CC BY
110
УДК 539.42
Fracture assessment of inclined double keyhole notches in isostatic graphite
H. Salavati1, Y. Alizadeh2, M.R. Ayatollahi3
1 Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, 76169-14111, Iran
2 Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, 15875-4413, Iran 3 Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, 16846-13114, Iran
In the present contribution, the static strength of isostatic graphite using keyhole notch specimens under mixed mode loading is investigated. An experimental program was performed and in total, 18 new experimental data are provided. In addition, different loading mode ratios are considered by varying the inclination angle of the notch with respect to the direction of the applied load. The criterion based on the averaged value of the strain energy density over a control volume at the notch edge is applied to assess the static strength of specimens. A sound agreement is found between experimental data and the results obtained from strain energy density criterion.
Keywords: brittle failure, keyhole notch, loading mode ratio, graphite
DOI 10.24411/1683-805X-2018-11003
Трещиностойкость образцов изостатического графита с наклонным двойным ключевидным вырезом
H. Salavati1, Y. Alizadeh2, M.R. Ayatollahi3
1 Керманский университет им. Шахида Бахонара, Керман, 76169-14111, Иран 2 Технологический университет им. Амира Кабира, Тегеран, 15875-4413, Иран 3 Иранский университет науки и технологии, Тегеран, 16846-13114, Иран
В статье экспериментально исследована статическая прочность образцов изостатического графита с ключевидным разрезом при нагружении смешанного типа. Рассмотрены различные режимы нагружения при изменении угла наклона разреза относительно направления приложенной нагрузки. Для оценки статической прочности образцов использован критерий средней плотности энергии деформации в контрольном объеме материала у вершины разреза. Экспериментальные данные хорошо согласуются с результатами, полученными с помощью критерия плотности энергии деформации.
Ключевые слова: хрупкое разрушение, ключевидный вырез, режим нагружения, графит
Nomenclature
v—Poisson's ratio, E—Young's modulus, X—loading mode ratio,
CTee and tree —stresses at distance r from the local frame origin,
CTut —ultimate tensile strength ,
Kj p and Kjj p —notch stress intensity factor,
R —control radius,
Kjc —fracture toughness,
Fcr —critical fracture load,
F —applied load,
SED—strain energy density, Wcr —critical SED,
Wap —averaged SED over the control volume related.
1. Introduction
This paper further develops the fracture assesment approach of notched components based on the strain energy density criterion proposed earlier [1]. In that paper, a comprehensive insight into the problem is offered, with detailed discussion of the literature. The SED approach is based on the evaluation of the averaged strain energy density over a control volume. The criterion was applied to assess the frac-
© Salavati H., Alizadeh Y., Ayatollahi M.R., 2018
Table 1
Mechanical properties
Material property Value
Elastic modulus E, MPa 8050
Shear modulus G, MPa 3354
Poisson's ratio v 0.2
Ultimate torsion strength, MPa 37
Ultimate compression strength, MPa 110
Ultimate tensile strength, MPa 30
Fracture toughness, MPa m05 1.06
Density, kg/m3 1.85
Porosity, % 7
Resistivity, pOhm m 11
Thermal conductivity, ^/(mK) 110
ture behaviour of different materials under mode I, mixed mode and torsion loading [2-5]. In this paper, a new set of experimental data was provided from notched specimens made of isostatic polycrystalline graphite with different values of notch opening angle and root radius, which should be useful to engineers engaged with static strength analysis of graphite components.
Brittle fracture in isostatic graphite is studied experimentally and numerically using keyhole notched samples under mixed mode loading (I+II) considering different combinations of the notch radius and the inclination angle of the notch. An experimental programme was performed to provide a new set of results. In total 18 new data are provided in the paper. Moreover, the SED criterion is applied to assess the critical fracture load and summarizes all the data in a single scatter band independent of the notch geometry and loading mode ratio.
2. Fracture experiments
2.1. Materials and geometry
In the present research, a commercial isostatic graphite is used due to its mechanical applications and its high performances. Table 1 summarises the main material properties. The experimental specimens are weakened by an eccentric double keyhole notch, as shown in Fig. 1. For all the specimens, the width, the thickness and the distance between the notches tips were 50, 10 and 10 mm, respectively. The effect of notch tip radius p and notch inclination angle P on the brittle fracture of specimens under tensile load was investigated. By varying the inclination angle P, different loading mode ratios can be produced. Three values of the notch radius (p = 0.5, 1.0, 2.0 mm) and two values of the angle P (P = 60° and 70°) were considered for the test specimens. Figure 2 shows some photos of the specimens and some details of the notches.
2.2. Experimental procedure
To prepare the test specimens (Fig. 2), some plates of 10 mm thick were obtained from a graphite block. The thick-
| a
/ / p
/ / J
1 11
W= 50 mm Fig. 1. The specimen geometry
b
Fig. 2. Different eccentric double keyhole notched specimens: P = 60° (a) and 75° (b), p = 0.5, 1.0, 2.0 mm
Fig. 3. Load-displacement curves for keyhole notched graphite specimens, p = 2 mm, P = 60°
ness was selected with the aim to get the plane strain conditions at the notch tip. Then, the specimens were accurately fabricated by using CNC water-jet machine. Before conducting the experiments, the cut surfaces of the graphite specimens were polished by using a fine abrasive paper to remove any possible local stress concentrations due to surface roughness. The fracture tests were performed by a universal tension-compression test machine under displacement control with constant displacement-rate of 0.5 mm/min. In total, 18 mixed mode fracture tests were performed. The load-displacement curves were recorded and used to obtain the critical fracture load. Figure 3 shows load-displacement curves for the case p = 2 mm, P = 60°. Moreover, Fig. 4 shows some broken specimens.
The values of the fracture loads and the corresponding mean value F are presented in Table 2. As evident in Table 2, the fracture load increases with increasing the notch radius.
The values of the crack initiation angles have been measured by using an optical microscope and a LAS software (Leica Application Suite). Figure 5 shows an example of the measurement. All the values of the initial crack angles with the corresponding mean values 9 are reported in Table 2.
Fig. 4. Some broken specimens
3. Evaluation of the loading mode ratio parameter
In order to quantify the loading mode ratio in the simulated specimens, some FE analyses were carried out based on the approach developed in Ref. [6]. For convenience, some formulas from Ref. [6] are given below. The generalized notch stress intensity factors (N-SIF) for mode I and mode II can be expressed as follows: 2^2nr ( g00 ) e=o
K
i,p " {2 +1.25p/r + 1.5(p/r)2 + 1.25(p/r)3}' K (Tr 0 )0=o x
x{1 +1.625p/r - 0.75(p/r)2 -1.875(p/r)3}-1,
(1)
(2)
where g00 and Tr0 are the stresses at a distance r from the local frame origin. Equations (1) and (2) are not expected to give a constant value for N-SIFs but slight variations are possible. In order to eliminate the weak dependence on the notch tip distance, the following expressions have been defined to calculate the mean values of the generalized NSIFs [7]:
_ 1 ro+np
K:,p=— I (Kj,p)dr, (3)
np r
_ 1 ro+np Kii,p = I (Kii,p)dr, p np r
(4)
Table 2
Experimental critical loads and initial crack angles
ß p, mm F1, N F2, N F3, N F N 91 92 93 9av
0.5 3846 3604 3890 3780 79.7o 83.70 74.20 79.20
75o 1.0 4696 4218 4076 4330 76.5o 74.7o 78.90 76.70
2.0 4465 4583 4032 4360 77.6o 70o 74.20 73.90
0.5 3546 3649 3684 3626 59.40 62.2o 57.7o 59.80
60o 1.0 3973 3901 3945 3940 64.00 65.1o 69.50 66.20
2.0 4452 4720 4631 4601 64.1o 66.2o 55.80 62.00
Fig. 5. Experimental measure of the initial crack angle by using the LAS software
n is between 0.2 to 0.3 [7] and set equal to 0.25 in the present paper. Moreover, r0 is equal to p/2, where p is the notch radius.
The loading mode ratio has been evaluated according to the following definition:
X = — arctan n
K
2,P
K
1, p
(5)
The value of % is equal to 0 under pure mode I loading and 1 under pure mode II. The values of K2 pjK1p as well as the loading mode ratio % of the simulated specimens are listed in Table 3. The trend of % for two constant values of the inclination angle (P = 60° and 75°) and for different notch radius is shown in Fig. 6.
4. Application of the SED criterion
The averaged strain energy density (SED) criterion as reported in [8] states that brittle or quasi-brittle failure occurs when the SED averaged over a control volume is equal
Fig. 6. Loading mode ratio % as a function of the notch radius p and the inclination angle P: 60° (1), 75°(2)
Table 3
Loading mode ratio of the investigated geometries
ß
75°
60°
p, mm
0.5
1.0
2.0
0.5
1.0
2.0
K 2,р/K
1,р
2.3
2.9
4.2
0.8
0.9
1.1
X
0.74
0.79
0.86
0.43
0.46
0.52
to the critical SED for the unnotched material, Wc. The SED approach is based both on a precise definition of the control volume and on the fact that the critical energy does not depend on the notch sharpness. Such a method was applied first to sharp (zero radius) V-notches and later extended to blunt U- and V-notches under mode I loading
[9].
The control volume in keyhole notched specimens under mode I loading conditions is centered in relation to the notch bisector line (Fig. 7, a). Under mixed mode loading, the critical volume is no longer centered on the notch tip, but rather on the point where the principal stress reaches its maximum value along the edge of the notch (Fig. 7, b)
[10].
The expression of the control radius Rc as a function of fracture toughness KIc, ultimate tensile strength aut, and Poisson's ratio v of the material is as follows [11]:
R
(1 + v)(5 -8 v)
4n
K
2
(6)
Fig. 7. Control volume in keyhole notched specimens mode I (a) and mixed mode (b)
a
b
s
s
7 6
5
,4-1
5
3 2 1
7
6 5
,4-1
3
3 2 1
\a Keyhole notch ß = 75°
t t t
1---- Wc = 0.056 MJ/mm3
^c = 0.41 mm
0.5 1.0 1.5 2.0 Notch radius, mm
b Keyhole notch ß = 60°
1 f
s— Wc = 0.056 MJ/mm3 Rc = 0.41 mm
0.5 1.0 1.5 2.0 Notch radius, mm
Fig. 8. Maximum principal stress and strain energy density contour lines for the case ß = 75° and p = 2 mm
Fig. 9. Comparison between experimental and predicted values of the critical load for the inclination angle of the notch P = 75° (a) and 60° (b), and different notch radius. Solid line is the fracture assessment based on strain energy density
The critical value of the SED can be determined as follows:
W =^
(7)
2E
In Eq. (7), Gut is the ultimate tensile strength of the material and E is the Young's modulus. In the present work, the values of Rc and Wc are equal to 0.406 mm and 0.0559 MJ/m3, respectively.
In the present work, for each geometry two models were created by FE code ABAQUS 6.13. The first model was
applied in order to determine the point where the maximum principal stress was located; the second one was used to obtain the averaged strain energy density over the well-defined control volume. All the analyses have been carried out under plane strain conditions and linear-elastic hypotheses. Figure 8 shows the maximum principal stress and SED contour lines for the configuration with p = 2 mm and P = 75°.
The values of the crack initiation angle 9 are summarized in Table 4 which shows a good with the FEM results.
Table 4
Comparison between numerical and experimental data
ß p, mm F N ± exp' fFEM, n FFEm/ Fexp 9exp 9FEM 9fEM/ 9exp
0.5 3780 3491 0.92 79.2° 78.3° 0.99
75° 1.0 4330 3672 0.85 76.7° 73.2° 0.95
2.0 4360 3926 0.90 73.9° 72.6° 0.98
0.5 3626 3647 1.01 59.8° 61.1° 1.02
60° 1.0 3940 3773 0.96 66.2° 62.7° 0.95
2.0 4601 3846 0.84 62.0° 62.5° 1.01
Fig. 10. Scatter band summarizing new data from keyhole notches
1.6 1.4 ^ 1.2
fe 1.0 w 0.8
0.4
O Present work
• Ref. [1]
s ®
¡S ! 1
h i I !
• o •
0 12 3 4
Notch radius, mm
Fig. 11. Scatter band in terms of strain energy density summarizing the new data and the data given in [1]
The critical fracture load could be evaluated by determining the mean value of SED over a control volume using the following expression:
F IW ap _ ap
F VW
cr V cr
where Fap is the applied load,
(8)
W„
, ap is the averaged SED o
fracture load, and Wcr is the critical SED. The values of
over the control volume related to Fap, Fcr is the critical
Fig. 12. Triangular (a, b) and quadrilateral (c, d) elements: fine (a, c) and coarse mesh (b, d)
the critical fracture load are summarized in Table 4 which shows a good agreement with the experimental data.
Figure 9 plot the experimental results and the theoretical predictions based on the SED approach as a function of the notch root radius for each value of the inclination angle P. However it is important to note that for all the cases taken into account, the containing the theoretical and experimental values is very narrow as shown in Fig. 10. Moreover, the average deviation between the theoretical and the experimental values of the critical fracture loads
Table 5
Effect of different kinds of elements on the averaged SED, MJ/m3
Triangular element Quadrilateral element
Fine mesh Coarse mesh Fine mesh Coarse mesh
Linear shape function Quad shape function Linear shape function Quad shape function Linear shape function Quad shape function Linear shape function Quad shape function
0.0046 0.0046 0.0047 0.0047 0.0045 0.0045 0.0047 0.0046
0.0041 0.0041 0.0042 0.0043 0.0041 0.0042 0.0043 0.0042
0.0036 0.0036 0.0036 0.0036 0.0036 0.0036 0.0036 0.0036
0.0042 0.0043 0.0042 0.0042 0.0042 0.0042 0.0042 0.0042
0.0039 0.0040 0.0037 0.0037 0.0039 0.0039 0.0037 0.0038
0.0038 0.0038 0.0035 0.0035 0.0038 0.0038 0.0035 0.0037
was found about 9%. Together with the new data, the normalised scatter band shown in Fig. 11 includes also previous data from keyhole notches with a different type of graphite [1] showing that the SED allows a synthesis that is completely independent of the notch shape and its sharpness.
5. Effect of mesh size and element type on the SED
The averaged SED over the control volume can be determined by coarse mesh as well as fine mesh [12]. This paper shows that size of the element (coarse/fine) besides the type of element (triangular/quadrilateral) and the type of shape function (linear/quadratic) do not have perceptible effect on the mean value of the SED. This result is of interest in the practical application of the SED approach to industrial components. Figure 12 shows the fine and coarse mesh for the triangular and quadrilateral elements, respectively. The effect of the element size, element type and the type of shape function on the average SED and the corresponding critical fracture load are well highlighted in Table 5. It can be found that the mean value of SED is insensitive to the element type and mesh density.
6. Conclusion
In the present work, the SED averaged over a well-defined control volume ahead at the notch edge was used to obtain the critical fracture load of keyhole notched specimens made of isostatic graphite under mixed mode loading.
The main findings of the present work can be summarized as follows:
1. The average deviation between the theoretical and the experimental values in terms of the critical fracture loads was found to be limited (9%). This shows the accuracy of the model.
2. The mean value of SED can be determined by using any size (fine/coarse) and shape of elements (triangular/ quadrilateral) as well as any shape function (linear/quadratic) for fracture assessments of notched graphite.
References
1. Lazzarin P., Berto F., Ayatollahi M.R. Brittle failure of inclined keyhole notches in isostatic graphite under in-plane mixed mode loading // Fatig. Fract. Eng. Mater. Struct. - 2013. - V. 36. - No. 9. - P. 942955. - doi 10.1111/ffe.12057.
2. Lazzarin P., Campagnolo A., Berto F. A comparison among some recent energy- and stress-based criteria for the fracture assessment of sharp V-notched components under mode I loading // Theor. Appl. Fract. Mech. - 2014. - V. 71. - P. 21-30. - doi 10.1016/j.tafmec. 2014.03.001.
3. Berto F., Elices M., Lazzarin P., Zappalorto M. Fracture behaviour of notched round bars made of PMMA subjected to torsion at room temperature // Eng. Fract. Mech. - 2012. - V. 90. - P. 143-160. - doi 10.1016/j.engfracmech.2012.05.001.
4. Berto F., Cendon D.A., Lazzarin P., Elices M. Fracture behaviour of notched round bars made of PMMA subjected to torsion at -60°C // Eng. Fract. Mech. - 2013. - V. 102. - P. 271-287. - doi 10.1016/ j.engfracmech.2013.02.011.
5. Berto F., Barati E. Fracture assessment of U-notches under three point bending by means of local energy density // Mater. Design. - 2011. -V. 32. - No. 2. - P. 822-830. - doi 10.1016/j.matdes.2010.07.017.
6. Zappalorto M., Lazzarin P. In-plane and out-of-plane stress field solutions for V-notches with end holes // Int. J. Fract. - 2011. - V. 168. -No. 2. - P. 167-180. - doi 10.1007/s 10704-010-9567-5.
7. Lazzarin P., Filippi S. A generalized stress intensity factor to be applied to rounded V-shaped notches // Int. J. Solids Struct. - 2006. -V. 43. - No. 9. - P. 2461-2478. - doi 10.1016/j.ijsolstr.2005.03.007.
8. Lazzarin P., Zambardi R. A finite-volume-energy based approach to predict the static and fatigue behavior of components with sharp V-shaped notches // Int. J. Fract. - 2001. - V. 112. - No. 3. - P. 275298.- doi 10.1023/a:1013595930617.
9. Lazzarin P., Berto F. Some expressions for the strain energy in a finite volume surrounding the root of blunt V-notches // Int. J. Fract. - 2005. -V. 135. - No. 1. - P. 161-185. - doi 10.1007/s10704-005-3943-6.
10. Berto F., Lazzarin P., Gomez F.J., Elices M. Fracture assessment of U-notches under mixed mode loading: two procedures based on the 'equivalent local mode I' concept // Int. J. Fract. - 2007. - V. 148. -No. 4. - P. 415-433. - doi 10.1007/s10704-008-9213-7.
11. Yosibash Z., Bussiba A., Gilad I. Failure criteria for brittle elastic materials // Int. J. Fract. - 2004. - V. 125. - No. 3-4. - P. 307-333. -doi 10.1023/B:FRAC.0000022244.31825.3b.
12. Lazzarin P., Berto F., Zappalorto M. Rapid calculations of notch stress intensity factors based on averaged strain energy density from coarse meshes: Theoretical bases and applications // Int. J. Fatig. -2010. - V. 32. - No. 10. - P. 1559-1567. - doi 10.1016/j.ijfatigue. 2010.02.017.
Поступила в редакцию 14.02.2017 г.
Сведения об авторах
Hadi Salavati, Assist. Prof., Shahid Bahonar University of Kerman, Iran, hadi_salavati@uk.ac.ir
Yoness Alizadeh, Assoc. Prof., Amirkabir University of Technology, Iran, alizadeh@aut.ac.ir
Majid R. Ayatollahi, PhD, Prof., Director, Iran University of Science and Technology, m.ayat@iust.ac.ir
