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УДК 539.422.22
Хрупкое разрушение закругленных V-образных надрезов в изостатическом графите в условиях статического многоосного нагружения
F. Berto1, A. Campagnolo1, M.R. Ayatollahi2
1 Падуанский университет, Виченца, 36100, Италия 2 Иранский университет науки и технологии, Тегеран, 16846, Иран
На сегодняшний день в литературе представлен большой объем экспериментальных данных для образцов из поликристаллического графита с трещиной в условиях простого нагружения (в частности нагружения типа I), в то время как результаты экспериментов для образцов с надрезом весьма немногочисленны. Авторам не удалось найти работ по исследованию образцов с V-образным надрезом в условиях нагружения смешанного типа (растяжение + кручение). В связи с этим, в работе впервые изучено хрупкое разрушение поликристаллического графита при нагружении смешанного типа (I + III). Рассмотрены цилиндрические образцы с кольцевыми надрезами разной остроты. Получены экспериментальные данные для различных геометрических конфигураций с разными значениями угла раскрытия и радиуса вершины надреза. Проведены многоосные статические испытания с разным соотношением режимов нагружения (соотношением между номинальным напряжением растяжения и номинальным напряжением кручения). Предложенный ранее критерий на основе плотности локальной энергии деформации, который применялся для простых режимов нагружения, использован в данной статье для случая смешанного нагружения растяжением и кручением. Указанный критерий позволяет выполнить эффективную оценку разрушающей нагрузки.
Ключевые слова: многоосное нагружение, изостатический поликристаллический графит, V-образный надрез, плотность энергии деформации, контрольный объем
Brittle fracture of rounded V-notches in isostatic graphite under static multiaxial loading
F. Berto1, A. Campagnolo1, and M.R. Ayatollahi2
1 Department of Management and Engineering, University of Padova, Vicenza, 36100, Italy 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
While a large bulk of experimental results from cracked specimens of polycrystalline graphite under pure modes of loading, in particular under mode I loading, can be found in the literature, only a very limited number of tests have been carried out on notches. At the best of the author knowledge dealing with the specific case of V-notches under mixed mode loading (tension + torsion) no results can be found in the literature. With the aim to fill this lack, the problem of mixed mode (I + III) brittle fracture of polycrystalline graphite is investigated systematically here for the first time. The present study considers cylindrical specimens weakened by circumferential notches characterized by different acuities. A new complete set of experimental data is provided considering different geometrical configurations by varying the notch opening angle and the notch tip radius. The multiaxial static tests have been performed considering different values of the mode mixity ratio (i.e. the ratio between the nominal stress due to tension and that due to torsion loading). A criterion based on the local strain energy density previously applied by the same authors only to pure modes of loading is extended here to the case of tension and torsion loadings applied in combination. The proposed criterion allows a sound assessment of the fracture loads.
Keywords: multiaxial loading, isostatic polycrystalline graphite, V-notch, strain energy density, control volume
1. Introduction
Isostatic graphite can be used in many industrial applications due to the good compromise between thermal and mechanical properties. The design of industrial products made of graphite is not focused only to structural applica-
tions. However, the large majority of components, although not thought as structural ones, are subjected to loads transferred by the other parts of the structure. For this reason, many studies in the past have been devoted to the investigation of the fracture strength of graphite. Brittle fracture
© Berto F., Campagnolo A., Ayatollahi M.R., 2015
is a typical behavior for this material and usually happens after the initiation of microcracks in the most stressed parts of the structure, combined in some cases with a very limited amount of plasticity [1].
The majority of the studies focused on structural integrity of graphite components have been devoted to the investigation of cracked components by quantifying the fracture toughness under prevalent mode I loading [2-7]. Dealing with isotropic graphite, innovative techniques have been proposed with this aim [8, 9]. Some researchers have also studied the fracture behavior of composite materials reinforced by graphite fibres at room temperature [10-13]. The problem related to the mechanical behavior at elevate temperature of graphite in presence of cracks is also a topic of active research [14-17]. Although the problem of brittle fracture of graphite components has been studied continuously for many years, only few predictive models have been proposed for the fracture assessment of cracked components. Some models are based on the microstructural properties [18-21]. A stress based criteria [22] has been recently proposed as an extension of the maximum tensile stress (MTS) criterion originally proposed in a pioneering study by Erdogan and Sih [23].
The papers briefly recalled in the first part of this introduction refer to the behaviour of graphite in the presence of cracks. A review of the recent and past literature shows that only very few papers are focused to the study of the notch sensitivity of graphite components. It is worth of mentioning here the pioneering study conducted by Bazaj and Cox [24] and Kawakami [25]. Only in the last years the topic of the fracture behavior of blunt notches has been faced by other researchers, who have investigated the case of pure mode I loading and in-plane mixed mode loading [26-32].
Dealing with static failure of notches under out-of-plane loading (i.e. pure torsion) the literature is very limited. There are few papers dealing with tension/torsion (mixed mode I/III) fracture in some other types of ceramics like Al2O3 [33] and inorganic glass [34].
Very recently Ayatollahi and Saboori [35] have proposed a new fixture for fracture tests under mixed mode I/III loading setting it with some PMMA specimens weakened by cracks.
Only one recent contribution by the present authors deals with V-notched specimens under pure torsion loading [36] while, at the best of the author knowledge, the case of static multiaxial loading (I + III) has never been investigated until now and no data are available for isostatic graphite. Due to this complete lack in the literature, the main aim of this research program is to systematically investigate the static behavior of isostatic graphite subjected to multiaxial loadings obtained as a combination of tension and torsion with different values of the mode mixity ratio (i.e. the ratio between the nominal stress due to tension and that due to torsion loading): 0.4, 0.5 and 1.0. A new complete set of experimental data from cylindrical specimens subjected to com-
bined tension and torsion loads is provided in the present contribution, considering a large variety of geometrical configurations obtained by varying the notch opening angle and notch depth. More than 40 new data from graphite specimens are summarized in the present paper with reference to different geometric configurations and various notch acuities. The notch radius has been varied from 0.3 to 2.0 mm and the notch opening angle from 30° to 120°.
A fracture model based on the strain energy density averaged over a control volume is used for the first time for the fracture assessment of notched specimens subjected to the multiaxial static loading case of tension and torsion applied in combination.
The strain energy density based approach allows a sound fracture assessment of the critical load for the specific material under investigation and it can be potentially extended to other types of graphite subjected to different combinations of mode I and mode III loading conditions.
The paper is divided in two parts: in the first one the experimental activity is presented (specimen geometry, test setting, details of experimental data). While the second part deals with the formulation and application of the averaged strain energy density criterion on the new data.
2. Fracture experiments
The details of the graphite material, the test specimens and the fracture experiments are presented in this section.
2.1. Material properties of EG022A
The fracture tests were conducted on a grade of iso-static polycrystalline graphite with commercial name of EG022A. The basic material properties of the tested graphite are listed in Table 1.
Nonlinear deformation sometimes is observed in the mechanical behavior of graphites, which makes the determination of Young's modulus rather complicated [37, 38]. How-
Table 1
Material properties
Material property Value
Elastic modulus E, MPa 8000
Shear modulus G, MPa 3300
Poisson's ratio v 0.2
Ultimate tensile strength, MPa 30
Ultimate torsion strength, MPa 37
Hardening, shore 53
Density, kg/m3 1830
Average grain size, |im 300
Resistivity, |iOhm • m 10.8
Thermal conductivity, W/(m • K) 119
Porosity, % 15
\â
) = 12.5
p = 40
7"
60
V-notched specimen
80
2a
> = 20
60
H
> = 20
Fig. 1. Geometry of unnotched (a) and notched (b) cylindrical specimens used in the experimental tests
ever, for simplicity the Young's modulus was obtained in this research from load-displacement graphs recorded by a universal tension-compression machine. The deviation observed from linear behavior was less than 0.01% at failure for the specimen used in the test. Young's modulus has been measured at a load where the deviation from linear behavior was less than 0.005%. The mean grain size was given in the material certify and measured by using the SEM technique while the density of the material was determined from the buoyancy method, submerging the tested graphite in a liquid of known density. The values have been checked and confirmed by the authors independently.
All tests were performed under displacement control on a servo-controlled MTS$ biaxial testing device (±100 kN/ ±1100 Nm, ±75 mm / ±55°). The load was measured by a MTS$ cell with ± 0.5% error at full scale. A MTS$ strain gauge axial extensometer (MTS$ 632.85F-14), with a gage length equal to 25 mm was used for measuring the tensile elastic properties on plain specimens while a multiaxis extensometer MTS$ 632.80F-04 (with a gage length equal to 25 mm) was used for measuring the torsional elastic properties on unnotched specimens.
Some load-displacement curves were recorded to obtain the Young's modulus E of the graphite using an axial
extensometer. The tensile strength at was measured by means of axisymmetric specimens with a net diameter equal to 12.5 mm on the net section and a diameter of 20 mm on the gross section (Fig. 1, a). Due to the presence of a root radius equal to 40 mm, the theoretical stress concentration factor is less than 1.03.
The torque-angle graphs recorded by the MTS device were employed together with the biaxis extensometer to obtain the shear modulus G and to measure the torsion strength Tt of the tested graphite. The ultimate shear strength Tt was found to be equal to 37 MPa.
2.2. Test specimens
As shown in Fig. 1, different round bar specimens were used for multiaxial (tension and torsion) static tests: unnotched specimens (Fig. 1, a) and cylindrical specimens with V-notches (Fig. 1, b). This allows us to explore the influence of a large variety of notch shapes in the experiments.
For V-notched graphite specimens with a notch opening angle 2a = 120° (Fig. 1, b), notches with four different notch root radii were tested: p = 0.3, 0.5, 1.0 and 2.0 mm. The effect of net section area was studied by changing the notch depth p. Two values were used, p = 3 and 5 mm, while keeping the gross diameter constant (20 mm).
For V-notched graphite specimens with a notch opening angle 2a = 60° (Fig. 1, b), four different notch root radii were considered in the experiments: p = 0.3, 0.5, 1.0 and 2.0 mm. With a constant gross diameter (20 mm), also the net section area was kept constant, such that p = 5 mm.
For V-notched graphite specimens with a notch opening angle 2a = 30° (Fig. 1, b), three different notch root radii were considered in the experiments: p = 0.5, 1.0 and 2.0 mm keeping constant the notch depth p = 5 mm.
At least three samples were prepared for each of the 15 specimen geometries described above, with a total number of 45 specimens. Figure 2 shows some specimens used in the tension-torsion tests and a typical fracture surface of
Fig. 2. V-notched specimens used in combined tension-torsion tests (a) and a specimen broken after a combined tension-torsion test (b)
Table 2
Geometrical parameters of the graphite specimens used in the tests
Notch opening Notch depth Notch radiu s Kt a n/ Tn
angle 2a p, mm p, mm Tension Torsion
0.3 4.29 2.09 0.5
3 0.5 1.0 3.54 2.75 1.85 1.59 0.5 0.5
120° 2.0 2.14 1.38 0.5
0.3 3.83 1.93 1.0
5 0.5 1.0 3.15 2.45 1.72 1.48 1.0 1.0
2.0 1.91 1.30 1.0
0.3 4.46 2.25 1.0
60° 5 0.5 1.0 3.52 2.59 1.90 1.55 0.4 0.4
2.0 1.96 1.32 0.4
0.5 3.53 1.94 1.0
30° 5 1.0 2.60 1.56 1.0
2.0 1.95 1.32 1.0
a notched graphite component after failure under combined tension and torsion loading.
In order to prepare the specimens, first several thick plates were cut from a graphite block. Then, the specimens were precisely manufactured by using a 2D CNC cutting 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 tests were conducted under three different combinations between the tensile and the torsional stresses, with the nominal mode mixity ratios of Gn/Tn = 0.4, 0.5 and 1.0. The details of applied loads are given in Table 2 as well as the stress concentration factors of each specimen under tension and torsion loading conditions, obtained by means of the finite element analysis adopting very refined meshes. Different nominal mode mixity ratios have been achieved by properly setting the torsional loading rate with
respect to the tensile loading rate. In particular the tensile loading rate was varied keeping constant the rotation control conditions with a loading rate of 1°/min.
Figure 3 shows two example of load-angle (torque versus 0) diagrams corresponding to some V-notched specimens characterized by the same notch radius and different values of the notch depth (p = 3 and 5 mm). The load-angle curves recorded during the tests always exhibited an approximately linear trend up to the final failure, which occurred suddenly. Therefore, the use of a fracture criterion based on a linear elastic hypothesis for the material law is realistic. The same trend has been observed for the tensile curves plotting the load as a function of the axial displacement.
All loads to failure (tensile load and torque) are reported in Tables 3-5 for each notch configuration and loading conditions. In particular, Table 3 reports the data for cn / Tn = 1.0 while Tables 4 and 5 summarize the data for the two ratios
Angle Angle
Fig. 3. Torque-angle curves related to V-notched graphite specimens with a notch depth equal to p = 3 (a) and 5 mm (b), 2a = 120°, p = 1 mm
Table 3
Experimental results in the case of on/Tn = 1.0, notch depth p = 5 mm
Specimen code Notch opening angle 2a Notch radius p, mm Tensile load, N Torque, N • mm an, MPa Tn, MPa °n/ Tn
1-01 1193 2659 15.19 13.54 1.12
1-02 0.3 1025 2280 13.05 11.61 1.12
1-03 1114 2500 14.18 12.73 1.11
2-01 1190 2690 15.15 13.69 1.10
2-02 0.5 1234 2631 15.71 13.40 1.17
2-03 120° 1200 2694 15.28 13.72 1.11
3-01 1302 2873 16.58 14.63 1.13
3-02 1.0 1251 2673 15.93 13.61 1.17
3-03 1283 2845 16.34 14.49 1.13
4-01 1497 3798 19.06 19.35 0.98
4-02 2.0 1451 3634 18.47 18.51 1.00
4-03 1532 3710 19.51 18.89 1.03
5-01 1073 2632 13.66 13.40 1.02
5-02 60° 0.3 1037 2867 13.20 14.60 0.90
5-03 1125 2883 14.32 14.68 0.98
6-01 1097 2852 13.97 14.53 0.96
6-02 0.5 1157 2704 14.73 13.75 1.07
6-03 1213 2917 15.44 14.86 1.04
7-01 1178 3038 15.00 15.47 0.97
7-02 30° 1.0 1112 2972 14.16 15.14 0.94
7-03 1214 3248 15.46 16.54 0.93
8-01 1302 3102 16.55 15.80 1.05
8-02 2.0 1319 3386 16.79 17.24 0.97
8-03 1486 3489 18.92 17.77 1.06
0.4 and 0.5, respectively. As visible from the tables the imposed mode mixity ratio is almost fulfilled with a variation of approximately ±10% with respect to the imposed value.
The variability of the loads to failure as a function of the notch opening angle is weak although not negligible. For a constant notch radius, the fracture load slightly increases for larger notch opening angles, although this effect is very low.
The main conclusion is that the stress concentration factors reported in Table 2 are not able to control the failure conditions due to a low notch sensitivity exhibited by the graphite specimens under combined tension and torsion loads.
3. Strain energy density averaged over a control volume: the fracture criterion
With the aim to assess the fracture load in notched graphite components, an appropriate fracture criterion is required which has to be based on the mechanical behavior of material around the notch tip. In this section, a criterion
proposed by Lazzarin and coauthors [30, 31] based on the strain energy density is briefly described.
Dealing with cracked components, the strain energy density factor S was defined first by Sih in a pioneering contribution [39] as the product of the strain energy density by a critical distance from the point of singularity. Failure was suggested to be controlled by a critical value of S, whereas the direction of crack propagation was determined by imposing a minimum condition on S.
Different from the Sih criterion, which is a point-wise approach, the averaged strain energy density criterion as presented in Refs. [30, 31, 40, 41] states that brittle failure occurs when the strain energy density averaged over a given control volume is equal to a critical value Wc. This critical value varies from material to material but it does not depend on the notch geometry and sharpness. The control volume, reminiscent of the Neuber concept of elementary structural volume [42], is considered to be dependent on the ultimate strength and on the fracture toughness, in the
Table 4
Experimental results in the case of on/Tn = 0.4, notch depthp = 5 mm
Specimen code Notch opening angle 2a Notch radius p, mm Tensile load, N Torque, N • mm an, MPa Tn, MPa °n/ Tn
9-01 636 3923 8.10 19.98 0.41
9-02 0.5 600 4010 7.64 20.42 0.38
9-03 630 4009 8.02 20.42 0.39
10-01 645 4449 8.21 22.66 0.36
10-02 60° 1.0 660 4634 8.40 23.60 0.36
10-03 631 4326 8.03 22.03 0.36
11-01 895 5164 11.40 26.30 0.43
11-02 2.0 811 5259 10.33 26.78 0.39
11-03 750 4700 9.55 23.94 0.40
Experimental results in the case of on/Tn = 0.5, notch depthp = 3 mm Table 5
Specimen code Notch opening angle 2a Notch radius p, mm Tensile load, N Torque, N • mm on, MPa Tn, MPa °n/ Tn
12-01 1482 11 606 9.63 21.54 0.45
12-02 0.3 1324 9657 8.60 17.92 0.48
12-03 1768 12 129 11.49 22.51 0.51
13-01 1701 11 768 11.05 21.84 0.51
13-02 0.5 1619 11 196 10.52 20.78 0.51
13-03 120° 1657 12 005 10.76 22.28 0.48
14-01 1739 12 150 11.30 22.55 0.50
14-02 1.0 1788 12 756 11.62 23.68 0.49
14-03 1816 12 611 11.80 23.41 0.50
15-01 2034 13 891 13.21 25.78 0.51
15-02 2.0 1756 12 500 11.41 23.20 0.49
15-03 1931 13 452 12.54 24.97 0.50
case of brittle or quasi-brittle materials subjected to static loads.
The method based on the averaged strain energy density was formalized and applied first to sharp (zero radius) V-notches under mode I and mixed mode I + II loading [30] and later extended to blunt U- and V-notches [31, 4347]. Some recent developments and applications are summarized in Refs. [41, 48-52] with some considerations also to three-dimensional effects [53-56], which have been widely discussed in Refs. [57-59].
When dealing with cracks, the control volume is a circle of radius R centered at the crack tip (Fig. 4, a). In the case of mode I loading under plane strain conditions, the radius R1c can be evaluated according to the following expression [41]:
R1c -
(1 + v)(5 - 8v)
- K
4%
(1)
where KIc is the mode I fracture toughness, v is the Poisson's ratio and is the ultimate tensile stress of an unnotched specimen.
For a sharp V-notch, the critical volume becomes a circular sector of radius Rc centred at the notch tip (Fig. 4, b). When only failure data from open V-notches are available, R1c can be determined on the basis of some relationships reported in [28], where KIc is substituted by the critical value of the notch stress intensity factors as determined at failure from sharp V-notches.
Dealing here with notched components under torsion loading, the control radius R3c can be estimated by means of the following equation [60]:
( r— ^ ya1-^)
R3c -
K
IIIc
1 + v tt
(2)
where KIIIc is the mode III critical notch stress intensity
Fig. 4. Control volume for crack (a), sharp V-notch (b) and blunt V-notch (c) under mixed mode I/III loading. Distance r0 = p(n - 2a)/(2n - 2a)
factor and Tt is the ultimate torsion strength of the un-notched material. Moreover, e3 is a parameter that quantifies the influence of all stresses and strains over the control volume and 1 -A3 is the degree of singularity of the linear elastic stress fields [61], which depends on the notch opening angle. Values of e3 and A3 are 0.4138 and 0.5 for the crack case (2 a = 0°).
For a blunt V-notch under mode I or mode III loading, the volume is assumed to be of a crescent shape shown in Fig. 4, c, where Rc is the depth measured along the notch bisector line. The outer radius of the crescent shape is equal to Rc + r0, being r0 the distance between the notch tip and the origin of the local coordinate system (Fig. 4, c). Such a distance depends on the V-notch opening angle 2a, according to the expression [40, 41]: n-2a
ro -W (3)
2n-2a
For the sake of simplicity, complex theoretical derivations have deliberately been avoided in the present work and the strain energy density values have been determined directly from the finite element models.
4. Strain energy density approach in fracture analysis of the tested graphite specimens
The fracture criterion described in the previous section is employed here to estimate the fracture loads obtained from the experiments conducted on the graphite specimens. In order to determine the strain energy density values, a finite element model of each graphite specimen was generated. A typical mesh used in the numerical analysis is shown in Fig. 5, a. In addition, Fig. 5, b shows the typical strain energy density contour lines under combined tension and torsion loading condition.
As originally thought for pure modes of loading the averaged strain energy density criterion states that failure occurs when the mean value of the strain energy density over a control volume W reaches a critical value Wc, which depends on the material but not on the notch geometry.
Under tension loads, this critical value can be determined from the ultimate tensile strength at according to the Beltrami expression for the unnotched material:
Wic =a2/(2E). (4)
By using the values of at = 30 MPa and E = 8000 MPa, the critical strain energy density for the tested graphite is W1c = = 0.05625 MJ/m3.
Under torsion loads, this critical value can be determined from the ultimate shear strength Tt according to the Beltrami expression for the unnotched material:
W3c =
2G
(5)
By using the values of Tt = 37 MPa and G = 3300 MPa, the critical strain energy density for the tested graphite is W3c = = 0.2074 MJ/m3.
In parallel, the control volume definition via the control radius Rc needs the knowledge of the mode I and mode III critical notch stress intensity factors KIc and KIIIc and the Poisson's ratio v, see Eqs. (1) and (2). For the considered material KIc and KIIIc have been obtained from specimens weakened by sharp V-notches with an opening angle 2a = 10° and a notch radius less than 0.1 mm. A precrack was also generated with a razor blade at the notch tip. The resulting values are KIc = 1.06 MPa • m0 5 and KIIIc = 1.26 MPa • m0 5 which provide the control radii R1c = 0.405 mm and R3c = = 0.409 mm, under pure tension and pure torsion, respectively. For the sake of simplicity, a single value of the con-
0.052543 0.047928 0.043313 0.038697 0.034082 0.029467 0.024852 0.020236 0.015621 0.011006
Fig. 5. Typical mesh used to evaluate the averaged strain energy density for a V-notched specimen with 2a = 120°, p = 1 mm, p = = 5 mm, Rc = 0.4 mm. Finite element mesh (a) and strain energy
density contour lines under combined tension and torsion loading (b)
T
trol radius was kept for the synthesis in terms of strain energy density setting Rc = R1c = R3c. As discussed in previous papers [40, 41, 62], the control radii under tension and torsion loading can be very different and this is particularly true when the material behavior differs from a brittle one: the difference is higher for materials obeying a ductile behavior. For this specific case, the values are so close to each other that a single value can be employed for the final synthesis. The strain energy density criterion has been applied here for the first time to mixed mode I/III loading conditions. Two different methodologies have been proposed. The two procedures will be described below.
4.1. Total critical energy: criterion 1
The first proposal for the application of the local strain energy density approach to the case of mixed mode I/III loading is based on the idea that the total critical strain energy density Wc is a function of the nominal mode mixity ratio at failure a nc/ t nc and it can be evaluated according to the following equation valid for both unnotched and notched components:
Wc = f(anc/Tnc) = Wi + W3. (6)
For unnotched specimens, Eq. (6) can be rewritten as follows:
_ 2 _ 2
Wc = ^SÇ + c 2E 2G
(7)
where anc and Tnc are the nominal critical stresses referred to the net area corresponding to different values of the mode mixity ratio. Then in this first criterion Wc is a function of the mode mixity ratio at failure anc/t .
Some tests on unnotched specimens have been made as a function of the nominal mode mixity ratio, Fig. 6 reports the trend of the critical strain energy density as obtained experimentally from unnotched specimens as a function of the nominal mode mixity ratio. When an/tn = 0, the case of pure torsion is achieved while when Tn tends to zero (a n/ t n tends to the case of pure tension is obtained. These two limits correspond to the left and right hand side limits of the diagram. By considering the critical energy as a function of the mode mixity ratio M = 2/ nx xarctan (an/ t n), as reported in Fig. 6, and numerically evaluating the averaged strain energy density for the notched graphite specimens tested in the present work, Table 6 was finally obtained. The values of the critical strain energy density as a function the mode mixity ratio are respectively 0.0833, 0.1173 and 0.1510 MJ/m3 for a J Tn = 1.0, 0.5 and 0.4, respectively as indicated in Fig. 6.
The averaged strain energy density has been calculated numerically by using the finite element code ANSYS 14.5$. The analysis has been carried out by using eight-node harmonic elements (PLANE 83) under axisymmetric conditions. Only one quarter of the geometry has been modeled in the positive X-Y quadrant. Being the averaged strain energy density value a mesh independent parameter [63, 64],
0.25-
^0.20-S
q-0.15-
w
ïo.ioH
CO
Limit of pure torsion
0.4
Van/Tn= 0-5
O
:0.05
n
Limit of pure tension
o.oo-
0.0 0.2 0.4 0.6 0.Í
M = 2/71 arctan(an/Tn)
1.0
Fig. 6. Trend of the critical strain energy density (SED) as obtained experimentally from unnotched specimens as a function of the nominal mode mixity ratio
a free mesh was used for all finite element models. Due to the fact that the averaged strain energy density is mesh insensitive, the similarity between the mesh patterns used to model different geometries would be unnecessary. The only point to consider is the correct definition of the control volume according to Fig. 4, c and 5. As stated above the size of the control volume is R = 0.4 mm for both tension and torsion loadings.
Table 6 summarizes for each case, the contributions to strain energy density due to mode I and due to mode III. The total strain energy density as obtained by the simple sum of the two contributions has been reported in the eighth column of the table. The last column presents the total strain energy density (SED) normalized by its critical value, as obtained from unnotched specimens for a specific nominal mode mixity ratio Gn/ Tn (Fig. 6). The normalized values have been given in the form of the square root (SED Wc )05 being this ratio proportional to the values of the critical loads. This representation has been used by the same authors also in the past [41].
As it can be seen in Table 6 for the tested graphite samples, the agreement between the experimentally obtained critical loads and the theoretical values based on the constant value of the critical strain energy density, is satisfactory; with the relative deviation (equal to the difference between the unity and the ratio (SED Wc)05) ranging from -15% to +15%. However, for very few test data, the deviation is slightly higher than 15% but still comparable with previous findings under in plane mixed mode loading, where the acceptable scatter was about ± 20%.
The most significant results have also been given in graphical form in Figs. 7 and 8 where the experimental values of the critical loads (open dots) have been compared with the theoretical predictions based on the constancy of the critical strain energy density in the control volume (solid line). The plots of Fig. 7 are given for the notched graphite specimens as a function of the notch radius p for V-notches with 2a = 30° and p = 5 mm both for the critical tensile load and for the critical torque. The theoretically predicted
Table 6
Experimental results: Overview of the data by using the method 1
p, mm 2a p, mm Tensile loading, N Torque, N • mm W1, MJ/m3 W3, MJ/m3 SED W1 + W3, MJ/m3 Wc, MJ/m3 (SED/ Wc)0'5
0.3 1193 2659 0.041 0.040 0.081 0.083 0.985
5 120° 0.3 1025 2280 0.030 0.029 0.060 0.083 0.846
0.3 1114 2500 0.036 0.035 0.071 0.083 0.923
0.5 1190 2690 0.040 0.040 0.080 0.083 0.981
5 120° 0.5 1234 2631 0.043 0.038 0.082 0.083 0.989
0.5 1200 2694 0.041 0.040 0.081 0.083 0.986
1.0 1302 2873 0.045 0.044 0.089 0.083 1.035
5 120° 1.0 1251 2673 0.042 0.038 0.080 0.083 0.979
1.0 1283 2845 0.044 0.043 0.087 0.083 1.022
2.0 1497 3800 0.046 0.072 0.118 0.083 1.192
5 120° 2.0 1451 3634 0.043 0.066 0.109 0.083 1.145
2.0 1532 3710 0.048 0.069 0.117 0.083 1.185
0.3 1073 2632 0.038 0.043 0.082 0.083 0.989
5 60° 0.3 1037 2867 0.036 0.051 0.087 0.083 1.023
0.3 1125 2883 0.042 0.052 0.094 0.083 1.062
0.5 1097 2852 0.030 0.050 0.080 0.083 0.980
5 30° 0.5 1157 2700 0.034 0.045 0.078 0.083 0.969
0.5 1213 2917 0.037 0.052 0.089 0.083 1.034
1.0 1178 3038 0.040 0.052 0.093 0.083 1.057
5 30° 1.0 1112 2972 0.036 0.050 0.086 0.083 1.018
1.0 1214 3248 0.043 0.060 0.103 0.083 1.112
2.0 1300 3102 0.041 0.047 0.087 0.083 1.023
5 30° 2.0 1319 3386 0.042 0.056 0.097 0.083 1.081
2.0 1486 3489 0.053 0.059 0.112 0.083 1.160
0.5 636 3923 0.011 0.097 0.108 0.151 0.845
5 60° 0.5 600 4010 0.010 0.101 0.111 0.151 0.857
0.5 630 4009 0.011 0.101 0.112 0.151 0.861
1.0 645 4449 0.011 0.107 0.118 0.151 0.885
5 60° 1.0 660 4634 0.012 0.116 0.128 0.151 0.920
1.0 631 4326 0.011 0.101 0.112 0.151 0.861
2.0 895 5164 0.020 0.132 0.152 0.151 1.005
5 60° 2.0 811 5259 0.017 0.137 0.154 0.151 1.009
2.0 750 4700 0.014 0.110 0.124 0.151 0.905
0.3 1482 11606 0.021 0.118 0.139 0.117 1.089
3 120° 0.3 1324 9657 0.017 0.082 0.099 0.117 0.917
0.3 1768 12129 0.030 0.129 0.159 0.117 1.164
0.5 1701 11768 0.027 0.120 0.148 0.117 1.121
3 120° 0.5 1619 11196 0.025 0.109 0.134 0.117 1.067
0.5 1657 12005 0.026 0.125 0.151 0.117 1.135
1.0 1739 12150 0.027 0.124 0.151 0.117 1.134
3 120° 1.0 1788 12756 0.028 0.137 0.165 0.117 1.187
1.0 1816 12611 0.029 0.134 0.163 0.117 1.179
2.0 2034 13891 0.027 0.156 0.182 0.117 1.247
3 120° 2.0 1756 12500 0.020 0.126 0.146 0.117 1.115
2.0 1931 13452 0.024 0.146 0.170 0.117 1.200
1600
rl200 *
o Ph
800-
0.5
4000 3000 2000
M
£
1000-
0.5
-a ? a i
o Experimental data — Theoretical estimation
0.5 1.0 1.5 2.C Notch radius p, mm
b i
8 Q 8____2>
y -o— o o Experimental data — Theoretical estimation
o Experimental data — Theoretical estimation
1.0 1.5
Notch radius p, mm
2.0
o Experimental data — Theoretical estimation
1.0 1.5
Notch radius p, mm
2.0
Fig. 7. Comparison between the experimental data and theoretical assessment for the graphite specimens weakened by V-shaped notches with 2a = 30° andp = 5 mm. Control radius Rc = 0.4 mm. Tensile (a) and torsion load (b)
2000-
<lT 15001 o «
5£ 1000-
500-
o Experimental data — Theoretical estimation
0.5 1.0 1.5 Notch radius p, mm
a F
? 12000 A Q
£
4000-
2.0
o Experimental data — Theoretical estimation
0.5 1.0 1.5 Notch radius p, mm
2.0
Fig. 8. Comparison between the experimental data and theoretical assessment for the graphite specimens weakened by V-shaped notches with 2a = 120° and p = 3 mm. Control radius Rc = 0.4 mm. Tensile (a) and torsion load (b)
loads are in good agreement with the experimental results. This holds true also for the other specimens. The same plots, in terms of tensile fracture load and critical torque, are presented in Fig. 8 dealing with V-notches with 2a = 120° and p = 3 mm.
Figure 9, a shows a synthesis in terms of the square root value of the local energy averaged over the control volume of radius Re, normalized with respect to the critical energy (as obtained from Fig. 6) as a function of the notch radius p. Indeed, the ratio on the vertical axis is proportional to the fracture loads. The aim of this figure is to investigate the range of accuracy of all SED-based fracture assessments for the tested graphite specimens. It is clear that the scatter of the data is very limited and almost independent of the notch opening angle. All experimental values fall inside a scatter band ranging from 0.8 to 1.2. Note that many of the results are inside a scatter band ranging from 0.85 to 1.15, which was also typical for the notched graphite specimens tested under in-plane mixed tension-shear loading [28, 29].
4.2. Normalized critical energies: criterion 2
The second proposed approach is a reminiscent of the work by Gough and Pollard [65] who proposed a stress-based expression able to summarize together the results obtained from bending and torsion. The criterion was extended in [66] in terms of the local strain energy density to V-notches under fatigue loading in the presence of combined tension and torsion loadings.
In agreement with [66] and extending the method to the static case, the following elliptic expression
Wle + W3 W3c = 1 (8)
is obtained. In Eq. (8) W1e and W3e are the critical values of strain energy density under pure tension and pure tor-
sion. For the considered graphite, W1e = 0.05625 MJ/m3 and W3e = 0. 2074 MJ/m3. The values of W1 and W3 have, instead, to be calculated as a function of the notch geometry and of the applied mode mixity ratio. Each specimen reaches its critical energy when the sum of the weighted contributions of mode I and mode III is equal to 1, which represents the complete damage of the specimen.
The detailed calculations employing this second criterion are reported in Table 7. The square root of the left-hand side term of Eq. (8), which is, in fact, proportional to the critical load, is given in the last column of this table.
.4-
+20%
I a
.0--
° n
W 0
x/i
>6
-20%
o.:
o.o
0.5 1.0 1.5
Notch radius p, mm
2.0
1
o
o £
+20%
2.5
m
1.0---
+
0.
-20%
1^0.:
O C7n/Tn= 1.0 ❖ Gn/Tn= 0.5 □ <T„/T„= 0.4
0.0 0.5 1.0 1.5
Notch radius p, mm
2.0
2.5
Fig. 9. Synthesis of the results from combined tension and torsion tests based on the averaged strain energy density: method 1 (a) and 2 (b)
Table 7
Experimental results: Overview of the data by using the method 2
p, mm 2a p, mm Tensile loading, N Torque, N • mm W1, MJ/m3 W3, MJ/m3 Wi/Wic + W,/W,0 Wi/Wic + W,/W3J0-5
0.3 1193 2659 0.041 0.039 0.935 0.967
5 120° 0.3 1025 2280 0.030 0.029 0.690 0.831
0.3 1114 2500 0.036 0.035 0.818 0.904
0.5 1190 2690 0.040 0.040 0.919 0.959
5 120° 0.5 1234 2631 0.043 0.038 0.965 0.982
0.5 1200 2694 0.041 0.040 0.932 0.965
1.0 1302 2873 0.045 0.044 1.028 1.014
5 120° 1.0 1251 2673 0.042 0.038 0.937 0.968
1.0 1283 2845 0.044 0.043 1.001 1.000
2.0 1497 3800 0.046 0.072 1.181 1.087
5 120° 2.0 1451 3634 0.043 0.066 1.101 1.049
2.0 1532 3710 0.048 0.069 1.203 1.097
0.3 1073 2632 0.038 0.043 0.899 0.948
5 60° 0.3 1037 2867 0.036 0.051 0.894 0.946
0.3 1125 2883 0.042 0.052 1.010 1.005
0.5 1097 2852 0.030 0.050 0.789 0.888
5 30° 0.5 1157 2700 0.034 0.045 0.824 0.908
0.5 1213 2917 0.037 0.052 0.921 0.960
1.0 1178 3038 0.040 0.052 0.985 0.993
5 30° 1.0 1112 2972 0.036 0.050 0.895 0.946
1.0 1214 3248 0.043 0.060 1.068 1.033
2.0 1300 3102 0.041 0.047 0.958 0.979
5 30° 2.0 1319 3386 0.042 0.056 1.023 1.012
2.0 1486 3489 0.053 0.059 1.241 1.114
0.5 636 3923 0.011 0.097 0.684 0.827
5 60° 0.5 600 4010 0.010 0.101 0.683 0.827
0.5 630 4009 0.011 0.101 0.701 0.838
1.0 645 4449 0.011 0.107 0.739 0.860
5 60° 1.0 660 4634 0.012 0.116 0.794 0.891
1.0 631 4326 0.011 0.101 0.701 0.838
2.0 895 5164 0.020 0.132 1.020 1.010
5 60° 2.0 811 5259 0.017 0.137 0.981 0.990
2.0 750 4700 0.014 0.110 0.800 0.894
0.3 1482 11606 0.021 0.118 0.967 0.983
3 120° 0.3 1324 9657 0.017 0.082 0.709 0.842
0.3 1768 12129 0.030 0.129 1.181 1.087
0.5 1701 11768 0.027 0.120 1.090 1.044
3 120° 0.5 1619 11196 0.025 0.109 0.987 0.993
0.5 1657 12005 0.026 0.125 1.089 1.044
1.0 1739 12150 0.027 0.124 1.097 1.047
3 120° 1.0 1788 12756 0.028 0.137 1.188 1.090
1.0 1816 12611 0.029 0.134 1.187 1.091
2.0 2034 13891 0.027 0.156 1.254 1.120
3 120° 2.0 1756 12500 0.020 0.126 0.985 0.992
2.0 1931 13452 0.024 0.146 1.159 1.076
A synthesis in terms of the square root value of the considered parameter, that is the sum of the weighted energy contributions related to mode I and mode III loadings, is shown in Fig. 9, b as a function of the notch root radius p. The obtained trend is very similar to that shown in Fig. 9, a.
Many of the results are inside a scatter band ranging from 0.9 to 1.1 with only few exceptions. This second criterion is surely more advantageous with respect to the first one because for its application only the critical values W1c and W3c related to the simpler cases of pure tension and pure torsion loadings from unnotched specimens are required. The first criterion requires, instead, the knowledge of the total critical energy from unnotched specimens as a function of the mode mixity ratio.
The fracture models proposed in this paper can be used for predicting the onset of brittle fracture in notched graphite components which are subjected to a combination of tension and torsion loadings. Such criteria would be very useful for designers and engineers who should explore the safe performance of graphite components particularly under complex loading conditions.
5. Conclusions
Brittle fracture in V-notched polycrystalline graphite specimens was investigated both experimentally and theoretically under combined tension and torsion loadings. Fracture tests were conducted on notched round bar graphite specimens. Different notch depths, notch radii and opening angles were considered in the test specimens as well as different combinations of the mode mixity ratio CTn/x .
The new set of data provided in the paper is unique because no previous paper has been devoted to similar topics dealing with graphite components.
The averaged strain energy density criterion was used for the first time in order to estimate the fracture load of notched graphite components under mixed mode I/III static loading. Two different approaches of the strain energy density criterion were proposed showing the capabilities of the suggested methods to assess the fracture behavior of poly-crystalline graphite under the considered loading conditions.
The results estimated by means of the strain energy density approach were found to be in good agreement with the experimental ones. The second criterion based on the elliptic expression described above seems to be very advantageous because it requires, as experimental parameters, only the critical energies from unnotched graphite specimens under pure tension and pure torsion loading conditions.
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Поступила в редакцию 06.07.2015 г.
CeeoeHUM ob aemopax
Filippo Berto, Prof., University of Padova, Italy, berto@gest.unipd.it
Alberto Campagnolo, PhD Student, University of Padova, Italy, campagnolo@gest.unipd.it
Majid R. Ayatollahi, PhD, Prof., Director, Iran University of Science and Technology, Iran, m.ayat@iust.ac.ir
