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High-speed Fracture Mechanics by Photography of Polypropylene Copolymers

M. BRAMUZZO

Monted ison/Hirnont Centro Richerche Giulio N a t t a

Ferrara. Italy The use of photography applied to stroboscopy in analyzing the processes

inherent to the starting and propagation of cracks in materials is a technique which has proved to be of great interest, especially since it enables one to check and “directly” study the evolution of such phenomena. Using fracture mechanics criteria this technique has been applied to the study of the impact behavior of some polypropylene copolymers at different rubber contents obtained either by blending or by synthesis. This technique makes it possible to determine numerous parameters of fracture mechanics, including C.O.D. (crack opening displacement), C.O.A. (crack opening angle), JIc. the plastic work parameter, determined from the resistance curve R, tearing modulus, and crack propagation velocity. Further- more, under high strain rate conditions, the value taken on by the coefficient “A” relating the J-integral to C.O.D. ( 1 , 2) were checked for those materials using the equation J = X. uy.C.O.D., Hayes and Turner (3) and Boyle (4). From analysis of the materials it was possible to note that the synergetic effect of the EP (ethylene- propylene) rubber increased, especially when present at percentages of more than 10 percent. Annealing the materials, on the other hand, produced a n increase in fracture toughness for those products having a low rubber content; however it did not have any effect on those with an elevated rubber content (26 percent).

INTRODUCTION 1) 2)

3) 4) 5) 6)

he use of polymeric materials is ever expanding T into new fields and ever greater performance is being demanded, especially in the field of high strain rate. Arising from the need to predict the processes which cause and propagate cracks in materials requiring great resistance reliability (i.e. metals), fracture mechanics can prove a valid support in the characterization of polymeric materials intended to withstand great deformation and absorb high impact energies. Little has been written in this field since most fracture mechanics or fracture propagation velocity measurements give less problems when performed at low strain rates.

In the present work extremely impact resistant materials such as polypropylene copolymers have been analyzed following the high speed test fracture mechanics principles using two main approaches: crack opening displacement (C.O.D.) (5) and the J- integral (6). This has been done in order to evaluate toughness as a function of EP (ethylene-propylene) rubber content and to experimentally verify the reliability of the photographic detection technique by stroboscopy.

MATERIALS AND APPARATUS Materials

Six polypropylene (PP) products were examined:

POLYMER ENGINEERING AND SCIENCE, AUGUST 1989, Vol. 29, No. 16

Homopolymer Blend A = 95 percent PP + 5 percent EPR (ethylene-propylene rubber) Blend B = 90 percent PP + 10 percent EPR Blend C = 85 percent PP + 15 percent EPR Blend D = 74 percent PP + 26 percent EPR ‘From synthesis’ copolymer = 74 percent PP + 26 percent EPR

The physical properties of the base materials used to obtain these products are given in Table I while Table 2 reports the main physico-mechanical properties according to the ASTM standards. The specimens were obtained from injection molded plates according to the diagram in Fig. 1.

Tests were also performed on specimens which had undergone a 1 4OoC annealing process for 2 h in an oven.

Apparatus

The machine used for this type of impact testing consisted of an instrumented falling weight which runs between two lateral guides and a support to house the specimen. The force-time curve was recorded with the following apparatus (see Ffg. 2):

0 PCB Piezoelectric power transducer with 500 N full

0 PCB load amplifier (A) scale (FT)

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M. Brarnuzzo

Table 1. Physical characteristics ot the Base Materials. ~~~ ~

Characteristics C.L..I--- in-. Polypropylene

homopolymer

Polypropylene copolymer

by synthesis cnlylwll=l rrdpylene

rubber - - 4.5 g/10 min

(230°C/2.16 Kg) ASTM D1238 (230°C/2.16 Kg)

ASTM D1646 Intrinsic Viscosity (7) 2 dl/g at 135% 2.9 dllg at 135OC 2.1 dl/g at 135OC

Molecular weight (Mw) . 1 0-3

Melt flow rate 3.4 9/10 min I

Mooney at 121 OC I 80 I

42% I 300 GPC /

Propylene content I

Index of molecular weight (MwlMn) 6.9 4.5 I 470 light scattering

obtained by gel permeation chro- matography

Density 0.905 g/cm3 0.865 g/cm3 0.887 g/cm3 ASTM-DI 505 Tacticity index 96.6 isotactic insol- I I

Crystallinity 58% I

uble in boiling heptane

X-ray analysis Cp = 14% c3 = 43%

X-ray analysis

Table 2. Physical Characteristics of Polypropylene Copolymers.

Pohrprowlene -. . - Blend A Blend B Blend C Blend D copolymer

Characteristics Method Unit Polwropy'ene 95% PP 90% PP 85% PP 74% PP by synthesis homopo'ymer 5% EPR iw0 EPR i50/~ EPR 26y0 EPR (26% Ethylene

content) Melt flow rate (230°C/2.16 Kg) Densitv

ASTM 1238 g/10 min 3.4 3.1 2.9 2.8 2.4 4.5

ASTM 1505 g/cm3 0.905 0.902 0.899 0.893 0.887 0.887 Notched lzod ASTM D-256 J/m 25 65 105 230 no break no break (23OC) Flexuial modulus ASTM 0-790 MN/m2 (23OC)

1660 1600 1460 1330 1000 740

Yield strength ASTM D-638 MN/m2 36 32 29 26 19 15 (23OC)

0 Could transient storage oscilloscope OS4040 (TR) 0 Linseis pen recorder LY 18 100 (PR)

The system set up by F. Polato (7) was used for photography and consisted of a Pentax MX camera (C), Gen Rad Stroboscope GR1538-A (SU), and Cam- era and stroboscope control unit (CSCU).

As outlined in Fig. 2, this apparatus functioned as follows:

Fig. 1 . Position for withdrawal of the specimen In respect to the plate injection direction.

I 1 Fig. 2. Schematic diagram of set up of the apparatus to record impact testing and single or multipleflash photographic detection.

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a) The hook freeing the fall weight, FW, is released b) The first switch, SW1, is turned on by passage of

the magnetic bar, MR, linked to the falling weight and timer, T, is activated which, in turn, controls the electromagnet, E. The latter opens the camera diaphragm, C. Timer, T, serves to keep the camera diaphragm open for approximately 500 ms.

c) The second switch, S2, is turned on by passage of the magnetic bar, MR, and the delay circuit, DT, for the strobe flash is activated.

d) The falling weight hits the specimen making impact and the storage circuit of the transient recorder-oscilloscope, TR, is activated by means of the power transducer, FT.

e) The delay circuit, DT, simultaneously sends the flash activating signal to the stroboscope unit, SU, and the transient recorder-oscilloscope.



f ) The impact force-time curve is recorded as well as the flash command and the image is recorded on film.

The delay circuit, DT, (of the camera and the stroboscope control unit, CSCU) is preset by means of a potentiometer in order to cut in the flash at the desired moment. Thus, the load level and image pho- tographed will coincide (see Fig. 3 ) .

The CSCU unit is also able to emit signals in sequence in order to produce the same number of flashes at times preestablished by the circuit PT; thus one may obtain several images in a single photogram (see Fig. 4 ) . The train of pulses is stopped by the digital counter, CT.

An Apple I1 Plus personal computer with 48 Kbytes of memory was used to calculate and diagram the data obtained.

Test geometry Three point bending geometry was used. Plane

strain conditions theory (8) requires that the follow-

I

1 ;!E FORCE-TIME

Pretrigger time1

Delay time SINGLE PULSE TO COMMAND STROBOSCOPIC

U N I T

X Fig. 3. Example of the force-time curve and single pulse to control the stroboscope unit (single image in the photogram).

I

I Pretrigger time

-~ X

Fig. 4. Example of the force-time curve and multiple pulses controlling the stroboscope unit [different images in the same photogram).


ing inequality be satisfied for specimen size:

Jlc a, B , (W - a) -> a*- u flow

where a = Crack length B W cr Jlc

= Breadth of specimen cross-section = Height of specimen cross-section = Coefficient variable between 25 and 50 = Critical strain energy release rate in the

plastic field o flow = (oy + uu)/2 uy = Yield strength u u = Tensile strength

If one assumes a value of Jlc = 20 KJ/m2 and u flow = 60 MN/m2 (under impact conditions), and taking a = 25, the result of the inequality (Eq 1 ) is 8.3 mm. For polymeric materials, molding of plates with a thickness of more than 3 to 4 mm leads to a progressive increase in inhomogeneity with an increase in thickness. In fact, if the thickness is great, the difference between the cooling time on the sur- face of the material and within the sample will lead to different crystallization kinetics and thus to a different number and size of the spherulites within the thickness. This is likewise influenced by the type of material used.

Keeping these problems in mind and considering that most injection molded polymeric items have a thickness on the order of 3 to 4 mm the following dimensions were chosen:

B = 3.2 mm

W = 1 5 m m

a = 7.5 mm

Notches, to a depth of 7 mm, were first made using a tool in accordance with the ASTM-D256 standard. A razor blade was then used to a depth of 0.5 mm. This was in agreement with the rules of fracture mechanics requiring the curve radius of the apex notch to tend toward 0.

Other test conditions were as follows:

Impact rate (Vo) = 2 m/s

Span (S) = 60 mm

S I W = 4

RESULTS AND DISCUSSION

Ftgures 5 and 6 exemplify the results obtained by application of the photographic technique to impact testing. The negatives of the photograms were en- larged by means of a slide projector in order to take measurements with the least possible margin of error.

G & J measurements The procedure proposed by Landes and Begley (8)

based on the method developed by Rice (10) for meas-

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M. Brarnuzzo

,001

Fig. 5. Photographic sequence of the fracturing of Blend A. Impact speed 2 m/s at +23"C.

urements on deeply cracked specimens was used to determine the J-integral:

In the case of specimens notched on only one edge (SEN) and under three point bending conditions, having an a/ W ratio > 0.15 it has been demonstrated that the coefficient, 8. under plastic conditions is equal to 2. This is so, even when these specimens

(2) 8 . A

B(W - a) J =

POLYMER ENGINEERING AND SCIENCE, AUGUST 1989, Yo/. 29, No. 16 1080

High-speed Fracture Mechanics b y Photography of Polypropylene Copolymers

200. 1

Y 0 . e , Y

11 timm hsl

f ] Y b

'2 timo h.1

t4 tima (msl

Fig. 6. Photographic sequence of the fracturing of the copolymer obtained from synthesis. Impact speed 2 mls at +23"C.

have been analyzed at a span greater than 4 W and

present case span = 4W and a / W = 0.5 have been

A is the absorbed energy corresponding to a certain displacement. In the case of impulsive stress, calcu-

POLYMER ENGlN€ER/NG AND SCIENCE, AUGUST 1!M9, YO/< 29, NO. 16

lation has been made using the following equation when they have 0.45 c a / W c 0.65 (11). In the (12):

used. A=U(l--&-) (3)

where U is the impact energy which has not been

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M . Bramuzzo

corrected by the deceleration effect of the falling mass which takes place during impact:

U = Vo s F ( t ) d t (4)

In fact, the impact speed Vo taken into consideration in this equation is constant.

K e = Yzm. Vo2

(kinetic energy of the falling mass) (5)

Vo = (impact speed at time to) (6) g = acceleration due to gravity

h = height of fall

Equat ion 2 was, thus, used to obtain the resistance curve (R) by plotting the J-integral as a function of the corresponding crack advancement, Aa.

One must recall, as has been shown in a previous work (1 3). that under plane strain conditions the crack begins in the point of maximum specimen stress; that is, on the inside. It then proceeds main- taining a propagation front similar to the initial one. Therefore the effective Aa value will be the value revealed externally through photography plus the internal value within the specimen. In order to measure the dimensions of the crack on the inside, some specimens of each material were subjected to impact with deflection preset by steel retainers. This was done so as to achieve slight, controlled advancement of the crack (13). Thereafter, the crack thus obtained was made more evident by staining the crack with red ink. The specimen was then opened (at the tem- perature of liquid nitrogen) thus making it possible to photograph the surfaces and take the necessary measurements.

Figure 7 gives an example of what has been de- scribed above. Internal advancement of the crack was approximately 0.5 mm for almost all the materials examined. In Fig. 8 an example of the J-integral trends is shown as a function of Aa for the nonan- nealed polypropylene obtained from synthesis and after annealing. The intersection point between the straight regression line of the experimental points and the “blunting line” (where J = 2uy. Aa) gives the Jlc value ( 13, 14).

Another important parameter describing the material’s ability to resist crack propagation is the tearing modulus (1 4):

dJ E da uy2

T = -.- (7)

This depends solely on the slope of curve R (dJ/da), on stiffness (modulus of elasticity, E ) and on tensile properties (yield strength, uy).

Observing the JZc values and tearing modulus data reported in Table 3 one will note that annealing no longer exerts any influence over the materials with a high EP rubber content (26 percent). This is most likely because, after the morphological resettling due to annealing, the contribution of the polypropylenic

a = a a by photography

b=Aa inside the specimen

c = a + b ( t r u e a a ) Fig. 7 . Example of the advancement front of the crack within a specimen.

140 , 7

I P o l y p r o p y l e n e C o p 0 l ymer

[ b y s y n f hesisl

0 1 2 3 4 5 6 7 Aa( m m 1

Fig. 8. Resistance curve R with “blunting line” and regression line. non annealed specimen, A annealed specimen.

matrix is of little relevance in comparison to the impact-resistant contribution of the rubber.

In the equations used, it is of utmost importance that the modulus of elasticity be determined at the same strain rate as the impact tests. Since the poly-

POLYMER ENGINEERING AND SCIENCE, AUGUST 1989, Vol. 29, No. 16 1082

High-speed Fracture Mechanics b y Photography of Polypropylene Copolymers

s

CQ S'z-00

mer is not perfectly elastic, but is rather a viscoelastic material, its mechanical properties change markedly with strain rate. This happens especially in determining the modulus of elasticity [ 15). The technique for determining this parameter under dynamic conditions, small ball rebounding (1 5), was widely used in the present work. The validity of this technique has been verified by applying the principles of linear elastic fracture mechanics to homopolymer polypropylene, a material which has demonstrated its brittle nature in the type of breaks it gives rise to.

By brittle break, one means those breaks which take place at stress values lower than the yield strength of the material without any appreciable plastic deformation (see Fig. 9).

To this purpose two different calculation methods were used which are related to two different hy- potheses. The first pertains to a criterion for the study of critical strains and defect dimensions ex- emplified by the well known ASTM equation (1 6) for calculating the critical stress intensity factor:

(8) KZC = UC- Ye & where

3 .Fmax. S 2 . B . W2 uc = (nominal critical stress) (9)

Y = the geometrical factor which depends on a /W and on SIW, that is, on the geometrical test conditions and on the way load is applied.

The second hypothesis is derived from energy consid- erations [ 18):

KIc = (10)

where GIc is the critical strain energy release rate from the body:

2A B.[W - a) Gic =

The only difference between the Klc data [see Table 3) obtained proved only to be 6 percent and

- 2 4 6 8

--- -- - 2 4 6 8 10 12 14 16 18 20

Displacement (mm)

Fig. 9. Force-displacement curves of the impact tests. 1- Homopolymer, 2- Blend A, 3- Blend B, 4- Blend C, 5- Blend D, 6- Copolymer obtained from synthesis.

POLYMER ENGlNEERlNG AND SCIENCE, AUGUST 1989, Yo/. 29, No. 16 1083

M . Bramuzzo

this is further evidence that the dynamic modulus of elasticity by rebound used in Eq 10 is exact.

Furthermore, this difference is annulled should a crack length correction be applied to Eq 8 by means of rp, the factor to be used in the presence of an extremely localized area of plasticity at the apex of the crack.

Different theories exist regarding the value which the coefficient 0 can take on; however, for polymers, it is preferable to use the model proposed by Dugdale

(1 9) where = - 8’

Again in the case of calculation of the critical strain energy release rate the three methods used have given quite similar results (see Table 3).

7r

2A B(W - a) method a: Glc =

KIc2 method b: GZc = - E

method c:

(131

4 = correction factor which depends on a/W and on S / W; that is on the geometrical conditions of the test

C.O.D. Measurements Crack Opening Displacement is defined as the dif-

ference between the opening taken on by the crack deformed by stress and the opening in the original position (see Fig. 10). Usually, C.O.D. measurements are performed with S E N type specimens under trac- tion or flexing by using special strain gage trans- ducers. By means of photographic detection one can avoid using such apparatus which, in the case of impact testing, often proves difficult to apply. Table 3 gives the C.O.D. data measured at maximum load “6m”.

(191.

(a)= razor notch(width< 0.05mrn)

x =blunting prior to tearing

&=cr i t ica l C.0.D Fig. 10. Examples of C.O.D. (b + d).

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d =C.O.D

X Measurements Following the path indicated by the work of J. N.

Robinson (2) who performed J and C.O.D. measurements on some steels, the value taken on by the X coefficient was verified by processing the data obtained from the impact tests. Theory (1) establishes that the product of this coefficient for the material yield strength represents the relation between J and C.O.D.:

J = X. ay.C.0.D. (14)

In Figs. 11, 12 and 13 some examples of J/uy us. C.O.D. diagrams are shown. One can see from the

1.5 c E E - h 0

= 1

.5

0 1.5 2 2.5 3 0 - 5 1

C.O.D[mml Fig. 1 1 . Determination of the coefficient X in the high strain rate tests.

- E E - h

\ 7

2

1.5

1

.5

0 0 .5 1 1.5 2 2.5 3

C.O. D f m m l Fig. 12. Determination of the coefficient h in the high strain rate tests.

POLYMER ENGINEERING AND SCIENCE, AUGUST 1989, Vol. 29, NO. 16


4

- 3

b”

E E v

\ 7

2

1

0 0 1 2 3 4

C. 0. D ( m m 1

Fig. 13. Example of determination of the coefficient X in the low strain rate tests.

work of Robinson and Tetelman (20, 2), performed on steels tested at low strain rates, that the value taken on by such coefficient is >1 up to 2.6

Given that in the present work the value of A is, instead, always less than 1, one may assume that this depends not so much on the strain rate to which the polymeric materials are subjected but rather to their intrinsic nature which is different from that of steel. In fact, tests performed at high strain rates (impact) and low strain rates (with Instron dynamom- eter at a speed of 5 cm/min) have given the same results (see Figs . 12 and 13). This parameter, moreover, also depends on the type of material (Table 3) as Robinson (2) has already demonstrated.

We can hypothesize that these results can be as- cribed to the type of deformation of the polymeric matrix and the microvoids around the rubber particle (Figs. 14, 15). In fact should we compare microvoid volumes developed in the craze phenomenon with equal volumes developed in the yielding phenomenon, we should notice that, in this latter case, a wider displacement of the polymeric matrix on the applied stress axis could be obtained. This brings about a n increase in the crack mouth opening and thus a lowering of the straight regression line slope on the J/ay us. COD diagram.

Crack Propagation Velocity Measurements

To measure this parameter a conductive grid is normally glued or designed directly on the specimen. This grid is then connected to an electronic circuit which reveals any step in voltage obtained at break of the conductive filaments while the crack is ad- vancing (2 1). Photographic detection, furthermore, makes it possible to use the image directly to perform the necessary measurements. The trend in the crack propagation velocity is obtained by diagramming the


stretching

P o l y p r o p y l e n e C o p o l y m e r b y s y n t h e s i c

( l o w s t r a i n r a t e )

direction

Fig. 14. Morphology of the etherophasic rubber particle after blushing (liquid nitrogen fracture).

..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . 4/ . . . . . . .

crazing yielding

Fig. 15. Deformation mechanism in the polymers: fa) rubber particle, fb) deformation mechanism in PPlEPR or PPI EPRIPE high impact copolymers, (cj deformatlon mechanism in other polymers.

space covered by the crack as a function of the time passed. The experimental points (Ftg. 16) were inter- polated by a second order polynomial ( y = a + bx + cx’) for the copolymer obtained from synthesis and by means of the curve thus obtained a simple deri- vation of A a was calculated in respect to t.

Figure 17 reports the crack propagation velocity a trends for all the materials analyzed, as a function of the crack space covered Aa. Obviously the more material that is able to absorb energy during the cracking process, the lower the crack propagation velocity itself will be. This is, moreover, in agreement with what has already been seen for the tearing modulus results.

The copolymer obtained from synthesis is the material for which the lowest velocity has been measured indicating that it is the most suitable material to be used in the creation of items requiring great resistance to cracks. Most likely the cracks did not reach the maximum velocity in the homopolymer and in Blend A since the curves did not reach a plateau.

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M. Bramuzzo

7

6 * - f s (D

4

3

2

1

0

P o l y p r o p y l e n e Copo lymer b y eynthesis

. 0 1 2 3 4 5 6 7

t(rn.1 Fig. 16. Example of polynomial interpolation of crack advancement as a function of time.

10

8 m . " E '* 6

4

2

0 0 1 2 3 4 5 6 7

Fig. 17. Crack propagation velocity trends in the materials analyzed.

Aa(mm)

CONCLUSIONS

Landes and Begley (9) assert that it would be ad- vantageous to obtain both JZc and C.O.D. measurements from the same test since both are quite important in delineating a material's toughness. The present work has been aimed at demonstrating that it is technically possible to obtain numerous fracture mechanics parameters which are, at times, difficult to obtain by using a method under impact conditions other than the photographic technique.

In regard to the specific topics dealt with in the present work the following conditions can be listed:

1) It is important to underline the fact that, when applied to impact testing, fracture mechanics is able to provide a more detailed description of the mechanical properties of the material than can other types of impact tests. In particular it can be a more selective

instrument than for example the Izod impact strength test. In fact, in analyzing the two materials with the highest rubber content the results achieved according to ASTM-D256 do not allow for any differ- entiation (in both cases the specimens did not break, see Table 2) .

On the other hand, with fracture mechanics it was ascertained that the 'from synthesis' copolymer pro- vides higher JIc and C.O.D. values than does the blend with the same rubber content (Table 3).

2) Three different methods for calculating the measurement of the critical stress intensity factor KIc and critical strain energy release rate GZc were verified on polypropylene homopolymer. The tests performed have shown that the three approaches are equivalent as the results obtained are in good agreement. Furthermore, computing of the equations used has highlighted the rebound method validity (1 5) in determining the dynamic modulus of elasticity. 3) Annealing does not influence the impact behav-

ior of copolymers having a high EP rubber content. 4) Besides determining fracture mechanics param-

eters with the two main approaches (i.e. J-integral and C.O.D.), it is likewise possible to measure the crack propagation velocity ci by deriving the space covered by the crack A a in respect to the time passed t.

5) The value of the coefficient X in the relation J = X.uy.C.0.D. depends on the material and can take on a value of less than 1 in polypropylene copolymers, because of the particular mechanism of the polymeric matrix deformation during the applied stress.

6) To summarize the results of the present work, the values of various parameters have been dia- grammed as a function of the EP rubber content within the polypropylene (Figs. 18 to 22). Observing

50

*- 40 E \ 7 x -

30 3

od

u 20

10

0 0 5 10 15 20 25 30

Fig. 18. Trend in the critical strain energy release rate and critical J-integral as a function of the percentage of EP rubber contained within the polypropylene. t Copoly- mer by synthesis.

EPR [ % I

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0 5 10 15 20 25 30 EPR [ % I

Fig. 19. Tearing Modulus trend as a function of the percentage of EP rubber contained within the polypropylene. * Copolymer by synthesis.

10 l 2 I-----

0 1 0 5 10 15 20 25 30

Fig. 20. Trend in the critical stress intensity factor as a function of the percentage of EP rubber contained within the polypropylene. * Copolymer by synthesis.

EPR (XI

the trends obtained one may assert that the synergetic effect of the rubber exerts a great influence on the mechanical properties, especially when the percentage is greater than 10 percent.

The KZc trend is delineated by a straight line (Fig. 20). This result may be of interest in the field of design since, once the Klc value the item will have to be able to bear has been established, one can easily interpolate the corresponding percentage of rubber the polypropylene will have to have in order, theoret- ically, to be able to withstand breaking.

POLYMEA ENGINEERING AND SCIENCE, AUGUST 1989, Vol. 29, No. 16

10 15 20 25 30

Fig. 21. Trend in crack opening displacement as a function of the percentage of EP rubber contained within the polypropylene. * Copolymer by synthesis.

0 5

EPR ( % I

10

8 - a \ E $ 6

4

2

0 0 5 10 15 20 25 30

EPR(%) Fig. 22. Trend in the crack propagation velocity as a function of the percentage of EP rubber contained within the polypropylene. * Copolymer by synthesis.

NOMENCLATURE

A = Absorbed energy. a = Crack length. a = Crack propagation velocity. a! = Coefficient (25 L 50). B = Breadth of specimen cross-section.

C.O.D. = Crack Opening Displacement. 6Zc = Critical C.O.D. 6m = C.O.D. at the maximum load.

P = */8.

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E 11 F 9 Glc 9 h Ke Klc

Jlc x rP S u flow = uy = uu = GC - -

- T - t u -

vo = w = X Y -

- - -

- - -

1. A.A.

M. Bramuzzo

Modulus of elasticity. Coefficient (it is 2 in plastic field). Force. Correction factor. Critical strain energy release rate. Acceleration due to gravity. Height of fall. Kinetic energy of the falling weight. Critical stress intensity factor of fracture toughness. Plastic work parameter. Coefficient. Plastic zone size. Span.

Yield strength. Tensile strength. Nominal critical stress. Tearing Modulus. Time. Energy uncorrected for deceleration during impact. Impact speed. Height of specimen cross section. Blunting prior to tearing. Geometrical factor.

(uy + au)/2.

REFERENCES Wells in Proceedings, Crack Propagation Sympo-

sium, Vol. I paper B4, Cranfield, England (1961). 2. J. N. Robinson, lnt. J. Fracture, 12,723 (1976).

3.

4.

5.

6. 7. 8.

9.

10.

11.

12.

13.

14.

15. 16.

17.

18.

19.

20.

21.

D. J. Hayes and C. E. Turner, Int. J. Fracture, 10. 17 (1 974). E. F. Boyle. “The Calculation of Elastic and Plastic Crack Extension Forces”, Ph.D. Dissertation, Faculty of Applied Science and Technology, Queen’s Univer- sity, Belfast, N. Ireland (1972). A. H. Cottrel, Iron and Steel Institute Special Report 69, 281 (1961). I. R. Rice, J. Appl. Mech., 379. June (1968). F. Polato, J. Mater. Sci. 20, 1455 (1985). J. D. Landes and J. A. Begley, Fracture Analysis ASTM-

J. D. Landes and J. A. Begley, Post-Yield Fracture Mechanics, p. 224, Applied Science Publishers, Lon- don (1 979). J. R. Rice, P. C. Paris, and J. G. Merkle, “Progress in Flow Growth and Fracture Toughness Testing,” ASTM-

M. G. Dawes, “Elastic Plastic Fracture,” ASTM STP668,

D. R. Ireland, “Intrumented Impact Testing,” ASTM- STP563, p. 7 (1973). A. Savadori, M. Bramuzzo, and C. Marega, Polymer Testlng, 4, 73 (1984). P. C. Paris, H. Tada, A. Zahoor. and H. Hernst, ”Elastic Plastic Fracture.” ASTM STP668. pp. 5-36 (1979). M. Bramuzzo, Polymer Testlng 5,439 (1985). Standard Method of Test for Plane Strain Fracture Toughness of Metallic Materials, E399-74, ASTM. Plane Strain Crack Toughness Testing of High Strength Metallic Materials, ASTM STP410. p. 13. G. R. Irwin, Fracture, in Encyclopaedia of Physics, Vol. 6, Springer, Heidelberg (1958). E. Plati and J. G. Williams, Polym. Eng. Scl.. 15, 470 (1975). J. N. Robinson and A. S. Tetelman, ”Fracture Tough- ness and Slow Stable Cracking,” ASTM-STP559. A. Kobayashi and N. Ohtani, J. Appl. Polym. Sci., 21, 2861 (1977).

STP560, pp. 170- 186 (1 974).

STP536, pp. 231-245 (1973).

pp. 307-333.

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