A chaos and fractal dynamicsapproach 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-Iloading and plane strainin elastodynamic crack
growth conditionsFig. 1 - Experimental Setup
Introduction
Dynamic fracture has been stimulating interest not only because of its fundamental importance inunderstanding fracture processes but also because ofthe challenges to mathematical analysis andexperimental techniques.
Fineberg-Gross and co-workers performed experiments on fast crack growth for different brittle materials, suchas PMMA, soda-lime glass, etc.,
revealing many new aspects, defiant to the fracture dynamic theory, related to unstable crack growth.
They observed the existence of a critical velocity starting from which instability begins
and measured the correlation between the fluctuation in crack growth velocity and the ruggedness of the generated surfaces.
Experimental Results
Crack branching generated by instabilities on different crack
growth velocities in the Fineberg
experiments
Fig. 2 – Crack branching on PMMA
FinebergExperimental Results
Fig. 3 - Physical aspect of the
fracture surface with fractal
ruggedness for diferent crack
growth velocities
They found a time delay (of the order of the stressrelaxation time) present during the whole fracture process in the materials used by them.
In spite of this experimental observation can be associated to stress relaxation properties in viscoelasticmaterial,
but they were unable to relate physically it to the onsetof the instability phenomenon in itself.
Therefore the necessity of a correct mathematical description of the instability process in crack growth represent one of the interesting challenge in dynamic fracture.
Introduction
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 3s for PMMA and 1.0s for soda-lime glass, for example, showing that it is
has given value for each material.
Introduction
How we can explain the instability process on fast crack growth?
We know that the fracture surface are fractals! Because its ruggednessAs in the low crack growthAs in the fast crack growth
Introduction
But, what is the difference between them?
The ruggedness on low crack growth is due the interaction of the crack tip with the microstructure of the material
Many, many papers treat with this!
Introduction
For us, we are interested on fast crack growth. Because only the interaction of the crack tip with the microstructure is not sufficient to explain their phenonmenons
Why? Because there is a intrinsec instability that
cannot explain by the classical fracture mechanics
How?
Theoretical development
Fracture criterion of the basical equations of the Classical Dynamic Fracture Mechanics
),(),( ooooooD vLvLG where
Elastodynamic energy released rate
Elastodynamic work of fracture
oDG
o
Theorethical foundations
We need to modify the classical equations using fractal theory into the elastic linear fracture mechancs, for example:
Classical Fracture Fractal Fracture
o
Lo dL
dLdL
UFdG )(
oo RG o
effo dLdLG 2
o
Lo dL
UFdG )(
Theorethical foundations
Where dL/dLo it is the ruggedness mathemathical term that must be used to explain thatfractal behaviour. Fig. 5 – Fractal fracture surface
model
Fracture surface fractal model
Fractal objects can be constructed with P-adics objects.
For the crack we will have P-adic solutions,perhaps? - Self-similar or self-affine mathemathical solutions.
Very good! Some papers already treat this! But we need more for dynamic fracture
case?
The foundations of quasi-static and dynamic fracture mechanics
o
oR
o
effDo dL
dL
dLdL
cv
G
1
2
Stationary problem and solution
oAoo
Do TUFdAdG
RoooooD c
vgLGvLG ).(),(
))(()(
)(1
2)( tLv
tdLdL
tdLdL
cv
t ooo
oR
o
effo
The foundations of non-stationary dynamic fracture mechanics
TUP
dAWUR
ij )(lim0*
dAuuT iR
i
21lim
0*dsuTP i
Si
)().(),,( tfcvgLGtvLGR
oooooD
Non-stationary problem: equation and solution
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 thetime 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
theenergy
flux at the crack tip
Fundamental Hipothesis
Fig. 8 - Transfer function with time delayat the crack tip
The time, t, required for acrack tip to advance a distance equal to lo, as the crack growsat 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".
Dynamic fracture model with a time delay
Fig. 9 - Energyflux to the crack
tip with time delay
The crack tip varies with the
time and moves with the crack
)()()(),,( tvtgLGtvL ooooo
))(())((),(()( tLvtLvtLt ooooooo
The crack tip feedback
feedback
tht ))(()(
Fig. 10 – The crack tip time delay with feeback conditions
))(()(
)())((1
2)(
tLv
tdLdL
tdLdL
ctLvt oo
o
oR
oo
effo
Advanced dynamic fracture mechanics considerations
The non-statinary solution it is not sufficient! We need a hypothesis more! But what? We need to add the chaos theory – using a
logistic map – for example, to explain the intrinsic instability phenonmenon
How? A self-similar or self-affine equation with a
self-similar or self-affine feedback solution to problem, explain the non-linear influence ofthe crack tip on the fast crack growth
We need a self-similar or a self-affine equation!
with p-adic solutions?
Fig. 11 – Different kinds of systems
Oscillators systems
))...)))((....((()( otfhhhhhntf
Advanced considerations based on fractal aspects of fracture surface for dynamic
fracture mechanics
Therefore we have: A self-affine crack growth velocity model
)()(
)())(())(())((
tdLtdL
tdLtLdhtLvtLv o
ooooo
)(/)())(())(( tdLtdLtLvtLv ooo
where we have
)))((())(( tLLhtLL oo
Anzatz solution : A self-affine function
A chaotic model fordynamic fracture
Logistic equations in fracture mechanics
oR
o
oR
o
o
o
oR
odLdL
c)t(v1
dLdL
c)t(v
dLdL2
GdLdL
c)t(v
oR
o1k dL
dLc
)t(vx
oR
ok dL
dLc
)t(vx
2
2
of
o
oc
o
o
o
LL
dLdL
G
kkk xxx 11
)())(()( 1 kk htht
Logistic-equation/map solutions
Use of the logistic-map on the fracture mechanics
Classical Solution Chaos Solution
Roo
oo
ooR
oo
cvdLdL
dLdL
G
where
tvdLdL
ctvtv
1
421
)()(1)(
Roo
oo
oRo
cvdLdL
dLdL
G
whereG
cv
1
12
)21(
Results of the Model
If the time delay was zero we recover the clasical solution
Therefore it is possible explain:
Comparison between theory and experiments
Experimental x Theoretical
0
0,2
0,4
0,6
0,8
1
0 0,3 0,6 0,9 1,2 1,5 1,8 2,1 2,4 2,7 3 3,3 3,6 3,9
tempo (s)
Com
prim
ento
pro
jeta
do d
a tri
nca
(mm
)
Seqüência1 Seqüência2 Seqüência3 Seqüência4
Experimental x TheorethicalThe existence of a critical velocity
Th inatingibility of the Rayleigh wave by the cracks
Comparison between theory and experiments
)21(o
Ro Gcv
Roc cv 34,0 Roc cv31
)()(1)( tvdLdL
ctvtv o
oR
oo
??odL
dL 1odL
dL
Ro cv Ro cv
Comparison between theory and experiments
Experimental x TheorethicalMaximmum crack growth velocity
Bifurcation of the crackRo cv 6,0max
Ro cv32
Discussion of the purposedsolution
We purppose of use a retard or time delay on a feedbacked system we can explain the instability in a general way
Time-delay x Chaos routeswith feedback
?Period-duplication
Intermittence
Quase-periodicity
Instability necessary conditions
Because, we have an instability when we have
Two situations equally probable on the output
Dissacord between the input and theoutput (a time delay, for example)
A feedback system
The instability process under the sight of time delay
Fig. 19 -Mathemathical criterion for the fracture and instability
Instability mathemathical region into thetime delay zone
Discussions The experiments performed by Fineberg and co-workers provide
evidence for instability in the brittle fracture of isotropic materials. On the other hand, theories based on conventional concepts such as
energy balance and quasi-static configurational forces at crack tipsshow no indication of strong oscillatory or branching instabilities by analyzing the stresses in the neighborhood of a crack tip growing athigh velocity hinted about the emergence of instabilities but the analysis is not a truly dynamic theory of forces and accelerations offracture surfaces.
The basic tenant of dynamic fracture mechanics is that the processesnear the fault tip occur at near wave velocities, and for this reason the crack tip is independent of the details of loading.
This is a famous theorem proven independently by Kostrov and Eshelby in 1964 and 1969, respectively, for the antiplane case and by Kostrov and Nikitin in 1970 for general loading.
Discussions
The logisitc solution depends
on theexperimental
setup Depending of the
experimentalsetup, other
logistic maps can be obtained
-1,5
-1
-0,5
0
0,5
1
-1,5 -1 -0,5 0 0,5 1 Seqüência1Seqüência2
21 1 kkk xxx
Fig. 19 – Logistic map
Discussions Observe that the Eq. (17) refers to a particular experimental set up. Therefore, in accord to the functional dependence of this equation
for the g(vo/cR) term or depending of the particular form of the experiments
other kinds of logistic maps can be obtained, since that the same procedure of calculations accomplished until now be done.
Eq. (46) developed in this paper is equivalent to Eq. (12), and the same improvements (finite size of the sample, influence ofboundary, etc.) proposed to Eq.(\ref{(12)}) can be incorporated into Eq. (\ref{(43)}) and Eq. (\ref{(46)}) without consequences tothe results presented in this paper.
This is corroborated by experimental evidence showing that the onset of instabilities is independent of the size of the sample and/orof the geometrical set up of the experiment (see also Ref.\cite{3}).
Conclusions This paper present arguments in favour of chaotic behavior of rupture. Its arguments are general and based on energy conservation principle which are
totally valid on fracture mechanics. The central hypothesis of this paper is that the energy flux to the crack tip is
converted at the crack tip to fracture energy with a time delay, , due to the development of a viscoelestic process zone in front of the crack tip.
It is tacitly assumed in this paper that such a delay exists and that this delay has a well-defined time scale, , being a characteristic property of the material.
Its magnitude is of the order of the viscoelastic relaxation time of the samplematerial under local fracture conditions.
A key assumption of the theory is that the onset of instability observed in the velocity of dynamic crack growth is due to the time delay.
This time delay factor, , in implies in the possibility to derive an equation forcrack growth the velocity in the form of a logistic map equation.
Conclusions It is necessary to remark that the hypothesis of linear energy transfer as given
is an oversimplified approach. The energy release rate, Go, is linearly dependent on the crack length whereas
the crack resistance, Ro, rises in a non-linear form. Based in this property will be used in a
forthcoming paper in which it is shown that the energy dissipation can also be written in the form of a logistic map having as consequence crack branching and other phenomena so far not explained by the classical fracture theory.
The purpose of this paper is to show that contrary to what has been thought previously,
the most familiar models in fracture mechanics are intrinsically unstable. Therefore, for this purpose it was used a more simple possible case of dynamic
crack growth of a semi-infinite body with plane strain condition, and well established concepts and results, to write
an expression for the crack velocity in the form of a logistic equation and map very well-known.
Conclusions From this map conclusions regarding instabilities of crack growth are drawn and
compared with the experimental results obtained by Fineberg and co-workers\cite{1}. This work shows that other logistic maps can be built, according to the particularity of
the experiment and according to the expression of its kinetic energy. The logistic map built have an interpretation capable to supply light to the understanding
of more complex situations of the phenomenon. From te above results it is concluded that the instabilities that abound in dynamic
fracture are consequences of the mathematical structure of chaos that underlies such
phenomena. In this paper it was able to write the straight line crack velocity in the form of a logistic map explaining the onset of instabilities observed by Fineberg et al. This achievement brings into fracture mechanics all the mathematical structure
developed for complex systems. This theoretical approach provides a single and concise tool to determine among others
properties the conditions under which crack growth becomes dynamically unstable and branching takes place as will be shown in a forthcoming paper.
Conclusions The literature scientific exhibition without doubt shade that there is fractals in
the fracture quasi-static in the fracture surfaces. Then without a doubt none in the dynamic fracture will must there will be
chaotic behavior in the formation of the same ones. Therefore, if the model proposed in the article is not the final answer for the
subject, at least as initial step, it lifts this subject and it opens a new proposal ofstudy of the phenomenon of the dynamic fracture.
Therefore, it is interesting to observe that the numbers that appear in the results of the calculations of this article resemble each other in a lot with theexperimental results and this that to say that "there is something there"!!!.Therefore, I want to say that, an experimental research is followed in our laboratories to illuminate the theoretical evidences more closely and you try lifted by this initial work.
Acknowledgments
This research work was in part supported financially by CNPq, FAPESP, CAPES and oneof the authors, Lucas Máximo Alves thanks the Brazilian program PICDT/CAPES andPROPESQ-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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