FRACTAL MODELLING OF UNSTABLE OR CATASTROPHIC

FRACTURE DYNAMICS

Lucas Máximo Alves

GTEME – Grupo de Termodinâmica, Mecânica e Eletrônica dos Materiais, Departamento de Engenharia de Materiais, Universidade Estadual de Ponta Grossa, Av. Gal.

Carlos Calvalcanti, 4748, Campus UEPG/Bloco L – Uvaranas – Ponta Grossa-PR, Brazil, CEP. 84030.000, Cx. Postal 1007,(lmalves@uepg.br).

SUMMARY FRACTAL MODELLING OF UNSTABLE OR CATASTROPHIC FRACTURE DYNAMICS..................................................................1

6. 1 - Introduction .................................................................................................................................................................................... 2

6. 2 - Theoretical development..................................................................................................................................................................3

6. 3 – Theoretical results...........................................................................................................................................................................3

6.3.1 – Inatingibility of waves Rayleigh velocity by cracks......................................................................................................................... 3

6.3.2 – Ruggedness of the crack fractal path ..............................................................................................................................................4

6.3.3 – Elastodynamic Energy Release Rate for crack rugged path..............................................................................................................4

6.3.4 – Crack growth velocity corrected by the ruggedness......................................................................................................................... 6

6. 4 – Comparison with experimental results............................................................................................................................................ 11

6. 6 - Discussions ................................................................................................................................................................................... 12

6. 7 - Conclusions .................................................................................................................................................................................. 13

6. 9 - Bibliografical References............................................................................................................................................................... 14

ABSTRACT

The fractal geometry is introduced in the dynamic fracture to describe the ruggedness

during the crack propagation characterizing the morphology of generated surfaces. It is shown that

the fractal dimension has a strong influence about the limit speed of fast crack propagation. It is also

shown that the inatingibility of speed of waves Rayleigh by cracks in some materials can be

correlated to the ruggedness of fracture surfaces by means of exponent Hurst, H, that measures the

ruggedness dimension. The experimental results of Fineberg et al [FINEBERG 1991; FINEBERG

1992] they were adjusted by the model. The unstability coefficient was calculated for Fineberg et al

measurements in agreement with the definition proposed by SANDER [1984]. One concluded that

exist an intrinsic unstability to the phenomenon that produces the ruggedness of surfaces, limiting

this way the maximum of crack propagation speed for a smaller value than the speed of waves

Rayleigh in the material.

Words key: speed of crack propagation, fractal dimension, self-affine

surface, speed of waves Rayleigh.

PACS numbers: 62.20Mk, 46.30Nz, 61.43H.

6. 1 - Introduction

- The elastodynamic classical theory of fracture (EDCTF) quantifies the growth, the speed and the

energy dissipation in crack propagation in terms of euclidian. Examples: lengths, areas and speeds

projected along the propagation direction geometry [EWALDS 1986; ANDERSON 1995].:

- The dynamics of crack growth is characterized by the curve GD ou JD [ANDERSON 1995] that

measures the rate of elastodynamic energy released in the propagation process in function of the

crack projected speed, vo, and also by the graph of speed crack propagation, vo , in function of its

projected length, Lo.

- However, it has been observed by different researchers that, the speed of crack propagation in

some materials, like PMMA for example, it doesn't reach the limit of Rayleigh elastic waves speed

in the material, as it is waited by EDCTF [DULANEY & BRACE 1960].

- Many authors have proposed different explanations to describe that impossibility [KOBAYASHI

1974, 1978; RAVI-CHANDAR 1984a, 1984b] and others have suggested special experimental

conditions where the speed of Rayleigh waves is reached by the cracks [GILMAN 1958; HULL 1966,

COTTERELL 1965; FIELD 1971].

- The explanation of this fact, it can simply be attributed the influence of the ruggedness of fracture

surface created in the crack propagation process which is a consequence of the several mechanisms

of energy dissipation by the crack including the dynamic unstability.

- The reason for the which this unstability appears is subject of a next paper.

- In this paper it will be seen as the equations of EDCTF for the unstable regime of cracks

propagation can be modified by means of fractal geometry, in way to include in the mathematical

description of fracture the ruggedness of surface.

- This way, the expression for the rate of the elastodynamic energy released, GD, as derived by

EDCTF, (eq. 6.66) it is corrected by the introduction of the fractality (ruggedness) of fractured

surface during the crack propagation (equation (6. 10)). This, procedure turns the classic expression,

eq. (6.53), of the projected crack speed, vo, in function of the projected length, Lo, in an equation

that contains the correction of the ruggedness (equation (6. 15)), showing the influence of this in the

superior limit of the speed of crack propagation.

- By means of this reformulations of the classic equations of the EDCTF proposals for IRWIN

[1948], FREUND [1990], SLEPYAN [1993] and others, it was possible better to understand the

geometric process of energy dissipation by the crack tip and its interaction with the microestructure

of the material, as well as the unstability process that can produce crack branches that searchs

liberate the excess of local energy.

6. 2 - Theoretical development

1 – Why the crack growth velocity don´t reaches the limit of Rayleigh Waves velocity

Because the rugedness formed in the process

2 – Why the ruggedness is formed during the crack growth?

Intrinsec instability process that ocurrs to fast crack growth conditions (subject of the other paper)

6. 3 – Theoretical results

Figure - 6. 1. Modelling fractal of the speed of growth of a self-affine fractal using the method Sand-Box to describe the speed of propagation of a crack.

H

lL

lH

LL2

0

0

2

0

00 1

(6. 1)

6.3.1 – Inatingibility of waves Rayleigh velocity by cracks

1dLdL

o (6. 2)

oo dL

dLvv

(6. 3)

Ro cv (6. 4)

6.3.2 – Ruggedness of the crack fractal path

o

ooo

o LLLLLL

dLdL

)()( , (6. 5)

22

22

1

)2(1

H

o

o

H

o

o

o

Ll

LlH

dLdL , (6. 6)

22

22

1

)2(1

H

o

o

H

o

o

o

Ll

LlH

vv . (6. 7)

6.3.3 – Elastodynamic Energy Release Rate for crack rugged path

o

Do dLdL)TU(F

dLdG . (6. 8)

ooR

oDo dL

dLdLdL

cv1GG

(6. 9)

2H2

o

o

2H2

o

o

R

o

o

o2

oDo

Ll1

Ll)H2(1

cv1

EL2G , (6. 10)

0 100 200 300 400 500

150

200

250

300

= 142 J/ m2

cR = 928 m/ s

Taxa

de

ener

gia

elas

todi

nâm

ica

liber

ada

GD (J

/m2 )

velocidade rugosa de propagação da trinca v (m/s)

Figure - 6. 2. Graph of the rate of energy elastodynamic in function of the speed of crack propagation for the rugged path, taking as example PMMA with = 142J/m2 e cR= 928m/s .

0.0 0.2 0.4 0.6 0.8 1.0

-80000

-60000

-40000

-20000

0

H = 0.2 H = 0.4 H = 0.6 H = 0.8 H = 1.0

Taxa

de

ener

gia

elas

todi

âmic

a lib

erad

a G

Do (

J/m

2 )

comprimento projetado da trinca Lo (mm)

Figure - 6. 3. Graph of the rate of energy elastodynamic for the path projected in function of the projected length of the crack, showing the effect of different propagation speeds

0.0 0.2 0.4 0.6 0.8 1.0-7000

-6000

-5000

-4000

-3000

-2000

-1000

0

1000

vo = 0 vo = 200 vo = 400 vo = 600 vo = 800 vo = 928

Taxa

de

ener

gia

elas

todi

âmic

a lib

erad

a G

Do (

J/m

2 )

comprimento projetado da trinca Lo (mm)

Figure - 6. 4. Gráfico da taxa de energia elastodynamic para o caminho projetado em função do comprimento projetado da trinca, mostrando o efeito de diferentes velocidades de propagação

6.3.4 – Crack growth velocity corrected by the ruggedness

ooR

oo R

dLdL

cv

1G

(6. 11)

oR

o

oo

dLdL

cv

dLdL

G1

2. (6. 12)

dLdL

dLdL

GR1cv o

ooRo

, (6. 13)

dLdL

LLcv o

o

ocRo

1 (6. 14)

2H2

o

o

2H2

o

o

o

ocRo

Ll

)H2(1

Ll

1

LL

1cv

, (6. 15)

0,0 0,2 0,4 0,6 0,8 1,00,0

0,2

0,4

0,6

0,8

1,0

H = 0,2 H = 0,4 H = 0,6 H = 0,8 H = 1,0

Cra

ck g

row

th v

eloc

ity (m

/s)

Projected crack length Lo (mm)

Figure - 6.5. Graph of the projected speed of propagation of a crack in function of its projected length, showing it influences of the ruggedness for different values of the exponent Hurst.

0,0 0,2 0,4 0,6 0,8 1,00,0

0,2

0,4

0,6

0,8

1,0

lo = 0,00001 lo = 0,0001 lo = 0,001 lo = 0,01 lo = 0,1

Cra

ck g

row

th v

eloc

ity (m

/s)

Projected crack length Lo (mm)

Figure - 6. 6. Graph of the projected speed of propagation of a crack in function of its projected length, showing it influences of the minimum size of the structure fractal.

0,0 0,2 0,4 0,6 0,8 1,00,0

0,2

0,4

0,6

0,8

1,0

Loc=0,00001 Loc=0,0001 Loc=0,001 Loc=0,01 Loc=0,1

Cra

ck g

row

th v

eloc

ity (m

/s)

Projected crack length Lo (mm)

Figure - 6. 7. Graph of the projected speed of propagation of a crack in function of its projected length, showing it influences of the critical size of Griffith for the fracture.

0,0 0,2 0,4 0,6 0,8 1,00,0

0,2

0,4

0,6

0,8

1,0

H = 0,2 H = 0,4 H = 0,6 H = 0,8 H = 1,0

Cra

ck g

row

th v

eloc

ity v

o (m

/s)

Projected crack length Lo (mm)

Figure - 6. 8. Graph of the projected speed of crack propagation in function of the projected length of the crack, for different dimensions fractais D = 2 - H.

0,0 0,2 0,4 0,6 0,8 1,00,00,10,20,30,40,50,60,70,80,91,01,11,21,31,41,51,61,7

lo =- 0.00001 lo =- 0.0001 lo =- 0.001 lo =- 0.01 lo =- 0.1

Cra

ck g

row

th v

eloc

ity v

o (m

/s)

Projected crack length Lo (mm)

Figure - 6. 9. Graph of the projected speed of crack propagation in function of the projected length of the crack, for different elementary sizes of the structure fractal.

0,0 0,2 0,4 0,6 0,8 1,00,0

0,1

0,2

0,3

0,4

0,5

Loc =- 0.00001 Loc =- 0.0001 Loc =- 0.001 Loc =- 0.01 Loc =- 0.1

Cra

ck g

row

th v

eloc

ity v

o (m

/s)

Projected crack length Lo (mm)

Figure - 6. 10. Graph of the projected speed of crack propagation in function of the projected length of the crack, for different critical sizes of Griffith

6. 4 – Comparison with experimental results

Figure - 6. 11. Lateral physical aspect of the fracture surface of PMMA of the experiments of Fineberg-Gross.

Figure - 6. 12. Medium speed of propagation in function of the length of the crack for PMMA

0 10 20 30 40 50 60 70 80 900

100

200

300

400

500

600

Data: Data1_BModel: Fineberg-Gross2Chi^2 = 1371.66532cR = 928m/slo = 1.4113E-19 ±4.3321E-17H = 0.99785 ±0.01433Loc = 15.03367 ±0.07204

Fineberg-Gross data Ruggedness Model

Cra

ck g

row

th v

eloc

ity (m

/s)

Projected crack length Lo (mm)

Figure - 6. 13. Adjust done in agreement with the model of developed ruggedness

6. 5 - Discussions

Observe that, although the inatingibility of the speed of the waves Rayleigh for the

cracks is explained by the influence of the ruggedness that appears in the fracture surface, according

shows the secção – 6.3.1, and the equations (6. 4) and (6.31) and the graphs of the Figure - 6. 2 and

of the Figure - 6. 14, the expression of the rate of energy released elastodynamic, given by the

expression (6. 10) and for the graph of the Figure - 6. 10, it cannot just be explained the theory

fractal, because she is not adjusted to the experimental results of Fineberg - Gross. Therefore, it is

necessary to run over some information the plus, in order to meet the true expression for the rate of

energy released elastodynamic. This will be made in the next papar when it will be proposed a

Principle of Maximum Energy Dissipation (PMDE) to explain the results of Fineberg-Gross

completely.

The Figure - 6. 7 shows a change in the behavior of the curves of the speed of crack

propagation, vo, with its projected length, Lo, starting from values of the minimum size of the

fracture, lo, much smaller than Loc (lo << Loc) An inverse behavior is observed in the Figure - 6. 3,

that is, for values of much smaller Loc than lo (Loc << lo). it Observes the presence of different

values for the initial speed, vo, in the point Lo = lo corresponding to the minimum length, lo in the

graph of the Figure - 6. 6. These values corresponding to the initial values of the speed of the crack

are also affected for the values of the critical length, Loc, in the graphs of the Figure - 6. 2.

The self-similar approach, although it is possible to find an as good as the self-affine

modelling adjustment, of the result shown in the Figure - 6. 14, with an almost imperceptible

difference in this adjustment, she is inferior to the self-affine model. This because this approach

introduces mistakes in the calculation underestimating the values of the ruggedness of the fracture

surface, dL/dLo, and of the minimum size of the microscopic fracture, lo, although she doesn't affect

the value of the exponent Hurst, H. However, this approach is only valid in I begin it of the crack

propagation

FINEBERG [1991, 1992] it registered the existence of a critical speed of crack, voc, where

the flow of energy for the crack tip stops of increasing when this value is surpassed. Starting from

there they begin the unstabilities.

It was proven that the flutuactions in the speed of the crack tip, that you/they result in

deflexões of the same, are provoked by the interaction of the reflected elastic waves of the border of

the material [GROSS 1993], or for the local thermal flutuactions [YUSE 1993], or due to intrinsic

phenomenons of dynamic unstability, whose processes dissipativos is not still inside very

understood of the Nonlinear Physics field. These flutuactions in the speed of the crack tip only

happen when this surpasses a critical value of propagation.

Feinberg and Gross showed that in the dynamic propagation of fast cracks, the

formation of a ramified structure, is a result of the unstability maid for the need of dissipating the

energy imposed in the crack tip, above a critical value. According to WILLIAMS [1987] the

unstability, in a general way, happens due to a desacoplamento among functions dynamic no-

harmonicas, as the speed of propagation of the phenomenon and the requested flow of energy. The

bifurcation happens when they exist at least two situations equally probable that the system can

occupy in a dynamic process. BOUDET [1996] it showed that the structure of the formed crack

follows an Scaling with potency law he/she doesn't inform among the ruggedness and it climbs of

measure. This characteristic demonstrates that the fracture surfaces follow an Scaling fractal as it

was already shown by MECHOLSKY [1989] and several other researchers. Slepyan [SLEPYAN 1993]

it formulated a Principle of the Rate of Maximum Energy Dissipation to explain to the unstabilities

maid for the desacoplamento among the flows of energy in the crack tip.

6. 6 - Conclusions

The model for the propagation of unstable or dynamic crack could explain

quantitativamente:

a) The inatingibility of the speed of Rayleigh for the crack.

b) The existence of the critical speed where the flow of energy for the crack tip stops of

increasing when this value is surpassed.

c) The values for the critical speed don't depend on the length critical lo used to measure

the crack. This length lo can be interpreted as a critical length to begin the fracture in the

microscopic scale of the material.

If it be considered that the fracture surface doesn't have only a dimension fractal, D, but

a small flutuaction exists about in the measured fractal of a medium value, because of the process

unstability, the graphs of the speed of crack propagation can exhibit a flutuaction in the speed,

similar the that registered by FINEBERG [1991, 1992] and SHARON [1995 and 1996].

In this paper it was obtained the explanation of the inatingibility of the speed of the

waves Rayleigh for the lowering of the speed limit of propagation of a crack due to ruggedness. In a

next paper the subject of the flutuaction of the speed produced by the unstability will be seen.

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