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(

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

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

Experimental x Theoretical

Chaotic nature of dynamic fracture

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

Fracture instability criterions

Fig. 18 – Physical aspect of the crack on instability phenomenon

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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