CC BY
34
УДК 539.3
Mode III notch fracture toughness assessment for various notch features
A.R. Torabi1, B. Saboori2, M.R. Ayatollahi2
1 Fracture Research Laboratory, Faculty of New Sciences & Technologies, University of Tehran, Tehran, 14395-1561, Iran 2 Fatigue and Fracture Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran, 16846, Iran
In the present paper, two stress-based failure criteria are proposed to predict the notch fracture toughness for three different notch features under pure mode III loading. These criteria are developed based on the two well-known failure concepts of the point-stress and the mean-stress previously used for predicting brittle fracture in notched members under various loading conditions. The validity of the criteria is verified through the comparison of their theoretical predictions with a bulk of test data reported in the open literature on mode III fracture of graphite notched round bars. Very good agreement is shown to exist between the experimental and theoretical results. Moreover, the comparison revealed that the mean-stress criterion is more accurate than the point-stress criterion in predicting mode III brittle fracture of V-notches and semicircular notches.
Keywords: brittle fracture, notch, mode III loading, failure criterion, fracture toughness
Оценка вязкости разрушения продольным сдвигом для надрезов разной формы
A.R. Torabi1, B. Saboori2, M.R. Ayatollahi2
1 Тегеранский университет, Тегеран, 14395-1561, Иран 2 Иранский университет науки и технологии, Тегеран, 16846, Иран
В работе предложены два критерия разрушения на основе напряжений для прогнозирования вязкости разрушения трех надрезов разной формы при продольном сдвиге. Критерии получены на основе известных концепций предельного и среднего напряжений разрушения, применяемых для описания хрупкого разрушения в деталях с надрезом при различных условиях нагружения. Для проверки выполнения предлагаемых критериев проведено сравнение теоретических оценок с использованием литературных данных испытаний на разрушение типа III графитовых стержней с надрезом. Показано хорошее соответствие между экспериментальными и теоретическими результатами. Сравнение также показало, что критерий средних напряжений дает более точную оценку при прогнозировании хрупкого разрушения типа III для V-образных и полукруглых надрезов, чем критерий предельного напряжения.
Ключевые слова: хрупкое разрушение, надрез, нагружение типа III, критерий разрушения, вязкость разрушения
1. Introduction
Various types of notches are often utilized in engineering components because of particular design requirements. As some of their practical applications, one can mention the existence of V- and U-shaped threads in screws, welded joints in structures and V- and U-notches in shafts. Similar to sharp cracks, notches can be subjected to three basic modes of deformation, namely mode I (opening), mode II (in-plane sliding) and mode III (out-of-plane tearing), or any combination of them known as mixed mode loading.
Fracture evaluation of notches of different shapes has been widely dealt with in the literature both experimentally and theoretically. In particular, theoretical failure criteria play a vital role in safety analysis of engineering components weakened by notches under various loading conditions.
Ductile and brittle failures of engineering components and structures in the presence of stress raisers like cracks and notches have been extensively studied theoretically and experimentally in different materials under various loadings [1-4]. As the simplest failure problem in notched do-
© Torabi A.R., Saboori B., Ayatollahi M.R., 2017
mains, the brittle fracture behavior of sharp and blunt notches subjected to pure mode I loading has been investigated in the past by many researchers. In general, there are a few main failure models in the literature for mode I brittle fracture assessment of notched components, namely the cohesive zone model [5], the strain energy density [6, 7], the generalized J-integral [8-11], the finite fracture mechanics [12], the point-stress and the mean-stress [13, 14] criteria, while relatively fewer failure models exist for notched domains subjected to mixed mode I/II loading. The most active failure criteria in the context of brittle fracture of notched members under mixed mode I/II and pure mode II loadings are the strain energy density [15-18], the point-stress and the mean-stress criteria [19, 20]. However, some limited papers have also been published on mixed mode I/
II brittle fracture assessment of notches by means of the cohesive zone model [21] and the finite fracture mechanics [22] criteria.
Compared with the in-plane loading conditions, there are very few papers dealing with brittle fracture for spatial (out-of-plane) loading cases including mode III loading. In this context, a fracture criterion based on the strain energy density averaged over a well-defined control volume which embraces the notch edge has been applied to the torsion loading case of U- and V-notches [23, 24]. This local strain energy density criterion has also been implemented recently for V-notches subjected to tension-torsion loading conditions [25]. In another study, Zheng et al. [26] used the fracture data of ceramic notched specimens under combined tension/torsion to assess the applicability of a failure criterion based on the critical normal stress. Additionally, a fracture model has been suggested for notched elements made of brittle materials under pure mode III loading [27], which is based on a combined normal stress/Griffith energy fracture criterion.
A comprehensive literature review revealed that no stress-based failure criteria have yet been reported for predicting brittle fracture in notched domains under pure mode
III loading. This is why in this research, the point-stress and the mean-stress criteria are formulated for various notch features under pure mode III loading. The main goal is to provide closed-form expressions for mode III notch fracture toughness of different blunt notches. Thereafter, the developed notch fracture toughness expressions are verified by means of a bulk of experimental results reported in Ref. [24] on various notched graphite round bars. It is shown that both the point-stress and mean-stress criteria could predict the experimental results successfully.
2. Elastic stress distribution around the notch tip under torsion
2.1. Blunt V-shaped notches
Figure 1 represents a typical blunt V-notch including its Cartesian and polar coordinate systems. The coordinate
origins are located at the distance r0 from the notch tip on the notch bisector line. As shown in Fig. 1, 2a is the notch opening angle and p is the notch tip radius.
According to a relation that exists between the Cartesian and the curvilinear coordinate systems, r0 can be written as [28]: q -1
q
2( n-a)
(1)
Using the reference frame shown in Fig. 1, in combination with the complex function approach, Zappalorto et al. [29] have developed an analytical solution for the stress field induced by circumferential blunt V-shaped notches in solid bars under torsion. In that problem, it was assumed that the material is isotropic and homogenous, and obeys the theory of linear elastic deformation. According to this solution, the local out-of-plane shear stress distributions close to the notch tip can be estimated by the following expressions [29]:
o_„ =
o>3
\X,-1
vr0y
sin(1 -X 3)0 +
M -X3
r3 V 3 J
sin(1 -|3 )0
(2)
a ^ =
Q3
/ \X 3 -1
r
Kr0j
cos(1 -X 3) 0 +
/ 3 -X3
r
r3 3
cos(1 -|3)0
(3)
where the constant coefficients 13 and o>3 (which depend on the notch opening angle) and also the following rela-
tions for the parameters X 3 Ref. [29]. In addition, t max stress at the notch tip n 1
X3 =
2(n-a) q
= (1 -13) P-
and r3 have been reported in is the maximum elastic shear
(4)
(5)
Using the transformation between the polar and Cartesian coordinate systems shown in Fig. 1, the out-of-plane shear stresses in polar coordinate system can be determined from
Fig. 1. A typical round-tip V-notch and its Cartesian and polar coordinate systems
x
Fig. 2. A typical VO-notch and its Cartesian and polar coordinate systems
ze
cos e sin e - sin e cos e
(6)
Then, the transformed out-of-plane shear stresses are obtained as
/ M -^3
sin (A3e)+
=
K V,p KIII
.—A.-}
G ze =
V^r1
k v,p
k iii
V2nr
1—A,
cos(A 3e) +
r3
v 3 y
f —A3
r
r3 3
sin(^3 e)
(7)
cos(|w3 e)
(8)
where the mode III notch stress intensity factor K^ has been defined as
KV'p = V2n
Tmax r0
l—Ao
(9)
2.2. V-notches with end holes
In Fig. 2, a V-notch with end hole (VO-notch) is schematically depicted together with its polar and Cartesian coordinate systems.
Zappalorto and Lazzarin [30] suggested that when the VO-notch (with a round-tip of radius p) is loaded by an out-of-plane shear load, the shear stress components close to the notch tip can be written in the form:
k vo
g _ = kl\ , sin(A3 e)
V2nr1
—A
l-
V2 A
K VO
g ze = cos( A 3 e)
42nrx
—A
l+
^ 2A 3
where the mode III notch stress intensity factor K been defined as [37]
VO III
KVO = jn Tmax
2 pA 3—1
(10)
(11) has
(12)
In Eq. (12), Tmax is the maximum elastic shear stress which can be obtained by setting r = p and e = 0 in Eq. (11).
3. Transformation of the notch stress field
It is known that dissimilar to mode I and mode II loadings, mode III loading generally results in twisting of the
Fig. 3. Different types of fracture propagation depending on the loading conditions
fracture surface (see Fig. 3). Thus, to make the formulation of the fracture criterion capable of predicting the out-of-plane fracture angle 6f due to mode III, it is assumed that in the initiation stage of fracture, the stress state in e—Z plane is equivalent to an element rotated by the angle 6 around the coordinate axis r (see Fig. 4).
The stress field in mode III loading conditions can be written in the form of the following stress tensor:
0 0 g
2 =
0 0
ez
ez 0
(13)
Using the transformation between the coordinate systems of (r, e, z) and (r\ e', z) shown in Fig. 4, the transformed stress tensor 2' can be calculated from
2'= ^2^T, (14)
where the transformation matrix ^ is expressed as "10 0
(15)
cos 6 sin 6
— sin 6 cos 6
Fig. 4. Stress state in e-Z plane
The transformed stress tensor can then be written as
0 ar0'
Jr 0'
w
0 ' z '
(16)
In the special loading case of pure mode III, the fracture surface does not kink (i.e. 0 = 0 as schematically shown in Fig. 3) and substituting the stresses given in Eqs. (7) and (8) into Eq. (14), the transformed tangential stress g0'0' for blunt V-notches is found as
M -X3
a0'0' (r> è) = -
1
K v,p
KIII
V2n
1 +
sin (2^). (17)
Similarly, the transformed tangential stress a0'0' for V-notches with end holes (VO-notches) can be obtained from Eq. (14) utilizing Eqs. (10) and (11) as follows
(r> è) = -
1 K
VO
iii
V2n "
1-u
1+
/pN2 X3
sin(2è). (18)
4. Brittle fracture criteria
4.1. Point-stress criterion
The point-stress criterion is the extension of the classical maximum tangential stress criterion originally proposed by Erdogan and Sih [31 ] for studying mixed mode I/II brittle fracture in cracked members. The main hypotheses of the point-stress failure criterion are:
1) Fracture initiates from a point on the notch border where the tangential stress becomes a maximum. The crack is propagated along a direction perpendicular to the maximum tangential stress.
2) The onset of fracture occurs when the tangential stress, at a critical radial distance from the origin of the polar coordinate system rcV, and along the direction above reaches the critical stress a c.
Under mode III loading conditions, using the hypotheses of the point-stress criterion and the mathematical theorem of extreme values of two-variable functions [32], the mathematical description of the point-stress criterion can be expressed. The necessary condition for the existence of local maximum for the continuous function is
^ (19)
dè
= 0
and the sufficient condition is
d 2 a.
dè2
< 0.
(20)
è=èf
Therefore, the first hypothesis of the point-stress criterion can be mathematically presented by Eqs. (19) and (20). The second hypothesis can also be written as
Fig. 5. Round-tip V-notch with critical distances associated with the point-stress criterion
a00 (rc,V f = ac- (21)
Note that the critical stress a c is assumed to be a material property and independent of the geometry and loading conditions. This parameter is commonly considered to be the ultimate tensile strength au for brittle and quasi-brittle materials [14].
4.1.1. Blunt V-shaped notches
Figure 5 shows schematically a blunt V-notch with the critical distances associated with the point-stress criterion. It is seen in this figure that the parameters rc and rc,V are the critical distances measured from the notch tip and from the origin of the coordinate system, respectively. Only rc is physically meaningful because it lies on the material. Since rcV is not measured from the notch tip, it is not a constant material property and depends on the notch geometry.
According to Fig. 5, we have
rc,V = r0 + rc- (22)
By substituting Eq. (17) into Eq. (19), the following simplified equation is derived for determining the out-of-plane fracture angle due to pure mode III loading:
cos(2 f = 0. (23)
The root of Eq. (23) that satisfies Eq. (20) is -n/ 4. It is seen that fracture in pure mode III loading conditions initiates along a constant out-of-plane fracture angle. In contrast, it has been previously found that the in-plane fracture angle due to pure mode II loading depends on the notch geometry (notch critical distance, notch opening angle and notch tip radius) [36].
As the second hypothesis of the point-stress criterion for mode III loading, Eq. (21) in combination with Eq. (17) and =-n/ 4, can be written as
1
K v,p
K IIIc
V2n "
1-à 3 rc.V
1 +
/ -X3
rcV
= a.
(24)
The parameter K^P is called the mode III notch fracture toughness of blunt V-notch. As is seen, is not a constant material property and depends on the notch opening angle and the notch tip radius. Equation (24), which is the governing equation for mode III brittle fracture of blunt V-shaped notches, can also be expressed as the following form:
kv,p i—
r 1-^3 rc,V
r- • (25)
1+(W^3
Equation (25) indicates that the determination of critical distance rc V and hence rc (see Eq. (22)) is necessary for predicting the mode III notch fracture toughness of blunt V-notches. In mixed-mode I/II fracture assessment of V-notched components, the following relation has been suggested for the critical distance rc which is assumed to be a material property [30-32]:
r = -
2 n
KT
\2
(26)
In the equation above, KIc is the plane-strain fracture toughness which is a material property. Equation (26) has also been used successfully in Ref. [37] for sharp V-notches under mixed-mode I/II loading. According to Eq. (26), the critical distance rc is a function of the mode I fracture toughness of material. For the loading cases containing mode III (e.g. mixed-mode I/III loading), one can suggest utilizing a relation for rc which is based on the mode III fracture toughness of material.
The tangential stress in its general form for a cracked component subjected to mixed-mode I/II/III loading has been obtained in Ref. [33]. In the case of plane strain, Ge'e' (r, e, 6) can be written as
Ge'e'(r> e> 6)crack =
1
V2nr
2 6 f 0 cos — KTcos—3 KTTsm—
2 2 TT 2
V J
x cos2
-K
TTT cos—sin(2^) +
+ 2v
KTcos-
- KTT sin-
sin
(27)
where KT, KTT
and KTTT are the mode I, II and III stress
intensity factors, respectively. According to the conventional maximum tangential stress criterion for sharp cracks [31, 33], fracture takes place when the tangential stress at the critical distance rc from the crack tip attains the critical stress ac. The fracture conditions under pure mode III loading are expressed as follows:
0f = 0,
(28)
r = r,
K j = 0,
K TT = 0,
KTTT = KTTTc ,
6f =— Ge'e'(r, e 6)crack =Gc .
where KIIIc is the mode III fracture toughness of material.
Fig. 6. VO-notch with critical distances associated with the point-stress criterion
Substituting Eq. (28) into Eq. (27) gives
O =
K
TTTc
2
(29)
Therefore, if Gc is considered to be equal to Gu for brittle and quasi-brittle materials, a new relation for calculating the critical distance rc is obtained from Eq. (29) as
r=
1
2n
K
2
TTTc
(30)
Equation (30) is expected to result in more accurate notch fracture toughness predictions under mode III and mode IIIdominant loading conditions, because it utilizes the mode III fracture toughness of material.
4.1.2. V-notches with end holes
Figure 6 represents schematically a V-notch with end hole (VO-notch) including the critical distances related to the point-stress criterion.
The critical distance rc is measured from the notch tip and is the same with that presented in Eq. (30) for blunt V-notches. The critical distance rc VO for VO-notches is measured from the hole center. The relation between rc and rc,VO for VO-notches is simply
rc,VO =P + rc- (31)
In a similar procedure to blunt V-notches, Eq. (19) together with Eq. (18) leads to the same Eq. (23) for determining the out-of-plane fracture angle associated with mode III loading. Consequently, the fracture of VO-notches subjected to pure mode III loading commences with the constant angle of 6f = — n/4 independent of the notch geometry.
To know about the mode III fracture threshold for VO-notches, substituting Eq. (18) into Eq. (21) gives the following relationship:
1K
VO TTTc
V2n
i-a3
rc,VO
i+
V c,VO J
= a.
(32)
Fig. 7. Round-tip V-notch with critical distances associated with the mean-stress criterion
where K^ is called the mode III notch fracture toughness (NFT) of VO-notches. Equation (32) indicates that the mode III NFT depends on the notch geometry as well as the material.
One can rewrite Eq. (32) in the following convenient form:
K
VO IIIc
a„
:V2n
1-X3 rc,VO
1 + (p/ rc,VO)2X3
(33)
4.2. Mean-stress criterion
In accordance with the mean-stress criterion, fracture occurs when the average value of the tangential stress over a specified critical distance from the notch border reaches the critical stress. Similar to the point-stress criterion, the mean-stress criterion can also be formulated based on the tangential stress distribution around a notch. The main difference in the mathematical procedures of the point-stress and the mean-stress criteria is that in the mean-stress model, the mean-stress distribution over a critical distance is used in the formulations instead of the tangential stress at a specific distance (point) from the notch border. Therefore, the mean value of the tangential stress over a specified critical distance should first be determined.
4.2.1. Blunt V-shaped notches
Figure 7 depicts a schematic blunt V-notch with critical distances associated with the mean-stress criterion. In this figure, dc and dc V are the critical distances measured from the notch tip and from the origin of coordinate system, respectively. Figure 7 implies that
d c,V = r0 + dc. (34)
For blunt V-notches, the mean stress over the critical distance can be written as
1 d c,V
— J a0'0' dr • d
(35)
In the case of mode III loading, substituting Eq. (17) into Eq. (35) and integrating gives
A 4>) = -
X3
K Vfp sin (2^) x dcV2rc
+ (r/r,)13 -X A 13
r =0 +d c
(36)
The mean-stress criterion formulation is completely the same as that of the point-stress criterion except that a0/0/ is replaced with a0'0>. Thus, the approach presented in Sub-sect. 4.1 can be repeated for a0'0' as follows:
9a0
- = 0 ^ ^ = provided that
32;
< 0- (37)
The angle is the out-of-plane fracture angle based on the mean-stress criterion. Substituting Eq. (36) into Eq. (37) gives
cos(2 f = 0. (38)
Accordingly, the mean-stress criterion predicts the mode III out-of-plane fracture angle identical to that found using the point-stress criterion (^f = -n/ 4).
The mean-stress criterion supposes that a0/0/ reaches the critical stress ac at brittle fracture instance. Thus, considering the fracture initiation angle of -n/4 we have
K V,P KIII
dcV2rc
_L + (r/r») |3~
X3 |3
r=r0+d c
= a„
(39)
To provide a relationship for the mode III NFT of blunt V-notches, one can express Eq. (39) as follows:
K
V,p III
a
_1 + (rl^f3 -
X3 13
r=r0 + d c
(40)
Equation (40) requires that the critical distance dc is known for predicting the mode III NFT. In some previous researches [13], it has been proposed and verified that the critical distance dc can be considered to be equal to the distance used in the past by Seweryn for sharp cracks as follows [34]:
d c =■
K
(41)
Similar to the critical distance of the point-stress criterion rc , proposing a relation for dc which makes use of the mode III fracture toughness of material might be more helpful and accurate for mode III and mode III-dominant notch fracture assessments. For this purpose, the mean-stress criterion is implemented herein for a sharp crack problem. Using Eq. (27), the mean stress over the critical distance dc (Eq. (35)) for a cracked body subjected to mixed-mode I/II/III loading can be written as
Fig. 8. VO-notch with critical distances associated with the mean-stress criterion
1
(ae'e' )crack ~~r\ (ae'e')crack dr = dc 0
cos
e
e ~ . e
KTcos—3 KTT sin — 1 2 11 2
2 e
xcos KTTTcos—sin(2+
+ 2v
r e . e
KT cos—KTT sin—
, T 2 TT 2
v y
sin
(42)
The mean-stress criterion proposes that fracture initiates when the mean stress over the critical distance attains the critical stress ac. Therefore, in pure mode III loading conditions expressed by Eq. (28), one can write
ac = KnicVVW (43)
Accordingly, by taking a c to be equal to au for brittle and quasi-brittle materials, a new suggestion for dc in the loading cases including mode III is obtained as
K
\2
TTTc
(44)
4.2.2. V-notches with end holes
A typical VO-notch together with the critical distances of the mean-stress criterion is illustrated in Fig. 8. The critical distances dc and dcVO are measured from the notch tip and from the hole center, respectively.
Similar to the point-stress criterion, dc is also considered to be equal to the critical distance of sharp cracks (Eq. (44) for mode III and mode III-dominant loading conditions). From Fig. 8, it is clear that
dc,vo =p + dc. (45)
For VO-notches, the mean stress over the critical distance is written as
_ 1 d c,VO
a00 = — f a00dr. (46)
= T Ï ae
d c p
Table 1
The main properties of the tested graphite material [24]
Material property
Elastic modulus E, MPa
Shear modulus G, MPa
Poisson's ratio v
Ultimate tensile strength, MPa
Ultimate torsion strength, MPa
Density, kg/dm3
Mean grain size, ^m
Porosity,
Value
8050
3354
0.2
28.5
30
1.85
Substituting Eq. (18) into Eq. (46) and integrating gives
-r=p+dc
=
k VO KTTT
sin (2^)
-1 +
V 2X 3
. (47)
r=p
Equation (47) in combination with the mean-stress criterion hypothesis of Eq. (37) results in the previously obtained Eq. (38). Consequently, based on the mean-stress criterion, the fracture initiation angle of VO-notches under pure mode III loading is the same as that of point-stress criterion.
Similar to the mean-stress criterion formulation for blunt V-notches, utilizing Eq. (47) and considering = -n/ 4, the governing equation for mode III fracture of VO-notches is obtained as
VO KTTTc
dcV2rcÀ3
1
/ p ^2À3
r=p+dc
= G,.
(48)
r=p
Therefore, the mode III NFT of VO-notches can be achieved in a convenient form as
dc V2rcX3
VO K TTTc
[ r À 3(1 - ( p/ r )2X 3) ]
ir=p+dc r=p
(49)
5. Experimental results reported in the literature
Berto et al. [24] conducted several fracture tests on graphite round bars weakened by U-notch, blunt V-notch
Fig. 9. Geometry of notched specimens used in fracture tests (dimensions in mm) [24]
n
Table 2
Geometrical parameters and the theoretical stress concentration factors of all notched models used in the tests [24]
Table 3
Fracture torques obtained for the specimens with V-notches [24]
Notch opening angle 2a Notch depth p, mm Notch radius p, mm Torsion Kt
V-notch 120° 5.0 0.1 2.43
0.3 1.93
0.5 1.72
1.0 1.48
2.0 1.30
2.0 0.1 2.76
0.3 2.13
0.5 1.89
1.0 1.62
30° 5.0 0.1 3.57
0.3 2.32
0.5 1.94
2.0 0.1 4.00
0.3 2.58
0.5 2.14
U-notch 0° 5.0 1.0 1.57
2.0 1.33
2.0 1.0 1.72
Semicircular notch - 0.5 0.5 1.79
1.0 1.0 1.64
2.0 2.0 1.44
4.0 4.0 1.21
and semicircular notch under torsion loading. A summary of the test material, the test specimens and the fracture experiments are presented in the next subsections.
5.1. Material
A commercial isostatic polycrystalline graphite was utilized in Ref. [24] to fabricate the notched round bars for the fracture tests. The basic material properties of the tested material are listed in Table 1.
5.2. Test specimens
Different round bar specimens were used in Ref. [24] for torsion tests: cylindrical specimens with U- and V-notches (Fig. 9, a) and cylindrical specimens with circumferential semicircular notches (Fig. 9, b).
For U-notched specimens, notches with two different notch root radii were tested: p = 1 and 2 mm. The effect of
Notch depth Notch radius Torque M t, N • mm
p, mm p , mm 2a = 120° 2a = 30°
6692 6592
0.1 6778 6808
6629 6936
6612 6230
0.3 6860 6995
6426 6643
5 6895 6474
0.5 6495 6719
6709 6605
6829
1.0 7064
6771
7198
2.0 7401
7408
25 221 24 688
0.1 25 628 23 469
26 027 24 054
25 535 23 076
0.3 24 509 22 408
2 25 053 23 293
25 135 22 749
0.5 23 930 23 145
24 764 24 860
24 746
1.0 23 445
26 399
the net section area was studied by changing the notch depth p. Two values were used, p = 2 and 5 mm, while keeping the gross diameter constant (20 mm).
For the V-notched specimens with a notch opening angle 2a = 30° (Fig. 9, a), three different notch root radii were used in the experiments: p = 0.1, 0.3 and 0.5 mm. Moreover, a larger opening angle (2a = 120°) was also considered, combined with five notch root radii, p = 0.1, 0.3, 0.5, 1.0 and 2.0 mm. With a constant gross diameter (20 mm), the net section area was varied in each specimen by changing the notch depth, p = 2 and 5 mm.
Notch depth p, mm Notch radius p, mm Torque M t, N • mm
U-notch 5.0 1.0 6148
6599
6801
2.0 6812
6995
6674
2.0 1.0 24578
22994
23200
Semicircular notch 0.5 0.5 45956
45012
43 504
1.0 1.0 33 590
36923
37533
2.0 2.0 25073
25087
28231
4.0 4.0 13317
12029
12511
Table 4
Semicircular notch
2.0
0.5
1.0
2.0
4.0
Notch radius p, mm
1.0
2.0
1.0
0.5
1.0
2.0
4.0
Torque M t, N • mm
6148
6599
6801
6812
6995
6674
24578
22994
23200
45956
45012
43 504
33 590
36923
37533
25073
25087
28231
13317
12029
12511
For semi-circular notches (Fig. 9, b), notches with four different notch root radii were tested: p = 0.5, 1.0, 2.0 and 4.0.
Geometric characteristics of the intended test specimens are summarized in Table 2. Also, included in Table 2 are
the theoretical stress concentration factors of each specimen under torsion loading [24].
With reference to the mechanics of materials, the nominal shear stress for each test sample is calculated from
16Mt
t-• <so)
where d, p and Mt are the diameter of the round bar specimen, the notch depth and the fracture torque, respectively.
Berto et al. [24] prepared at least three samples for each of the 24 specimen geometries described above. A total number of 80 torsion tests were performed under rotation control conditions with a loading rate of 1°/min.
As reported in [24], the torque-angle curves recorded during the torsion tests always exhibited an approximately linear trend up to the final failure, which occurred suddenly. Therefore, the use of fracture criteria based on a linear elastic material behavior is realistic. Therefore, it is acceptable to use the experimental results [24] for evaluating the theoretical results of the point-stress and the mean-stress criteria developed in Sect. 4 based on the linear elastic notch fracture mechanics. All torsion loads to failure Mt are reported in Tables 3 and 4 for various notch configurations.
6. Results and discussion
In this section, the theoretical results of the point-stress and the mean-stress criteria in predicting the mode III NFT of blunt V-notches and VO-notches are compared with the experimental results provided in Sect. 5. In order to make this comparison, the recorded fracture loads of the notched graphite specimens (listed in Tables 3 and 4) should be converted to the corresponding values of the mode III NFT. For this purpose, one should employ Eqs. (9) and (12) for blunt V-notches and VO-notches, respectively. In these equations, Tmax is the maximum elastic shear stress at the notch tip, as previously defined. Considering Eqs. (8) and (9) at the notch tip, Tmax for blunt V-notches is determined from the following relation:
Fig. 10. The variations of the theoretical normalized mode III NFT versus the notch tip radius together with the experimental results for V-notch with 2a = 120°
Fig. 11. The variations of the theoretical normalized mode III NFT versus the notch tip radius together with the experimental results for V-notch with 2a = 30°
Fig. 12. The variations of the theoretical normalized mode III NFT versus the notch tip radius together with the experimental results for U-notch
Fig. 13. The variations of the theoretical normalized mode III NFT versus the notch tip radius together with the experimental results for semicircular notch
_0>3(a z0 )r=r0,0=0
(51)
maX"1 + ( „) * For VO-notches, Eqs. (11) and (12) at the notch tip result
in Tmax _ (az0)r_p,0_0.
To compute the stress component a z0 at the notch tip for each notched graphite sample, the nominal shear stress and the stress concentration factor are required which are available from Eq. (50) and in Table 2, respectively.
In view of Eqs. (25), (33), (40) and (49), calculating the theoretical values of the mode III NFT for the point-stress and mean-stress criteria needs the critical distance related to each criterion. Therefore, with reference to Eqs. (30) and (44) for mode III loading conditions, the value of the parameter KIIIc for the tested graphite material is required. However, the mode III fracture toughness of the tested graphite material has not been reported in Ref. [24]. In the absence of the needed data from cracked components, the parameter KIIIc can be estimated considering the test results of two blunt V-notches with the minimum available radius, p = 0.1 mm, and 2a = 30°. For each notch depths ofp = 2 and 5 mm, Tmax is first calculated using the related test data for the mentioned notch geometry (presented in Table 3) from Eq. (51) and then, the parameter K^ is determined by using Eq. (9). By means of the average value of the two obtained notch stress intensity factors, solving Eq. (24) for rc V and then using Eq. (30) results in an estimation of KIIIc for the graphite material based on the point-stress criterion. In a similar way, solving Eq. (39) for dc and thereafter utilizing Eq. (44), another estimation of KIIIc for the graphite material will be available which is based on the mean-stress criterion. Now, the mean value of the two KIIIc estimated on the basis of the experimental data can be used for calculating the critical distances needed for the point-stress and mean-stress notch fracture toughness predictions. The mode III fracture toughness obtained for the tested graphite through the described procedure is KIIIc = 1.9 MPa • m1/2.
Figures 10 and 11 display the variations of the theoretical values of the normalized mode III NFT provided by the point-stress and mean-stress criteria versus the notch tip radius for the blunt V-notched graphite specimens with the notch opening angles of 120° and 30respectively. Also shown in these figures are the experimental values of the normalized mode III NFT calculated from the data provided in Table 3.
In Figs. 12 and 13, similar curves are presented for the U-notches and semicircular notches, respectively, together with the corresponding experimental results reported in Table 4. Note that the theoretical predictions of the point-stress and mean-stress criteria for U-notches are provided using the blunt V-notch formulations in the particular case of 2a = 0°. Moreover, the mode III NFTs of the graphite specimens weakened by semicircular notches are predicted by means of the point-stress and mean-stress criteria for VO-notches having 2a = 180°. All of the theoretical values of the normalized notch fracture toughness together with the average experimental results are also presented in Table 5 including the relative discrepancies.
It is clear in Fig. 10 that the theoretical mode III NFTs of the point-stress and mean-stress criteria for V-notches with 2a = 120° enhance gradually, as the notch radius increases. Furthermore, this behavior is also seen generally in the experimental results. The growth of mode III NFT is more considerable for V-notches with 2a = 30° and for U-notches (see Figs. 11 and 12). This trend can be due to a decrease in the stress concentration near the notch tip and therefore an enhancement in its load-carrying capacity. On the other hand, Fig. 13 represents a different trend for semicircular notches. As observed in Fig. 13, both the curves related to the point-stress and mean-stress criteria as well as the test results indicate that by increasing the notch root radius, the mode III NFT decreases. This is because for semicircular notches, the notch depth is identical to the notch tip radius and hence, as the notch tip radius increases, the specimen net section area decreases, which affects signifi-
Table 5
The theoretical values of the normalized mode III NFT together with the mean values of the experimental results including the mean discrepancies
Notch opening angle 2a Notch radius p, mm Notch depth p, mm Mean experimental value Point-stress criterion Mean-stress criterion
Value Discrepancy, % Value Discrepancy, %
30° (V-notch) 0.1 5.0 0.0571 0.0536 6.0 0.0589 3.2
0.3 0.0597 0.0541 9.3 0.0616 3.2
0.5 0.0627 0.0557 11.2 0.0641 2.1
0.1 2.0 0.0554 0.0536 3.2 0.0589 6.3
0.3 0.0561 0.0541 3.5 0.0616 9.8
0.5 0.0604 0.0557 7.7 0.0641 6.1
Avg. 6.8 Avg. 5.1
120° (V-notch) 0.1 5.0 0.2536 0.2369 6.6 0.2559 0.9
0.3 0.2625 0.2304 12.2 0.2518 4.1
0.5 0.2685 0.2293 14.6 0.2519 6.2
1.0 0.2824 0.2315 18.0 0.2550 9.7
2.0 0.3142 0.2400 23.6 0.2632 16.2
0.1 2.0 0.2690 0.2369 11.9 0.2559 4.9
0.3 0.2669 0.2304 13.7 0.2518 6.4
0.5 0.2646 0.2293 13.3 0.2519 6.4
1.0 0.2724 0.2315 15.0 0.2550 5.2
Avg. 14.3 Avg. 6.7
0° (U-notch) 1.0 5.0 0.0508 0.0450 11.5 0.0523 2.8
2.0 0.0638 0.0527 17.5 0.0601 5.8
1.0 2.0 0.0492 0.0450 8.6 0.0523 6.2
Avg. 12.5 Avg. 4.9
180° (semicircular notch) 0.5 0.5 2.6199 2.1404 18.3 2.1799 16.8
1.0 1.0 2.2683 1.8672 17.7 1.9882 12.3
2.0 2.0 2.0575 1.6223 21.1 1.7732 13.8
4.0 4.0 1.9790 1.4560 26.4 1.5813 20.1
Avg. 20.9 Avg. 15.8
Total Avg. 13.7 Total Avg. 8.1
cantly the load-carrying capacity of the notched round bar. Another visible point in Figs. 10 to 13 is that the notch fracture toughness curves of the mean-stress criterion locate always beyond those of the point-stress criterion meaning that the point-stress criterion generally provides more conservative estimates to the experimental results.
Table 5 reveals that the mean-stress criterion gives generally better predictions to the experimental results than the point-stress criterion. As presented in Table 5, the maximum values of the mean discrepancies between the experimental and theoretical results are achieved for the semicircular notch for both the point-stress and mean-stress crite-
ria. In addition, the minimum average discrepancies between the experimental and theoretical results are found for the U-notches for the mean-stress criterion. It is also seen from the data associated with 2a = 30° and 120° in Table 5 that by increasing the notch opening angle, the accuracy of both the point-stress and mean-stress criteria decreases.
A review of Figs. 10 to 13 indicates that the deviation between the curves of point-stress and mean-stress criteria and the experimental results increases as the notch tip radius enhances. The total average discrepancies of the point-stress and mean-stress criteria are obtained as 13% and 8%,
respectively, demonstrating that both criteria are accurate enough to be used in predicting the mode III NFT for blunt V-notches, U-notches and semicircular notches.
7. Conclusions
The main conclusions of the present study are summarized as follows:
Two stress-based failure criteria, namely the point-stress and the mean-stress, were formulated for predicting mode III brittle fracture in VO-notched and blunt V-notched components.
The mode III notch fracture toughness predicted by the two criteria were verified by using the experimental results reported in the literature on isostatic graphite round bars weakened by blunt V-notches and U-notches as well as those weakened by semicircular notches. The verification was performed for notches of different depths, opening angles and tip radii.
Good agreement was found to exist between the results of the point-stress and mean-stress criteria and the experimental results under mode III loading conditions for various notch geometries.
Under mode III loading, the mean-stress criterion was found to be generally more accurate than the point-stress criterion in predicting brittle fracture of U-, V-, and semicircular notches.
The point-stress criterion was found to be more conservative than the mean-stress criterion in predicting brittle fracture of U-, V-, and semicircular notches.
Generally, by increasing the notch tip radius, the accuracy of the point-stress and mean-stress criteria decreases, particularly for V-notches with larger notch opening angles.
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Поступила в редакцию 18.11.2016 г.
CeedenuM oö aemopax
Ali Reza Torabi, PhD, Assist. Prof., University of Tehran, Iran, [email protected]
Behnam Saboori, PhD, Iran University of Science and Technology, [email protected]
Majid R. Ayatollahi, PhD, Prof., Director, Iran University of Science and Technology, [email protected]
