Brittle fracture of sharp and blunt V-notches in isostatic graphite under torsion loading

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Brittle fracture of sharp and blunt V-notches in isostatic graphite under torsion loading F. Berto a, P. Lazzarin

a,* ,

M.R. Ayatollahi

b

a

Department of Management and Engineering, University of Padova, Stradella S. Nicola, 3, 36100 Vicenza, Italy 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 16846, Tehran, Iran

b

A R T I C L E I N F O

A B S T R A C T

Article history:

Brittle fracture of polycrystalline graphite under torsion loading is studied experimentally

Received 27 September 2011

and theoretically using axisymmetric specimens weakened by sharp and rounded-tip V-

Accepted 21 December 2011

notches. The main purpose is twofold. First, to provide a new set of experimental data from

Available online 30 December 2011

notched samples made of isostatic polycrystalline graphite with different values of notch opening angles and root radii, which should be useful to engineers engaged with static strength analysis of graphite components. At the best of authors’ knowledge, data from notch specimens under torsion are not available in the literature for this material. Second, to apply to the torsion loading case a fracture criterion based on the strain energy density (SED) averaged over a well-defined control volume surrounding the notch tip, extending what was made by the present authors for in-plane tension-shear loading conditions in notched graphite specimens. Good agreement is found between the experimental data related to the critical loads to failure and the theoretical assessments based on the constancy of the mean SED over the material-dependent control volume.  2011 Elsevier Ltd. All rights reserved.

1.

Introduction

Isostatic graphite is manufactured by using cold isostatic pressing technique and is often known for its homogeneous structure and excellent isotropic electrical, thermal and mechanical properties. It is sometimes purified in special-designed graphitization furnace to remove non-carbonaceous inclusions and impurities. Isostatic graphite is extensively used in various industrial applications such as: moulds in continuous casting systems for making shaped steel, cast iron and copper; crucibles for melting precious metals or alloys; moulds for making shaped glass; heating elements, heat shields, crucibles, etc. Some of the advantages of isostatic graphite are: high thermal and chemical resistance, isotropic and homogeneous properties, high resistance to thermal shock and proper electrical conductivity.

However, isostatic polycrystalline graphites are prone to brittle fracture under mechanical or thermal loads, particularly in the presence of stress concentrators like cracks or notches. Graphite has been considered a brittle material in a large body of research. This, in particular, is true for older investigations, for example in 70s when the AGR reactors with graphite moderators were under construction and there was a need for fast and simple design procedures. However, it has long been suggested that assessing the integrity of graphite structures can benefit from more realistic material models based on physical observations. It is believed that graphite best fits a class of materials called ‘quasi-brittle’. Limited plasticity in the form of microcracking is a sign of such materials, which has been reported in many investigations. Utilising loading-unloading procedures on the basis of nonlinear energy principles, Sakai et al. [1] established an empirical

* Corresponding author: Fax: +39 0444 998888. E-mail address: [email protected] (P. Lazzarin). 0008-6223/$ - see front matter  2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2011.12.045

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method for evaluating the nonlinear fracture mechanics parameters, i.e. the nonlinear energy toughness Gc, the crack growth resistance R, the Jc value, and the plastic energy dissipation rate. Although graphite components are not usually designed for load bearing purposes, they are sometimes subjected to tensile, bending or torsion loads transferred from the adjacent components. Therefore, it is important to study the structural integrity of cracked and notched isostatic graphite components under different loading conditions. As a measure of material resistance against brittle fracture, fracture toughness is conventionally used by engineers to assess the structural integrity of cracked specimens. Cracks and notches may experience two major types of deformation under in-plane loading: mode I (notch opening) and mode II (notch sliding). Any combination of these two modes is called mixed mode I/II deformation. Several researchers have attempted to measure fracture toughness of polycrystalline graphite either under mode I or under mixed mode I/II loading conditions. For example, Awaji and Sato [2], Li et al [3], Yamauchi et al. [4,5], and Shi et al [6] conducted a series of experiments to determine mode I and mixed mode I/II fracture toughness of polycrystalline graphite using various cracked test specimens. Recently, Mostafavi and Marrow [7] investigated crack nucleation in poly-granular graphite under uniaxial and bi-axial loading conditions and suggested a fictitious crack model to predict the site and onset of crack nucleation in graphite components. Wang and Liu [8] also proposed an innovative technique called the spiral notch torsion fracture toughness test for measuring the mode I fracture resistance of graphite. In another work, Etter et al. [9] made use of single-edge notched beam specimens to measure mode I fracture toughness KIc of isotropic polycrystalline porous graphite in addition to the graphite/aluminium composite. The extensive applications of graphite fibres in the composite materials (such as graphite/epoxy composites) have also prompted some researchers to study the fracture resistance of these materials under pure mode I and mixed mode I/II loading conditions [10–13]. Graphite is known for its excellence performance in high temperature environments. Therefore, the thermal shock behaviour of graphite has been investigated by Bruno and Latella [14] through the arc-discharge method and also by Jae et al [15] via the laser irradiation technique. In another work, Sato et al [16] studied the thermal shock fracture toughness for two different types of carbon as a function of heat treatment temperature during the graphitization process. Sato et al [17] investigated also the high temperature fracture toughness of graphite. On the other hand, some theoretical models have been presented in the literature for estimating brittle fracture in graphite based mainly on its microstructural properties (e.g. [18–21]). For instance, an energy release rate criterion was employed by Lomakin et al [19] for analysing the fracture initiation in cracked graphite specimens under pure mode I loading. A prominent model suggested by Burchell [21] relates brittle fracture of graphite to the crack nucleation and the subcritical crack growth from pre-existing pores or defects under the influence of the local stress field due to those stress raisers. Dealing with crack propagation phenomena, the

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fracture behaviour of anthracite-based carbon materials was analysed by Allard et al. [22] who investigated the influence of the microstructure, especially the particle type and orientation, and determined the crack growth resistance curves (R-curves). The R-curves exhibited a typical shape characterised by three stages: an increase in growth energy due to process zone development; a plateau value corresponding to steady-state crack propagation; finally, a sharp increase of energy due to specimen edge effects. Microstructure with higher anthracite fraction was found to lead to easier crack growth. There are also some models for mixed mode I/II fracture in polycrystalline graphite. For example, Ayatollahi and Aliha [23] made use of a modified version of the maximum tensile stress (MTS) criterion [24] and demonstrated that significantly improved estimates for the onset of mixed mode fracture can be achieved for two grades of polycrystalline graphite containing sharp cracks. The modified MTS criterion was later used by Ayatollahi and Aliha [25] to provide very good estimates for the experimental results obtained from sharp cracks embedded in some ceramic materials like Sialon, Mullite, SiC, Glass, Sintered Carbide, Porcelain, Zirconia, and Alumina. The papers reviewed above all deal with the mechanical behaviour of graphite components in the presence of sharp cracks. Whereas cracks are viewed as unpleasant entities in most engineering materials, U- and V-notches of different acuities are sometimes deliberately introduced in design and manufacturing of products made from graphite. Graphite moulds, graphite heating elements and graphite chucks are only some examples for industrial components that contain U- or V-shaped notches. A review of literature shows that in spite of extensive studies on mode I and mixed fracture in cracked graphite specimens, there are very few papers e.g. [26,27] focused on brittle fracture of notched graphite components and the notch sensitivity problem. Stress-concentration factors as well as fatigue strength reduction factors were determined by Bazaj and Cox [26] for grooved tensile specimens made of a fine-grained graphite (Great Lakes Carbon, grade H205). Some equations were derived to relate the fatigue-strength-reduction factors to the theoretical stress-concentration factors and the radius of the groove providing a sound basis to the analysis of the notch sensitivity index. A decrease in the observed stress concentration effect was also documented for an increasing of the grain size [26]. Kawakami [27] tested IG-11 (fine grained isotropic graphite), Stackpole 2020 (SP2020, fine grained isotropic graphite) and PGX (moulded anisotropic graphite). Experiments included uniaxial tensile tests, compressive tests as well as thermal shock tests by water quenching. In water quenching tests, the specimens were heated in N2 gas atmosphere and rapidly cooled by water quenching, which induced strong thermal stresses. In the tensile and the compressive tests, notch effects were evaluated by comparing the results of notched specimens with those of the smooth specimens. Especially in the compressive tests, specimens with notches both transverse to the stress direction and parallel to the stress direction were used. Recently, with reference to pure mode I loading conditions, the effect of stress concentrations has been analysed by

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Ayatollahi and Torabi [28] who conducted a series of mode I fracture tests on three different V-notched test specimens made of a polycrystalline graphite. They also proposed a mean stress criterion which was found able to assess their experimental results with very good accuracy. By proceeding on parallel tracks, brittle fracture in V-notched graphite specimens under mixed mode I/II loading has been recently investigated theoretically and experimentally by the authors using centrally notched Brazilian disc specimens made of graphite [29]. The aim was to determine the fracture loads when the specimens were subjected to different combinations of mode I and mode II loading conditions. They employed a theory based on the averaged strain energy density [30,31] to summarise all those experimental data from sharp and blunt Vnotches in a unified way. While a lot of work has been devoted to characterise fracture in cracked and notched graphite components under in-plane (mixed mode I/II) loading conditions, the current literature lacks any information concerning brittle failure of graphite components under mode III (or tearing mode) of deformation such as mode III fracture in notched graphite bars subjected to torsion loading. There are very few papers dealing with tension/torsion (mixed mode I/III) fracture in some other types of ceramics like Al2O3 [32] and inorganic glass [33]. In the former case, the fracture data have been used to assess the applicability of a failure criterion based on the critical normal stress [32]; while in the latter case, a torsion fracture model was proposed for notched elements of brittle materials [33], which is based on a combined normal stress/ Griffith energy fracture criterion. The purpose of the present research is twofold: a. To provide a large body of experimental data from static fracture of notched graphite specimens subjected to torsion loads with varying notch root radii, notch opening angles and notch depths. Such data should be helpful to engineers engaged in static strength analysis of graphite components. More than 70 new experimental results are summarised in the paper with reference to various notch configurations. b. To provide a fracture model to estimate the critical loads to failure in notched graphite components subjected to torsion loading. Recent work by the present authors has revealed that the strain energy density averaged on a control volume was successful for brittle failure assessments of notched graphite samples in mixed mode I/II loading conditions. Dealing with torsion loads, a synthesis of the new results based on the SED approach is one of the main aims of the present paper. The paper is structured in the following way: in the first section, the experimental work is described including: specimen geometries, testing procedures and experimental results. The second section is focused on the discussion of the new experimental results. Finally, the third section presents a synthesis of all data in terms of averaged SED using a control volume based on the basic material properties under torsion.

2.

Fracture experiments

The details of the graphite material, the test specimens and the fracture experiments are presented in this section.

2.1.

Material

The fracture tests were conducted on a commercial isostatic polycrystalline graphite. The mean grain size was measured by using the SEM technique and the density was determined from the buoyancy method. The basic material properties of the tested graphite are listed in Table 1: mean grain size is 2 lm, porosity 7%, bulk density of 1850 kg/m3, mean tensile strength of 28.5 MPa, Young’s modulus of 8.05 GPa and shear modulus 3.350 GPa. The compressive strength is equal to 110 MPa, whereas the flexural strength is 49 MPa. As mentioned in the introduction, graphites sometimes show nonlinearity in their mechanical behaviour, which makes defining a single value as elastic modulus questionable (see for example [34,35]). Losty and Orchard [34] observed that neutron irradiation at low temperatures increased both Young’s modulus and strength of graphite. Similarly, they proved that the heat treatment of ungraphitized carbon reduced Young’s modulus and the strength while maintaining a constant strain energy at failure. These points were also highlighted in the survey by Greenstreet [35]. In the present work, for simplicity, a single value for elastic modulus is used. This value was obtained from the load–displacement graphs recorded by a universal tension-compression machine. The deviation observed from linear behaviour was less than 0.02% at failure for the specimen used in the test. Young’s modulus has been measured at a load where the deviation from linear behaviour was less than 0.01%. All tests were performed under load control on a servocontrolled MTS bi-axial testing device (±100 kN/±110 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 multi-axis extensometer MTS 632.80F-04 (with a gage length equal to 25 mm) was used for measuring the torsion elastic properties on plain specimens.

Table 1 – Material properties. Material property

Value

Elastic modulus E (MPa) Shear modulus G (MPa) Poisson’s ratio m Ultimate tensile strength (MPa) Ultimate torsion strength (MPa) Ultimate compression strength (MPa) Ultimate bending strength (MPa) Hardness (Shore) Density (Kg/dm3) Porosity (%) Resistivity (lohm m) Thermal Conductivity (W/(m K))

8050 3354 0.2 28.5 30 110 49 58 1.85 7 11 110

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Plain specimens φ 12.5

φ 20

(a)

φ 20

(b)

ρ 40 60

80

60

2α U and V-notches

p

ρ Semicircular notch

ρ

φ 20

(c) 200

Fig. 1 – Geometry of specimens used in torsion experiments.

Some load–displacement curves were recorded to obtain the Young’s modulus (E) of the graphite using an axial extensometer. The tensile strength (rt) was measured by means of axis-symmetric specimens with a net diameter of 12.5 mm on the net section and a diameter of 20 mm on the gross section (see Fig. 1a). Due to the presence of a large root radius, 40 mm, the theoretical stress concentration factor is less than 1.05. The torque-angle graphs recorded by the MTS device were employed together with the bi-axis extensometer to obtain the shear modulus (G) and to measure the torsion strength (st) of the tested graphite. The ultimate shear strength st was found to be equal to 30 MPa.

2.2.

Test specimens

As shown in Fig. 1, different round bar specimens were used for torsion tests: plain specimens (Fig. 1a), cylindrical specimens with U- and V-notches (Fig. 1b), and cylindrical specimens with circumferential semicircular notches (Fig. 1c), making it possible to explore the influence of a large variety of notch shapes in the experiments. In more detail: •

•

For U-notched specimens in Fig. 1b, notches with two different notch root radii were tested; q = 1 and 2.0 mm. The effect of 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 V-notched specimens with a notch opening angle 2a = 30 (Fig. 1b), three different notch root radii were used in the experiments: q = 0.1, 0.3 and 0.5 mm. Moreover, a larger opening angle (2a = 120) was also considered, combined with five notch root radii, q = 0.1, 0.3, 0.5, 1.0, 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.

•

For semicircular notches (Fig. 1c), notches with four different notch root radii were tested: q = 0.5, 1.0, 2.0 and 4.0 mm.

At least three samples were prepared for each of the 24 specimens described above. All in all, a total number of 80 tests were carried out. Fig. 2a shows some samples of the specimens used in the torsion tests, whereas Fig. 2b shows a notched component after failure. 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 2-D 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. For each geometry shape, three torsion tests were performed under rotation control conditions with a loading rate of 1/min. Fig. 3 shows three sample load–angle (Mt versus h) curves corresponding to one of the U-notched specimens. The load–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 a fracture criterion based on linear elastic hypothesis for the material law is realistic. In Fig. 3 the deviation from linearity (Dh/h) is shown for two notched specimens whereas Table 2 gives the same ratio Dh/h for all the geometries. The linearity is better approximated for larger values of the notch depth as well as for greater values of the notch tip radius. All torsion loads to failure (Mt) are reported in Tables 3–5 for each notch configuration. Torque-angle curves related to a V-notched graphite specimen (2a = 30) are shown in Fig. 4. A review of the experimental data presented in these Tables shows a strong increase in the fracture load as the

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30000

Torque [Nmm]

25000

U-Notch (ρ =1mm p=2mm)

Test 1

average Δθ/θ = 16%

Test 2 Test 3

20000 15000 10000

Δθ

5000 0

0

1

2

3

4

5

6

Angle θ [degrees]

Fig. 3 – Torque-angle curves related to a U-notched graphite specimen.

Table 2 – Values of the theoretical stress concentration factor for all notched models. 2a ()

p (mm)

q (mm)

V-notch

120

5

0.1 0.3 0.5 1 2

8.2 6.3 5.1 4.8 4.4

2.43 1.93 1.72 1.48 1.30

V-notch

120

2

0.1 0.3 0.5 1

16.2 14.3 11.2 10.6

2.76 2.13 1.89 1.62

V-notch

30

5

0.1 0.3 0.5

13.1 10.1 6.2

3.57 2.32 1.94

V-notch

30

2

0.1 0.3 0.5

20.9 14.1 13.4

4.00 2.58 2.14

U-notch

0

5

0

2

1 2 1

7.4 7.0 16.3

1.57 1.33 1.72

0.5 1 2 4

0.5 1 2 4

22.0 18.0 14.0 9.5

1.79 1.64 1.44 1.21

Semi-circular

Fig. 2 – Notched specimens used in torsion tests (a) and a sample specimen broken after the test.

notch depth decreases from p = 5 mm to p = 2 mm. When the notch angle is kept constant, the fracture load slightly increases for larger notch tip radii but this variation is much lower than that due to the net section area variation. The variability of the loads to failure as a function of the notch opening angle is also weak. For a constant notch radius, the fracture load slightly increases as the notch opening angle increases, 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 torsion load.

Dh/h (%)

Kt

3. Fracture criterion based on the strain energy density averaged over a control volume In order to estimate the fracture load in notched graphite components, engineers need an appropriate fracture criterion based on the mechanical behaviour of material around the notch tip. In this section, a strain-energy-density based criterion is briefly described, which allows us to assess the fracture loads for notched specimens with good accuracy. Dealing with cracked components, the strain energy density factor S was defined first by Sih [36] as the product of the strain energy density by a critical distance from the point

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Table 3 – Fracture loads obtained from the specimens with semicircular notches.

Table 5 – Fracture loads obtained from the specimens with V-notches (2a = 30 and 2a = 120). p (mm)

q (mm)

2a = 120

2a = 30

Mt (N mm)

Mt (N mm)

q (mm)

Mt (N mm)

0.5

45,956 45,012 43,504

5

0.1

6692 6778 6629

6592 6808 6936

1

33,590 36,923 37,533

5

0.3

6612 6860 6426

6230 6995 6643

2

25,073 25,087 28,231

5

0.5

6895 6495 6709

6474 6719 6605

4

13,317 12,029 12,511

5

1

6829 7064 6771

5

2

7198 7401 7408

2

0.1

25,221 25,628 26,027

24,688 23,469 24,054

Table 4 – Fracture loads obtained from the specimens with U-notches. Mt (N mm)

p (mm)

q (mm)

5

1

6148 6599 6801

2

0.3

25,535 24,509 25,053

23,076 22,408 23,293

5

2

6812 6995 6674

2

0.5

25,135 23,930 24,764

22,749 23,145 24,860

2

1

24,578 22,994 23,200

2

1

24,746 23,445 26,399

9000 V-Notch 30° (ρ=0.1mm - p=5mm)

8000

Δθ/θ = 21%

7000

Torque [Nmm]

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 Sih’s criterion, which is a point-wise criterion, the averaged strain energy density criterion (SED) as presented in Refs. [30,31,37] states that brittle failure occurs when the mean value of the strain energy density 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 Neuber’s concept of elementary structural volume [38], is thought of as dependent on the ultimate tensile strength and the fracture toughness KIc in the case of brittle or quasi-brittle materials subjected to static loads. The method based on the averaged SED was formalised and applied first to sharp, zero radius, V-notches under mode I and mixed, I + II, loading [30] and later extended to blunt Uand V-notches [31]. Applications of the method are reported in [39,40] where it was used to assess the critical loads to failure from U- and V-notched specimens subjected to mixed mode conditions. Contrary to some values integrated in the local criteria (e.g. maximum principal stress, hydrostatic stress, deviatoric stress), which are mesh dependent, the SED averaged over a control volume is not sensitive to the mesh size. As widely

6000 5000 4000 3000

Δθ Test 1

2000

Test 2

1000 0

Test 3

0

0.5

1

1.5

2

2.5

3

Angle θ [degrees]

Fig. 4 – Torque-angle curves related to a V-notched graphite specimen (2a = 30).

discussed in [41,42], refined meshes are not necessary for determining the mean value of the SED over a control volume, because this parameter can be determined via the nodal displacements, without involving their derivatives. As soon as the average SED is known, the notch stress intensity factors (NSIFs) or the stress concentration factors (SCFs) can be determined a posteriori on the basis of very simple expressions

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linking the local SED and NSIFs or SCFs in plane problems. The extension of the SED method to three-dimensional cases is also possible as well as its extension to notched geometries exhibiting small scale yielding [43]. Modelling the material according to a linear elastic law or, alternatively, a Ramberg– Osgood law, the constancy of SED was documented for sharp V-notches under plane strain conditions (large constraint effects). Under torsion loads, however, small scale yielding conditions are difficult to maintain, both under static and medium cycle fatigue loads, and then SED depends on the material law [43]. In the SED approach the determination of the control volume is based on the mean values of some material properties (which are typically fracture toughness and the ultimate tensile stress of the plain specimens, as shown later by Eq. (1) in this section 4). The concept of control volume is the basis also of some probabilistic approaches which describes the proximity of cleavage fracture by using a scalar Weibull stress as suggested in Beremin’s model [44]. In this model the Weibull stress is calculated by integrating a weighted value of the maximum principal stress r1, rather than the SED, over the plastic zone ahead of the stress concentration. For a sound discussion on the Beremin’s model and the use of two or three-parameter Weibull distribution functions for critical load assessments of notched components, the readers should refer to a recent paper by Horn and Sherry [45] and the reference list reported therein. For a review of statistical models of fracture relevant to nuclear grade graphite see also a very recent contribution by Nemeth and Bratton [46]. Dealing with SED approach applied to cracked components, the critical volume is a circle of radius Rc centred at the tip (Fig. 5a). Under plane strain conditions, the radius Rc can be evaluated according to the following expression [47]:  2 ð1 þ mÞð5  8mÞ KIc ð1Þ R1c ¼ 4p rt where KIc is the fracture toughness, m the Poisson’s ratio and rt the ultimate tensile stress of a plain specimen. For a sharp V-notch, the critical volume becomes a circular sector of radius Rc centred at the notch tip (Fig. 5b). When only failure data from open V-notches are available, Rc can be determined on the basis of some relationships reported in [30], where KIc is substituted by the critical value of the notch stress intensity factors (NSIFs) as determined at failure from sharp V-notches.

2α

2α=0 2α

R 3c

(a)

r0 R 3c R 2=R 3c+r0

(b)

2a (rad) 0 p/6 p/3 p/2 2p/3 3p/4

k3

e3

0.5000 0.5455 0.6000 0.6667 0.7500 0.8000

0.4138 0.3793 0.3448 0.3103 0.2759 0.2586

Dealing here with sharp notches under torsion loading, the control radius R3c can be estimated by means of the following equation [48]: R3c ¼

rffiffiffiffiffiffiffiffiffiffiffi  1 e3 K3c 1k3  1þm st

ð2Þ

where K3c is Mode III critical notch stress intensity factor and st is the ultimate torsion strength of the unnotched material. Moreover, e3 is the parameter that quantifies the influence of all stresses and strains over the control volume and (1  k3) is the degree of singularity of the linear elastic stress fields [48,49], which depends on the notch opening angle. The values of e3 and k3 are reported in Table 6 for different opening angles. The mean value of the elastic deformation energy under torsion is [37,48]: W¼

e3 K23  2ð1k 3Þ 2ð1 þ mÞG R3c

ð3Þ

where K3 is the mode III notch stress intensity factor and G is the transverse shear modulus. For a blunt V-notch under mode III loading, the volume assumes the crescent shape shown in Fig. 5c, where R3c is the depth measured along the notch bisector line. The outer radius of the crescent shape is equal to R3c + r0, being r0 the distance between the notch tip and the origin of the local coordinate system. Such a distance depends on the V-notch opening angle 2a, according to the expression [31,37] r0 ¼ q

ðp  2aÞ ð2p  2aÞ

ð4Þ

Stress fields for a variety of notch configurations under torsion loading are reported in the literature [50–54]. On the basis of those theoretical solutions, it is possible to evaluate the SED over the control volume. However, for the sake of simplicity, complex theoretical derivations have deliberately been avoided in the present work and the SED values have been determined directly from the FE models.

ρ γ

R3c

Table 6 – Values of the parameters k3 and e3 as a function of the notch opening angles [37].

Ω

(c)

Fig. 5 – Control volume for crack (a), sharp V-notch (b) and blunt V-notch (c) under mode III loading. Distance r0 = q · (p2a)/(2p2a). For a U-notch r0 = q/2.

4. SED 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 SED values, first a finite element model of each graphite specimen was generated. A typical mesh used in the numerical analyses is shown in Fig. 6a. The averaged

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In parallel, the control volume definition via the control radius Rc needs the knowledge of the mode III critical NSIF K3c and the Poisson’s ratio m, see Eq. (2). In the absence of specific data from cracked components, the parameter K3c can be estimated considering the results from two geometries with the minimum available radius, q = 0.1 mm, and 2a = 30. This simplified procedure should be verified a posteriori by comparing the value of the estimated control volume and the real notch tip radius used in substitution of q = 0. The procedure can be accepted only if the results related to the radius Rc are much greater than the notch tip radius q. According to Ref. [48], for p = 5 mm, the following equation can be used for K3c: K3c ¼ k3 R1k3 snom ¼ 0:786  50:4545  34:6 ¼ 56:5 MPa mm0:4545 ¼ 2:44 MPa m0:4545

ð6Þ

In the case p = 2 mm, the relevant expression is: K3c ¼ k3 R1k3 snom ¼ 0:726  80:4545  30:0 ¼ 56:0 MPa mm0:4545 ¼ 2:42 MPa m0:4545

Fig. 6 – Mesh (a), control volume (b) and iso-strain energy density contour lines (c) for a V-notch with 2a = 30, q = 0.3 mm, p = 2 mm.

strain energy density criterion (SED) states that failure occurs when the mean value of the strain energy density over a control volume, W, is equal to a critical value Wc, which depends on the material but not on the notch geometry. Under torsion loads, this critical value can be determined from the ultimate shear stress st according to Beltrami’s expression for the unnotched material Wc ¼

s2t 2G

ð5Þ

Using the values of st = 30 MPa and G = 3354 MPa, the critical SED for the tested graphite is Wc = 0.134 MJ/m3.

ð7Þ

In Eqs. (6) and (7) snom is the nominal stress to failure referred to net sectional area, as determined from the mean values of the torsion loads reported in Table 4 for the angle 2a = 30. By using Eq. (2), with st = 30 MPa, e3 = 0.379, m = 0.2 and (1  k3)=0.4545, the radius of the control volume is obtained as R3c = 1.1 mm. For the sake of simplicity, we assume in the following R3c = 1 mm. Note that R3c is 10 time greater than the minimum notch tip radius, q = 0.1 mm. Dealing with V-notches under linear elastic hypothesis, the theoretical load to failure can be easily obtained by a simple proportion between the applied load M in the FE model and theffi square root values of averaged SED, i.e. Mth =M ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Wc =W. These values are given in Table 7 together with the mean values of the critical loads to failure, hMi, of all tested graphite specimens. The values of the SED for the same loads are also given in Table 7. It is possible now to compare the SED values at failure with the theoretical value, Wc = 0.134 MJ/m3, as determined a priori for polycrystalline graphite by means of Eq. (5). As can be seen in Table 7 for the tested graphite samples, the agreement between the experimental results of the critical loads and the theoretical values based on the constant value of the averaged SED, 0.134 MJ/m3, is satisfactory; with the relative deviation ranging from 14% to +12%. However, for sixteen out of eighteen data, the deviation is less than 10%. The numerical values of SED have been calculated numerically by using the FE code ANSYS 11.0. All the analyses have been carried out by using eight-node harmonic elements (plane 83) under axis-symmetric conditions. Only one quarter of the geometry has been modelled in the positive quadrant. Being the SED value substantially mesh insensitive [41,42], a free mesh was used for all models. There is no need to assure the similarity among the meshes used to model different geometries. Attention should be paid only to the correct definition of the control volume according to Figs. 5c and 6. Fig. 6c shows the iso-strain energy density contour lines for a graphite specimen with 2a = 30, q = 0.3 mm and p = 2 mm. Due to torsion loading condition, the SED is, obviously, symmetric with respect to the bisector line.

1950

CARBON

5 0 ( 2 0 1 2 ) 1 9 4 2 –1 9 5 2

Table 7 – Values of the averaged SED ðWÞ and the maximum elastic shear stress smax as obtained from the FE analyses; comparison between theoretical and experimental torques to failure. W (MJ/m3)

2a ()

p (mm)

q (mm)

Mtheor. (N mm)

hMexpi (N mm)

D%

V-notch

120

5 5 5 5 5 2 2 2 2

0.1 0.3 0.5 1.0 2.0 0.1 0.3 0.5 1.0

6591 6576 6523 6459 6367 23,441 23,351 23,283 23,173

6699 6633 6699 6888 7335 25,625 25,032 24,609 24,863

1.6 0.8 2.6 6.2 13.2 8.5 6.7 5.4 6.8

82.8 65.2 58.6 52.1 48.7 87.9 66.5 57.9 50.1

0.138 0.136 0.141 0.152 0.178 0.160 0.154 0.150 0.154

V-notch

30

5 5 5 2 2 2

0.1 0.3 0.5 0.1 0.3 0.5

7545 7342 7208 26,304 25,837 25,524

6778 6622 6600 24,070 22,926 23,585

11.3 10.9 9.2 9.3 12.7 8.2

123.4 78.3 65.1 119.6 73.5 62.8

0.108 0.109 0.112 0.112 0.106 0.114

U-notch

0

5 5 2

1.0 2.0 1.0

7217 6885 25,907

6516 6827 23,590

10.8 0.8 9.8

52.2 44.9 50.4

0.109 0.132 0.111

0.5 1.0 2.0 4.0

44,888 37,605 24,873 11,213

44,824 36,015 26,130 12,619

0.1 4.4 4.8 11.1

59.6 51.8 46.6 44.9

0.134 0.123 0.148 0.170

60 Semi-circular notches

50

Torque [N m]

Table 7 also gives the maximum value of the shear stress at the notch tip (smax) as obtained from the FE models of the graphite specimens by applying to the model the mean value of the critical loads to failure. It is worth noting that the maximum shear stress at the notch tip is much greater than the ultimate shear stress 30 MPa determined from the plain (unnotched) sample (e.g. about four times for q = 0.1 mm and 2a = 30). The material is then characterised by a low notch sensitivity, as documented also by the large value of the control volume radius. The most significant results have also been given in graphical form in Fig. 7 where the experimental values of the critical loads (open dots) have been compared with the theoretical predictions based on the constancy of SED in the control volume (solid line). The plots are given for the notched graphite specimens as a function of the notch radius q for semicircular notches (Fig. 7a) and for V-notches with 2a = 30 and p = 2 mm (Fig. 7b). The theoretically predicted loads are in good agreement with the experimental results. This holds true also for the other specimens, although the relevant plots have been omitted here for the sake of brevity. A synthesis in terms of the square root value of the local energy averaged over the control volume of radius Rc, normalised with respect to the critical energy of the material, is shown in Fig. 8 as a function of the ratio q/R3c. Indeed, the ratio on the vertical axis is proportional to the fracture load. The aim is to investigate the range of accuracy of all SED-based fracture assessments for the tested graphite specimens. From the figure, it is clear that the scatter of the data is very limited and almost independent of the notch opening angle. Note that 68 out of 70 experimental values fall inside a scatterband ranging from 0.85 to 1.15. Note also that many of the results

40 30 20 10 0

0

1

2

3

4

Notch radius, ρ [mm]

(a) 50 V-Notch 120° , p=2 mm

40

Torque [N m]

Semi-circular

smax (MPa)

30 20 10 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Notch radius, ρ [mm]

(b) Fig. 7 – Comparison between experimental data and theoretical assessment (solid line) for the graphite specimens; semicircular notches (a) and V-shaped notches with 2a = 120 and p = 2 mm (b).

CARBON

5 0 ( 20 1 2 ) 1 9 4 2–19 5 2

2 Graphite EG022A Notch radius from 0.1 to 4 mm Notch opening angle from 0 to 120° 70 Experimental data

(W/W3c)0.5

1.6

1.2

0.8

τt

= 30 MPa R3c = 1 mm

0.4

W3c = 0.134 MJ/m

0 0.1

1

3

10

ρ/R3c

Fig. 8 – Synthesis based on SED of the results from torsion tests.

(about 75%) are inside a scatter ranging from 0.9 to 1.1, which was typical for the notched graphite specimens tested under in-plane mixed tension-shear loading [29].

5.

Conclusions

Brittle fracture in U- and V-notched polycrystalline graphite specimens was investigated both experimentally and theoretically under torsion loading. Fracture tests were conducted on notched round bar specimens. Different notch depths, notch radii and opening angles were considered in the test specimens. The SED criterion was used for the first time in order to estimate the fracture load of notched graphite components under mode III static loading. It was shown that the proposed method is suitable for the polycrystalline graphite, being the experimental results in good agreements with the results estimated by the SED approach. From the sound agreement between the theoretical and experimental results, it can be deduced that for the polycrystalline graphite the torsion critical energy and the radius of the control volume are both constant material properties not influenced by the geometrical parameters.

Acknowledgements This work was carried out in the ambit of the Italian Research Programs CPDA100715 and PRIN 2009Z55NWC. The Authors would like to express their gratitude for financial support.

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