FRACTURE OF HETEROGENEOUS SOLIDS

baconossifiedMechanics

Oct 29, 2013 (4 years and 15 days ago)

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Elisabeth Bouchaud

GROUPE FRACTURE

S
ervice de
P
hysique et
C
himie

des
S
urfaces et des
I
nterfaces

CEA
-
Saclay

The Chinese University of Hong
-
Kong, September 2008

FRACTURE OF

HETEROGENEOUS SOLIDS

Cindy Rountree

Laurent Ponson

Daniel Bonamy

Gaël Pallarès

Akshay Singh

Claudia Guerra

The Fracture

Group

Montpellier University

Matteo Ciccotti

Mathieu Georges

Christian Marlière

Bordeaux University

Stéphane Morel

Orsay University

Harold Auradou

Jean
-
Pierre Hulin

CEA
-
Saclay

Jean
-
Philippe Bouchaud

Stéphane Chapuilot

Caltech

G. Ravichandran

Onera

Denis Boivin

Jean
-
Louis Pouchou

Leonardo da Vinci’s fracture experiments on metallic wires

The Chinese University of Hong
-
Kong, September 2008


Compromise of mechanical properties:


The importance of being imperfect…



Pure metals are too «

soft

»




Alloys:

solid solution

atoms







dislocations (atomic)







intermetallic inclusions

(1
-
50

m
m)






& interphase boundaries







grains & grain boundaries (up
~
0.1mm)






Polymers rigid but brittle




reinforced by soft

rubber particles (

1nm
-
1µmF






Glasses?

Amorphous structure (1nm)





The Chinese University of Hong
-
Kong, September 2008

Composite material: epoxy matrix, graphite fibers

(Columbia University)

The Chinese University of Hong
-
Kong, September 2008

Balsa wood (Vural & Ravichandran, Caltech)

The Chinese University of Hong
-
Kong, September 2008

Ni
-
based alloy


grain size 20 to 80 mm

(Onera)

The Chinese University of Hong
-
Kong, September 2008

Ni
-
based alloy


grain size 2 to 30 mm

(Onera)

The Chinese University of Hong
-
Kong, September 2008

Polyamide reinforced

with rubber particles

(L. Corte, L. Leibler,

ESPCI)

The Chinese University of Hong
-
Kong, September 2008

Polymeric foams

(S. Deschanel, ENS LYON
-
INSA)

The Chinese University of Hong
-
Kong, September 2008

Polymeric foams

(S. Deschanel, ENS LYON
-
INSA)

Tomographic images

during deformation

Silica tetrahedron

Silica tetrahedra sharing an oxygen atom:

membered rings

O

O

O

O

Si

AM
ORPHOUS

SILICA

The Chinese University of Hong
-
Kong, September 2008

How to estimate the properties

of a composite ?


Young’s modulus:

s
=
E
e

s

s

E
composite



F
E

+

F
E

Except if… cracks develop !

Why ?

The Chi nese Uni versity of Hong
-
Kong, September 2008

GENERAL OUTLINE

1
-

What is so specific about fracture?

2
-

Elements of Linear Elastic

Fracture Mechanics

3
-

Fracture mechanisms in real materials

4
-

Statistical characterization of fracture

5
-

Stochastic models

1. What is so specific about fracture?




A crude estimate of the strength to failure



Stress concentration at a crack tip



Damage zone formation in heterogeneous materials:



rare events statistics

2. Elements of Linear Elastic Fracture Mechanics



Griffith’s criterion



Fracture toughness and energy release rate



Weakly distorted cracks



Principle of local symmetry

OUTLINE

The Chinese University of Hong
-
Kong, September 2008

1
-

What is so special about fracture?

s

s



A crude estimate of the strength to failure

s
=E

D
x

a

Failure :
D
x≈a

s
f
≈ E

s
f
≈ E/100

Presence of flaws!

The Chinese University of Hong
-
Kong, September 2008

1
-

What is so special about fracture?

Stress concentration at a crack tip
(Inglis 1913)

s

s

2b

2a

A

s
A
>
s
:⁳tr敳scnc敮trt楯n

The Chinese University of Hong
-
Kong, September 2008

1
-

What is so special about fracture?

Infinitely sharp tip:

s

s

Irwin (1950)

K=stress intensity factor

Sample geometry

s

(r)

r

Strong stress gradient

Crack mostly sensitive at tip!

1
-

What is so special about fracture?

Mode II

In
-
plane, shear,

sliding

K
II

Mode I

Tension, opening

Mode III

Out
-
of
-
plane, shear

Tearing

K
I

K
III

Mixed mode, to leading order:

1
-

What is so special about fracture?

Heterogeneous material:


Fracture of a link if
s
(r,

)>
s
c_local

P(
s
c_local
)

s
c_local

s
c_min

s
c_max

Length R
C
of the damaged zone?

Statistics of rare events

The Chinese University of Hong
-
Kong, September 2008

2
-

Elements of fracture mechanics

Griffith’s energy balance criterion

Elastic energy

Surface energy

Total change in potential energy:

Propagation at constant applied load:

2a

B

s

d
a

Happens for a critical load:

Stress intensity approach:

Elastic energy per unit volume:

Crack increment
d


The Chinese University of Hong
-
Kong, September 2008

2
-

Elements of fracture mechanics

r

At the onset of fracture:




=1/2



Fracture toughness

Energy release rate

2
-

Elements of fracture mechanics

2
-

Elements of fracture mechanics

T
-
stress:


-

Stability of the crack


-

SIF variation due to out
-
of
-
plane


meandering

The Chinese University of Hong
-
Kong, September 2008

(Cotterell & Rice 80)

WEAKLY DISTORTED 2D CRACK

2
-

Elements of fracture mechanics

The Chinese University of Hong
-
Kong, September 2008

(Cotterell & Rice 80; Movchan, Gao & Willis 98)

Weight function (geometry)

Infinite plate:1/√
-

x

2
-

Elements of fracture mechanics

The Chinese University of Hong
-
Kong, September 2008

WEAKLY DISTORTED PLANAR CRACK

(Meade & Keer 84, Gao & Rice 89)

2
-

Elements of fracture mechanics

The Chinese University of Hong
-
Kong, September 2008

Weakly distorted 3D crack front

(Movchan, Gao & Willis 98)



K
II
=0



2
-

Elements of fracture mechanics

The Chinese University of Hong
-
Kong, September 2008

Crack path:

principle of local symmetry

Summary

-
LEFM (Linear Elastic Fracture Mechanics):


∙ Fracture toughness K
Ic



K
I
<K
Ic
: stable crack




K
I
≥K
Ic
: propagating crack


∙ Weak distorsions: change in SIFs




rough cracks and fracture surfaces

-
In real life…


∙ Dissipative processes



Plasticity




Brittle damage (microcracks)


∙ Subcritical crack growth


due to corrosion, temperature, plasticity…

The Chinese University of Hong
-
Kong, September 2008

Process zone size

V (m/s)


Rc

(nm)

Along the direction

of crack propagation


Perpendicular to the direction

of crack propagation




ln(V*/V)


The Chinese University of Hong
-
Kong, September 2008

3
-

Fracture mechanisms in real materials

1.5 nm

-
1.5 nm

x

Image 146

Kinematics of cavity growth

Image 50

x

A

B

C

x

Image 1

A

2

4

6

t (h)

100

200

300

x (nm)

A

B

C

The Chinese University of Hong
-
Kong, September 2008

3
-

Fracture mechanisms in real materials

Front arrière de la cavité

V = 8
±

5 10
-
12

m/s

Intermittency of propagation

C (foreward front cavity)

V = 9
±

8 10
-
12

m/s

A (main crack front)

V = 3
±

0.8 10
-
12

m/s

Positions of fronts A, B, C (nm)

B (rear front cavity)

V= 8
±

5 10
-
12

m/s

“Macroscopic” velocity 3 10
-
11

m/s!

The Chinese University of Hong
-
Kong, September 2008

3
-

Fracture mechanisms in real materials

Position of the main crack front (A)

Time

1
st

coalescence

2
nd

coalescence

Velocity 3 10
-
12

m/s

Velocity 3
10
-
11

m/s

3
-

Fracture mechanisms in real materials

The Chinese University of Hong
-
Kong, September 2008

3
-

Fracture mechanisms in real materials

(J.
-
P. Guin & S. Wiederhorn)

No plasticity, but what about nano
-
cracks?

…Fracture surfaces…

Summary

-

Dissipative processes: damage formation



∙ Fracture of metallic alloys:




the importance of plasticity






Quasi
-
brittle materials: brittle damage




∙ Stress corrosion of silicate glasses:




brittle or quasi
-
brittle?

-

From micro
-
scale mechanisms to a

macroscopic description:

∙ Morphology of cracks and fracture surfaces

∙ Dynamics of crack propagation

The Chinese University of Hong
-
Kong, September 2008