 = ,,  aaKaFct Stress Intensity Factor K for 3D cracks

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A. Zeghloul Fracture mechanics, damage and fatigue – Stress intensity factor 1

Stress Intensity Factor K for 3D cracks

Through crack

Front crack Surface

crack Corner

crack

The through cracks, with a well defined crack front, are usually treated as a two- dimensional problem. The previous methods allow the calculation of the SIFKin a many simple cases.

When the cracks are three dimensional, such as corner or surface cracks, the SIF K changes along the crack front : for these cracks, the SIF K is calculated by using finite element method.

In practice, when the cracks initiated, they propagate in the directions where the SIFKis higher, but they tend most often to a configuration where the SIFKstabilizes along the crack front, so that they can be studied as a two-dimensional problem.

I S

, ,

K aF a a

σ π

^∞

^ c t ϕ ^

=  

 

For the surfaces cracks or the corner cracks, elliptical in shape (a/c<1), in plates loaded in opening mode, Raju et Newman^9,10 have proposed empirical relationship for calculation of the SIFK_I. These empirical formulas are obtained by digitally adjusting the results of finite element calculations.

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(

²

)

³

3 1 0, 08 0,15( ) (1 cos )

g = + + a t − ϕ

(

²

)

³

2 1 0, 08 0, 4( ) (1 sin )

g = + + a t − ϕ

Critical value of SIF - Toughness

Being given a crack and a loading mode (mode I, II or III), experience shows that sudden fracture occurs when the SIFK reaches a critical value calledK_c. This critical value, which characterizes the ability of a material to withstand the abrupt propagation of a crack, is calledtoughness. Since the opening mode is the most damaging, it is the critical value K_Ic obtained in mode I, which is generallyused as a measure of toughness.

The role of the toughnessK_Icin linear elastic fracture mechanics, is analogous to the role of the yield strength σE in classical mechanics. Toughness K_Ic depends, as do the elastic limitσE, of the test temperature and the strain rate.

K_icdepends also of the thickness of the test specimen.

Characteristic K_Ic evolutions, obtained from standardized tests, are shown schematically in the following figure.

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Plane stress

Plane strain

Thickness

Thickness affects the measurement of toughness. In the thin plates, loaded in mode I, the plane stress state is predominant and the critical value of SIFK_I is high, especially in ductile materials.

When the thickness increases, a transition towards a plane strain state is observed. The critical value of the SIF decreases and no longer evolves beyond a certain thickness - it is this stabilized minimum value ofK_Ic, which defines the toughness of the material.

Brittle

Temperature

In metal alloys such as steels, the temperature dependence is characterized by a very significant transition between a fragile area at low temperature and low toughness, and a ductile area at high temperature and high toughness.

This ductile-brittle transition zone moves (shifts) to higher temperature as strain rate increases. This behavior makes the sizing of structures very difficult when there is a risk of explosion – because an explosion causes a sudden increase in the strain rate.

The aging of materials influences, as the strain rate, the toughness. As the materials age, the fragile area extends at the expense of ductile area, with displacement of the ductile- brittle transition zone to the higher temperatures.

Because of this, some old steel structure bridges are closed in winter when it freezes,

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http://nte.mines-albi.fr/SciMat/co/SM6uc2-2.html Material Toughness (MPa.m^1/2) σR(MPa)

2014 T651 40 500

TA6V 85 1020

40CrMoV20 42 1850

35NiCrMo16 95 1850

Ceramic 5 800

Some values of the material toughness

Aluminium alloy Titanium alloy Hardened steel

Polymer Wood Concrete

Toughness measurements are made on standardized specimens, fatigue pre-cracked. For a good measure of toughness, the ASTM standards impose the following conditions :

Toughness measurement

Fatigue crack

Where σE, a, eand Ware respectively the elastic limit of the material, the crack length, the thickness and the width of the specimen.

W-a is the length of the uncracked ligament.

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i.The crack lengths measured on each side of the surface of the test specimen do not differ by more than 10% of the average length. The average length is measured at the heart of the test specimen after fracture.

ii.The maximumK_Iapplied to the specimen in the pre-cracking phase does not exceed 0.6KIc, and during the last stage of crackingK_I/E<0,32. 10^-3m^1/2.

iii. P_u/P_Q<1,1 where P_u and P_Q loads are determined according to the graphical procedure shown in the .figure below.

iv.0,55MPa.m^1/2/s <∆KI/t< 2,75MPa.m^1/2/s where∆KI/t is the load rate.

In addition to the above conditions, the ASTM standards require, during the K_Ic measurement test, to ensure that :

Load-displacement diagram and graphical procedure to determine P

_Q

et P

_u

.

IC Q

K = K

Load

slope (OQ)=0,95 slope (OA)

Displacement

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Tutorial 10 : Measurement of toughness K

_Ic

3-point bending specimen (figure above) is tested in accordance with the ASTM standards.

The values of the yield strength and the Young modulus areσE=1200MPa andE=210GPa.

The specimen is loaded at 100kN/mn, and fatigue pre-cracked atP₁^max=45kN andP^min=0.

The last cycles are carried out atP₂^max=30kN.

The measured dimensions are :

1 2 3

8 4 30

3, 996 4, 007 3, 997

3,915 3, 952

left right

surface surface

W cm e cm L cm

a cm a cm a cm

a cm a cm

= = =

= =

Fractured surface

The recording of the load-displacement diagram gave the following values :

Calculate the K

_Ic

toughness and verify that the found value meets the requirements of ASTM.

SIF K

_I

for the tested specimen is calculated by the following equation :

Load

slope (OQ)=0,95 slope (OA) Displacement

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Condition i. is satisfied

1 2 3

Averaging length 4

moy

3 a a a

a = + + = cm

3 2 3 2

80 0,3

2, 663 70, 6 0, 04 (0, 08)

Q Q

P L a kN m

K g MPa m

eW W m

⋅

 

=   = =

 

1 2, 663

2 a a

W g W

 

=    =

 

We then check all the conditions of ASTM

Condition ii. is satisfied Condition iii. is satisfied

Condition iv. is satisfied

70, 6

IC Q

K = K = MPa m

All ASTM conditions are satisfied

and the last cycles are carried out at From where

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Griffith’s energy approach

In a material, a ∆a extension of a crack length a, is accompanied by the following energy variations :

ext el

W W U

∆ = ∆ + ∆

∆W_ext is the applied energy change due to external forces

∆W_el is the variation of stored elastic energy

∆U is the energy expended to propagate the crack over a∆alength

In Griffith's theory, which applies to brittle fracture,∆Uis the energy needed to create new surfaces in the material. Griffith introduces, from∆U, a crack propagation energy per unit area, denoted G and defined by :

lim

_A 0

U U

G

^{∆ →}

A A

∆ ∂

= =

∆ ∂ where ∆ = ∆ A e a

Generally, considered the unit of thickness (e=1), so that the energyGper unit thickness, becomes :

lim

_a 0

U U

G

^{∆ →}

a a

∆ ∂

= =

∆ ∂

EnergyG is also called energy release rate. To understand this meaning, we will consider the crack propagation (in a specimen of thick unit), in the following two classic cases :

 Crack propagation with imposed displacement (figure b)

 Crack propagation with imposed force (figure c)

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Crack propagation with imposed displacement (b) or with imposed force (c)

Load-displacement diagram

Crack propagation

2 2

2

1 2 2 2

el el

u

u u C

W CP W a

C C a

∂

 

= =  ∆ = −   ∆

∂

 

∆u=0 ∆W

_ext

=0 with W

_el

=Pu/2, either, introducing the compliance, i.e. the opposite of stiffness C=u/P :

2 2

2

2 _u

2

_u

u C P C

G C a a

∂ ∂

   

=   =  

∂ ∂

   

i. Crack propagation with imposed displacement (u=constant)

It is thus observed that the elastic energy stored decreases.

As ∆ W

_ext

= = ∆ + ∆ 0 U W

_el

 ∆ = −∆ U W

_el

and is then given by G

In this case, is the decrease of the elastic energy, which was used to propagate the crack

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2

_P

P C

G a

∂

 

=   ∂  

i. Crack propagation with imposed force (P=constant)

0 u 0 or u C

P C u C

∆ ∆

 

∆ =  ∆   = =

 

2 2

1 from where

2 2 2

el el

P

P P C

W Pu C W a

a

∂

 

= = ∆ =   ∂   ∆

2 ext

P

Pu C

W P u C P a

C a

∂

 

∆ = ∆ = ∆ ≈   ∆

∂

 

2

and the Griffith's energy is then written :

ext el

2

P

P C

U W W a

a

∂

 

∆ = ∆ − ∆ =   ∂   ∆

Previous expressions of Griffith G energy, for crack propagation at imposed displacement or at imposed force, can be put into a single form:

2

_u or _P

P C

G a

∂

 

=  

∂

 

If the specimen thickness e, is not equal to unity, the expression of Griffith energy G, is :

2

or

2

_u _P

P C

G e a

∂

 

=  

∂

 

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In the two cases examined (crack propagation at imposed displacement or at imposed force), it appears that the energy∆U spent during the propagation, is equal to the area of the triangle OAB in the load-displacement diagram.

To achieve this type of loading, the testing machine must be either infinite stiffness to impose a displacement, or infinite flexibility to impose a force.

In practical tests, it is between these two extremes cases. The load-displacement diagram for different lengths ofa_icrack, looks as follows :

Load-displacement diagram for different cack lengths

Load

Displacement

Consider now two crack lengths, a

_i

and a

_j

. The energy ∆ U corresponds to the area of the triangle OA

_i

A

_j

:

' ' ' '

( )

( ) ( ) ( )

1 1 1

( )( )

2 2 2

i j

i i i i j j j j

i i i j j i j j

U Area OA A

Area OA A Area A A A A Area OA A

Pu P P u u P u

∆ =

= + −

= + + − −

1 ( )

2

ⁱ ^j ^j ⁱ

U Pu P u

 ∆ = −

1 2

i j j i

j i

Pu P u G e a a

= −

−

and the propagation energy of Griffith, is

then written :

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Critical Griffith’s Energy G

_C

As SIFK, experience shows that for a given loading mode (mode I, mode II or mode III), the sudden propagation occurs when the energy of Griffith reaches a critical value, denoted G_Ic,G_IIcorG_IIIc.In practical, theG_Icvalue is used as measurement of toughness. The unit of toughnessG_Icis thekJ/m².

Tutorial 11 : Measurement of toughness G

_Ic

These results are represented on the load-displacement curves of the following figure :

A series of tests on 1mm thick specimens, pre cracked in mode I at different crack length, has been carried out to determine the toughness of a steel. These tests gave the following results :

Crack length Critical load Critical displacement

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Load-displacement curve at failure of a steel

Calculate the mean value (in KJ/m²) of the toughness of the steel by the two methods :

1 2

i j j i

I

j i

Pu P u

G e a a

= −

−

2

ou I

u P

P C

G e a

∂

 

=  

∂

 

Load

Displacement

G

_IC

calculation from the areas in the load-displacement diagram

1 2

i j j i

I

j i

Pu P u

G e a a

= −

−

30, 2 0, 6 /

2

5

C mean IC I

G G KJ m

 =  = ±

2

1 2 3 3

1 4 0, 5 3, 5 0, 4

Triangle 30, 0 /

2 10 (40 30) 10

IC

OA A  G =

₋

× − ×

₋

= KJ m

⋅ − ⋅

Areas

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The values of compliance C=u/P are directly derived from the data :

G

_IC

calculation from the variations of the compliance

2

ou I

u P

P C

G e a

∂

 

=  

∂

 

2 I

2 P C

G e a

∂

 

=   ∂   Toughness is calculated by the relationship

( ) ( ) ( ) ( ) ( )

7

6 1

3 7

6 1

3 7

6 1

3 7

6 1

3 7 3

(1, 43 1, 00) 10

4, 30 10 10 10

(2, 02 1, 43) 10

5, 62 10 10, 5 10

(2, 79 2, 02) 10

6, 94 10 11,1 10

(3,59 2, 79) 10

7, 92 10 10,1 10

(4, 26 3, 59) 10 7,3 10

D

C a N

C a

− − −

−

− − −

−

− − −

−

− − −

−

− ⋅

∂ ∂ = = ⋅

⋅

− ⋅

∂ ∂ = = ⋅

⋅

− ⋅

∂ ∂ = = ⋅

⋅

− ⋅

∂ ∂ = = ⋅

⋅

− ⋅

∂ ∂ =

⋅

6 1

9,18 10⁻ N⁻

= ⋅

( )

⁶ ¹

40 ^mean 4, 96 10

a= mm  ∂ ∂C a = ⋅ ⁻ N⁻

( )

⁶ ¹

50, 5 ^mean 6, 28 10

a= mm  ∂ ∂C a = ⋅ ⁻ N⁻

( )

⁶ ¹

61, 6 ^mean 7, 43 10

a= mm  ∂ ∂C a = ⋅ ⁻ N⁻

( )

⁶ ¹

71, 7 ^mean 8, 55 10

a= mm  ∂ ∂C a = ⋅ ⁻ N⁻

( )

² ⁶ 6 2

3

3, 5 10

( 40 ) 4, 96 10 30, 38 /

IC 2 10 IC

G a mm ⋅₋ ⁻ G KJ m

= = ⋅  =

⋅

For greater clarity, G

_IC

is determined from mean values of ∂ C/ ∂ a

calculated left and right of each crack length value a.

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a (mm) 40 50,5 61,6 71,7 G _IC (KJ/m ² ) 30,38 30,57 29,13 29,35

29,86 0, 39 /

2

4

mean IC IC

G G KJ m

 =  = ±

Both methods give close G

_IC

toughness values

Crack length Crack length

Compliance

Relationship between Griffith G Energy and SIF K

Consider a crack solicited in mode I - Mode II or III loads are similarly studied.

The crack with initial length a, propagates over a distance

∆ a.

The stress field, downstream of the end A of the crack, is given by :

( , 0)

2

I y

r K σ θ r

= = π

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The filed of displacement of the lips of the crack, upstream of the end A’, is written :

The force applied to the lips of the crack, is σ

y

(r)edx with r=x-a.

The displacement at r’=a+∆a-x is u

_y

(r’).

The regression work is given by :

= in plane strain state ( ', ) 2 ' (1 ) with

*= in plane stress state 1+

I y

K r

u r

π µ π υ υ υ υ υ

υ

 

= − 



To determine the rate of energy release (i.e. Griffith energy G), it is more convenient to calculate the regression work of the crack from position A’(x=a+∆a) to position A(x=a), in other words, to consider the work needed to close the lips of the crack.

( ) ( ')

' 2

2

a y y

a a

r u r

W U σ dx

∆ = −∆ = 

+∆

( , 0) 2

I y

r K σ θ r

= = π 2 '

( ', )

^I

(1 *)

y

K r

u r π υ

µ π

= −

2

(1)

( ) ( ') 1 *

' 2

2

a y y I a

a a a a

r u r K a a x

W U dx dx

x a

σ υ

µ π

+∆ +∆

− + ∆ −

∆ = −∆ = =

  −

2

and at

1 x a X

a dX

X dx a

x a a X

x a X

=  = ∞

∆ =  = − ∆  

= + ∆  =

− 

To calculate the integral (1), we choose the following change of variable :

1 2

The integral (1) becomes then 1 dX which is integrated in parts, by putting :

I a X

X

∞  

= ∆ − − 

 



1 1 2

1 2 1 1

1 2 1

X d dX

X dX

X I a

dX X X X

d X X

α α

β β

∞ ∞

 = −  =  

  − 

 −  = ∆  − 

    

  −

 = −  =    





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2

1 *

2 K

I

U υ a

µ

∆ = − ∆

²

0

1 *

lim 2

I

I a

K G U

a

υ µ

∆ →

∆ −

 = =

∆

( )

2

1

2

P strain I

I

G K

E υ

−

= −

2

P stress I

I

G K

E

−

=

2

' in Plane stress state

or with

' ' in Plane strain state

1

I I

E E

G K E

E E

υ

=

 

= 

 = −



1 1

1

2 1

X dX

I a

X X X

∞ ∞

 −  

 

= ∆   − 

  −

 







For a plane strain state *=

υ υ

For a plane stress state *=

1+

υ υ υ

Mode II

2

' in Plane stress with

' ' in Plane strain

1

II II

E E

G K E

E E

υ

=

 

= 

 = −



Mode III

2

III III

G K

= µ

Similar calculations in mode II and III, give

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35

Master Mécanique-Matériaux-Structures-Procédés

Chapter 5 – Material fatigue

Prof. Abderrahim Zeghloul Université de Lorraine

Fracture Mechanics, Damage and Fatigue

 Introduction

 Fatigue endurance

 Fatigue damage

 Fatigue crack growth

 Damage in fatigue propagation

 Fatigue crack closure

 Small fatigue crack growth

 Fatigue life calculations

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A. Zeghloul Fracture mechanics, damage and fatigue – Material fatigue 37

Introduction

Fatigue corresponds to the deterioration of the mechanical properties of the materials resulting from the application of cyclic stresses.

Fatigue damage, associated with a large number of failures of mechanical systems (automotive, railway, aerospace, naval industries ...), is manifested by the appearance of cracks that then propagate to failure.

In the practice of fatigue characterization procedures, the crack initiating period (occurrence of crack) is treated with tests on smooth specimens (sometimes on notched specimens) and it then determines a lifetime, according to a stress or strain amplitude.

The crack propagation phase, for its part, is essentially studied from fatigue tests on notched specimens. The fatigue crack growth rate is then described in terms of

∆K, the amplitude of the stress intensity factor.

The development of fracture mechanics has been a major contribution in the study and the description of the propagation of fatigue cracks.

Fatigue propagation

zone

Brutal failure

zone

Stop lines cracking

Radial line Crack initiation

Failure of a cylinder of an jackhammer

Crack initiation

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The derailment occurred when a pre-existing fatigue crack reached a critical size and caused the failure of the rail under the weight of the train.

Transportation Safety Board of Canada

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flight 32 of Qantas, an Australian company

The accident occurred on November 4, 2010 at 10:01 am local time as the aircraft was re-sailing from Singapore to Sydney

Causes of the accident

Australian Transportation Safety Agency and Rolls-Royce Engineers Investigate the Accident.

They established that it is a manufacturing defect of the high pressure turbine, which is involved : this defect then caused fatigue cracks.

So Rolls-Royce that was held responsible, and had to pay compensation of EUR 72 million to the Australian company.

This aircraft has resumed commercial flights in April 2012.

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Fatigue endurance

To describe the results, we often use the amplitude ∆σ and the stress ratio R,

defined by :

min

max 1

R σ

=σ < ∆ =σ σ^max−σ^min =σ^max(1−R)

max min 1

1 1 2(1 )

R moy R

R R R

σ σ

σ = ^∆ σ = ^∆ σ = ⁺ ∆σ

− − −

The fatigue curves are often presented according to σa, the alternating stress which corresponds to half the stress amplitude.

(

^max ^min

)

1

2 2

a

σ =^∆σ = σ −σ

σa

A stress cycle is characterized by a stress amplitude ∆σ, and mean stress σ^moy^.

When a set of test specimens, is subjected to cyclic loading, at fixed frequency and at different stress amplitudes, a curve is obtained describing the alternating stress as a function of the number of cycles to rupture.

The shape of this curve, in semi logarithmic scale, is :

This curve is Whöler curve. It is also called the fatigue endurance curve, orS-Ncurve (Stress-Number of cycles).

There are three areas on this curve.

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A zone with a low number of cycles (zone 1) called low cycle fatigue zone. The alternating stress is high (σa>σE the yield strength) in this zone and a significant cyclic plastic deformation is observed.

A limited endurance zone (zone 2), where the rupture occurs at a number of cycles, which increases as the applied cyclic stress decreases.

In this area, one of the many relationships givingN_Rthe number of cycles to failure according toσathe alternating stress, is the relationship Weibull :

An endurance zone unlimited, called safety zone (zone 3), where the failure does not occur for à high number of cycles, 10⁶cycles in steels and 10⁷cycles in aluminium alloys). This last area is bounded by an asymptote that gives the fatigue limit or endurance limit, denotedσD.

( )

/ ⁿ pour

R a D a D

N =A σ σ− σ >σ

The zone 1 corresponds to the low cycle fatigue area, where the stresses applied are greater than the elastic limit of the material. In this zone, the stabilized stress- strain curve, takes the form of a hysteresis loop

( )

' 2 Manson-Coffin relationship 2

' and are the coefficient and the exponent of fatigue ductility

p c

f R

f

N c

ε ε ε

∆ = ⋅

( )

' 2 Basquin relationship 2

' and are the coefficient and the exponent of fatigue strength

f b e

NR

E b

ε σ σ

∆ = ⋅

The empirical laws, the most used to describe the results of low cycle fatigue, are :

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In zone 3 unlimited endurance, it is sometimes difficult to assess the fatigue limit or endurance limit σ

D

.

Then introduced the notion of conventional fatigue limit which is the greater stress amplitude for which the probability of failure is 50%, after N cycles of loading (N varies from 10

⁶

to 10

⁸

cycles).

Probabilistic aspect of the Wohler curve

Given the dispersion observed on endurance test results, it is

necessary to construct equiprobability curves.

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The Wohler curve is the median curve at 50%, it is verified by experiment that : - The distribution ofLog(N_R) follows a normal distribution for a given stress ; - The distribution of the stress follows a normal distribution for a given number of cycles.

Influence of the average stress on the Wöhler curve When the fatigue tests are carried out at

a non-zero average stress σ

m

, the lifetime is modified :

- An average tensile stress decreases the service life ;

- An average compressive stress increases it ;

- The endurance limit σ

D

varies in the same direction.

Various diagrams used to represent these variations : - Haig diagram ;

- Straight curve of Söderberg or Goodman;

- Parable of Gerber.

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Haig Diagram

This diagram, deduced from the Wöhler curves, represents the variation of the amplitude of stress σa=∆σ/2 as a function of average stress σm, at a givenN_R. Points Aand Brespectively correspond toσDobtained at R=-1 (σm=0) and at R_mthe ultime tensile strength. The experimental construction of the Haig digram, results from :

- Dynamic test atσm= 0 to determine the pointA; - Dynamic tests atσmi≠0 to determine the pointsA_i.

Construction of Haig diagram

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Several representations of the AB curve have been proposed for the construction of the Haig endurance diagram. These empirical representations require knowledge of the endurance limit atσm=0, of the yield strengthσEand ultimate tensile strengthR_m(orσu) of the material.

Söderberg straight curve

Goodman straight curve Goodman straight curve

Goodman straight curve

Gerber parable

- Söderberg straight curve and, to a lesser extent, Goodman straight curve are too penalizing forσm> 0 and too optimistic forσm< 0 ;

- The parable of Gerber is fair enough forσm> 0 , but penalizing forσm< 0 : it does not take into account the increase in endurance atσm< 0.

 = ,,  aaKaFct Stress Intensity Factor K for 3D cracks

Stress Intensity Factor K for 3D cracks

, ,

K aF a a

σ π

 c t ϕ 

=  

 

(

)

(

)

Critical value of SIF - Toughness

Some values of the material toughness

Toughness measurement

Load-displacement diagram and graphical procedure to determine P

et P

.

IC Q

K = K

Tutorial 10 : Measurement of toughness K

The recording of the load-displacement diagram gave the following values :

Calculate the K

toughness and verify that the found value meets the requirements of ASTM.

SIF K

for the tested specimen is calculated by the following equation :

Averaging length 4

3

a a a

a = + + = cm

80 0,3

2, 663 70, 6 0, 04 (0, 08)

P L a kN m

K g MPa m

eW W m

⋅

 

=   = =

 

1 2, 663

2

a a

W g W

 

=    =

 

We then check all the conditions of ASTM

70, 6

K = K = MPa m

All ASTM conditions are satisfied

Griffith’s energy approach

In a material, a ∆a extension of a crack length a, is accompanied by the following energy variations :

W W U

∆ = ∆ + ∆

lim

U U

G

A A

∆ ∂

= =

∆ ∂ where ∆ = ∆ A e a

lim

U U

G

a a

∆ ∂

= =

∆ ∂

1

2 2 2

u u C

W CP W a

C C a

∂

 

= =  ∆ = −   ∆

∂

 

∆u=0 ∆W

=0 with W

^ c t ϕ ^