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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.

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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

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Introduction Dynamic fracture has been stimulating interest not only because of its fundamental importance in understanding fracture processes but also because of the challenges to mathematical analysis and experimental techniques. Fineberg-Gross and co-workers performed experiments on fast crack growth for different brittle materials, such as 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.

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Experimental Results Crack branching generated by instabilities on different crack growth velocities in the Fineberg experiments Fig. 2 – Crack branching on PMMA

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Fineberg Experimental Results Fig. 3 - Physical aspect of the fracture surface with fractal ruggedness for diferent crack growth velocities

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They found a time delay (of the order of the stress relaxation 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 viscoelastic material, but they were unable to relate physically it to the onset of 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

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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, v o (t), and the response in the form of the fracture surface at A o (t + ) takes place (having its notation changed, in the present text, to L o (t) instead A o (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.

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Introduction How we can explain the instability process on fast crack growth? We know that the fracture surface are fractals! Because its ruggedness As in the low crack growth As in the fast crack growth

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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!

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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?

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Theoretical development Fracture criterion of the basical equations of the Classical Dynamic Fracture Mechanics where Elastodynamic energy released rate Elastodynamic work of fracture

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Theorethical foundations We need to modify the classical equations using fractal theory into the elastic linear fracture mechancs, for example: Classical FractureFractal Fracture

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Theorethical foundations Where dL/dL o it is the ruggedness mathemathical term that must be used to explain that fractal behaviour. Fig. 5 – Fractal fracture surface model

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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?

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The foundations of quasi-static and dynamic fracture mechanics Stationary problem and solution

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The foundations of non-stationary dynamic fracture mechanics 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

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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

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Physical phenomenons at the crack tip Fig. 7 - The time delay effect on the energy flux at the crack tip

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Fundamental Hipothesis Fig. 8 - Transfer function with time delay at the crack tip The time, t, required for a crack tip to advance a distance equal to l o, as the crack grows at crack growth velocity, v o, introduces a process time, given by t = l o /v o, that must be compared to a characteristic relaxation time, ~ t, of the material to determine if the process is "fast" or "slow".

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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

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The crack tip feedback Fig. 10 – The crack tip time delay with feeback conditions

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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 of the crack tip on the fast crack growth

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We need a self-similar or a self-affine equation! with p-adic solutions? Fig. 11 – Different kinds of systems Oscillators systems

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Advanced considerations based on fractal aspects of fracture surface for dynamic fracture mechanics Therefore we have: A self-affine crack growth velocity model where we have Anzatz solution : A self-affine function

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A chaotic model for dynamic fracture Logistic equations in fracture mechanics

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Logistic-equation/map solutions Use of the logistic-map on the fracture mechanics Classical Solution Chaos Solution

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Energy flux to the crack tip Fig. 12 - Logistic solution for the fracture problem

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Results of the Model Fig. 13 - Logistic Map for diferent periods or cycles

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Results of the Model Fig. 14 - Different solutions for differents stages k

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Results of the Model If the time delay was zero we recover the clasical solution Therefore it is possible explain:

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Comparison between theory and experiments Experimental x Theoretical

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Experimental x Theorethical The existence of a critical velocity Th inatingibility of the Rayleigh wave by the cracks Comparison between theory and experiments

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Experimental x Theorethical Maximmum crack growth velocity Bifurcation of the crack

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Experimental x Theoretical Chaotic nature of dynamic fracture

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Discussion of the purposed solution We purppose of use a retard or time delay on a feedbacked system we can explain the instability in a general way Time-delayx Chaos routes with feedback ? Period-duplication Intermittence Quase-periodicity

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Instability necessary conditions Because, we have an instability when we have Two situations equally probable on the output Dissacord between the input and the output (a time delay, for example) A feedback system

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

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The instability process under the sight of time delay Fig. 19 - Mathemathical criterion for the fracture and instability Instability mathemathical region into the time delay zone

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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 tips show no indication of strong oscillatory or branching instabilities by analyzing the stresses in the neighborhood of a crack tip growing at high velocity hinted about the emergence of instabilities but the analysis is not a truly dynamic theory of forces and accelerations of fracture surfaces. The basic tenant of dynamic fracture mechanics is that the processes near 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.

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Discussions The logisitc solution depends on the experimental setup Depending of the experimental setup, other logistic maps can be obtained Fig. 19 – Logistic map

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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 of boundary, etc.) proposed to Eq.(\ref{(12)}) can be incorporated into Eq. (\ref{(43)}) and Eq. (\ref{(46)}) without consequences to the 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/or of the geometrical set up of the experiment (see also Ref.\cite{3}).

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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 sample material 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 for crack growth the velocity in the form of a logistic map equation.

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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.

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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.

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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 of study 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 the experimental 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.

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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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References - J. Fineberg, S. P. Gross, M. Marder and H. L. Swinney, Instability in dynamic Fracture, Phys. Rev. Lett. 67, p. 457, (1991). - J. Fineberg, S. P. Gross, M. Marder and H. L. Swinney, Phys. Rev. B, 45, n. 10, p. 5146, (1992-II). - B. V. Kostrov and L. V. Nikitin,. Archiwum Mechaniki Stosowanej, 22, 749-775, (1970). - R. M Christensen, Theory of Viscoelasticity: An Introduction, Academic Press, New York, (1982). - J. R. Willis, A comparison of the fracture criteria of Griffith and Barenblatt, J. Mech. Phys. Solids, 15, 151, (1967). - J. R. Walton, The dynamic energy release rate for steadily propagating antiplane shear crack in a linearly viscoelastic body, J. App. Mech., 54, 635, (1987). - G. Golenievski, International Journal of Fracture, 33, 39-44 (1988). - L. B. Freund, Dynamic Fracture Mechanics, (Cambridge University Press, 1990). - Brian Lawn, Fracture of Brittle Solids, Cambridge University Press, (1995). - H. L. Ewalds and R. J. H. Wanhill, Fracture Mechanics, (Edward Arnold Publishers, 1993).

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References - Melvin F. Kanninen and Carl H. Popelar, Advanced Fracture Mechanics, (CRC Press, Oxford 1985 p. 437, Chapter 7). - T. L. Anderson, Fracture Mechanics, Fundamentals and Applications,Chapter 4, section 4.1.2, Eq. s (4.14), (4.19) and (4.20), p. 215-218, 2nd. Edition, (CRC Press, 1995). - Eron Sharon and Jay Fineberg. Confirming the continuum theory of dynamic brittle fracture for fast cracks, Nature, 397, 28 January, p. 333-335,(1999). - P. D. Washabaugh and W. G. Knauss, A reconciliation of dynamics crack velocity and Rayleigh wave speed in isotropic brittle solids. Int. J. of Fracture, 65, 97-114, (1994)). - L. L. Mishnaevsky Jr., Int. J. of Fracture, 79, p. 341-350, (1996). - A. A. Griffith, Phil. Trans. R. Soc. London, A221, p. 163, (1920). - Alves, Lucas Máximo Rosana Vilarim da Silva, Bernhard Joachim Mokross, The influence of the crack fractal geometry on the elastic plastic fracture mechanics. Physica A: Statistical Mechanics and its Applications, 295, n. 1/2, p. 144-148, 12 June (2001). - Huajian Gao, Surface roughening and branching instabilities in dynamic fracture, J. Mech. Phys. Solids, 41, n. 3, p. 457-486, 1993; A theory of local limiting speed in dynamic fracture 44, n. 9, p. 1453-1474, 1996. - N. F. Mott, Brittle fracture in mild steel plates. Engineering, 165, 16-18, (1948).

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References - L. I., Slepyan, Principle of Maximum Energy Dissipation Rate in Crack Dynamics, J. Mech. Phys. Solids, 41, p. 1019-1033, (1993). - W. G. Knauss and K. Ravi-Chandar, Int. J. of Fracture, 27, 127-143 (1985). - Steven Paul Gross, Dynamics of fast fracture, Doctor Philosophy Dissertation, Faculty of the Graduate School of the University of Texas at Austin, August (1995). - M. P. Marder and Steven P. Gross, Origin of crack tip instabilities, Journal Mechanics and Physics Solids, 43, n. 1, p. 1-48, (1995). - Alves, Lucas Máximo, A new fractal thermodynamic principle to the energy dissipation in the fracture dynamics, to be published - L. B. Freund; Energy Flux into Tip of an Extending Crack in an Elastic Solid, Journal of Elasticity, 2, p. 341-349, (1972). - Eron Sharon; S. P. Gross and J. Fineberg, Local Crack Branching as a Mechanism for Instability in Dynamic Fracture, Phys. Rev. Lett., 74, 5096, (1995). - J. F. Boudet; S. Ciliberto and V. Steinberg, Dynamics of crack propagation in brittle materials, J. Phys. II France, 6, p. 1493-1516, (1996).

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References - J.G. Williams, and A. Ivankovic, Limiting crack speeds in dynamic tensile fracture test, International Journal of Fracture, 51, p. 319-330, (1991). - N. Fiedler-Ferrara and C. P. C. do Prado, Caos, Uma Introducção, (Ed. Edgard Blücher Ltda, Brazil, 1994). - S. Neil Rasband, Chaotic Dynamics of Non-Linear Systems, (John Wiley, New York p. 23, 1990). - C. Beck and F. Schlögl, Thermodynamics of Chaotic Systems, (Cambridge University Press, Cambridge, England, 1993). - Peter Gumbsch. An atomistic study of brittle fracture: Toward explicit failure criteria from atomistic modeling. Journal of Materials Research, 10, 11, p. 2897- 2907, (1995). - Peter Gumbsch, S. J. Zhou and B. L. Holian, Molecular dynamics investigation of dynamic crack stability, Phys. Rev. B, 55, n. 6, 1 February (1997). - Peter Gumbsch, In: Computer Simulation in Materials Science, 227-244, edited by H. O. Kirchner, L. Kubin and V. Pontikis (Kluwer Academic Publishers, Dordrecht, Netherlands, 1996).

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References - B. Cotterell and J. R. Rice, Int. J. Fract.Mech., 16, p. 155, (1980). - E. Yoffe, Phil. Mag., 42, p. 739, (1951). - Eron Sharon; Steven Paul Gross and Jay Fineberg, Energy dissipation in dynamic fracture, Physical Review Letters, 76, n. 12, p. 2117-2119, 18 March (1996). - Honglai Tan and Wei Yang, Nonlinear motion of crack tip atoms during dislocation emission processes, J. Appl. Phys., {\bf 78}, n.12, 15 December (1995). - R. Mohan; A. J. Markworth and R. W. Rollins, Effects of dissipation and driving on chaotic dynamics in an atomistic crack-tip model, Modeling Simul. Mater. Sci. Eng., 2, p. 659-676, (1994).

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