size effect on fracture

Available online at www.sciencedirect.com

Procedia Engineering 10 (2011) 1348–1353

ICM11

Notch Radius and Specimen Size Effects on Fracture Toughness of
Low Alloy Steel
Abdel-Hamid I. Mourada, *, Aly El-Domiatya
a
Mechanical Engineering Department, Faculty of Engineering, United Arab Emirates University, P.O.Box 17555, UAE

Abstract

The present work is amid to investigate the effect of notch root radius and specimen size (thickness B and ligament b) on the
apparent fracture toughness KI,app. Series of fracture tests on non standard compact tension CT specimen of AISI 4340 low alloy
steel with different values of notch root radius ( =0.08 mm-3 mm) and ligament size/crack length to the specimen width ratio
(a/W=0.1-0.9) have been conducted to measure the apparent fracture toughness KI,app. The experimental results show that KI,app
depends on and specimen size (B and b). There is a critical notch root radius c under which the apparent fracture toughness
becomes independent of the notch radius . Two empirical models have been proposed to transfer the data among specimens of
different notch radii and predict the experimental values of KI,app considering the effect of value. Also, Two models have been
developed to transfer the data among specimens with different sizes and predict KI,app considering the effect of specimen
thickness B, specimen ligament band the combined effect of B and b values. The combined effec –based model is developed by
modifying the E 1921 standard equation (that is usually used to predict the thickness effect only). The models predictions are in
good agreement with the present work experimental results and results reported in the literature.

© 2011 Published by Elsevier Ltd. Selection and peer-review under responsibility of ICM11

Keywords: Fracture toughness predictions, Specimen size effect, Notch root radius effects.

1. Introduction

In many situations, it is necessary to correlate between the measured apparent fracture toughness KI,app from non
standard specimen and the plain strain fracture toughness KIc obtained by testing standard one. The size scale effects on
the mechanical and fracture properties of the material are very important topic in engineering design. In recent years, a
great deal of research [1-2] on size scale effects has been carried out in order to gain a precise description of this
phenomenon and to high light this effect in quantitative method. Several investigators tried to use the two parameters
fracture mechanics approach to describe the effect of constrains on ductile tearing resistance and resolve the issue of
transferability [2]. The two parameters fracture mechanics approach [2] is used to quantify the level of stress triaxiality.
Studies by several investigators (e.g. Chao et al. [3]) have illustrated that the apparent fracture toughness Kc varies with
specimen size and loading configuration for brittle materials.
Study of the transferability of the fracture toughness measurements from laboratory specimens to structural
components has been performed by some researchers. Some researchers correlated the Charpy impact energy to the
fracture toughness of steels. Scibetta, et al. [4] have conducted experimental study on the fracture toughness transferability

* Corresponding author. Tel.: +971-3-7133574; fax: +971-3-+971-3-7623158.
E-mail address: ahmourad@uaeu.ac.ae.

1877-7058 © 2011 Published by Elsevier Ltd. Selection and peer-review under responsibility of ICM11
doi:10.1016/j.proeng.2011.04.224

Abdel-Hamid I. Mourad and Aly El-Domiaty / Procedia Engineering 10 (2011) 1348–1353 1349

to pressurized thermal shock (PTS) representative loading conditions. In the work carried by Ono et al. [5] four different
sizes of compact tension (CT) specimens made of steel, were tested by unloading compliance method according to ASTM
E 1820-99a in order to measure the fracture toughness JQ. Their experimental results indicated that JQ value increased
with decreasing the specimen thickness. Toshiro et al. [6] have reviewed specimen size effect which significantly affects
fracture toughness. They have discussed controversial scale problems on fracture and fracture toughness. The work by
many investigators, [e g. 7-10], show that the apparent fracture toughness increases linearly with the notch root radius .
A critical notch root radius below which fracture toughness is independent of the notch root radius has been found by
some researches [11, 12].
A real life component, on the other hand, undergoes a substantial amount of stable crack growth SCG. However, it does
depend on the size of the structure or the mechanical component and the shape of the pre-existing crack, i.e., its sharpness.
Mourad et al. [13-15] have studied the effect of the notch root radius on the SCG parameters through CT of AISI 4340 low
alloy steel. The effect of in-plane and out-of plane constraints (specimen geometry, crack length and loading
configuration) on SCG behavior through different specimens’ geometries of different materials under different modes of
loading have been also investigated extensively by Mourad et al [13- 17.]The correct methodology of extrapolation of the
fracture properties for large component is consequently an important issue in many fields such as design of the piping
system in nuclear power plants and power stations. These systems have major problem because of using material
properties obtained from specimens with laboratory scale sizes.
In the light of the above discussion, many studies have been conducted to investigate how the apparent fracture
toughness and fracture tests (e.g. stable crack growth tests) were dependent on notch root radius and specimen size (in
plane and out of plane constraints). Nevertheless, more research is demanded on such important issues. The present study
is a continuation of the previous research work aiming to investigate, the effect of specimen size and notch root radius of
the machined crack (non-sharp crack) on the fracture toughness and the mixed effect of in-plane (ligament length) and out-
of-plane (thickness) constraints on the apparent fracture toughness. This may contribute to resolving the transferability
issue of the fracture toughness from laboratory specimens to structural components.

2. Experimentation

Low alloy steel EN34NiCrMo6, which is equivalent to AISI 4340/4330 steel, is used in this investigation. The
chemical compositions of the material are given in Mourad et al. [13-15] The major mechanical properties obtained by the
tensile test are: the modules of elasticity (E = 217 GPa), the yield strength ( y=551MPa), the ultimate strength ( u=779
MPa), percent elongation (% L=8) and the Ramberg–Osgood constitutive equation ( / o= / o+ { / o}n) constants are
=0.473and n=8.114. More details about the material are given in Mourad et al. [13-17]. CT specimens were machined out
from the supplied material according to the geometry shown in Mourad et al [13-17]. The wire electric discharge
machining technique (WEDM) is used to prepare the machined crack with different notch radius . The selected notch root
radiuses are 0.08, 0.16, 0.25, 0.50, 1.00, 2.00 and 3.00 mm. After machining, the notch radius is measured by the aid of
profile projector with an enlargement of 50 x. The size effect of the specimen is introduced by selecting different values
for the initial crack length to specimen width ratio (a/W). These values are 0.1, 0.2, 0.3, 0.4, 0.45, 0.5, 0.55, 0.60, 0.70,
0.8, and 0.9 (i.e., initial crack length, a, of 10, 20, 30, 40, 50, 55, 60, 70, 80, 90 mm). A total number of 110 different
combinations of the two parameters and a/W are considered. To ensure the reproducibility of the results 3 specimens
were test for each condition. Therefore, a total number of 330 specimens are machined and tested in the experimental
program.

3. Experimental Results

3.1. Notch Radius Effect on KI,app
The measurements of the load-load line displacement are conducted on compact tension specimens. The maximum
load for each specimen is obtained and plotted as a function of the notch root radius for different a/W ration in Fig. 1
The data show that the maximum load Pmax increases linearly with the notch radius and decreases as a/W increases.
Similar trend of variation has been obtained by Akourri et al. [18] using three point bending TPB specimen of mild steel
with .different (up to =2 mm) and a/W ratio (0.2: 0.7).

1350 Abdel-Hamid I. Mourad and Aly El-Domiaty / Procedia Engineering 10 (2011) 1348–1353

The maximum loads are used to calculate the apparent fracture toughness KI,app by using the following equation
KI,app=f1(a/W) Pmax/ (B(W-a) (1)
where f1(a/W) is the geometrical function, B and W are the specimen thickness and width. The results are used to plot
KI,app vs. and its reciprocal 1/ for different values of (a/W) as shown in Fig. 2. The results show that KI,app is dependent
on and a/W ratio. KI,app is almost constant in the range from = 0.08 mm up to 0.16 mm for all a/W ratio. Then it
increases almost in a linear relationship up to 0.6 mm for a/W ratio from 0.1 up to 0.6, prior it starts to increase
nonlinearly with up to = 3 mm. However, for a/W = 0.7 up to 0.9, KI,app increases linearly in the range from >0.16 up
0.25 mm prior it increases nonlinearly with up to =3 mm. That linear increase has been reported by some
researchers ( [7-10]. KI,app reaches a minimum value (KIc,app) when reaches its minimum value ( =0.08 mm), however,
there is a critical value c for each a/W below which KI,app becomes almost independent of . Therefore, the curves of Fig.
2 consist of three regions I, II and III. In the first region there is a rapid decrease in KI.app followed by less rate of decrease
in the 2nd region prior the curve becomes almost a horizontal line (or reaches a lower plateau) in the 3rd region, i.e. KI,app
reaches KIc.app, ( =0.16 up to 0.08 mm). A critical notch root radius below which fracture toughness is independent of
has been reported [11, 12].

Fig. 1 Variation of Pmax (kN) with for different a/W ratio Fig. 2 The apparent fracture toughness vs. 1/ and vs. for a/W ratio
KI,app as a function of a/W ratio for all tested notch radii are shown in Fig. 3. Similar trends were obtained by Xiao-zhi
Hu [19], where the fracture energy decreases as the ratio a/W increases. The behaviour shown in Fig. 3 can be divided into
three regions, (I, II and III) or 3 line segments with different slopes. For the first region the value of a/W is in the range of
0.1 a/W 0.4, region II is for the range of 0.4 a/W 0.6 (through which fracture mechanics approach is suitably
applicable) and region III for the range of 0.6 a/W 0.9.
Akourri et. al. [19] have measured the work done to fracture using TPB specimen with different ( =0.05, 0.25, 1, and
2 mm) and different a/W ratio (a/W=0.2, 0.3, 0.4, 0.5, 0.6 and 0.7). Their results were digitized and re-plotted, not
presented here) which demonstrate the variation of the experimental critical non–linear energy Ucnl as a function of for
different a/W ratio. Their data have been used, in the present investigation, to calculate Japp integral as a function of 1/ for
different a/W ratio and as a function a/W for different . The trends of variations of Japp vs. 1/ and Japp vs. a/W of are
similar to that of Figs. 2 and 3 regardless the specimen type. (CT and TPB).

Fig. 3 KI.app vs. a/W ratio for different Fig. 4 KI,app vs. yield strength for AISI 4340 low alloy steel


3.2. Ligament length effect on KI,app

The variation of the K,app with a/W for =0.08 curve is one of the curves presented in Fig. 3 This curve
represents the variation of the critical apparent fracture toughness KIc,app with a/W, as at =0.08mm the variation of
KI,app with for all a/W ratios reaches the lower plateau (Fig. 2). The upper and lower bounds of region II, are KIc,app
= 200 MP m at a/W=0.4 or (b=60 mm) and 150 MP m at a/W = 0.6 (or b=40 mm), respectively. The plane strain
fracture toughness (according to standards E339 or E1828) requires ligament length b in the range from 40 mm up
to 60 mm (the range of region II). Therefore the nominated value for the tested material KIc is expected to be within
the values of region II, providing that the thickness requirement by the standards is also met.
Figure 8.31 of “ Dowling N, Mechanical Behaviour of material, 2nd edition, 1999) show that the AISI 4340 steel
has KIc ranges from 40 MP m up to 190 MP m depending on the yield strength. According to yield strength of
the used AISI 4340 steel, y=552 MPa, the corresponding plane strain fracture toughness KIc is between 180 and 190
MPa m. This range is within the expected range according to Fig. 3 (150 MP m - 200 MP m).
Figure 4 shows the variation of the KI,app,with b. The variations based on the plane strain fracture toughness KIc
requirements [b 2.5 (KIc,app/ y)2 or b = 2.5 z (KIc,app/ y)2, z >1] is also included for different values of z. It is shown
that the critical apparent fracture toughness increases KI,,app with b. The variation based on three different values of z
are presented (z =1, 2.0, 2.3 and 2.5). The curve for z =2.3 is the closest one and almost superimposes with the
experimental results over the range from b=40 up to b=60 mm and fracture toughness range from 150 up to 200
MPa m. This is the range within which the standard specimen ligament and the KIc of AISI 4340 steel (180 up to
190 MPa m) is expected. Similar trend of variation of Fig. 4 has obtained by Ono et al. [6] using CT specimen of
JLF-1LN steel. Their results show that the fracture toughness Japp increases with the b and decreases as B increases
[6] . Also the variation of Japp with b based on the results obtained by Akourri et al [18] on the variation of work
done to fracture with for different a/W ratios using TPB specimen resembles the variation of KI,app with b in Fig. 4.
KI,app(MPa m)

KI,app(MPa m)

1/ (m)-1 1/ (m)-1

Fig. 5 Experimental and pedicted (Model I) variation. Fig.6 Experimental and pedicted (Model II) variation.

4. Fracture Toughness Modeling

4.1. Notch Root Radius Effect

Model I: Two models are developed for predicting the KI,app for specimens of varying and with given a/W ratio
and B. In model 1, the variation of the KI,app with 1/ (Fig. 2) is found to be predicted by the following proposed
empirical
KI,app = KIc,app [C.e1/ +1] (1)
where C and KIc,app are constants for each KI,app vs. 1/ curve of a specimen geometry with specific a/W ratio and
B. These two constants can be determined by conducting two tests using two specimens with two different . This
makes it possible to use Eqn. 1 to predict the KI,app vs. curve for a given specimen geometry (a/W ratio and B).
Note, KIc,app is the minimum value of KI,app and corresponds to c for a specimen with specific a/W ration and B.
The applicability of Eqn. 1based-model is verified using the experimental measurements of the present work. For
instance, KI,app values at = 0.08 mm and 3 mm are used here for each a/W to determine C and KIc,app for all a/W

1352 Abdel-Hamid I. Mourad and Aly El-Domiaty / Procedia Engineering 10 (2011) 1348–1353

ratios. The predicted KI,app vs.1/ curves for all experimental curves of Fig. 2 are obtained and plotted. Fig. 5 (Only
results of a/W = 0.4 is presented) shows good agreement between the experimental and predicted results.
Model II: KI,app values have been also found to be predicted using the following proposed Eqn.:
( rp / ) m
K I ,app K Ic ,app ( Pmax / B W ). e . f (a / W ) (2)
where W is the specimen width, m is a constant, f (a/W) is the geometry factor and rp is the plastic zone size
[rp=(1/2 ) (KI,app/ y)2]. Unlike the model I (Eqn. 1), KI,app is calculated in terms of specimen geometry(W, B and
a/W), rp and Pmax. The KI,app vs curve can be obtained knowing the two constants KIc,app and m that can be
determined by testing only two specimens with different . The capability of Eqn. 2 is verified using the
experimental results of the present work. Substituting KI,app and and other data (Pmax, B, W, rp, f(a/W)) for any two
tests in the Eqn. 2 and solve for the constants m and KIc,app. The predicted KI,app values using Eqn. 2 are plotted
versus 1/ along with the experimental ones for all a/W ratios. Fig. 6 shows the case of a/W =0.4. The good
agreement reflects the potential of Eqn. 2 in predicting KI,app vs 1/

4.2. Thickness Effect

The thickness effect on the fracture toughness can be predicted by using the equation of the standard E1921.
KJcE1921(1T) =Kmin + (KJcxT-Kmin)*(BxT/B1T)1/4 (3 )
Eqn 3 can be rewritten in the following form
Ky =Kmin + (Kx-Kmin)*(Bx/By)1/c (4 )
Eqn 4 predicts Ky for a specimen (y) knowing Kx of a specimen (x). Also it can predict KIc for standard specimen,
knowing Kx of a non standard specimen. Kmin is the lower bound fracture toughness and c is the Weibull slope.
The predicted values of Ky as function of thickness ratio Bx/By according to Eqn 4 have been obtained. The
data of the specimen x were a/W=0.5, Bx=8 mm and Kx=259.2 MPa m, Kmin was taken =20 MPa m. The results
show that, as By increases the thickness ratio Bx/By decreases and consequently the fracture toughness (Ky) and
vice versa. The prediction also depends on the c value. for c = 4, the required specimen ratio to meet the plane strain
fracture toughness of tested material (180:190 MPa m is Bx/By= 0.2:0.25 or By=32:40. This range is within the
expected range of the standard specimen thickness. Eqn 12 may also be presented in terms of J-integral as follows:
Jy =Jmin + (Jx-Jmin) x (Bx/By)1/c (5)
Fig. 7 demonstrates the variation the experimental (Ono et al. [5]) and the predicted (Eqn. 5) Jy with Bx/By. The
prediction accuracy is sensitive to c, Jx and Jmin. Eqn. 5 has the potential to predict the apparent fracture toughness
for 0.5T-1CT specimen of By= 12.7 mm knowing the plane strain fracture toughness (Kx= KIc = 568 MP m) of 1T-
1CT specimen of Bx= 24 with good accuracy (7.2% deviating) using c=4.

Fig. 7 Experimental [5] and predicted (Eqn. 5) Jy vs. Bx /By. Fig 8. Experimental [5] and predicted (Eqn. 7) Jy vs. Bxby/By bx

4.3. Ligament Effect

Eqn 5 has been modified by replacing the thickness ratio Bx/By ratio by the ligaments (by/bx) as follows:
Ky =Kmin + (Kx-Kmin). (by/bx)1/c ( 6)
In Eqn 6, the fracture toughness Ky is directly proportional to by. Fig. 7 shows the experimental and predicted variations of
the fracture toughness vs. by/bx (and vs. by) are in good agreement between especially in the range of by=40: 90mm
(a/W=0.1:0.6) for different c values. The maximum deviation is at by=90mm.


4.4. Combined Effect of Thickness and Ligament

It is noticed from the results of sections 4.2 and 4.3 that the apparent fracture toughness increases with band
vice versa with B Therefore, a controlling parameter, that is the thickness to ligament ratio B/b, is introduced to
study the combined effect of both B and b on the apparent fracture toughness. The experimental results obtained by
Ono et al [5] on the fracture toughness J using different CT specimen ( in terms of B, b and a/W) have been plotted
(i.e. J vs B/b; is not presented here) in order to study the simultaneous effect of B and b sizes. The results show that
the fracture toughness decreases as the B/b ratio increases. Therefore equation 3 (of E 1921 standard, KJcE1921(1T)
=Kmin + (KJcxT-Kmin) (BxT/B1T)1/4) which consider only the thickness effect is modified here to consider the
combined effect of both B and b sizes. The proposed new equation is presented by the follows Eqn:
Jy=Jmin + (Jx-Jmin) (Bxby/Bybx)1/c (7)
Fig. 8 demonstrates the experimental and predicted variation of Jy vs. (Bx.by/By.bx). In Eqn. 7 the specimen x of
Jx=404 kJ/m2 and Bx/bx = 1.2834 (Ono et al. [5] results is considered to be the reference specimen and Jmin = 20
kJ/m2. Good agreement between the results is observed for c=2 and 4. The good agreement between the
experimental and predicted results in Fig. 8 reflects the potential of the proposed model (given by equation 6) in
predicting the combined effect of ligament and thickness dimensions on the fracture toughness.

5. Conclusion

The effect of and specimen size (thickness B and ligament b) on fracture toughness has been studied. Critical c
was found at which the fracture toughness is completely independent of . The experimental results of the present work
and reported in the literature show also the dependence of the fracture toughness on B and b. Four different models
(empirical equations) have been developed to evaluate the apparent fracture toughness considering the effect of and
specimens size. The 1st model predicts the apparent fracture toughness as a function of . The 2nd model considers the rp
size as well as . The 3rd model predicts the apparent fracture toughness considering the effect of b In the fourth model
the equation of the standard E 1921has been modified to account for the combined effect of both B and b effect. The
predicted fracture toughness using the developed models is in good agreement with the measured one. The developed
models may contribute to the solution of the transferability problem.

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