cosr

K I

xy

yy

xx

2

2

2

cos

sin

r

K III

yz

xz

The stresses near the crack tip are defined with the help of K (KI, KII, KIII)

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P

D

uncracked

cracked

D1

P1

P2

Force displacement diagram

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P

D

energy of uncracked plate

P

D

energy of cracked plate

P

D

change in energy as crack develops

U U + dU dU

P1

P2

D1 D1

P1

P2

D1

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da

dU

BG

1

P

D

change in energy as crack develops

dU

P1

P2

D1

The rate of change in potential energy with change in crack area is

da

W

Bs

s

1

Work per unit area required to form new crack surface

Note that dU < 0 is the surface energy of a crack sW

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Stress Intensity Factor

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If the potential energy released as the crack grows is greater than the energy needed to create new crack surface,

then crack growth will become unstable

sG

Critical stress intensity factor

G is related to the rate of energy release for a growing crack.

K is related to stresses and displacements, which can also be solved for energy. Thus there should be a relation

2222 11

IIIIII KE

KKE

G

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Stress Intensity Factor

16

The stress intensity factor always has the form

=

stress geometry function crack length

In fatigue we consider the range of the stress intensity factor

=

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Stress Intensity Factor Geometry functions

17

=

F

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Stress Intensity Factor Geometry function

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Plastic zone

19

Plane stress Plane strain

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Phases of fatigue life and relevant factors

cyclic slip

crack nucleation

micro crack growth

macro crack growth

final failure

initiation period crack growth period

Kt

stress concentration

factor

K

stress intensity

factor

KIC, KC

fracture

toughness

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Crack growth of macro cracks

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Region 1: Threshold

Depends on the R-ration

For welds Kth = 2 MPam

Region 2: Linear stable crack growth

a = crack length N = number of cycles da/dN = crack growth/cycle K = SIF range C and n are material dependent constants

Paris law

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Region 3: Instability final failure

Fracture toughness Kc depends on: Material quality (increases with increase quality) Thickness (decrease with increased thickness) Temperature (decrease with lower temperature)

Threshold region Paris region

Instability region

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Region 3: Instability final failure

Fracture toughness Kc

Plane stress Transition area Plane strain

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Fracture toughness Kic testing

CT-specimen

3 point bending

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The test is carried out in two stages: First stage is fatigue to develop a fatigue crack. The crack length is

determined after the fatigue test (0.45W < a < 0.55W) A slowly increased load P is applied. The Crack tip Opening

Displacement (COD) is measured and plotted vs. P.

Tensile test, CT (Compact Tension)

Bending test

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Stress intensity factor - CT

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Stress intensity factor 3 pt bending

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Example

Calculate the fatigue life of the welded joint. The fatigue failure is assumed to occur trough the throat thickness (a). Final failure is assumed when the ligament is 1.8 mm. The stress is varied between 25 to 75 MPa.

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Example

Material parameters for crack growth are C = 1.8 10-13 and n = 3 in mm and N.

Approximate the case with center crack in plate with axial loading

The elementary case for this SIF

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Example

The initial crack length is a0 = 0 mm and the final value is a0 = 4-1.8 mm = 2.2 mm

We get for different crack length

The stress intensity factor range:

Finally the life

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Example

Incremental integration

Final fatigue life: 1.38 million cycles

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Example

Same example but with FE Analysis

Detailed mesh around the crack tip

Crack tip elements

Crack tip displacements

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Example

Stress intensity factors

Fatigue life

Final fatigue life: 1.36 million cycles (same as analytical)

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Case study #1

Residual stress effect on fatigue of welded steel structures using LEFM

Barsoum Z. and Barsoum I., Engineering Failure Analysis, Volume 16, Issue 1, pp. 449-467, January 2009.

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Case study #1

Development of a FE subroutine for welding simulation

Predict the residual stresses due to the welding process

Development of a LEFM subroutine for automatic simulation of fatigue crack growth (FCG)

LEFM Model Fillet weld root cracking

Stress intensity

factors: KI , KII

Res. stresses: Kresidual

Crack size: a0 and af

Deflection angle: i

Crack increment: ai

Material parameters: C & m

Paris law/Forman

Numerical integration

Paris law

maximum circumferential stress criterion (max).

Nf IWSD M2.5

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-1,5

-1

-0,5

0

0,5

0 1 2 3r / R (-)

re

sid

ua

l st

ress

/

yie

ld s

tres

s

(-)

Radial residual stres - Pavier et alTangential residual stress - Pavier et alRadial residual stress - FE simulationTangential residual stress - FE simulationRadial residual stress -mappingTangential residual stress - mappingRadial residual stress - after 10 cyclesTangenital residual stress - after 10 cycles

r (mm)

-1,5

-1

-0,5

0

0,5

0 1 2 3r / R (-)

resi

du

al

stress

/

yie

ld s

tress

(-

)

No crack a=0.3 mm a=3 mma= 8 mm

Radial Residual Stress - y

0.3 mm

3 mm8 mm

-1,5

-1

-0,5

0

0,5

0 1 2 3r / R (-)

resi

du

al

stress

/

yie

ld s

tress

(-

)

No cracka=0.3 mma=3 mma=8 mm

Tangential Residual Stress - x

0.3 mm3 mm

8 mm

Cold expanded hole in aluminium plate Residual stresses no cracks

Case study #1

Residual stress relaxation due to crack growth

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Case study #1 - MAN B&W Diesel

Weld B

Weld A

Welded engine frame box

Objective:

Influence of stress reliefing

Fatigue cracking weld root

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Case study #1 - MAN B&W Diesel Welded engine frame box

WELDED ENGINE FRAME BOX

10

100

1,E+04 1,E+05 1,E+06 1,E+07

Nf (Cycles)

Rig

fo

rce

(k

N)

PWHT experiment

PWHT prediction

As Welded experiment

As Welded Prediction

run outs

Fatigue life - Experiment vs. Prediction

Welding simulation

-400

-300

-200

-100

0

100

200

300

400

0 5 10 15 20 25

Crack path (mm)

No

rm

al

resi

du

al

stress

(M

Pa

)

WELD A

WELD B

Crack path

design root error 5 mm

design root error 8 mm

Residual stresses weld root

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Example: T-joint structure with several welds Structural steel Domex 355

Plate thickness = 12 mm

R-ratio = -1

Kth = 6 MPam

KIC = 120 MPa m

Crack growth: C = 5e-12, m=3

30 MPa

12

a6i1

a6i1 S6, w=4

W = 500 mm

250

250 200 200

UY = 0

UX = 0

Task: Predict the fatigue life of the welded structure (butt and fillet welds) using LEFM

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Example: T-joint structure with several welds

Linear Elastic Fracture Mechanics (LEFM)

Fracture mechanics assumes the existence of an initial crack ai.

It can be used to predict the growth of the crack to a final size af.

For cracks starting from weld toe, an initial crack depth of a = 0.15 mm and an aspect ratio of a:c = 1:10 should be assumed.

For root cracks e.g. at fillet welds, the root gap should be taken as an initial crack

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Fatigue Crack Growth

Region I,

threshold of slow crack growth rate region

Region II

stable Paris law region

Region III,

fast / unstable crack growth region

Example: T-joint structure with several welds, LEFM

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Fatigue Crack Growth


In region I, R has a big effect

In region II, R has a moderate effect

In region III, R has a big effect

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Mean stress effects / Crack closure


time

K

Kcl Kop

Keff

Kmin

K Kmax

Closure ratio

Aswelded condition: Crack is fully open during the whole range of cyclic loading due to assumption of tensile residual stresses at the crack tip!

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Initial root crack 6 mm Initial root crack 10 mm

Toe crack, 0.15 mm

Toe crack, 0.15 mm

FE model with cracks; ANSYS / FRANC2D

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-30

-10

10

30

50

70

90

110

130

150

0 2 4 6 8 10 12

crack (mm)

K

I (M

Pa

?m

)

fillet toe (ai=0.15mm)

butt toe (ai=0.15mm)fillet root (ai=10mm)

butt root (ai=6mm)?Kth

KIC

SIF / Crack propagation

No crack growth KI < Kth

Crack growth from butt weld root Nf = 20 000

cycles

6

7

8

9

10

11

12

0 10000 20000 30000 40000 50000

Accumulated cycles

cra

ck

le

ng

th (

mm

)

No mean stress effects

Crack closure / Mean stress effects

Nf = 47 000 cycles

Butt weld root failure


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Comparsion between fatigue design methods for welded joints

Weld root failure butt weld

Nominal mtd 119 000 cycles

Hot spot mtd not possible

Eff notch mtd 58 000 cycles

LEFM no mean stress effects 20 000 cycles

LEFM mean stress effects 47 000 cycles

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Case study #2

Analysis of Rear frame of a loader using fracture mechanics and crack growth analysis Volvo Construction Equipment

Geometry

Section through the sub model

Inside frame Single sided fillet weld without weld prep.

Outside frame Double sided fillet weld without weld prep.

Weld root LOF

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Case study #2 - Analysis of Rear frame of a loader using fracture mechanics and crack growth analysis

Results for symmetric load conditions

Global model without notch

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Sub model 2 Section through weld with crack

Crack

Crack tip

Sub model 2 swept to 3D

Sub model 2 merged into sub model 1

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Result from cross section

Stress intensity along the whole sub model, at the weld root

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3 different sub models are analysed for crack growth analysis

Crack length (mm) SIF (MPam) Slpoe

8 12 (16-12)/(12-8) = 1

12 16 (40-16)/(16-12) = 6

16 40 -

Assume bilinear functions

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LEFM crack growth calculations

Paris law Crack growth rate =

=

Where C = 5 10-12 [MPa,m ] and m = 3

K

a

ao af

So if, K = A + aB, we can integrate analytically

= =

0

0

= 1

1

+

0

=1

+ +1

1

+ 11

af

a0

=

a = 8 12 mm: B = 1 N 304 000 cycles a = 12 16 mm: B = 6 N 55 000 cycles

359 000 cycles

This agrees well with the notch stress analysis = 420 000 cycles

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Comparison of calculated fatigue lifes

Conservative fatigue life estimation could be to safe Notch stress method agrees well with LEFM crack growth analysis

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Case study #3

Failure investigation of Dipper arm in Volvo CE excevator

Field crack section view

Field fatigue failure

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Case study #3 - Failure investigation of Dipper arm in Volvo CE excevator LEFM crack growth analysis Case A

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Case study #3 - Failure investigation of Dipper arm in Volvo CE excevator

Definition of crack direction in sub model 1

Principal max stress direction check

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Sub model analysis result

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FE models Case B

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Bench test loading Design in Case A & B

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Problem 2.5.1

Change to a High Strength Steel Assume that a component in the shape of a large sheet is to be fabricated C-Mn Steel. It is required that the critical flaw size be greater than 2 mm, the resolution limit of available flaw detection procedures. A design stress of one half the tensile strength is indicated. To save weight, and increase in the tensile strength is suggested, from 1520 to 2070 MPa. Is such a strength increment allowable ? (assume plane-strain conditions in all computations)

iwsd -module 2-2_5 introduction to fracture mechanics

Documents

stress concentrations

terms of stress

mpa nominal stress

fracture toughness iwsd

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kiii iwsd m2

ideal model iwsd m2