Dynamic fracture - ESPCI Paris · 2017-06-21 · 16/46 Ravi-Chandar, CENTER FOR MECHANICS OF SOLIDS, STRUCTURES AND MATERIALS nucleus active crack front d n, nucleation distance s,

$Page 1: Dynamic fracture - ESPCI Paris · 2017-06-21 · 16/46 Ravi-Chandar, CENTER FOR MECHANICS OF SOLIDS, STRUCTURES AND MATERIALS nucleus active crack front d n, nucleation distance s,$
Center for Mechanics of Solids, Structures and Materials

Department of Aerospace Engineering and Engineering Mechanics

“Measure what is measurable, and make measurable what is not so.” - Galileo GALILEI

Dynamic fracture

Krishnaswamy Ravi-Chandar

Lecture 4 presented at the

University of Pierre and Marie Curie

May 21, 2014

Ravi-Chandar, CENTER FOR MECHANICS OF SOLIDS, STRUCTURES AND MATERIALS 2/46

Outline

• Crack surface roughness development

• Crack front waves

• Crack branching

• Tape peeling


Dynamic crack evolution


Surface roughening

Mirror Mist Hackle

Branch

Ravi-Chandar and Knauss, Int J Fract, 1984


Dynamic crack front evolution

Mirror Mist Hackle

Branch

Ravi-Chandar and Knauss, Int J Fract, 1984


Microcracking model for dynamic fracture

Ravi-Chandar and Knauss, Int J Fract, 1984 – Homalite - 100


Surface roughening and limiting speed

• Microcracking, instability

– R-C and Knauss, IJF, 1984, R-C and Yang, JMPS, 1996

• Dynamic instability and microbranching

– Sharon and Fineberg, PRL, 1996

• Crack twisting – mode III perturbations

– Hull, J Mat Sci, 1997

• Microcracking, roughening

– Bonamy and coworkers, 2010,…

Lp


Surface roughening - PMMA

Ravi-Chandar and Yang, J Mech Phys Solids, 1997 – PMMA


Formation of conic marks

a

100 mm

b

5 mm

Smekal, 1953 Ravi-Chandar and Yang, J Mech Phys Solids, 1997 ; Ravi-Chandar, Int J Fract, 1998 – PMMA


‘Plane’ of the microcracks - microbranches



Nucleus of the conic

Ravi-Chandar, Int J Fract, 1998 – PMMA


Statistics of conic marks - 1

0

500

1000

1500

2000

2500

3000

3500

0 100 200 300 400 500 600 700 800

Region aRegion bRegion c

Den

sity

of

Co

nic

Mar

kin

gs

- m

m-2

Position - mm


Focal Length - microns

Fre

qu

en

cy

0

1

2

3

0 1 2 3 4 5 6 7 8


Fre

qu

en

cy

0

1

2

3

0 1 2 3 4 5 6 7 8


Fre

qu

en

cy

0

1

2

3

0 1 2 3 4 5 6 7 8


Statistics of conic marks - 2

Scheibert, Guerra, Celarie, Dalmas and Bonamy, PRL, 2010


Speed of microcracks?

• Assume that microcracks grow with the same speed

– If microcrack speeds are of equal, then the shape of the conic is independent of the speed (R-C and Yang, 1997)

– Dalmas et al (IJF, 2013) show this by examining the detailed shapes of conics

• How fast do the microcracks grow?

– It is not really possible to examine this experimentally. To be determined by other means!

– This is still an open issue

• Can we simulate this? – just the kinematics!


Nucleation and growth model: Flaw Nuclei

• Nucleation is by cavity growth

• Spacing and density are governed by the stress level

• Measured densities are in the range of 500 to 2500 nuclei per mm2.

Ravi-Chandar and Yang, J Mech Phys Solids, 1997

0 100 200 300 400 500 600 700 800 900 10000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


nucleus

active crack front

dn, nucleation distance

s, spacing between nuclei

Nucleation and growth model: Nucleation

• When the stress at a flaw nucleus reaches a critical value, the nucleus becomes an active microcrack; calculation of this requires detailed knowledge of the flaw dimensions and the detailed stress field in its vicinity.

• Here we assume that when an active crack front approaches a nucleus to within a critical nucleation distance, dn the flaw is nucleated into a microcrack.



Nucleation and growth model: Growth

• Constant microcrack speed

– all active microcracks grow with the same speed regardless of the density of microcracks

• Constant energy flux

– microcrack speed depends on the number of active microcracks and the available energy



Constant microcrack speed



Constant power input

Position

No

rma

lize

d e

ns

em

ble

cra

ck

sp

ee

d

0

1

2

3

4

0 5 10 15

Mac

rocr

ack

sp

eed

Position

Position

Mic

roc

rac

k s

pe

ed

0

0.5

1

1.5

0 5 10 15

Mic

rocr

ack

sp

eed

Position



Comments…

• Differences in density of conic marks between different types of PMMA need to be reconciled

– Bonamy et al have one to two orders of magnitude lower conic marks per unit area

• Quantitative modeling of the deformation within the fracture process zone – cavitation, crack initiation, growth, coalescence,…

• Continuum modeling based on heterogeneity?

– Line models and quantitative calibration/comparison

– Crack front waves


Crack front waves – a quick summary

Morrissey and Rice, JMPS, 2000

0

ˆ ( , ) ˆˆ( , ) ( , )G k

P k A kG

0( , ) ( , )A z t a z t v t

… obtained from perturbation solution of Willis and Movchan; has a simple zero corresponding to a propagating mode with speed Cf ; this is the crack front wave.

ˆ( , )P k

f 0.96 0.97 RC C


Source

Wallner lines

Field, J Contemporary Physics, 1971


Wallner lines

Photographs: Jill Glass, Sandia National Laboratories


Mode III perturbations

Bonamy and Ravi-Chandar, 2003

x z

y

12.7 mm

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

6.00E-06 6.50E-06 7.00E-06 7.50E-06 8.00E-06 8.50E-06

f

MHz

l

mm

t

ns

0.5 6880 2000

5 688 200

20 172 50

Wavelength and duration for glass

-3

-2

-1

0

1

2

3

0.00 0.05 0.10 0.15 0.20 0.25

Distance traveled - m

Pu

lse

am

pli

tud

e -

V

Glass


25

Specimen P

v= 890 m/s

f = 5 MHz

10 mm


26

Specimen AC

v= 440 m/s

f = 5 MHz

12.7 mm


27

Specimen AI

v= 440 m/s

f = 20 MHz

10 mm


Crack-ultrasonic pulse interaction

l

vl/Cs

Shear wave

Crack front

Cs

v

q

x z

y

x

z

v

m/s

f

MHz

lx l v/Cs

mm

480 5 96

867 5 174

895 20 45 Attenuation is similar to that

of bulk glass


Results of Fineberg et al


Summary on crack fronts

• Theory predicts that they exist – Important in roughness generation; bulk waves can also

deliver energy to the cracks and cause roughness, but these CF waves are more effective.

– Difficult to distinguish from Wallner lines

• Small amplitude plane wave perturbation – Perfectly linear response to mode III perturbations

– Responds to wavelength of input, speed of crack – no inherent characteristic length!

– No persistent interaction between different pulses

– No persistence when perturbation is removed

• Need larger amplitude perturbations?


Dynamic crack branching



Crack branching

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0 50 100 150 200 250

Time - ms

Dy

na

mic

Str

es

s In

ten

sit

y F

ac

tor,

KI -

MP

a m

1/2

0

40

80

120

160

200

Cra

ck

po

sit

ion

, a

- m

m

a

KI


Microcracking model for dynamic fracture

Ravi-Chandar and Knauss, Int J Fract, 1984 – Homalite - 100


Control of surface roughening and branching

Ravi-Chandar and Knauss, IJF, 1984


Fracture mechanics

Lp

Inner Problem: Are the details of the failure mechanisms and fracture processes important only in determining the fracture energy, G?

Outer Problem: Knowledge of constitutive laws and balance equations is adequate to calculate the energy release rate, G


Crack branching analysis

• Stress field based

– Yoffe (1951): v~0.65 CR

• Energy-based

– Eshelby (1970)

– Freund (1972): v~0.53 CR

– Adda-Bedia et al. (2005, 2007) -0.25

0

0.25

0.5

0.75

1

1.25

1.5

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Angle (Radians)

Ho

op

Str

ess

- f

qq

v/Cd=0.0

0.1

0.2

0.3

0.35

0.4

221

( ) dynamic energy release rateI IG A v KE


Energy balance calculations of Adda-Bedia et al

0( , ) ( ) ( ,0), , ,p p pK t v k v K t p I II III

0( , ) ( ) , ', ( ,0)p l pl l

l

K t v k v H v v K tl

Adda-Bedia et al. 2005, 2007,…

Not all the Hpl are available, but H33

was calculated by Adda-Bedia et al, corresponding to the mode III problem.

0

33( , ) ( ) , ', ( ,0)III III IIIK t v k v H v v K tl

2

0

33

1( ) , ( ,0)

2IIIG g v H v K tl

m

;G v G v G G

0.39 ; 0.22(39.6 ); 0sv C vl


In-plane modes

Adda-Bedia et al. 2005, 2007,…

2 2

0 0

11 21

1( ) , ( ,0) ( ) , ( ,0)

2I I II IIG g v H v K t g v H v K tl l

m

;G v G v G G

H11 and H21 are not available, but Adda-Bedia argues that in the limit of zero crack speed, these should behave similarly to H33, corresponding to the mode III problem.

0.518 ; 0.13(23.4 ); 0Rv C vl

influences the critical speed, but not the branching angle

vG


Dynamic lifting of a tape

• Inextensible, flexible tape with tension T, mass per unit length r

• … transverse deflection

• … transverse velocity

• … slope (assumed small)

,w w

w wt x

( , )w x t

T

l ct c 2 21

2L w Twr

2 22

2 20 speed of transverse waves

w wc c T

x tr

Expect piecewise linear solutions:

In the absence of adhesion in the string, you can show that

If is prescribed, then

0w w

s w wt x

s c

ds

sdt

f f s f s

( , )w x t

( , )w x t

( , )w x t

/w w c w

Burridge and Keller, 1978, Freund, 1990, Duomochel et al, 2008,…


Dynamic peeling of a tape with adhesion

Now let us consider adhesion of the tape to the substrate with an adhesive energy G

,w w

w wt x

( , )w x t

T

l st sExpect piecewise linear solutions:

But another equation is needed to determine

This is derived from energy balance: dynamic energy release rate

0w w

s w wt x

Consider pulling one end of a semi-infinite tape at a constant speed, . As a result, a peel front moves with a speed s

s

2 2 2/2

TG L s L s w w c

w

22

21

2

T sG w

c


Dynamic peeling of a tape with adhesion

,w w

w wt x

( , )w x t

T

l st s

Impose Griffith condition: G = G

122

21 qs

sw w

c

2 /qsw T G

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

Normalized Velocity

No

rma

lize

d S

lop

e

22

21

2

T sG w

c

G

2 2 2

2 2 22 /

s w c

c T w c

G


Let us peel quasi-statically:

At this point, the peel front encounters a weak patch; let us fix d

Quasi-statically, crack will extend until

Peeling front encountering a weak patch

2 1

1 2 1

0 x l

l x

G G

G G

2 2 2

1 2

2 / ; 0

2 /

qsw T w

d l T

G

G

2 1 12 /qs qsw w T G

2

1 1

equilibrium crack length

qs

al

l

G

G

1 12 /qsw T G

1l

d 2qsw

qs

al

1qsw

2G 1G

However, the crack will propagate dynamically


But, the crack will advance dynamically, at a speed

A kink wave will propagate towards the fixed end in the peeled portion of the tape

Dynamic peeling front encountering a weak patch

2 1

1 2 1

0 x l

l x

G G

G G

ct

1l

h

c

s

1 1,w w

2qsw

1 1crack tip: 0sw w

1 2 1 2kink: 0w w c w w

2 2 1 2 1 1

dynamic energy release rate

2 / 2 /cw w w w w s T G

1 2 1

1 2 1

1

2qs

w

w

G G

G G

s

1l

c

s c

t

x

c

2 1

2 1

/ 1

/ 1

s

c

G G

G G


Next, we consider reflection from the fixed end of the kink wave

This will arrive at the location at

Dynamic peeling front encountering a weak patch

1l

c

s c

t

x

c

al

t

ax l 1 /at l l s

1l

d 2qsw1 1,w w

s

0,r rw w

c

1 1kink: 0r rw w c w w

11 1 1

2

1r qs qs

sw w w w

c

G

G

The peel front arrests and everything becomes quiescent!

21

1

qs

a al l lG

G


Extensions to this one-d problem

• Continue loading at x = 0; must consider multiple reflections

• Introduce speed dependence to the energy G(v)

• Consider flexibility – straightforward extension to the bending problem

• Consider extensible strings – more complex due to the presence of longitudinal and transverse waves

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Dynamic fracture - ESPCI Paris · 2017-06-21 · 16/46 Ravi-Chandar, CENTER FOR MECHANICS OF SOLIDS, STRUCTURES AND MATERIALS nucleus active crack front d n, nucleation distance s,