A new critical velocity in the dynamic fracture of brittle ...sdm.isc.cnr.it/GFF/presentations/Scheibert.pdf · New critical velocity: v a = 160m/s ~ 0.2 c R. Dynamic fracture energy

A new critical velocity

in the dynamic fracture

of brittle amorphous materials

J. Scheibert1,2,3, C. Guerra1, F. Célarié1, D. Dalmas2, D. Bonamy1

1. Grp. Complex Systems & Fracture, SPCSI, CEA Saclay, France

2. SVI, UMR CNRS/Saint-Gobain, Aubervilliers, France

3. PGP, University of Oslo, Norway

Dynamic: crack tip velocity v ~ speed of sound cR

Brittle: material is elastic up to fracture

crack propagation = bond breaking

� silica glass and polymethylmethacrylate (PMMA)

Amorphous: no long range cristalline order

heterogeneities: Glass ~ 0.1nm ; PMMA ~ 10nm

In the following: only linear elastic materials

Dynamic fracture of brittle amorphous materials

v

Equation of motion (Freund, 1990)

E

KvAGvAv

2

)()()( ==Γ

cR : Rayleigh speed (predicted limiting crack speed)

−≈

Rc

vvA 1)(

K = quasi-static stress intensity factor !

Energy balance between: - energy release rate G = K²/E

- dynamic fracture energy Γ(v)

- kinetic energy (velocity v)

Linear Elastic Fracture Mechanics (LEFM, Freund 1990)

� single smooth crack in linear elastic medium

Micro-branching: a first critical velocity

Micro-branching occurs over vb ~ 0.4 cR

Sharon et al

PRL 1995

Crack propagation

100µm

Equation of motion valid only for v < vb

PMMA Glass

vb vb

Dynamic fracture energy under vb ?

Derived from

Dissipating mechanisms at low velocities ?

E

KvAv

2

)()( =Γ

Γ(v=0)

(Sharon and Fineberg, Nature 1999)

Experimental setup: Wedge Splitting test

Hole: adjustable stored energy U0

Quasi-static loading

Rollers: no friction

Measurements:

1. Load when crack initiates (3-20 kN)

2. Velocity profile

3. Fractography

Load Wedge

Jaw

Roller Groove

Seed crack

Hole

Sample

(PMMA)

Support

140mm

125mm

Thickness: 15mm

Decelerating configuration

E

KvAv

2

)()( =Γ

R

25 lines

350µsU

UmesRmes

R >> Rmes :

Umes varies linearly with crack length

NR

RUU mes

mes =

Velocity measurement

90 lines

E

KvAv

2

)()( =Γ

Stress intensity factor: finite elements calculations

Ingredients: linear elasticity

plane stress

elastostatics

θθθθw

F

Hyp : constant wedge position

during crack propagation

E

KvAv

2

)()( =Γ

Dynamic fracture energy vs velocity

E

K

c

v

R

2

1

−≈Γ

v from measurements

K from FEM

Complete Γ curve from v=0 to vb

vb ~ 350m/s (micro-branching)

New critical velocity: va = 160m/s ~ 0.2 cR

Dynamic fracture energy vs stress intensity factor

E

K

c

v

R

2

1

−≈Γ

2 linear regimes:a = 0.77

b = 0.48

Origin of the transition at va ? � fractography analysis

va

vb

R

Ra

cva

cv

vv−−

−=<Γ

1

1

)( α

R

Ra

cvb

cv

vv−−

−=>Γ

1

1

)( β

2 critical velocities:

(1 – a) cR = 0.23 cR ~ va

(1 – b) cR = 0.52 cR ~ effective limiting crack speed

Fractography: parabolic marks

velocity threshold at va

�Same transition as on Γ

v < vavb < vva < v < vb

A

B

C

500µm

Simple model (Smekal 1953, Ravi-Chandar 1997)

Nucleation center +

Nucleation distance d

vmaincrack = vmicrocrack

� intersection = parabola

Micro-cracking damage

d

For v > va, the crack goes faster

because many micro-cracks

open simultaneously

Γ

vva

Conclusions

• New dynamic transition in the brittle fracture of amorphous solids…

…at a well-defined critical velocity va ~ 0.2cR

• Transition corresponds to the onset of micro-cracking damage

� Brittle to quasi-brittle transition

• Generic transition:

Microcracks in many brittle amorphous materials

(polymers, glass, rocks,…)

Ref: J. Scheibert, et al., arXiv:0906.2877