effect of thickness and notch radius on fracture toughness of polycrbonate

8/14/2019 Effect of Thickness and Notch Radius on Fracture Toughness of Polycrbonate

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Int J Fract (2013) 181:112

DOI 10.1007/s10704-013-9808-5

ORIGINAL PAPER

Three dimensional finite element investigations

into the effects of thickness and notch radiuson the fracture toughness of polycarbonate

Brunda Kattekola Abhishek Ranjan

Sumit Basu

Received: 9 August 2012 / Accepted: 10 January 2013 / Published online: 26 February 2013

Springer Science+Business Media Dordrecht 2013

Abstract Fracture toughness of polycarbonate (PC),

a commercially important glassy amorphous polymer,

is known to be sensitively dependent on a number of

factors including molecular weight, ageing time, load-

ing rate and specimen geometry. In this work, we ana-

lyze the effect of notch radius and specimen thickness

on the near tip fields and the consequence of these on

the mode I fracture initiation. To this end, we have

performed extensive three dimensional Finite Element

simulations within the framework of large deformation

elasto-plasticity based on a realistic constitutive modelthat hasbeencarefullycalibratedforPC. Usinga simple

set of criteria for fracture initiation by void nucleation

or ductile tearing, we are able to reproduce experimen-

tally observed brittle to ductile transitions that occur in

PC with decreasein thickness and increasein notch

radius.

Keywords Ductile to Brittle transition Thickness Notch tip radius Polycarbonate

1 Introduction

Glassy amorphouspolymers likepolymethylmethacry-

late (PMMA), polystyrene (PS) and polycarbonate

(PC) etc. form a large class of important structural

B. Kattekola A. Ranjan S. Basu (B)Department of Mechanical Engineering, Indian Institute

of Technology, Kanpur 208016, Uttar Pradesh, India

e-mail:[email protected]

materials. Exhaustive investigations by Kramer and

co-workers(Kramer 1983;Kramer and Berger 1990)

on thin polymer films have established that crazing

and shear yielding are the two important localization

micromechanisms in these materials and often, failure

is a result of either or a competition between the two

(Estevez et al. 2000). The polymer is considered brit-

tle when crazing is the dominant failure mechanism

and ductile whenshearyielding dominates. However,

whether a polymer behaves in a brittle or ductile man-

ner depends on a host of factors. For example, PC, oneof the most widely used materials of this type, shows

brittle to ductile transitions with temperature (Ravetti

etal. 1975), molecularweight (PitmanandWard1979),

ageing time (Ho and Vu-Khanh 2004) and rate of load-

ing (Rittel and Maigre 1996). Moreover, the fracture

toughness of PC is significantly lower in plane strain

compared to plane stress (Parvin and Williams 1975;

Fraser and Ward 1977), though the specific mecha-

nism of failure depends sensitively on the notch radius

(Nisitani and Hyakutake 1985).

It is clear from the above discussion that the frac-

ture mechanism and hence fracture toughness KI cof

PCdepends ona largenumberof factors.Consequently,

the numerical prediction ofKI cbecomes difficult and

typical constitutive models of polymers, even when

micromechanically motivated, (Boyce et al. 1988;Wu

and Van der Giessen1993;Anand and Ames 2006)

involve a large number of fitting parameters. Local-

ization of the deformation into shear bands occurs

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2 B. Kattekola et al.

naturally in simulations with these constitutive mod-

els(Lai and Van der Giessen 1997;Basu and Van der

Giessen2002) but incorporation of crazing requires

the use of yet another set of phenomenological laws

(Estevez et al. 2000;Basu and Van der Giessen 2002).

Fidelity of these largely phenomenological predictions

require careful calibration of the many model parame-ters and may vary sensitively with any of the factors

listed in the previous paragraph.

Typicaluniaxial stressstrainresponseof PC consists

of a yield drop or intrinsic softening right after reach-

ingyield, analmost perfectlyplastic response thereafter

followed by a steep hardening with strain till a maxi-

mum stretch is attained (see Fig.1). Consequently, the

stress field ahead of a notch in PC (for that matter in

any material with a similar uniaxial response) is sig-

nificantly different from those in power law harden-

ing metals. Finite Element (FE) simulations using ratesensitive finite deformation constitutive models have

shown this clearly in two dimensional settings.Lai and

Van der Giessen(1997) used a small scale yielding

simulation to investigate the effect of variations in the

uniaxial response on the nature of the crack tip plastic

zone. For materialswith no hardening, theplastic zones

are qualitatively similar to that in metals. For strongly

softening materials, the plastic zone consists of multi-

ple families of shear bands that keep multiplying with

load. On the other hand, with both softening and hard-

ening (as in the case of PC), a competition betweenthe two counteracting factors ensues and a single set

of shear bands initiated at the notch tip grow almost

0.0 0.2 0.4 0.6

25

50

75

Strain

Stress

(MPa)

Fig. 1 Typical uniaxial true stress versus true strain response

of PC

self similarly with load. The nature of the plastic zone

ahead of the tip of a notch affects the initiation of crazes

as shown byEstevez et al.(2000).

Further, Basu and Van der Giessen (2002)) looked at

the rate and temperature effects on the fracture initia-

tion toughness of a PC-like polymer using a craze initi-

ation criterion due to Sternstein and Ongochin (1969).Gearing and Anand(2004) attempted to capture the

transition from brittle fracture in high triaxiality condi-

tions prevalent ahead of sharp notches to ductile modes

in low triaxiality situations ahead of blunt ones using a

failure mechanism that involved competition between

the elastic volumetric strain and an effective plastic

stretch. It should be emphasized that in these simula-

tions, crack propagation was not modelled and except

in GearingandAnand(2004), quantitativecomparisons

with experiments were not made.

While two dimensional simulations of PC provideimportant insights into the interplay of various phys-

ical parameters and imposed conditions on initiation

of fracture, they necessarily pertain to infinitely thick

specimens. Experiments show that the effect of thick-

ness in fracture testing of PC is rather strong. In fact,

thickness and notch radius seem to sensitively gov-

ern the nature of failure (see, Parvin and Williams

1975;Fraser and Ward 1977;Nisitani and Hyakutake

1985)a fact that two dimensional simulations are not

equipped to capture. Experiments typically show that

thick samples of PC fail by crazing which starts at thecenterof thespecimen. Thin specimensundergo ductile

tear, through a lip-like thin zone that propagates ahead

ofthe tip.Onthe other hand,sharpnotches tend tocause

brittle and blunt notches ductile failure. As shown by

Nisitani and Hyakutake(1985), the shape of the plastic

zone ahead of the tip also changes considerably with

notch radius.

The purpose of the present paper is to see if existing

constitutive models of PC are capable of qualitatively

and quantitatively predicting the effects of thickness

and notch radius on the fracture toughness of PC. To

this endweperform finitedeformationbasedfully three

dimensional FE simulations using a constitutive model

due toWu and Van der Giessen(1993) which has been

calibrated against careful uniaxial tensile tests reported

elsewhere (Kattekola et al. 2012). All the simulations

reported herein are performed on single edge notched

plate (SEN) specimens with varying thickness 2Band

notch radius , as shown in Fig.2.

1 31 3


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Three dimensional finite element investigations 3

Fig. 2 Single edge notch plate specimen

In thefollowing, tensors aredenoted by bold capitals

while bold lower case denotes vectors. Scalar product

of two tensors are denoted as A: Bwhile the dyadic

product is denoted by A B and( )denotes the devi-atoric part of a tensor.

2 Modelling details

2.1 Constitutive model

When crazing does not take place or is suppressed, as

in compression or shear tests, amorphouspolymers like

PC can undergo quite large strains (up to about 100%).

Their response shows softeninguponyielding followed

by progressivehardeningas thedeformation continues.

In a numerical investigation of inelastic deformation

and localization in PC,Lu and Ravi-Chandar(1999)

pointed out that macroscopic softening of thespecimen

does not necessarily imply that softening is an intrinsic

property of the material. However, in an analysis of

the stress and strain fields around the tip of a blunted

crack under mode I,Lai and Van der Giessen(1997)

showed that intrinsic softening is necessary to capture

the localized strain fields observed experimentally as

inIshikawa et al.(1977). Also, it is important to note

that softening is seen in uniaxial compression tests.We start out with the constitutive description of

amorphous polymers at large plastic strains for tem-

peratures below the glass transition Tg . The constitu-

tive model is based on the formulation ofBoyce et al.

(1988) but we use a modified version introduced by

Wu and Van der Giessen(1993). Details of the gov-

erning equations and the computational aspects can be

found inWu and Van der Giessen(1996). The reader is

also referred to the review byVan der Giessen(1997)

together with a presentation of the thermomechanical

framework inBasu and Van der Giessen(2002).Theconstitutive model makes useof thedecomposi-

tion of therate of deformationD into anelastic (De) and

a plastic part (Dp) asD = De+Dp. Prior to yielding, noplasticity takes place and Dp= 0. In this regime, mostamorphous polymers exhibit visco-elastic effects but

these are neglected here since we are primarily inter-

ested in the effect of the bulk plasticity. Assuming the

elastic strains to remain small, the constitutive model

takes the form,=

Le:D

e, (1)where

is the Jaumann rate of the Cauchy stress given

in terms of the spin tensor Was

=W+ W, (2)andLethe usual fourth-order isotropic elastic modulus

tensor given by

Le= I2 I2 + I4. (3)Here,and are Lames constants while I2and I4are

the symmetric second order and fourth order identity

tensors. Assuming that the yield response is isotropic,

the isochoric visco-plastic strain rate

Dp=p2

, with=

12

:, (4)is specified in terms of the equivalent shear strain rate

p=Dp: Dp, the driving stress= b and the

related equivalent shear stress . The back stress tensor

bdescribes the progressive hardening of the material

as the strain increases and will be defined later. The

equivalent shear strain rate pis taken from Argons(1973) expression

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p = 0expAs0

T

1

s0

5/6 forT< Tg,

(5)

where 0 and A are material parameters and T theabsolute temperature [note that plastic flow is inher-

entlytemperature dependent through (5)]. InEq. (5) theshear strengths0is related to elastic molecular proper-

ties in Argons original formulation but is considered

here as a separate material parameter. Further in order

to account for the effect of strain softening and for the

pressure dependence of the plastic strain rate,s0in (5)

is replaced bys +p, whereis a pressure sensitivitycoefficient and p= 1

3tr . The shear strength sis

taken to evolve from the initial values0with the plastic

strain rate through

s= h(1 s/sss)p, (6)so as to incorporate strain softening in a simple way.

Here, h controls the rate of softening while sss repre-

sents the final, steady state value ofs .

Completion of the constitutive model requires the

description of the progressive hardening of amorphous

polymers upon yielding due to deformation-induced

stretch of the molecular chains. This effect is incorpo-

rated through theback stressb in thedriving shearstress

in Eq.(4). Its description is based on the analogy

with the stretching of the cross-linked network in rub-

ber elasticity, but with the cross-links in rubber being

replaced with the physical entanglements in an amor-

phous glassy polymer (Boyce et al. 1988). The defor-

mation of the resulting network is assumed to derive

from the accumulated plastic stretch (Wu and Van der

Giessen1993) so that the principal back stress compo-

nentsbare functions of the principal plastic stretches

as

b =

b(ep ep) , b= b(), (7)

in which epare the principal directions of the plastic

stretch. In a description of the fully three-dimensional

orientation distribution of non-Gaussian molecular

chains in a network,Wu and Van der Giessen(1993)

showed that b can be estimated accurately with the fol-

lowing combination of the classical three-chain model

and the eight-chain description (Arruda and Boyce

1993):

b= (1 )b3ch + b8ch , (8)

where the fraction= 0.85/

Nisbased onthe max-

imum plastic stretch= max(1, 2, 3)and on N,the number of segments between entanglements. The

use of Langevin statistics for calculating bimplies a

limit stretch of

N. The expressions for the principal

components ofb3ch and b8ch contain a second mate-

rial parameter: the initial shear modulus CR= nkBT,in whichnis the volume density of entanglements (kBis the Boltzmann constant).

For the 3-chain network model the principal back

stretches in terms of the accumulated plastic stretches

() are given by the expression (Boyce et al. 1988):

b3ch =1

3CR

NL

1

N

. (9)

For the 8-chain network model the principal back

stretches in terms of the accumulated plastic stretches

() are given by the expression(Arruda and Boyce1993):

b8ch =1

3CR

N

2

cL1

c

N

, (10)

with

2c=1

3

3=1

2 . (11)

The model parameters have been carefully fitted

using results from standard uniaxial tension experi-

ments on commercially available PC. These results

are discussed elsewhere (Kattekola et al. 2012) but,

the parameters used in the simulations are summa-

rized in Table1. Here, for PC, s0 = 77.5 MPa and0= 8.7 1026 s1 have been used at T= 293 K.

2.2 Numerical model

The rate tangent formulation of the above constitutive

model closely follows the procedure outlined inPeirce

et al.(1984). Details of the formulation are available in

Wu and Van der Giessen(1993). The forward gradientexpression for the plastic strain rate is a linear interpo-

lation between its values at times tand t+ tand isgiven by,

Table 1 Material properties for PC

E/s0 sss/s0 As0/T h/s0 N CR/s0

27 0.38 0.82 125 5.80 0.08 3 0.20

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p= (1 )p (t) + p (t+t), (12)where 0 1. Further, using a Taylors seriesexpansion forp(t+t), we obtain,

p = p(t) + tp

. (13)

Following Wu andVanderGiessen(1996)therateformof the constitutive equation for the back stress tensor

can be expressed as,

b= R : D. (14)Using the above, the Jaumann rate of the Cauchy stress

tensor is finally expressed as

= L : D+ , (15)

where the tangent modulus L is defined as

L = Le M M. (16)Further, the vectorcan be expressed as

= p1 + M, (17)

where,

M= 2G2

. (18)

Also,

=1

2

g(1 + ) , and (19)and gare given as

g= 12

2G + 1

2: R:

, and,

= gtp

.

The rate dependent incremental constitutive model

hasbeen implemented inABAQUS v6.9 using a UMAT

routine. Eight noded brick elements were used in all

the analyses reported. The J integral (Rice 1968) is

calculated using a domain integral representation suit-

able for FE computations due to Nakamura et al.

(Shih et al. 1986).

2.3 Boundary conditions

All the simulations reported herein are performed on

SEN specimens (see Fig. 2)of width W, height 2L

and thickness 2B. The circular notch with center at the

origin has a radius of. The crack length a is cho-

sen such that a/W=0.52. In the analyses reported,L=29 and W= 25 mm are kept fixed while 2Bandare varied.

Using symmetry along the length and the thickness

of the specimen, only one-fourth (X2

0 andX3

0)

of the specimen is analysed and symmetry boundaryconditions are applied as:

u2(X1 >a, 0,X3) = 0 (20)and

u3(X1,X2, 0) = 0. (21)The top surface of the specimen is pulled at a constant

velocity so that

u2(X1,L ,X3) = 0(t), (22)where

0(t) =0t. (23)Intheabove, tis the timeanda fixedvalue of1 mm/min

is used for0in all the runs.Further, the following boundary condition is applied to

constrain the rigid body motion:

u1(25, 0,X3) = 0. (24)For a load controlled specimen, the applied load P

on a SEN specimen can be related to the stress intensity

factor KIthrough a geometric factor f(a/W)as

KI= P2BW

a f aW

. (25)

For a displacement controlled situation, the appliedKIcan be related to the displacement using a simple

scheme that involves the crack length dependent com-

plianceC(a)of the sample. An estimate of the applied

stress intensity factor KIis thus obtained as

KI=

E

4BC2C

a. (26)

The method of computation of compliance C(a)of a

SEN specimen is given in the Appendix. Note that

the value of KIobtained from Eq. (26) gives a refer-ence value that we use to normalize certain quantities.

This may be thought of as the far field KIexperienced

by the sample under plane stress small scale yielding

conditions.

Typically about 1015 brick elements were used

across the thickness. The crack circumference was

divided into elements whose sides were approximately

/10 in the X1 X2plane. The total number of ele-ments used ranged from about 20,000 to 100,000.

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Fig. 3 Contours of normalized accumulated plastic strain in

specimens with = 200m and thickness, 2B =(a) 1, (b)4, (c) 10mm and the (d) plane strain scenario. In ac, theupper

half shows the plastic zone at center(X3/B= 0) and thelowerhalf at (X3/B= 1) the surface of the specimen

contours for a 2D plane strain case is also shown for

comparison (Fig.3(d)).

The contours of mean stress m shown in

Fig.4(a)(c) for the same cases as in Fig. 3 further

underlines the importance of thickness. For all the

cases, the regions of highest mean stress are located

in the interior. Evidently, the surface of the specimenhas lower mean stress values due to plane stress condi-

tions on the surface. However, for the thinest sample,

the levels ofmon the surface and in the interior are

very close. In the thickest specimen on the other hand,

the mean stress on the surface is considerably lower. In

the interior of the thickest specimen, a region of high

mdevelops ahead of the crack tip. This is the region

where the void nucleation criterion is likely to be sat-

isfied. For thinner specimens such a region is not seen

ahead of the notch even in the interior of the speci-

men.Shape and size of the plastic zone also varies with

the notch radius. This is shown in Fig.5(a), (b) where

contours of normalizedphave been plotted for thick

specimens (2B = 10 mm) with notch tip radius, = 50 and 500m. The contours have been plot-ted in the interior of the specimen (X3/B = 0) atan instant when one of the failure criteria was met.

Clearly, at the instant of failure, the size of the plas-

tic zone is the smallest for the specimen with the

sharpest notch. The plastic zone size as well as the

intensity of the maximum accumulated plastic strainp at failure increases with notch radius. In sum-

mary, Figs.3,4and5demonstrate how thickness and

notch radius affect the nature of the plastic zone ahead

of a notch in PC. Firstly, thin samples tend to have

banded plastic zones running parallel to the line of

symmetry almost throughout their thickness. As thick-

ness increases, the bands on the surface become wider

and the plastic zone in the interior turns away from

the line of symmetry. With further increase in thick-

ness, the plastic zone both at surface and the interior

ceases to be banded and grows mainly in the X2direc-tion.

At a constant thickness, very sharp notches give

rise to small plastic zones. The extent to which they

are banded depends on the thickness. With increase in

notch radius, the plastic zones get larger. Thus, for both

small values ofand large values ofB , the plastic zone

tends to be smaller compared to the cases with larger

and smaller B.

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Fig. 4 Contours of m in specimens with

= 200m and

thickness, 2B= (a) 1, (b) 4 and (c) 10 mm. In each figure, theupperhalf shows the plastic zone at center(X3/B= 0) and thelowerhalf at (X3/B= 1) the surface of the specimen

3.2 Variations of the Jintegral and stresses through

the thickness

The value of the J integral at X3/B = 0 has beenplotted against the applied load KIin Fig.6. We have

Fig. 5 Contours of accumulated plastic strainat failure forspec-

imens with 2B= 10 mm and notch radius =(a) 50 and (b)500m

chosen = 200m for these plots while 2B= 1,4 and 10 mm have been used. These correspond to

2B/ = 5, 20 and 50 respectively. The circles onthe plot indicate the values of Jcat which one of the

competing failure criterion is met. Clearly, the value

of J(X3/B = 0) increases monotonically with theapplied load KIas expected. The thickest specimen

(2B= 10 mm or 2B/= 50) fails in brittle mannerwith the lowest value ofJc

=5.46 N/mm compared to

the other two specimens which fail in ductile fashionsatisfying the ductile tearing criterion.

As indicatedin theprevious section, thickspecimens

with sharp notches seem to develop plastic zones sim-

ilar to thinner specimens with even sharper notches. In

fact, the ratio 2B/seems to be a good indicator of the

failure modes in PC. For large 2B/, the failure mode

is brittle irrespective of the sample thickness. Simi-

larly, for smaller values of 2B/, the failure mode is

ductile.

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0 0.5 1 1.5 20

20

40

60

80

X1(mm)

m(MPa)

X3/B = 0

X3/B

X3/B = 1

0 0.5 1 1.5 20

20

40

60

80

X1(mm)

m(MPa)

X3/B = 0X3/B = 0.4

= 0.4

X3/B = 1

0 0.5 1 1.5 20

20

40

60

80

X1(mm)

m

(MPa)

X3/B = 0X3/B = 0 .4X3/B = 1

(a)

(b)

(c)

Fig. 8 Variation ofmwith X1for (a) 2B= 1mm and =250m (b) 2B=1 mm and=50 m and (c) 2B=10mmand= 200mtriaxiality levels are low and the ductile tearing crite-

rion will need to be satisfied in order that fracture may

initiate.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

50

100

150

200

250

300

(mm)

Jc(N

/mm)

0 2 4 6 8 10

5

10

15

20

25

2B (mm)

Jc(N/mm)

(a)

(b)

Fig. 9 Variation of Jintegral with (a) at constant thickness

2B= 10mm and (b) thickness at constant= 200m

3.3 Ductile-brittle transitions with thickness

and notch radius

In this section, we quantitatively summarize the find-

ings from this investigation. In Fig.9(a), and (b), we

have plotted the values of Jcwhich is the value of the

Jintegral at the point where either the void nucleation

or the ductile tearing criterion was met. In Fig9(a), thethickness has beenkeptconstantat10mm andvaried,

while in Fig.9(b), the notch radius was 200m.

With notch radius, a gradual transition from brittle

behavior characterized by very low values ofJcto duc-

tile tearing is seen. It should be noted that whenever the

voidnucleationcriterionis satisfied, initiationof failure

is likely to happen at the center of the specimen. On the

contrary, whenever ductile tearing happens, the mean

stress across the thickness is relatively constant. At the

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0 4 8 12

0.2

0.4

0.6

2B (mm)

(m)

Ductile

Brittle

Fig. 10 Mode of failure map on the 2Bplane. Thedashedlineapproximately divides the plane into brittle () and ductile() parts

thickness shown, the critical value ofat which the

brittle-ductile transition happens is between 400 and

500m which is commensurate with the experimental

trends inNisitani and Hyakutake(1985).

On the other hand, a much sharper ductile to brit-

tle transition occurs with the sample thickness. For

= 200m, samples with thickness below about3mm exhibit ductile failure.A failuremapforcommer-

cial PC can be constructed on the 2B plane indicat-ingthe failuremode fora given thicknessnotchradius

combination. Using the failure criteria adopted in this

work, such a map has been shown in Fig.10.Combi-

nations that fail in brittle manner have been denoted

by , while ductile ones are denoted by . Theline 2B/ 25.5 approximately separates the duc-tile region from the brittle one.

4 Conclusions

We have conducted extensive three dimensional FE

simulations on displacement controlled SEN samples

of PC with a range of thicknesses and notch radii with

a view to understand the ductile to brittle transitions

that are commonly seen in fracture experiments on

this material. A well calibrated constitutive model is

used within the framework of large deformation elasto-

viscoplaticity. Simple but representative criterion for

failure initiation are devised assuming that the brittle

behavior is associated with crazing, which in turn is

facilitatedby high triaxiality. On theother hand, ductile

failure takes over in low triaxiality geometries where

crazeinitiation is suppressed andlarge opening stresses

are required to cause tearing. Note that we are looking

at the initiation toughness of the specimen and steady

state crack growth may require higher loads.

We have analyzed the changes in plastic zone and

mean stress distributions that occur with changes ineither thickness or notch radius. Suitable combination

of these two geometric parameters can cause the triax-

iality levels all over the thickness of the specimen to

drop leading to suppression of void nucleation.

Our simulations are able to showthat with thickness

a ductile- brittle transition occurs in PC. Similarly, a

ductile-brittle transition occurs with notch radius. For

a given thickness, the critical notch radius at which

the transition happens depends on the thickness of the

specimen. A critical value of the ratio of the thickness

to the notch radius of about 25.5 seems to separate thebrittle zone from the ductile in a failure map.

Appendix: Determination ofK for a displacement

controlled SEN specimen

Consider that C(a)is the compliance of a specimen

with a crack of lengtha, and a displacementhas been

applied to it in a manner shown in Fig.2. The force P

is related to the displacement by

= C(a)P, (28)and the total strain energy stored is

U= 12

P= 12

2

C(a). (29)

Theenergy release rateG is related to the rateof change

of the strain energy per unit crack area and thus

G= 12B

U

a, (30)

which, in terms of the compliance becomes (using

Eq.28),

G= 14B

P2C

a. (31)

Further, since we want to use the value of KI as a

reference, we choose

G= K2

I

E, (32)

which gives the plane stress (small thickness) approx-

imation to KI.

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Textbooks of fracture mechanics (see, eg.Kumar

2009) routinely give Mode I stress intensity factors for

standard specimens like the SEN, in terms ofP . For a

SEN specimen in particular, the relation reads:

KI=P

2BW

a f

a

W , (33)where the function f(x) = 1.12 0.23x+ 10.55x2 21.72x3 + 30.39x4. Comparing Eqs.33and31givesC

a= a

BW2 f2 a

W

, (34)

which, on integration, gives

C(a) C(0) = aBW

a/W0

f2(x)d x. (35)

Further, note thatC(0), the compliance when there is

no crack, is given by

C(0) = L2BWE

. (36)

KnowingC(a), G and henceKIcan be obtained from

Eq.(31).

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effect of thickness and notch radius on fracture toughness of polycrbonate

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