International Journal of Fracture 44: 167-178, 1990. ©1990 Kluwer Academic Publishers. Printed in the Netherlands'.

167

Crack layer analysis of fatigue crack propagation in rubber compounds H. A G L A N , (1~ A. C H U D N O V S K Y , (21 A. M O E T , C1) T. F L E I S C H M A N (2) and D. S T A L N A K E R (2) ~l)Department of Macromoleeular Science, Case Western Reserve University, Cleveland, Ohio 44106-2699, USA; (2~Firestone Tire and Rubber Company, Akron, Ohio 44325, USA

Received 3 October 1988; accepted in revised form 13 February 1989 Abstract. A methodology to characterize the resistance of rubber compounds to crack propagation (fracture toughness) is presented. A constitutive model based on the crack layer theory is utilized to extract the specific energy of damage y*, a material parameter characteristic of the material's resistance to crack propagation and the dissipative characteristic, ft. The model expresses the rate of crack propagation as da dN

flJ~ '/*R1 - Jl'

where da/dN is the cyclic rate of fatigue crack propagation (FCP), Jl is the energy release rate (tearing energy) and R t is the resistance moment which accounts for the amount of damage associated with the crack advance. Microscopic examination revealed that crack tip microcracking is the dominant damage mechanism. Hence, R~ was evaluated as the area (m2) of microcracking surfaces per unit crack advance. Fatigue crack propagation data for a particular rubber compound have been analyzed using the present model. The proposed equation describes the entire FCP history in the compound. According to this model, 7* and fi for the compound investigated, are found to be 9.3 kJ m 2 and 9.7 x 10 9m4/J_cycle, respectively.

1. Introduction

The resistance o f the material to crack p r o p a g a t i o n depends on the energy expended on irreversible d e f o r m a t i o n (damage) in the vicinity o f the crack tip. The objective of crack p r o p a g a t i o n studies is to identify and determine material p a r a m e t e r s responsible for its resistance to crack p r o p a g a t i o n , i.e., fracture toughness. It is thus h o p e d to establish predictive relationships to aid in lifetime assessment of load bearing structural c o m p o n e n t s and to guide the d e v e l o p m e n t o f crack resistant material. Classical fatigue crack p r o p a g a t i o n (FCP) laws express the cyclic crack extension in terms of some function o f the applied load. P a r a m e t e r s such as the stress intensity factor K and the stress intensity factor range A K are useful as correlative tools particularly since they possess an invariant nature. The energy release rate is m o r e a p p r o p r i a t e to correlate the rate o f crack p r o p a g a t i o n for highly strained and inelastic materials; energy release rate is also m o r e a p p r o p r i a t e f r o m t h e r m o d y n a m i c considerations. The logarithmic linear relationship between the incremental change in crack length d a / d N and A K p r o p o s e d by Paris and E r d o g a n [1] describes a linear stage o f m o n o t o n i c crack p r o p a g a t i o n . This and other empirical a p p r o a c h e s m a y oiler a facile m e t h o d to a p p r o x i m a t e crack p r o p a g a t i o n b e h a v i o r but do not assist a lucid understanding o f the associated p h e n o m e n a .

168

H. Aglan, A. Chudnovsky, A. Moet, T. Fleischman and D. Stalnaker

Fatigue crack propagation in rubber and reinforced rubber compounds, based on these empirical fatigue models, presents more difficulties. Rivlin and Thomas [2] stated that the solution of the mathematical problem of determining the stress distribution in a specimen of a highly elastic material is intractable and that the stress concentration at the tip of the crack is limited to a very small region in which measurement cannot be readily carried out. These workers calculated the tearing energy T for a tensile rubber specimen to be correlated with da/dN,

T = 2kWa,

(1)

where W is the strain energy density (obtained from the area under the tensile stress-strain curve) and _ais the crack length in the unstrained state. The proportionality constant k varies slightly with strain as experimentally found by Greensmith [3]. Many workers in the field, for example, Lake and Lindley [4, 5] studied the FCP behavior of natural rubber gum vulcanizates. These workers recognized the difficulty of explaining the rate of FCP in terms of the empirical fatigue models. They proposed that the behavior may be described by four empirically fitted equations. It is also reported [6-10] that there are several cases in which FCP in polymers does not obey the empirical models. This phenomenon is attributed to the rate at which damage accumulates in the crack tip region in resistance to crack propagation. The crack layer model relates the rate of crack propagation to material and fracture process parameters controlling damage accumulation in the crack tip region [11, 12]. The rate of FCP, based on the crack layer approach, is expressed as

da

dD/dN

dN

Y'R1 -- J l '

(2)

where dD/dN is the cyclic rate of energy dissipated on submicroscopic processes leading to damage formation and growth within the active zone (Fig. 1). The value of dD/dN may be expressed as dD dN

d Wi dN'

fl'--

(3)

where dW~/dN is the cyclic rate of the irreversible work done on the active zone associated with crack propagation and fl' represents the portion of dW~/dN expended on damage accumulation within the active zone. The rest of the irreversible work is converted into heat. In (2), J1 is the energy release rate (tearing energy), 7* is the specific energy of damage (material parameter), and RI is the resistance moment. The latter is a measure of the extent of damage expressed as the density of damage integrated over the trailing edge of the active zone. In this paper, a methodology based on the crack layer theory is presented to characterize the resistance of rubber compounds to crack propagation. This constitutive model is utilized to extract the specific energy of damage 7", a material parameter characteristic of its resistance to crack propagation (fracture toughness), and a dissipative characteristic parameter fi'.

Crack layer analysis of fatigue crack propagation in rubber compounds

169

m~ActJv(~ ZOfle

Fig. 1. Schematic illustration of a crack preceded by an active zone. 2. Material and experimental procedures The rubber compound used in this investigation was 100 percent natural rubber, sulfur cured, with 60 pph of carbon black (Type N326) and a variety of antioxidants and processing aids. The cure was 20 min at 149°C. Specimens of the rubber compound were cut to the dimensions 178 x 28.5mm from a 1.22mm thick compression molded sheet. The specimen length between the grips was 125 mm. A cut of 1 mm depth was introduced in the center of one edge with a razor blade. Fatigue crack propagation tests were performed using a 10001bf load cell on an MTS hydraulic testing machine. The tests were conducted in a laboratory atmosphere using a sinusoidal waveform at a frequency of 1 Hz. The maximum strain range was chosen as 21 percent. The specimen was stretched to the required maximum strain and relaxed to zero strain on each cycle. Crack propagation was monitored using a traveling optical microscope attached to the loading frame. A video system was attached to the microscope in order to obtain interval records of crack propagation. Load-displacement curves were simultaneously recorded at regular intervals of the crack growth using a Techmar digitizer connected to an IBM AT computer. The load-displacement curve up to the required maximum strain for one uncut specimen was established. The area under the load-displacement curve divided by the volume of the test piece is the strain energy density W. A damage mechanism in the form of microcracks was observed in this rubber compound. Microscopic analysis was carried out on the fractured specimens to measure the density of the observed microcracks.

3. Fatigue crack propagation data The crack layer model, (2) and (3), describes the rate of crack propagation as a function of the energy release rate J (or T), the resistance moment R~, and the cyclic rate of the irreversible work d W~/dN. The specific energy of damage 7" and the dissipation coefficient /3' are material and loading history dependent constants which can be extracted from FCP data using the proposed model. In the following sections, FCP data obtained from the rubber compound specimens are presented.

170

H. Aglan, A. Chudnovsky, A. Moet, T. Fleischman and D. Stalnaker 30

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No OF CYCLES

Fig. 2. Crack length vs. number of cycles in the rubber compound specimen.

3.1. Crack growth rate The actual crack length is obtained from the video playback and plotted against the number of cycles N. This is shown in Fig. 2. The crack grows slowly, approximately 5 mm in the first 800 000 cycles. Then it advances very rapidly reaching the ultimate failure in less than 50 000 additional cycles. The slope of this curve is the average crack speed at the corresponding crack length.

3.2. Energy release rate The energy release rate was evaluated experimentally based on the load-displacement curve obtained at increments of crack length. The area under the load-displacement curve was measured, thus determining the stored energy U. The difference between this and the stored energy just before crack initiation, U0, gave the total energy loss AU due to the total amount of crack growth. Values of A U were plotted against the crack length _a. This is shown in Fig. 3. The energy release rate is calculated from the slope of this curve divided by the initial thickness of the sample t. Thus

J1 -

ldU . t da

(4)

The energy release rate calculated from (4) was then compared with that obtained from Rivlin and Thomas [Eqn. (1)], namely T = 2kWa. The proportionality constant K has a value of 2.6 at an extension ratio 2 of 1.21 [3]. Plots of the energy release rate according to plots of (t) and (4) are shown in Fig. 4. It is evident that our experimental evaluation of Jl agrees reasonably well with the Rivlin and Thomas method within the limited region in which it is applicable. This region represents 20 percent of the sample width as stated by Rivlin and Thomas [2] and shown in Fig. 4. This comparison lends confidence to our method of energy release rate measurement.

Crack layer analysis of fatigue crack propagation in rubber compounds

171

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CRACKLENGTH(a) [ram] Fig. 4. The energy release rate, J1, (and the tearing energy) vs. crack length _a, for the rubber compound specimen.

3.2. Resistance moment The resistance moment R 1accounts for the amount of damage increment associated with the crack advance. In the present case, microcracks constitute the dominant damage mechanism, Fig. 5. Quantitative microscopy [6] provides a useful technique for measuring the distribution of microcracking associated with crack propagation. The amount of microcracking per unit crack advance was measured from reflected light micrographs similar to that of Fig. 5 but of higher magnification (20 x ). These microcracks were generally found to cut through the specimen thickness, hence measurements were made for the average length of microcracks and multiplied by the initial thickness of the specimen to obtain microcracking density as area per unit area of crack extension. Equidistant lines were drawn parallel to the crack path at 2 mm intervals. The area of microcracks within each interval divided by the area of crack extension within the same interval was taken as the resistance moment RI. These measurements were conducted on both sides of the crack trajectory. The relationship between the resistance moment and the crack length is shown in Fig. 6. It is noticed that R~ has units of (m2/m2), and based on this, ?* in (2) has units of kJ m -2.

172

H. Aglan, A. Chudnovsky, A. Moet, T. Fleischman and D. Stalnaker

Fig. 5. Damage in the form of microcracks in the rubber compound specimen.

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0

I

5

I

I

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10 15 20 CRACK LENGTH (a) [ram]

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25

30

Fig. 6. The resistance moment, R~, as a function of crack length, _a, in the rubber compound specimen.

Figure 7 is an SEM micrograph of a section of the main crack with a branched off microcrack (arrow). A higher resolution examination of an area S adjacent to the main crack reveals finer and more profuse microvoids (Fig. 8) which could not be resolved in the optical microscope examination. Further high resolution examination of the two halves of the fractured specimen within and outside the active zone was then performed. The results show that these profuse microvoids of Fig. 8 are very isolated surface voids which can be ignored. Thus the value of R1 is still a representative measure of the resistant moment.

3.4. Irreversible work d W~/dN can be evaluated from the load-displacement relationship. Experimentally, d W~/dN can be evaluated from the area within the hysteresis loop associated with each loadingunloading cycle during a fatigue experiment.

Crack layer analysis of fatigue crack propagation in rubber compounds

173

Miorocrack

Fig. 7. An SEM micrograph of the main crack with a branched off microcrack for the rubber compound.

Direction of Crack

Fig. 8. Profuse microcracking adjacent to the main crack at a higher magnification of the area S in Fig. 7. Prior to crack propagation (a = a0, where a0 is the notch length) a finite a m o u n t o f irreversible work is observed ~ ( 0 ) . This is obtained from the area o f the loop at the first recorded cycle. If it could be assumed that the irreversible work Wi(0) which is expended on the bulk o f the material remains constant during crack propagation, the total irreversible work done on the active zone per cycle would be given by

dW~/dN =

W ~ ( a ) - Wi(0 ).

(5)

174

H. Aglan, A. Chudnovsky, A. Moet, T. Fleischman and D. Stalnaker

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Crack tength a = I

/7

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q 25.3

DI SPLACEMENT

Fig. 9. Typical series of hysteresis loops in a displacement controlled fatigue experiment on the rubber specimen.

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Fig. 10. The relationship between the irreversible work and crack length for the rubber compound specimen.

A typical series of hysteresis loops of the rubber compound is shown in Fig. 9. As seen in Fig. 9, the area of the hysteresis loops decreases with the increase in the crack length. The relationship between d W~/dN and the crack length for this material is shown in Fig. 10. The total irreversible work, d W~/dN, expended on damage accumulation within the active zone is extremely small in comparison with that dissipated by the bulk of the material. Thus the rate of energy dissipation given by (3) cannot be employed in the present investigation on rubber samples. This is due to the invalidity of the assumption that the irreversible work done on the bulk of the material is constant. In order to overcome this problem the following treatment has been adopted. The rate of dissipation, for linear viscoelastic behavior, has been shown proportional to the active zone length times the energy release rate [13]. Following the Dugdale-Barenblatt model for linear elasticity [14, 15], the active zone length is proportional to the energy release rate. It seems to be useful to approximate the rubber behavior by these notions and, on this basis, the rate of energy dissipation [3] can be expressed as

dD/dN = flj2,

(6)

Crack layer analysis of fatigue crack propagation in rubber compounds

175

where fl is a phenomenological constant, different from fl', which characterizes the dissipative nature of the material. Substituting (6) into (2) gives

da

fiJ~

dN

7"R1 - Jl'

(7)

Thus (7) can be employed instead of (2) to extract the specific energy of damage 7" and the phenomenological coefficient fi in the rubber compound under investigation.

4. Discussion

The logarithm of the average crack propagation speed for this rubber compound as a function of the logarithm of the energy release rate is shown in Fig. 11. Fatigue crack propagation kinetics of this rubber compound display the familiar S-shaped character. Three stage crack propagation kinetics is obvious from Fig. 11. The threshold stage is followed by a stage of reduced acceleration approaching the stage of critical propagation. The behavior is indicative of crack tip damage (active zone) evolution. The new methodology developed through the work outlined above will be used to determine the parameters which characterize the resistance of the rubber compound to crack propagation, i.e., fracture toughness. These parameters 7* and fi, which control the fracture process, can be evaluated by (7). Thus, J1

j2 -

R 1

(da/dN)

(8)

tR 1'

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for the rubber compound specimen.

176

H. Aglan, A. Chudnovsky, A. Moet, T. Fleischman and D. Stalnaker 10 ")'*=9300 J/m 2

n,.

I

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f

I

1

2

3

4

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J2/[lda/dNlt

R l = 1613 [kj2-cYCle/m6l

Fig. 12. Fatigue crack propagation behavior of the rubber compound sample plotted in the form of (8).

where _t is the initial thickness of the sample. If the experimental results are in accord with the proposed model a plot of (J~/R 1) vs. [J2/(da/dNtRl)] should give a straight line with 7* being the intercept and/~ the slope. Indeed, when the results of the present experiment were plotted on the basis of (8) it was found that the average of the experimental points makes a good straight line. This is shown in Fig. 12 from which 7* = 9.3 kJ m -a and/? = 9.7 x 10 .9 m4/j-cycle are obtained. A curve plotted on the basis of (7), with the obtained values of 7* and/~, together with the experimental results of Fig. 11, are shown in Fig. 13. It would appear from Fig. 13 that the curve gives a very good fit to the experimental points. In order to verify the reproducibility of the experimental results another experiment was conducted on the same compound, under similar testing conditions. Optical examination of this specimen [16] reveals a similar crack path and damage mechanism (microcracks) to those of the specimen analyzed in the current investigation. Moreover, the values of 7* and/~ obtained above have been used to predict the behavior of the specimen in [16], which was tested under similar loading conditions; the only differences were the frequency and the loading frame used. As can be seen in Fig. 14, the predicted relationship between the crack speed and the energy release rate (solid line) closely fits the experimental results (points). Thus, it appears to be safe to conclude that the theory under test is in accord with the experimental results.

5. Conclusions

Fatigue crack propagation kinetics in rubber compounds display the familiar S-shaped behavior reflecting crack tip damage evolution. Based on the evolution of this damage (microcracks in the present study) the crack layer theory introduces a methodology to describe fatigue crack propagation behavior in rubber compounds over the entire range of

Crack layer analysis of fatigue crack propagation in rubber compounds Frequency-1

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ENERGY RELEASE RATE (J) [k Jim 2]

Fig. 14. The predicted fatigue crack propagation speed based on (7) (using ?* and/~ from the present work) vs. the energy release rate, together with the experimental data for a different specimen from the same rubber compound.

the elastic energy release rate. Invoking the crack layer concept, a specific energy of damage 7* and a dissipative coefficient/3 are extracted from the experimental results. The values of 7* and /3, for the rubber compound under investigation, are found to be 9.3 kJ m -2 and 9.7 x 10 .9 mg/(J-cycle) respectively. Thus, rubber compounds can be classified based on their values of 7* and/3. A compound with a higher value of 7* and a lower value of 13 is more resistant to crack propagation.

178

H. Aglan, A. Chudnovsky, A. Moet, T. Fleischman and D. Stalnaker

Acknowledgement This research is s p o n s o r e d b y the F i r e s t o n e C o m p a n y t h r o u g h E P I C ( E d i s o n P o l y m e r Innovation Corporation).

References 1. P. Paris and F. Erdogan, Transactions ASME (1963) 528-534. 2. R.S. Rivlin and A.G. Thomas, Journal of Polymer Science 10 (1953) 291-318. 3. H.W. Greensmith, Journal of Applied Polymer Science 7 (1963) 933-1002. 4. G.L. Lake and P.B. Lindley, Rubber Journal, Part 1 (1964) 24-30. 5. G.L. Lake and P.B. Lindley, Journal of Applied Polymer Science 9 (1965) 1233-1251. 6. A. Chudnovsky, A. Moet, R. Bankert and M. Takemori, Journal of Applied Physics 54, 10 (1983) 5562-5567. 7. E. Andrews and B. Walker, Proceedings Royal Society, London, Set. A 325 (1971) 57-79. 8. Y. Mai, Journal of Materials Science Letters 9 (1974) 1896-1898. 9. A. Sandt and E. Hornbogen, Journal of Materials Science 16 (1981) 2915-2919. 10. P. Bretz, R. Hertzberg and J. Manson, Polymer 22 (1981) 575-576. 11. A. Chudnovsky and A. Moet, Journal of Materials Science 20 (1985) 630-635. 12. A. Chudnovsky and A. Moet, Journal of Elastomers and Plastics 18 (1986) 50-55. 13. J. Botsis, A. Chudnovsky and A. Moet, International Journal of Fracture 33 (1987) 277-284. 14. D.S. Dugdale, Journal of the Mechanics and Physics of Solids 8 (1960) 100-104. 15. G.I. Barenblatt, in Advances in Applied Mechanics, Academic Press, New York (1962) 55-125. 16. H. Aglan and A. Moet, International Journal of Fracture 40 (1989) 285-294.

R6sum6. On pr6sente une m6thodologie pour caract6riser la r6sistance de composbs de caoutchouc A la propagation des fissures du point de vue de la t6nacit6 fi la rupture. Un mod61e constitutif bas6 sur la th6orie de la couche de fissuration est utilis6 pour obtenir l'6nergie sp6cifique d'endommagement y*, un param6tre du mat6riau repr6sentatif de sa r6sistance ~ la propagation d'une fissure, et une caract6ristique de dissipation ft. Le mod61e exprime la vitesse de propagation d'une fissure de fatigue par cycle da/dN en fonction de ces deux param+tres, de la vitesse de relaxation de l'6nergie de cisaillement J~, et du moment r6sistif R 1 qui tient compte de 6tat de l'endommagement associ6 fi la progression de la fissure. Un examen microscopique r6v6le quc la microfissuration l'extr6mitb de la fissure est le m6canisme d6terminant de l'endommagement. D6s lors, on 6value R~ en fonction de l'aire de microfissuration (en m2) par unit6 de progression de la fissure. Des donn~es de propagation de fissure de fatigue sont analys6es fi 1'aide du prbsent mod61e pour un composb de caoutchouc particulier. L'6quation propos6e d6crit l'enti6ret6 de la propagation de la fissure duns le compos& Des valeurs num6riques pour y* et pour fl de respectivement 9,3 kJm -2 et 9,7 x 10-9 m-4/J-cyclesont trouv6es.