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CH-3 Energetic of fracture

HUMBERT Laurent

Thursday, march 11th 2010

laurent.humbert@epfl.chlaurent.humbert@ecp.fr

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FRACTURE MECHANICS PROJECTS

Group 1: Experimental project #1 Berglund NiklasFogelfors OscarTunçer Gözde

Group 2: Experimental project #2 Farmand Ashtiani EbrahimKhoushabi Azadeh

Group 3: Numerical project #3 Baader JakobBernet AdelineUriarte Amaia

RESPONSIBLES

Laurent HUMBERT ROOM ME C1 365Samuel STUTZ ROOM ME C1 375 (group 2)

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SESSIONSS1Thursday, March 11th 10h-12h group1 and 3Friday, March 12th 10h-12h group2S2Thursday, March 18th 10h-12h group1 and 3Friday, March 19th 10h-12h group2S3Thursday, March 25th 10h-12h group1 and 3Friday, March 26th 10h-12h group2S4Thursday, April 1st 10h-12h group1 and 3? Friday, April 2nd 10h-12h group2 Good FridayS5Thursday, April 15th 10h-12h group1 and 3Friday, April 16th 10h-12h group2S6Thursday, April 22nd 10h-12h group1 and 3Friday, April 23rd 10h-12h group2S7Thursday, April 29th 10h-12h group1 and 3Friday, April 30th 10h-12h group2S8Thursday, May 6th 10h-12h group1 and 3Friday, May 7th 10h-12h group2S9? Thursday, May 13th 10h-12h group1 and 3 Ascension DayFriday, May 14th 10h-12h group2S10Thursday, May 20th 10h-12h group1 and 3Friday, May 21st 10h-12h group2S11Thursday, May 27th 10h-12h groups 1 , 2 and 3 Oral presentation

Proposed work schedule ...

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3.1 Theoretical strength (recalls)

Ideal crystal structure Equilibrium at inter-atomic distance a=a0

Atomic scale

Bond energy

where l assumed to be approximately equal to twice the atomic spacing a0

P = applied forcec2πP P sin Δal

=

Force-displacement relationship idealized as one half of the period of the sine wave

Cohesive force Pc

Derivative of the potential energy interaction °

12 6

D CUa a

= −

Lennard-Jones potential

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- Lattice deformation:

0

0 0

a aΔa =a a

ε −=

: critical stress (theoretical strength)cσ

c2πsin Δaσ σ=

• Approximate Stress-strain relationship:

- Dividing P by the number of bonds per unit area,

- Assuming small strains (linear elastic), expanding the sin function in Taylor series:

02 acπl

σ σ ε≈

0

dEd ε

σε =

⎛ ⎞= ⎜ ⎟⎝ ⎠

• Young’s modulus:

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And by taking ,02l a=

c0

E (1)2π a

σ =From the definition of the Young’s modulus :

cEπ

σ ≈

cE10

σ ≈ with more accurate calculations

2

0

1 2sin (2)2 2

l/

s c cπ x ldxl π

γ σ σ⎛ ⎞= =⎜ ⎟⎝ ⎠∫

Introducing the surface energy :sγ

Surface energy of a solid = energy it costs to make it

2 new surfaces → s2γ

sc

0

Eaγσ =(1) (2)+ →

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In reality :In practice, real materials do not achieve the ideal value c

E10

σ ≈

Why ?

Because all crystals contain defects :

Such as vacancies, dislocations, imperfect grain boundaries depending on the environmental conditions

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3.2 Griffith’s analysis of strength

• Stresses concentration appears in holes, slots, threads and huge change in section

c : characteristic dimension associated with the defect

→ lower the global strength by magnifying the stress locally

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• Maximum local stress ?maxσ

1/ 2

maxmin

1 cσ σ αρ

⎛ ⎞⎛ ⎞⎜ ⎟= + ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

: minimum radius of curvatureminρ

maxtK σ

σ= : stress concentration factor (dimensionless !)

for tension2α

for torsion and bending1 2α

F

F

0

FS

σ =

S0 : nominal section

Equation inexact but adequate working approximation for many design problems

→

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• Examples : Circular hole of radius R in a large plate loaded in tension

min Rρ =c R=

3tK =

Local yielding occurs when !3Yσ σ=

Yσ

ε

σ

Elliptical cavity (semi-axes a > b) traversing a plate loaded in tension

Rmaxσ

σ

σ

a

σ

σ

bmaxσ

2min b aρ =

c a= 1 2taKb

= +

Result given by Inglis (1913)

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a

a

• Cracked material

Cracks highly concentrate stress !

σ

a

σ

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The engineering problem of a crack in a structure

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1 2Aaσ σρ

⎛ ⎞= +⎜ ⎟

⎝ ⎠

• Non uniform local stress :Aσ

Cracks are sharp :

- previous equation for holes, notches of finite radius of curvature not relevant here !- their radius at the crack tip are essentially zero

rises deeply at the crack tipAσ

2Aaσ σρ

σ

σ

Aσ2

, ba ba

ρ =

With the minimum radius at crack tip 0aρ ≈ sf

0

E4aγσ =

Assuming failure when A cσ σremote stress at failure

→ rough estimate : continuum approach of Inglis not valid at the atomic level

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Griffith energy balance:

Slender ellipse 2 2a b

• Total energy of the plate : sWΨ Π= +

U FΠ = − : potential energy = strain energy U - work of external forces F

sW : work required to create two new surfaces

• Griffith energy balance (equilibrium) : 0sWA A AΨ Π ∂∂ ∂

= + =∂ ∂ ∂

for an increment in the crack area dA

2A Ba=

B

= Through crack in a plate of thickness B

σ∞

σ∞

σ∞

σ∞

model based on a global energy approach

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2 2

0a BΠ Π

Eπσ∞= −

For the cracked plate configuration (plane stress conditions),

Moreover with a constant thickness B and a through crack of length 2a

2s sW . 2aBγ=

: surface energy per unit area of the materialsγ

Crack area = projected area of the crack = 2aB (here)

Surface area = two matching surfaces = 4aB

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0 sBΠ a 4 Ba

EπσΨ γ∞= − +

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0BΠ Π a

Eπσ∞= −

2

s2 B a 4 B

a EΨ πσ γ∞∂

→ = − +∂

Solving for a , 2

2 sc

Ea γπσ∞

=

equilibrium

Energy

Ψ

Π

sW

crack length a

ca

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1/ 2

sf

2πEaγσ σ∞

⎛ ⎞≡ = ⎜ ⎟⎝ ⎠

Solving for the stress :

Type of equilibrium ?

→ sign of the second derivative of the energy

2 2

2

2 B 0a EΨ πσ∞∂

= − <∂

→ stable or unstable propagation

Always unstable configuration !

fσ fracture stress

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• Griffith approach applicable to other crack configurations :

Ex: penny-shaped crack configuration

( )

1/ 2

sf 2

E2 1 aπγσν

⎛ ⎞⎜ ⎟=⎜ ⎟−⎝ ⎠

Fracture stress :

: Poisson’s ratioν

E : Young’s modulus

a

σ∞

σ∞

circular crack of radius a

Remarks :

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( )1 2

s pf

2a

/

E γ γσ

π

⎛ ⎞+= ⎜ ⎟⎜ ⎟⎝ ⎠

plastic work per unit area of surface createdp sγ γ

• Large underestimation of the fracture strength for metals :

→ modified Griffith equation for materials that exhibit some plastic deformation (Irwin and Orowan)

account for additional energy dissipation (e.g. dislocation motion)

only valid for brittle solids (e.g. glass) → total energy of broken bonds in a unit areasγ

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General effect of temperature on the fracture energy of structural metals :

• Generalization for any type of energy dissipation :

1 2

ff

2 wa

/

Eσπ

⎛ ⎞= ⎜ ⎟⎝ ⎠

wf : fracture energy

Limited to elastic global behavior (because of Π) and wf assumed to be constant !

→ plastic, viscoelastic, viscoplastic effects

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The force P does no work during a virtual crack increment : 0F =

Glass wedge

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3 2

3

Ed hUa

=Strain energy :

calculated by considering the thin layer as a cantilever beam ( of unit thickness, B=1)

3.3 Obreimoff’s experiment

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3 2

38Ed h

a a aΨ ⎛ ⎞∂ ∂

= ⎜ ⎟∂ ∂ ⎝ ⎠

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3 2

3

Ed hΠ U Fa

= − =Thus,

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3 2

4

Ed ha

= −2

2

3 03 2

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Ed ha 2aΨ∂

→ = >∂

s2 aγ=

1 430

/3 2s

cs

W Ed hd ada a a 16Ψ Π

γ⎛ ⎞∂∂

= + = ⇒ = ⎜ ⎟∂ ∂ ⎝ ⎠

Always stable configuration !

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3 2

4

Ed hUa a

−∂=

∂

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Cracked specimen under load :

3.4 Energy analysis – compliance method

Π U F= −Potential energy :

Total energy :sWΨ Π= +

Work F done by the applied forces ?

F PΔ=

Linear elastic behaviour :

U ?

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0

U Pd Δ

Δ= ∫Compliance :

PCΔ

=

Strain energy expressed by

2 12 2

U PCΔ Δ= =From the linear law,

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Π PΔ= −Clearly,

s1Ψ P 2 a2

Δ γ= − + Total energy (with unit thickness B=1)

Crack propagation when Ψ 0A∂

=∂

Sign of 2

2

Ψ ?a

∂∂

Positive : stable propagationNegative : unstable propagation

sWa aΠ ∂∂

⇒ − =∂ ∂

2 2

2 2

Ψa a

Π∂ ∂=

∂ ∂Note that when Ws is a linear function of a

A Ba=

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Energy release rate (ERR):

By definition (Irwin 1956),

GAΠ∂

= −∂

Crack extension occurs when :

2 sG γ=

Stability of the crack growth ?

0dGdA

< Stable crack propagation (controllable manner)

0dGdA

> Unstable crack propagation (uncontrollable manner)

Ex : For the plate of with a center crack of length 2a

G 0A∂

>∂

2aGE

πσ= Always unstable !

“crack driving force”

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ERR under constant load :

12

U P

F P

Δ

Δ

⎫= ⎪⎬⎪= ⎭

Π U→ = −

Π U F= −

B

1

PB aΠ∂⎛ ⎞= − ⎜ ⎟∂⎝ ⎠

1

P

UB a

∂⎛ ⎞= ⎜ ⎟∂⎝ ⎠G

AΠ∂

= −∂

Thus,2 P

PB a

Δ∂⎛ ⎞= ⎜ ⎟∂⎝ ⎠Graphical interpretation :

OD E

G dA dU dF= − + ( )OBE OAD ABED= − − +OAC CBED ABED= − + OAC ABC OAB= + =

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Expression with the compliance C :

PCΔ =

2

2P CGB a∂

=∂

By reporting in the expression of G, one has

differencing with respect to a,

P CC Pa a aΔ∂ ∂ ∂= +

∂ ∂ ∂

P

CPa aΔ∂ ∂⎛ ⎞ =⎜ ⎟∂ ∂⎝ ⎠

P constant :

Δ constant : P P Ca C aΔ

∂ ∂⎛ ⎞ = −⎜ ⎟∂ ∂⎝ ⎠

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ERR under constant displacement

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U P

F

Δ ⎫= ⎪⎬⎪= ⎭

Π U→ =

1 UB a Δ

∂⎛ ⎞= − ⎜ ⎟∂⎝ ⎠G

AΠ∂

= −∂

Thus,

2P

B a Δ

Δ ∂⎛ ⎞= − ⎜ ⎟∂⎝ ⎠

Because P P Ca C aΔ

∂ ∂⎛ ⎞ = −⎜ ⎟∂ ∂⎝ ⎠and PCΔ =

2

2P CG

B a∂

=∂

expression of G at constant load recovered !

Graphical interpretation :

O

A

C

G dA dU= − OAC=

Difference with load control2

dPd ABCΔ→ negligible

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Generalization :

O

( )P ,aΔ

aa +da

Δ dΔ Δ+

P

: possible relationship between the load P and the displacement Δ for a moving crack

( )P ,aΔ

Identification of the crack driving force G

AB

CD

G dA dU dF= − + dU OBC OAD OAE EBCD→ − = − +

dF ABCD→

E

G dA OAE EAB OAB= + =Thus,

Having measured and knowing the loading curves for neighbouring crack areas, the development of the ERR can be followed

( )P ,aΔ

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Example : Double Cantilever Beam (DCB)

Double cantilever beam specimen at fixed load (a) or fixed grips (b)

δ

LP

P

2 H

B(a)

(b) L

2 H

B

δ

2δ

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PGBLδ

=

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Cracked structure with a finite system compliance :

M MP k C PΔ = δ + = δ + δ1M Mk C=

δ and G are assumed to depend only of ( )A,P

Structure held at a fixed remote displacement

Differentiating, 0M

P A

d dA dP C dPA P

Δ ∂δ ∂δ⎛ ⎞ ⎛ ⎞= + + =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠(1)

P A

G GdG dA dPA P

∂ ∂⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

P A

G G G PA A P AΔ Δ

∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠Dividing by dA and fixing Δ , (2)

( )1

P MA P A

G G GA A P A C PΔ

∂ ∂ ∂ ∂δ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ + ∂δ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(1) + (2) implies

cf Anderson’s book