Experimental Analysis of Glassy Polymers Fracture Using a Double Notch Four Point-Bending Method

Fracture of Polymers, Composites and Adhesives II B.R.K. Blackman, A. Pavan and J.G. Williams (Eds) © 2003. Published by Elsevier Ltd. and ESIS.

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EXPERIMENTAL ANALYSIS OF GLASSY POLYMERS FRACTURE USING A DOUBLE NOTCH FOUR POINT-BENDING METHOD

N. SAAD, C. OLAGNON, R. ESTEVEZ, J. CHEVALIER GEMPPMINSA Lyon, France ABSTRACT A twin notch specimen under four point bending is designed to analyse the mechanisms and the properties of glassy polymers fracture. The two notches are submitted to an identical bending moment so that one will fail and provides a measure of the toughness while the other serves as a snap-shot of the strain fields prior to unstable crack propagation. The evolution of the toughness with the loading rate and the influence of the notch radius is analyzed for both PMMA and PC. KEYWORDS Polymer fracture, plasticity, crazing, toughness INTRODUCTION Failure of amorphous polymers in the glassy state results from the competition between shear yielding a nd crazing. W hen c razing c an b e s uppressed, for i nstance u nder c ompression, t he bulk material shows a localized plastic deformation through shear bands related to soflening upon yielding followed by progressive strain hardening as the deformation continues. Crazing involves also some localized plasticity [1], albeit at a smaller scale, and precedes crack propagation. After initiation for a critical stress state, the crazes widen by the growth of fibrils which break down for a critical width resulting in the nucleation of a crack. In a numerical study [2] featuring a viscoplastic model for shear yielding and a viscoplastic cohesive zone for crazing, it was demonstrated that the competition between shear yielding and crazing is governed by the time scales involved in each mechanism. This competition together with the condition for craze fibrils breakdown and crack propagation determines the level of toughness and governs the ductile to brittle transition with increasing loading rate. A ductile response is related to the development of some plasticity in the bulk prior to crack propagation while a brittle response corresponds to the development of crazing only, the bulk remaining elastic.

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CHEVALIER

Although the failure properties for metals have been widely investigated, for which standard experimentations are available, attention to glassy polymers has been focused later on. Due to their intrinsic softening response, no analytical results are available for such materials. For a properly designed specimen and configuration test, linear elastic conditions need to be ensured while non linear deviations have to remain confined at a small scale. Under these conditions, classical linear elastic solution can be used for the analysis of failure and the estimation of the toughness. The characterization of the fracture features has then required the development of appropriate test configurations and specific preparation rules of the specimen, and related standards are now emerging. The present experimental study is connected to a recent modeling of crazing [2] within a cohesive surface methodology which incorporates the three stages of initiation, widening and breakdown of crazes; and this mechanism is assumed to precede failure. The motivation of the present experimental study is to define a protocol to calibrate the parameters involved in this theoretical description and attention is devoted here to the craze widening. This mechanism is viscoplastic so that the energy dissipated by crazing for the nucleation of a crack is time dependent and the parameters involved in the kinetics of craze widening are related to the toughness. Therefore, the evolution of the toughness with the loading rate is one of the key features for the calibration of the parameters used in the description of craze widening. A complete calibration requires additional experiments to analyze craze initiation and craze fibril breakdown which are out of the scope of the present work. A twin notch four point-bending configuration is developed to analyse fracture as shown in Fig. la. The two notches are on the same side of the specimen and located in the region in which a constant bending moment prevails. The sample is designed to fulfil both small scale yielding and plane strain conditions. As the load increases, crack propagation takes place at one of the two notches. The critical stress intensity factor is then estimated. The remaining notch is used as a snap-shot of the strain fields prior to crack propagation. Its observation under crossed poralizers indicates whether or not shear yielding has developed prior to crack propagation and serves to verify the condition for small scale yielding. An example of such test is reported in Fig. lb for two glassy polymers. 1^

P/2 p W

1

n P/2

11A M : II' A1 V

1

^

Si

Fig. la : Twin notch configuration under bending

Fig. lb : samples of glassy polymers after testing,

Experimental Analysis of Glassy Polymers Fracture

29

The first part of the paper is devoted to the design of the notched specimen and the analysis of the resulting stress intensity factor from a finite element calculation. The second part deals with the practical preparation of the samples. In a third part, the evolution of the toughness and the analysis of the crack tip fields with the loading rate are presented for PMMA and PC. The influence of the notch radius is also considered. DESIGN OF THE TWIN NOTCH SPECIMEN The dimensions of the specimen need to meet theoretical and practical size requirements. From a dimensional point of view, the classical size criterion for plain strain and small scale yielding conditions needs to be verified [3]: N2

a,B,(W-a)>2.5

K ic 'y

0) ;

The parameters a and W are the crack length and the specimen width indicated in Fig.la, B is the thickness and Gy is the yield stress of the material when no crazing takes place as in compression. A compromise between these requirements and practical aspects implies that the size of the specimen remains not too large. Thus, we fix the thickness to 10 mm to ensure plane strain conditions while Si=90 mm and S2= 40mm. Several geometries are considered and indicated in table 1. The geometry of PI follows the recommendations for mode I under pure bending according to Tada et al. [4] (S2/2W larger than two in Fig. la) which are close to that of the TC4 committee of ESIS [3]. For the specimen Gl, the ratio S2/W is more compact and the conditions of mode I and pure bending are verified numerically in the sequel as well as for the configuration with the twin notches G2. For a single notch specimen under pure bending and an elastic isotropic material, the stress intensity factor is [4] Kj =aoV7ia'F(a) , (2) where a = a/W , F(a) = 1.122~1.40a + 7.33a^-13.08a"^+14.0a'* and the reference stress ao(P) = 3P(Si - S2)/2W^B with P the applied load. Configuration

1

.

iPi

1

1

Gl 1

1 i I 1

W(mm) B(mm)

10 10

20 10

20 10

a (mm)

5

10

10

Tab.l Dimensions characteristics of the three configurations

G2

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N. SAAD, a OLAGNON, R. ESTEVEZANDJ.

CHEVALIER

First, the elastic stress distributions of the un-notched specimens are obtained from a finite element analysis. For the PI un-notched specimen, the discrepancy between the finite element and the analytical result is very small (about 0.01%), thus validating the finite element calculation in terms of accuracy through the meshing and the type of element used. Therefore a similar calculation is conducted on the Gl un-notched specimen where the span to height ratio is smaller. The mismatch on the maximum stresses at the bottom and at the top of the beam between the finite element calculations and the analytical solution is 0.74% in tension and 0.79% in compression (and remains constant upon further mesh refinement). This estimation of the stress distribution is then used for the following evaluation of the stress intensity factor. We use the weight function method to evaluate the stress intensity factor. Basically, it consists in a superposition problem conducted in two steps: the estimation of the stress distribution along a fictive crack in an un-notched configuration and the knowledge of a displacement solution along the crack of a similar problem involving an edge crack under mode I. The weight function method [5] for the calculation of any the stress intensity factor involves: r^ n I— Ki=faoV7ia

uu with

r faCT(x/a)m(a,x/a) , f=J—^^ .— -dx ,

,.. (3)

where a is the crack length and x the coordinate along the crack. The term a(x/a) is the stress distribution along the fictive crack in the un-notched specimen and CTO(P) is a reference stress taken as the remote stress related to the applied load P. The definition of the weight function m(a,x) is: m(A,X)=^'^^-^^^-"^"\ (4) Kref(a) da where E'=E for plane stress and E'=E/(l-v^) for plane strain, Kref is the stress intensity factor for the same load and the reference edge crack problem. The function Uref is a known reference elastic displacement along the crack which is differentiated with respect to the crack length. Approximated and simplified expressions of the weight functions are found in Wu and Carlsson [5] for the edge crack problem. It suffices here to indicate the expression we used for the calculation of the function fin Eq. (3):

f

5

1 jfH^ip,.(l-x/ay-^'^dx. ; ;-'"

(5)

i=l

where the five coefficients pi derive from a series expansion of the elastic displacement and are given in [5]. The calculation in Eq. (5) involves the longitudinal component axx(y) along the domain [0,a] of the stress distribution from the un-notched specimen which has been approximated with a multi-linear function to further simplify the calculation in (5). Then, the S.I.F. derives from the expression of Eq. (3). From the analytical solution of a(x) for pure bending, we compared the SDF estimated with the weight function method [5] with that of Tada et al.'s handbook [4] in Eq. (2). The discrepancy

Experimental Analysis of Glassy Polymers Fracture

31

is about 1%, so that the weight function methodology can be adopted. The SIF has therefore been calculated for the Gl configuration at different positions x of the crack (Fig.2), i.e. for different shifts from the centre. The deviation of the calculated SIF from the pure bending solution remains very small (< 2%), even for cracks strongly off-centre. Since the experimental errors in measurements of the toughness are generally higher, this allows the use of the simple expression of Ki (Eq. 2) for the configuration Gl. In the configuration Gl and the position of the crack corresponding to G2, the domain dominated by the stress singularity is smaller than the ligament and the off-centre abscissa of the crack so that no overlap between the singularities from the two notches is assumed. This suggests that the SIF calculation of Gl can be used for G2. However, we verify experimentally that the toughness is similar with both configurations Gl and G2 for an elastic material. 2.0

1.0 \

f. 0.0 I

2x/S2 = 0 2x/S2 = 0.25 2x/S2 = 0.5 2x/S2 = 0.75

^ •1.0 \

-2.0

0.2

0.3

0.4 aAV

0.5

0.6

Fig.2. Comparison of the SIF of a single notch under pure bending (Ki^) and that from finite element calculation of the unotched Gl configuration (Ki). The different x values represent increasing off centre, normalised by the inner span.

A^. SAAD, C OLAGNON, R. ESTEVEZANDJ. CHEVALIER

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EXPERIMENTAL Materials The fracture process is investigated for two glassy polymers: polymethyl methacrylate (PMMA) and polycarbonate (PC) which are generally thought to show a brittle and a ductile response respectively and thus selected to illustrate the method. These materials consist of commercial sheets (from Goodfellow) of 10 mm thickness which ensures plane strain conditions for both materials. Caution about plane strain conditions concerns primarily PC which is prone to develop plasticity and a 10 mm thickness appears reasonable according to analysis of the influence of the thickness on its toughness found in [6, 7]. In order to check experimentally the size criterion given in Eq. (1), we analysed the evolution of the yield stress ay with strain rate for both materials. Compression tests were carried out with cylinders of 10 mm height and 8 mm diameter. An Instron tensile/compression test machine was used with prescribed clamp speeds of 6.10'"^ - 60 mm/min, , resulting in initial strain rates of lO'Vs lO'Vs. The resulting yield stress varied from 60 to 130 MPa for PMMA and 50 to 65 MPa for PC.

Micrometric thrust

J

Razor blade (fixed)

15=

±Moving platei

-> Sample (transverse motion)

Fig. 3. Device for machining automatically a sharp notch with a razor blade. Notch preparation A first notch of 250 micrometers radius at the tip was mill cut with a rotary cutter. In order to prevent heating while machining, specimens were cooled with fresh compressed air during cutting. A sharp notch was fiirther introduced at the tip of the first notch with a razor blade. The displacement of the razor blade was controlled by a micrometric thrust so that slow and careful control of the blade advance could be monitored. Figure 3 shows the device used to machine the sharp notches automatically in order to improve reproducibility. Examples of the blunt and sharp notches are shovm in Fig.4. From the first blunt notch of 250 jim radius (Fig. 4a), the machined sharp notches for PMMA (Fig. 4b) and for PC (Fig. 4c) are similar and their crack tip is about few micrometers. The comparison between Figs. (4b) and (4c) suggests that some plasticity is induced by the machining in PC and not in PMMA.

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(a) (b) (c) Fig. 4. Observations of the notch tip obtained by (a) milling a slit in the material, (b) razor blade for PMMA and (c) razor blade for PC This is confirmed from an analysis under crossed polarizers reported in Fig. 5. The observations in Fig. 5 correspond to a region close to the center of the specimen wddth, where the extension of these non linear effects is larger. A non linear zone is also observed when focusing at the surface but its extension is smaller. This indicates that some initial stresses are induced in the preparation of PC while these effects are negligible for PMMA.

(a) (b) Fig. 5. Photoelasticity of PC for (a) 250 jam notch radius and (b) a sharp notch radius.

Bending test Four point bending tests are used to investigate the evolution of the toughness with the loading rate for the two materials (see Fig.l and table 1). We used an Listron servohydraulic tensile test machine in which a force rate was prescribed from 12N/mn to 5200N/mn. We choose to represent the influence of the loading rate with the variable Kj which is derived from Eq. 3. The stress rateCTQis then involved and estimated from the prescribed force rate. This variable K J is preferred to the prescribed force rate to provide data for the material fracture under mode

N. SAAD, C OLAGNON, R. ESTEVEZANDJ. CHEVALIER

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I from the present investigation and to allow comparisons with data reported from other devices and with an accompanying numerical investigation [2]. However, Kj is actually directly proportional to the force rate. The specimens were tested at room temperature, in air and for loading rates from 10"^ MPa Vm/s to 0.2 MPa Vm/s for PC and to 0.4 MPa Vm/s for PMMA. The load at fracture was used to calculate the toughness Kic from Eq. (2). RESULTS Influence of the configuration on the toughness For PMMA, the evolution of the toughness with the loading rate is reported in Fig.6. We notice that all the configurations (PI, Gl and G2) provide the same estimation of Kic. The observations of the unbroken notch under crossed polarizers do not show any birefiingence so that the material response remains i sotropic and elastic until failure. S ince the experimental toughness is similar for all configurations and the material is elastic, the similar toughness observed for the three configurations is in agreement with the calculation presented in the design section where a similar SIF was predicted between PI and Gl. Moreover, this justifies experimentally that the expression of Ki from Eq. (2) can be used for both Gl and especially the twin notch specimen G2. The toughness of PMMA is observed to increase slightly with increasing loading rate. For this material, only crazing takes place so that the variation of the toughness with the loading rate reflects the influence of the time dependent craze mechanism in the failure process.

OG2 PMMA DG! PMMA OP1 PMMA • G 2 PC BGI PC • P I PC 6.00 5.00 ^4.00

I

•••

•

f

t

•

2.00

0.00 0.0001

0.001

^

o^o e

O

1.00

0.01

B

0.1

Fig. 6. Evolution of toughness versus loading rate for PMMA and PC for the three configurations Pi, Gi and G2 and sharp notches

Experimental Analysis of Glassy Polymers Fracture

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The corresponding evolution for PC is also reported in Fig. 6. The level of toughness is observed to be about three to four times larger than that of PMMA and shows more scattering. The results are dependent on the configuration. The highest values of the toughness are observed for PI. By taking the toughness approximately to 4.5MPavm/s, the ligament (W-a) in PI is of 5 mm, the size criterion of Eq. (1) would be fulfilled for a yield stress about 100 MPa which is unrealistic for PC. Therefore, the value of Kic obtained with the configuration PI is questionable and overestimated. For the configurations Gl and G2, by taking a mean value of the toughness to 3.8MPaVm/s, the size requirements from the criterion (1) involve a yield stress larger than 60 MPa. The strain rate at the crack tip is difficult to estimate due to the stress and strain concentration but one can reasonably consider that its value is larger than the strain rate used for the compression tests. Therefore, a yield stress about or larger than 60 MPa is likely to be observed at the crack tip so that the size criterion is fulfilled for these geometries. The estimation of the toughness for Gl and G2 were observed to be similar and most of the experiments have been conducted with the twin configuration G2. The evolution of the toughness with the loading rate shows some scattering, however. The preparation of the sharp notch resulted in similar geometries so that the scatter is unlikely due to variations of the notch size. However, it probably originates in initial stresses subsequent to the notch machining.

(a) (b) (c) Fig.7. Deformation zone of the remaining crack tip for PC with (a) Ki =10"^MPaVm/s ,(b) Kj =10-^MPaVii5^/s and(c) Kj =10"^MPaVm/s . The snap-shots prior to crack propagation of the crack tip region (Fig.7) show stress induced birefringence for all the loading rates considered. Along the crack symmetry plane, a craze appears at the crack tip while localized plasticity has developed out of the crack plane. Therefore, the measured toughness does not involve only the energy dissipated during crazing but accounts for energy dissipation due to plasticity.

N. SAAD, C. OLAGNON, R. ESTEVEZANDJ. CHEVALIER

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Influence of the notch tip radius on the fracture toughness The crack tip radius recommended for the ASTM standard is 250 ^m (O.Olin). The machining is thought to be easier and expected to introduce less plasticity than the razor blade procedure so that the results are expected to be more reproducible. We investigated the influence of this notch radius on the toughness and the strain fields around the crack tip with our twin configuration. These samples will be designed by GO (having the same dimensions as G2) and the related toughness is compared with that of 02 specimens with sharp notches. -GOPC # 6 2 PC XGO PMMA OG2 PMMA

: X

X

0.001

0.01

S. 3

0.0001

I

i

•t

0.1

Fig. 8 Toughness versus loading rate for blunted and sharp twin notch configuration. The evolution of the toughness with the loading rate is reported for both notch types in Fig. 8. For PMMA, the toughness of the blunt notches specimens GO is about two to three times higher than for the sharp notch specimens 02. For the configuration GO, we notice a drop in the toughness over Ki=10-^ MPa. vm/s. For loading rates smaller that this value, the observations of the crack tip region under crossed polarizers reported in Fig. 9 show a non linear region (Figs. 9a-b) while the material appears elastic for higher loading rates (Fig. 9c). Although PMMA is generally thought to be brittle and to remain primarily elastic, a ductile to brittle transition can be evidenced when blunt notches are used for these low loading rates.

Experimental Analysis of Glassy Polymers Fracture

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(a) (b) (c) Fig.9 Non linear zone around the notch tip for GO PMMA at different loading rates: (a) Kj =10"^MPaVm/s ,(b) Ki =10"^MPaVm/s and(c) Kj =10"^MPaVm/s . For PC, we also reported in Fig. 8 the corresponding comparison for the toughness resulting from blunt (GO) and sharp (G2) notch specimens. The toughness of the blunt cracks configurations is higher than the corresponding value for sharp cracks and remains at a constant level with increasing loading rates. The observations of the crack tips under crossed polarizers (Fig. 10) show that shear yielding has developed for all loading rates. The craze is observed at the tip of the plastic zone and not at the crack tip as demonstrated in Estevez et al. [2], with this location coinciding with that of maximum hydrostatic stress^ where the shear bands intersect. The scatter in the results is smaller than for the sharp cracks configurations G2 but not negligible.

(a) (b) (c) Fig. 10 Non linear zone (shear bands) around the notch tip for GO PC at different loading rates: (a) Ki =10~^MPaVm/s ,(b) Kj =10"^MPaVm/s and(c) Kj =10"^MPaVm/s

DISCUSSION The twin notch configuration presented here allows for an analysis of polymers fracture at two scales: the toughness is measured at the macroscopic level while at a micro scale, the deformation fields at the onset of crack propagation can be observed. The observations at the latter scale provide additional information about the fracture process than the usual analysis of the fracture surface or the analysis of the crack path. For the glassy polymers investigated here.

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N. SAAD, C OLAGNON, R. ESTEVEZAND J. CHEVALIER

the twin notch configuration indicates whether or not plasticity accompanies failure of the material by crazing, for the loading rate under consideration. We compared the estimation of the toughness from the twin configuration with blunt (crack tip of 250 micrometers) and sharp cracks, with a crack tip of few micrometers. The sharp cracks are machined automatically so that its geometry is reproducible. For a brittle glassy polymer like PMMA, the results for both sharp and blunted cracks are reproducible and show negligible scattering. The sharp cracks give a lower estimation of the toughness but this is not only related to a notch root effect. In the case of a blunt crack with notch radius of 250 micrometers, stress induced birefringence related to plasticity is observed for loading rate smaller than Kj = 10~^MPaVm/s while the response is fully elastic for larger values. Thus, a brittle to ductile transition is observed for these low loading rates. Therefore, machining a sharp notch is recommended to avoid non linear effects for the loading rate domain considered here. For PC, which is more ductile, the configuration with sharp notches generates noticeable scattering in the measure of the toughness. The machining of the sharp crack introduces some stresses which affects the failure process and related measure of the toughness. The scatter is significantly reduced when blunt cracks are used but then, plasticity is enhanced and is observed for the whole domain of loading rates. Thus, the toughness is not representative of the failure process by crazing. To analyse crazing only, the preparation of the sharp notch could be improved by performing the notching at a lower temperature in order to suppress any plasticity and the development of initial stresses. This is currently under investigation. This work is a contribution to the definition of an experimental protocol which aims in identifying the parameters involved in a description of crazing within a cohesive surface methodology. The results obtained for PMMA are valuable for the calibration to perform in connection to the numerical work of Estevez et al. [2]. The method of preparation needs to be improved for more ductile material in order to characterize the failure by crazing only. While restricted here to glassy polymers, such configuration test may be extended to other polymeric materials such as polymer blends and could help in determining the mechanism involved before unstable crack propagation. REFEFIENCES 1. Kramer H.H. and Berger L.L. (1990). Advances in Polymer Science. Springer, Berlin. 2. Estevez R., Tijssens M.G.A. and Van Der Giessen E. (2000) J. Mech. Phys. Sol 48,2585. 3. WilHams J.G., Moore D.R. and Pavan. A. (2001). Fracture Mechanics Testing Methods for Polymers Adhesives and Composites. Elsevier, Oxford. 4. Tada H., Paris P.C. and Irwin O.K. (2000). The Stress Analysis of Cracks Handbook. Professional Engineering Publishing, London. 5. Wu X.R. and Carlsson A.J. (1991). Weight Functions and Stress Intensity Factor Solutions. Pergamon Press, Oxford. 6. Parvin M. and Williams J.G. (1975) Int. J. Fracture, 11, 963. 7. Liberg J.P.F. and Gaymans R.J. (2002) Polyme, 43, 3767