Transcript A chaos and fractal dynamics approach to the fracture mechanics Lucas Máximo Alves GTEME - Grupo de Termodinâmica, Mecânica e Eletrônica dos Materiais, Departamento de.

A chaos and fractal dynamics
approach to the fracture mechanics
Lucas Máximo Alves
GTEME - Grupo de Termodinâmica,
Mecânica e Eletrônica dos Materiais,
Departamento de Engenharia de Materiais,
Universidade Estadual de Ponta Grossa
Caixa Postal 1007, Av. Gal. Carlos Calvalcanti, 4748,
Campus UEPG/Bloco L - Uvaranas, Ponta Grossa,
Paraná,CEP. 84030.000 Brasil.
The Experimental Problem
Dynamic Fracture
problem on fast crack
growth in PMMA for the
Fineberg-Gross
experiments
 A semi-infinite plane
plate under Mode-I
loading and plane strain
in elastodynamic crack
growth conditions

Fig. 1 - Experimental Setup
Experimental Results

Crack branching
generated by
instabilities on
different crack
growth velocities
in the Fineberg
experiments
Fig. 2 – Crack branching on PMMA
Fineberg
Experimental Results

Fig. 3 - Physical
aspect of the
fracture surface
with fractal
ruggedness for
diferent crack
growth velocities
A time delay phenonmenon betwen the
crack growth velocity and the fracture
surface
They showed that as the crack
speed reaches a critical value a
strong temporal correlation
between velocity, vo(t), and the
response in the form of the
fracture surface at Ao(t + )
takes place (having its notation
changed, in the present text, to
Lo(t) instead Ao(t) to designate
the fracture surface length). The
time delay measured between
this two greatness present a
value about of   3s for PMMA
and 1.0s for soda-lime glass,
for example, showing that it is
has given value for each
material.
Theoretical development

Fracture criterion of the basical equations of
the Classical Dynamic Fracture Mechanics
GoD  ( Lo , vo )  o ( Lo , vo )
where
GoD
o
Elastodynamic energy released rate
Elastodynamic work of fracture
Theorethical foundations
We need to modify the classical equations
using fractal theory into the elastic linear
fracture mechancs, for example:
 Classical Fracture
Fractal Fracture
d (F  U L )
Go 
dLo
Go  Ro
d ( F  U L ) dL
Go 
dL
dLo
Go  2 eff
dL
dLo
Theorethical foundations

Where dL/dLo it
is the
ruggedness
mathemathical
term that must
be used to
explain that
fractal
behaviour. Fig. 5 – Fractal fracture surface
model
The foundations of quasi-static and
dynamic fracture mechanics

Stationary problem and solution
d
F  U Ao  To 
GDo 
dAo
 v 
GoD ( Lo , vo )  Go ( Lo ).g  
 cR 
2 eff
dL
GDo 
 vo dL  dLo
1 

 cR dLo 
2 eff
dL
o (t ) 
vo ( Lo (t ))
 vo dL  dLo (t )
1 

 cR dLo (t ) 
The foundations of non-stationary
dynamic fracture mechanics

Non-stationary problem: equation and solution
P  U  T  
P   Ti ui ds
S
GoD

U  lim W ( ij )dA
*0
R
1
ui ui dA

*0 2
R
T  lim
 v 
 ( Lo , vo , t )  Go ( Lo ).g   f (t )
 cR 
Therefore the solution for dynamic fracture problem
must be a kind of “P-adic differential equation” with
invariance by scale transformation
Crack tip problem of the
process zone formation

The origin of the
time delay from
Fineberg-Gross
experimental
evidences is due
the instability of the
atoms during the
breaking of chemical
bonds at the crack
tip
Fig. 6 – IBM computational simulations of crack
tip on fracture phenomenon
Physical phenomenons at
the crack tip
Fig. 7 The time
delay
effect on
the
energy
flux at the
crack tip

Fundamental Hipothesis
The time, t, required for a
crack tip to advance a distance
equal to lo, as the crack grows
at crack growth velocity, vo,
introduces a process time,
given by t = lo/vo, that must
be compared to a
characteristic relaxation time, 
~ t, of the material to
determine if the process is
"fast" or "slow".

Fig. 8 - Transfer function with time delay
at the crack tip
Dynamic fracture model with a
time delay

Fig. 9 - Energy
flux to the crack
tip with time
delay
 The crack tip
varies with the
time and moves
with the crack
t
 ( Lo , vo , t )  Go ( Lo ) g ( )vo (t )

 o (t   )  o (Lo (t ), vo (Lo (t ))vo (Lo (t   ))
The crack tip feedback

(t   )  h( (t ))



feedback
Fig. 10 – The crack tip time delay with feeback
conditions
2 eff
dL
 o (t   ) 
vo ( Lo (t   ))
vo ( Lo (t )) dL dLo (t )
1
cR
dLo (t )
Oscillators systems


We need a self-similar
or a self-affine
equation!
with p-adic solutions?
f (t  n )  h(h(h(h....h( f (to ))...)))

Fig. 11 – Different
kinds of systems
Advanced considerations based on fractal
aspects of fracture surface for dynamic
fracture mechanics

Anzatz solution : A self-affine function
L(Lo(t   ))  h(L(Lo(t )))

Therefore we have: A self-affine crack growth
velocity model
dh( L(t )) dLo (t )
vo ( Lo (t   ))  vo ( Lo (t ))
dLo (t ) dL (t )
where we have
vo ( Lo (t ))  v( L(t ))dLo (t ) / dL(t )
A chaotic model for
dynamic fracture

x k 1
Logistic equations in fracture mechanics
vo ( t   ) dL
Go vo ( t ) dL 
vo ( t ) dL 
 1 


dL c R dLo 
cR
dLo
c R dLo 
2
dLo
vo ( t   ) dL
vo ( t ) dL   G o   Lo
xk 

dL
Loc
c R dLo
cR
dLo
2
xk 1  xk 1  xk 
dL o
 (t   )  h( (t ))  k 1  h( k )
 o


 of




2
Logistic-equation/map solutions

Use of the logistic-map on the fracture
mechanics
Classical Solution
2
vo  cR (1 
)
Go
where

2 dL
1
Go dLo
dL
1
dLo
vo  cR
Chaos Solution

vo (t ) dL 
vo (t )
vo (t   )   1 
cR dLo 

where
1  
2 dL
4
Go dLo
dL
1
dLo
vo  cR
Energy flux to the crack tip

Fig. 12 - Logistic solution for the fracture
problem
Results of the Model

Fig. 13 - Logistic Map for diferent periods or
cycles
Results of the Model

Fig. 14 - Different solutions for differents
stages k
Comparison between theory
and experiments
Experimental x Theoretical
1
0,8
0,6
0,4
0,2
3,
9
3,
6
3
3,
3
2,
7
2,
4
2,
1
1,
8
1,
5
1,
2
0,
9
0,
6
0
0
0,
3
Comprimento projetado da
trinca (mm)

tempo (s)
Seqüência1
Seqüência2
Seqüência3
Seqüência4
Comparison between theory
and experiments
Experimental
x
Theorethical
The existence of a critical velocity
voc  0,34cR
voc 
1
cR
3
Th inatingibility of the Rayleigh wave by the cracks
vo  cR (1 
2
)
Go
dL
 ??
dLo
vo  cR

vo (t ) dL 
vo (t )
vo (t   )   1 
cR dLo 

dL
1
dLo
vo  cR
Comparison between theory
and experiments
Experimental
x
Theorethical
Maximmum crack growth velocity
vo max  0,6cR

vo 
Bifurcation of the crack
2
cR
3
Chaotic nature of dynamic
fracture

Experimental x Theoretical
Fracture instability
criterions

Fig. 18 – Physical aspect of the crack
on instability phenomenon
Discussions
The logisitc
solution depends
on the
experimental
setup
Depending of the
experimental
setup, other
logistic maps can
be obtained
1


0,5
0
-1,5
-1
-0,5
0
0,5
1
Seqüência1
Seqüência2
-0,5
-1
-1,5
Fig. 19 – Logistic map

xk 1  xk 1  xk
2

Acknowledgments
This research work was in part supported
financially by CNPq, FAPESP, CAPES and one of
the authors, Lucas Máximo Alves thanks the
Brazilian program PICDT/CAPES and
PROPESQ-UEPG for concession of a
scholarship.
 The authors thanks your supervisor Prof. Dr.
Bernhard Joachim Mokross and too the Prof.
Leonid Slepyan for helpful discussions
 Prof. Benjamin de Melo Carvalho
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