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УДК 539.422.22
Хрупкое разрушение смешанного типа I/II «бразильских» дисков с V-образным надрезом в условиях отрицательной нагрузки типа I
A.R. Torabi1, B. Bahrami2, and M.R. Ayatollahi2
1 Тегеранский университет, Тегеран, 14395-1561, Иран 2 Иранский университет науки и технологии, Тегеран, 16846, Иран
Проведены испытания на разрушение смешанного типа I/II «бразильских» дисков с закругленным V-образным надрезом из полиметилметакрилата в условиях отрицательной нагрузки типа I для различных значений угла и радиуса надреза с целью экспериментального определения разрушающей нагрузки и угла, при котором начинается разрушение. Методом конечно-элементного анализа выявлено, что несмотря на действие отрицательной нагрузки типа I одна сторона надреза находится под действием растягивающих касательных напряжений, что указывает на вероятность начала разрушения с данной стороны надреза. Результаты конечно-элементного анализа подтверждены результатами экспериментов, в ходе которых начало разрушения наблюдалось со стороны надреза, находящейся под действием растягивающих напряжений. Проведена оценка экспериментальных результатов с помощью двух критериев хрупкого разрушения на основе напряжений: критерий максимальных касательных напряжений для закругленного V-образного надреза и критерий средних напряжений. Показано, что оба критерия позволяют с высокой точностью предсказывать результаты экспериментов, получаемые в условиях отрицательной нагрузки типа I.
Ключевые слова: хрупкое разрушение, V-образный надрез, нагружение смешанного типа, отрицательная нагрузка типа I, ««бразильский» диск с V-образным надрезом
Mixed mode I/II brittle fracture in V-notched Brazilian disk specimens under negative mode I conditions
A.R. Torabi1, B. Bahrami2, and M.R. Ayatollahi2
1 Fracture Research Laboratory, Faculty of New Sciences and Technologies, University of Tehran, Tehran, 14395-1561, Iran 2 Fatigue and Fracture Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, 16846, Iran
The well-known round-tip V-notched Brazilian disk specimen is utilized for conducting mixed mode I/II fracture tests on PMMA under negative mode I conditions for different notch angles and various notch radii with the aim to measure experimentally the fracture load and the fracture initiation angle. It is shown by the finite element analysis that although the notch is under negative mode I loading, one side of the notch border still experiences tensile tangential stresses suggesting that fracture would take place from the same side of notch border. Experimental observations also indicated that fracture occurs from the tensile side of the notch border confirming the finite element results. The experimental results are then theoretically estimated by means of two stress-based brittle fracture criteria, namely the round-tip V-notch maximum tangential stress and the mean-stress criteria. It is shown that both criteria provide very good predictions to the experimental results obtained under negative mode I conditions.
Keywords: brittle fracture, V-notch, mixed mode loading, negative mode I, V-notched Brazilian disk
Nomenclature
E—Young's modulus, p—notch radius,
K Ic —plane-strain fracture toughness of material, ctu —ultimate tensile strength, CTee —tangential stress,
(ct66 )c or ctc—critical stress,
âee —mean value of tangential stress,
RV-BD—round-tip V-notched Brazilian disk,
RV-MTS—round-tip V-notch maximum tangential stress,
RV-MS—round-tip V-notch mean stress,
© Torabi A.R., Bahrami B., Ayatollahi M.R., 2016
d^—theoretical critical distance of the RV-MS criterion measured from the notch border,
dce —experimental critical distance of the RV-MS, criterion measured from the notch border,
rcth —theoretical critical distance of the RV-MTS criterion measured from the notch border,
rce —experimental critical distance of the RV-MTS criterion measured from the notch border.
1. INTRODUCTION
Due to the catastrophic failure, designers are often concerned with those structural components which are made of brittle and quasi-brittle materials. For such components, fracture takes place suddenly and hence, they are often designed in a way they do not sustain high level of stresses. However, indirect loads from adjacent members may be transmitted to them and finally cause fracture.
In the presence of discontinuities, like cracks, notches, scratches, etc., brittle fracture in engineering components becomes more critical. These discontinuities are the source of stress concentration and hence, decrease significantly the strength of the components. Unlike cracks, scratches and similar stress raisers which are normally undesirable, notches are widely used in engineering structures because of design necessities. For example, blunt notches are extensively used in machine design, like making a path of movement for reciprocating elements, designing bolts, screws, etc., or joining two or more components. Therefore, engineers employ notches of various shapes in design of components and structures in a way the stress concentration does not lead to failure, particularly to sudden fracture in brittle and quasi-brittle members. From the statements above, it is obvious that appropriate brittle fracture criteria are essentially needed for designing notched brittle components subjected to different loading conditions.
V- and U-shaped notches have been investigated more than the other notch features because of their wide applications in engineering structures. In the past 15 years, extensive investigations have been performed on brittle fracture of V-and U-notched components under mode I, mode II, mode III and mixed mode loadings. For example, Refs. [1-17] can be quoted under pure mode I loading conditions. The number of papers published under mixed mode I/II loading is less than that under mode I loading. Some of the research results on mixed mode I/II brittle fracture of notched members can be found in Refs. [18-39]. As an extreme for mixed mode I/II loading, pure mode II loading has also been investigated in brittle fracture of notched domains, but very limited compared to mode I and mixed mode I/II loadings because of limited engineering applications [40-42]. Brittle fracture in notched components under out-of-plane loading conditions has also been studied experimentally and theoretically. For example, Susmel and Taylor [43] studied mixed mode I/III
brittle fracture in notched polymethylmethacrylate (PMMA) specimens by means of the theory of critical distances (TCD). Moreover, Berto et al. [44-46] have investigated mode III brittle fracture in notched round bars made of isostatic graphite and PMMA.
In the entire references quoted above, mode I loading is referred to as the pure opening mode under which V- or U-notch bisector line experiences tensile stresses. However, V-and U-notches may be subjected to closing mode (herein after referred to as negative mode I loading) under which the notch bisector line experiences compressive stresses. Some of the works performed on brittle fracture of notched members under negative mode I loading conditions are briefly reviewed herein. Hoek and Beniawski [47] reported some test results on failure of elliptical notches of various inclination angles. Cotterel [48] re-considered and analyzed the results of Ref. [47] and found that although brittle fracture in elliptical notches are explained by two separate models under tension and compression, the failure loads always depend on the notch radius. The attribution of dominant inclined failure plane in uniaxial test to Coulomb has been stated by Bell [49]. Lajtai [50] assessed the mechanism of failure as material progresses through a few stages of microfracture nucleating from the intact state and finishing with the state of residual strength under compression. On the basis of the experimental results achieved from rock materials, Lajtai et al. [51] proposed an empirical failure model under compression. Wang and Shrive [52] reviewed the crack initiation and growth, as well as brittle fracture criteria under compression, with the main result that brittle fracture initiation in both tension and compression is basically by mode I crack initiation mechanism. The onset of fracture in circular cavities loaded under compression has been assessed by Dzik and Lajtai [53] by showing that brittle fracture may initiate from three positions around the cavity; at the tensile stress concentration, at a position inside the material far from the cavity and at the compres-sive stress concentration. By testing several graphite specimens weakened by V-notches, Kawakami [54] investigated the compressive brittle fracture in V-notched samples under uniaxial tension, compression and thermal shock. He also made a comparison between notched and unnotched specimens for investigating the effect of notch on fracture strength of the specimens. Churchman et al. [55] obtained a first order approximation to the closure extent based on the asymptotic solution for a semi-infinite V-notch. By means of the average strain energy density (SED) criterion, Berto et al. [56] obtained fracture loads for graphite plates weakened by double V-notches with end holes (VO-notches) and subjected to pure compressive stresses. Recently, Ayatollahi and Torabi [57] have developed two closed-form expressions for predicting the negative mode I notch fracture toughness (NFT) for VO-notches based on two well-known brittle fracture criteria, namely the point-stress and the mean-stress criteria. They successfully verified their expressions by using the test results
reported in Ref. [56]. The most recent works on compressive brittle fracture of notched components are those published by Ayatollahi and Torabi [58, 59]. They suggested and used two new test specimens, called V-notch stepped cottage (VSC) and flattened V-notched semi-disk (FVSD), made of PMMA for preparing some new experimental results on the negative mode I NFT of blunt V-notches. They also predicted theoretically the experimental results by means of two stress-based brittle fracture criteria, namely the point-stress and the mean-stress criteria. It was demonstrated that the experimental results could satisfactorily be predicted by using these two criteria [58, 59].
A comprehensive literature survey indicated that in the entire papers published on mixed mode I/II brittle fracture of notches, positive mode I loading (i.e. the opening mode) has been combined with mode II loading. No paper or technical report was found in which negative mode I loading is mixed with mode II loading. Consequently, it is attempted in the present investigation to study brittle fracture in blunt V-notches under mixed negative mode I + mode II loading.
Ayatollahi and Torabi [37] investigated brittle fracture in round-tip V-notched Brazilian disk (RV-BD) specimens made of PMMA under mixed mode I/II loading. In their investigation, positive mode I loading (i.e. the opening mode) has been mixed with mode II loading leading to brittle fracture by tensile tangential stresses which initiates from the right half border of the notch (i.e. the half border that locates at the right side of the notch bisector line). It was found in Ref. [37] that for the entire values of the loading angle P between zero and PII (the mode II loading angle), the right and the left half borders of the blunt V-notch experience tensile and compressive tangential stresses, respectively and brittle fracture takes place from the right border that experiences tensile stresses.
To assess mixed mode I/II brittle fracture in blunt V-notches under negative mode I loading conditions, the same RV-BD specimens made of PMMA (different from the PMMA reported in Ref. [37]) are tested by considering some loading angles which are larger than PII. The experimental observations indicated that although mode I loading is negative and the notch bisector line experiences compressive stresses, brittle fracture still takes place from the right notch border.
Table 1
The properties of the tested PMMA
Material property Value
Elastic modulus E, GPa 1.816
Poisson's ratio 0.38
Ultimate tensile strength, MPa 68.5
Ultimate compressive strength, MPa 100.5
Plane-strain fracture toughness, MPa • m05 1.71
Such observations were confirmed by finite element results which revealed that the right halfborder of the V-notch is affected by relatively high tensile tangential stresses but the left border by low compressive stresses. The experimentally obtained fracture loads and fracture angles are theoretically predicted by means of two well-known brittle fracture criteria, namely the round-tip V-notch maximum tangential stress and the round-tip V-notch mean-stress criteria, as well as the fracture initiation angles. It is shown that both criteria provide successful predictions to the experimental results.
2. Experiments
A large bulk of mixed mode I/II brittle fracture tests are performed on disk-type specimens weakened by blunt V-notches under negative mode I conditions for different notch angles and various notch radii. Details are presented below.
2.1. Material
Polymethylmethacrylate, which is the most common material in brittle fracture tests, is selected to conduct fracture experiments at room temperature on blunt V-notches. Because there is a wide range of PMMA materials which are manufactured by different methods resulting in various mechanical properties, experiments were first performed to determine some necessary material properties. For tensile, fracture toughness and compressive properties, the standard codes ASTM D638-99 [60], D5045-99 [61] and D695-10 [62] were used, respectively. The properties of the tested PMMA are presented in Table 1. The true stress-strain curves of PMMA under tension and compression are also shown in Fig. 1.
0.00 0.04 0.08 0.12 0.16 True strain
Fig. 1. The true stress-strain curves of PMMA under tension (a) and compression (b)
2.2. Specimen
The well-known round-tip V-notched Brazilian disk specimen is used in this study for performing mixed mode I/II fracture tests on blunt V-notches under negative mode I loading conditions. The RV-BD specimen is schematically shown in Fig. 2.
In Fig. 2, P is the angle between the loading direction and the notch bisector line. The parameters 2a, p, t, d, D and P are the notch angle, the notch radius, the disk thickness, twice the notch length (i.e. the overall slit length), the disk diameter and the applied compressive load, respectively. The disk diameter, the overall slit length (i.e. the tip-to-tip distance) and the disk thickness are constant and equal to 80, 40 and 8 mm, respectively. The values of the notch angle and the notch radius are equal to 2a = 30° and 60° and p = 0.5, 1.0, 2.0 and 4.0 mm, respectively.
When the direction of the applied load P is along the notch bisector line (i.e. P = 0), the two V-shaped corners of the central rhombic slit is subjected to pure positive mode I loading conditions. When P increases gradually from zero, the type of loading varies from pure mode I towards pure mode II. For a particular angle, called PII, pure mode II loading is obtained. The angle PII is always less than 90° and depends on the notch length, the notch angle, and the notch radius [37, 63]. If again P increases gradually from PII, the loading type varies from pure mode II towards negative mode I, and when it becomes 90°, the notch experiences pure negative mode I loading. Therefore, for 0 < P < PII, the notch is subjected to mixed mode I/II loading under positive mode I conditions, while for PII < P < 90 ° under negative ones.
To perform mixed mode I/II fracture tests under negative mode I conditions, the angle PII should first be determined in order to select appropriate intermediate angles between PII and 90°. The angles PII for different values of the relative notch length (RNL) and the relative notch radius (RNR) have been presented by Torabi and Taherkhani [63]. Considering the RNL value (i.e. the ratio of the overall slit length to the disk diameter) equal to 0.5 for the present RV-BD specimens,
PII can be obtained from Ref. [63] to be between 26° and 27° for 2a = 30° and between 28° and 30° for 2a = 60°, for various notch radii.
Note that the P values for successful tests should be carefully selected because for high values of P between PII and 90°, the lateral notches are simultaneously subjected to mixed mode I/II loading under positive mode I conditions. Therefore, fracture may occur from the lateral notches instead of the main notches which is not desirable. To avoid undesirable fracture from the lateral notches, many finite element analyses were performed. For each notch angle 2a and notch radius p, P was gradually increased from PII to 90° by 5° increments and for each P, the principal stresses on the borders of lateral and main notches for a constant load P were computed and compared. Since the material properties are the same, the RV-BD specimen is expected to fail from the notch that experiences greater stresses. The specific value of P between (PII and 90°) below which fracture takes place from the main notch is called the critical loading angle Pc. The finite element analyses indicated that the value of Pc depends on both the notch angle and the notch radius. For 2a = 30° and 60°, Pc values were found to be in the range of 55° to 60° and 50° to 55°, respectively. Therefore, some appropriate values of P between PII and Pc were selected for conducting mixed mode I/ II brittle fracture tests on the main notches under negative mode I conditions. Those P values are P = 30°, 40° and 50° for 2a = 30° and P = 30°, 40° and 45° for 2a = 60°.
A high-precision 2D CNC laser cutting machine was utilized to fabricate the RV-BD specimens from a PMMA sheet of 8 mm thick. For each of the notch geometries, three specimens were tested for checking the repeatability of the results. To provide monotonic loading conditions on the specimens, the test speed was considered to be equal to 1.0 mm/min. All in all, 72 fracture tests were performed under displacement-control conditions. Some of the RV-BD specimens are represented in Fig. 3 during and after the fracture tests, respectively. The experimental observations (see for example Figs. 3, c, d) indicated that the entire specimens fractured
Fig. 2. The RV-BD specimen
Fig. 3. The RV-BD specimens during and after the fracture tests: 2a = p = 1 mm, P = 40° (c); 2a = 60°, p = 0.5 mm, P = 30° (d)
from the right half border of the notch. It is demonstrated later in Sect. 4 that this half border experiences tensile tangential stresses during loading while the notch bisector line and the left half border of the notch sustain compressive stresses. The experimentally obtained fracture loads and the fracture initiation angles for the RV-BD PMMA specimens are presented in Tables 2 and 3, respectively. In Table 2, Pi (i = 1, 2, 3) denotes the fracture loads for the three repeated tests. The parameter Pav denotes also the average of the three experimental fracture loads. Similar definitions can be found in Table 3 for the fracture initiation angle where 60i- and 60av are used. The fracture initiation angle 60 is defined in the next section.
The load-displacement graphs recorded from the tests on the RV-BD specimens are completely linear up to final fracture and for each specimen; fracture is seen to occur suddenly. Therefore, brittle fracture criteria based on the linear elastic notch fracture mechanics (LENFM) can generally be used to predict the experimental results. Figure 4 displays two samples of the load-displacement curves for the RV-BD specimens.
3. Mixed mode I/II brittle fracturecriteria
3.1. The round-tip V-notch maximum tangential stress criterion
The conventional maximum tangential stress (MTS) criterion is a well-known failure criterion frequently used for investigating mixed mode brittle fracture in cracked domains [64]. According to the MTS criterion, brittle fracture occurs
p = 1 mm, P = 40° (a); 2a = 60°, p = 0.5 mm, P = 30° (b); 2a = 30°,
when the tangential stress aee at a specified critical distance rc ahead of the crack tip attains the critical stress ctc. The parameters rc and ctc are often considered to be the material properties and independent of the geometry and loading conditions. Ayatollahi and co-researchers were the first who extended systematically the MTS criterion to sharp and blunt notched domains (e.g. the U-notch MTS (UMTS) [32], the sharp V-notch MTS (SV-MTS) [34] and the round-tip V-notch MTS [37] criteria). The first hypothesis of the round-tip V-notch MTS criterion suggests that brittle fracture initiates from a point on the notch round border where the tangential stress is a maximum and propagates radially along a direction which is perpendicular to the maximum tangential stress. The angular position of the fracture initiation point on the notch round border is called hereafter the fracture initiation angle e0 (Fig. 5, a). The RV-MTS criterion also proposes that brittle fracture in a round-tip V-notched component takes place when the tangential stress CTee at a critical radial distance rc from the notch border attains the critical stress ac. Figure 5 depicts a blunt V-notch with some definitions related to the notch geometry and the RV-MTS criterion.
It has been shown in several references by Ayatollahi and Torabi (see for instance Refs. [10, 11, 37]) that for brittle and quasi-brittle fracture assessment of V-notched components, the critical stress ac could successfully be considered to be equal to the ultimate tensile strength of material au. Moreover, if the notch fails by tensile stresses, the critical distance from the notch border can be considered to be equal to the
Table 2
The experimentally recorded fracture loads for the RV-BD PMMA specimens
2a p, mm ß P1? N P2, N P3, N P , N
30° 0.5 30° 3379 3862 3512 3717
40° 4263 4760 5055 4692
50° 6376 6863 7677 6972
1.0 30° 3630 3689 3232 3517
40° 4009 4084 4651 4248
50° - 6674 7334 7004
2.0 30° 3803 - 3936 3869
40° 5211 4106 4244 4520
50° - 5605 6314 5960
4.0 30° 4729 4376 4474 4526
40° 4779 6028 5681 5496
50° 5818 6647 5589 6018
60° 0.5 30° 4228 4502 4082 4271
40° 8374 8362 - 8368
45° 10942 11252 - 11097
1.0 30° 4149.6 4146 4285 4194
40° 7380 6073 8434 7296
45° 8859 8865 8890 8871
2.0 30° - 4215 4252 4234
40° 6360 6525 7017 6634
45° 7179 7324 - 7252
4.0 30° 5359 4436 5088 4961
40° 5401 6158 5335 5631
45° 7255 6293 - 6774
Table 3
The experimentally measured fracture initiation angles for the RV-BD PMMA specimens
2a p, mm ß ® 01 ® 02 ® 03 ® 0av
30° 0.5 30° 62° 60° 62° 61.3°
40° 73° 69° 72° 71.3°
50° 78° 81° 79° 79.3°
1.0 30° 63° 64° 62° 63.0°
40° 66° 64° 68° 66.0°
50° 76° 74° 75° 75.0°
2.0 30° 59° 56° 62° 59.0°
40° 61° 64° 66° 63.7°
50° 68° 72° 72° 70.7°
4.0 30° 54° 53° 52° 53.0°
40° 61° 58° 62° 60.3°
50° 64° 58° 65° 62.3°
60° 0.5 30° 53° 55° 53° 53.7°
40° 64° 62° 64° 63.3°
45° 65° 63° - 64.0°
1.0 30° 47° 48° 50° 48.3°
40° 58° 55° 57° 56.7°
45° 61° 61° 60° 60.7°
2.0 30° 48° 52° 51° 50.3°
40° 56° 55° 62° 57.7°
45° 58° 62° - 60.0°
4.0 30° 51° 50° 45° 48.7°
40° 54° 59° 54° 55.7°
45° 57° 56° - 56.5°
critical distance for sharp cracks as follows [10, 11, 37, 6567]:
In Eq. (1), Klc and au are the plane-strain fracture toughness and the ultimate tensile strength of material, respectively, which both are the material properties. According to Table 1 and Eq. (1), the value of the critical distance rc is equal to 0.1 mm for the tested PMMA.
-8000
-0.4 -0.8 Displacement, mm
-0.5 -1.0 -1.5 -2.0 Displacement, mm
-2.5
Fig. 4. Two samples of the load-displacement curves for the RV-BD PMMA specimens
3.2. The round-tip V-notch mean-stress criterion
Brittle fracture occurs in accordance with the round-tip V-notch mean-stress failure concept when the average value of the tangential stress over a specified critical distance dc ahead of the notch border attains the critical stress ac. The mean-stress failure concept has been frequently utilized to predict mode I, mode II and mixed mode I/II brittle fracture in engineering components containing notches of different features (see for example Refs. [6, 7, 17-19, 40]). A blunt V-notch with the critical distance of the RV-MS criterion is depicted in Fig. 5, b.
To predict the fracture load of a blunt V-notched component by means of the RV-MS criterion, the point on the notch border corresponding to the maximum tangential stress should first be determined. Then, a line is drawn from the notch center of curvature to this point and continued, and the average value of the tangential stress over the critical distance dc (Fig. 5, b) is computed. Finally, the load on the V-notched component is increased till the average stress attains the critical stress Gc (normally equal to the ultimate tensile strength of material a u). The corresponding load is known as the fracture load.
Seweryn [13] has suggested Eq. (2) to compute the critical distance dc for cracks and sharp notches that fail by tensile stresses (e.g. the tangential stress). Equation (2) has also been employed several times by other researchers for brittle fracture assessment of sharp and blunt notches under various loading conditions (see for instance Refs. [6, 7, 17-19, 40]):
dc
n
K,
(2)
is four times
Comparing Eqs. (1) and (2) indicates that rc. Therefore, the value of dc for the present PMMA material is simply computed to be equal to 0.4 mm.
It is shown in the next section by finite element analysis that although V-notches in the RV-BD specimens are loaded
under negative mode I conditions and the bisector line of the notch sustains compressive stresses, the right half border of the notch experiences tensile stresses. As mentioned in Sect. 2, the fracture in the RV-BD specimens occurs also from the right half border of the notch. Therefore, the obtained rc and dc values are allowed to be used in predicting the fracture load of the RV-BD specimens by means of the RV-MTS and RV-MS criteria, respectively.
4. Finite element analysis
A plane-stress model was created for each RV-BD specimen and the finite element analyses were performed by using the commercial code ABAQUS. Numerous 8-node quadratic elements were used at the notch neighborhood in a structured manner. Figure 6 shows a typical mesh pattern used for the RV-BD specimen with the notch angle of 30° and the notch radius of 4 mm. As shown in Fig. 6, refined meshes are used at the notch tip vicinity due to the high level of stress gradient. A polar coordinate system is defined for each finite element model in which the coordinate center locates on the notch center of curvature and all stresses are reported from this coordinate system.
To check the mesh-independency for the finite element analyses, various numbers of elements were used and the analyses were repeated to achieve a reasonable convergence. It was found that a total number of69 000 elements resulted in a desired convergence. The size of the elements at the first row close to the notch border is equal to about 0.008 mm. The boundary conditions for the finite element model of the RV-BD specimens are in a way that the point of the specimen at which the concentrated load P is applied can solely move in the radial direction and the opposite point is completely fixed. The concentrated load P (see Fig. 2) is also applied to the model at the loading angle P.
Figure 7 represents sample contours of tangential stresses at the vicinity of the notch border corresponding to 4520 N
Fig. 5. A blunt V-notch with its geometrical parameters and some defenitions related to the RV-MTS (a) and RV-MS criterion (b)
Fig. 6. The mesh pattern of the RV-BD specimen with the notch angle of 30° and the notch radius of 4 mm. The whole specimen (a) and the notch border vicinity (b)
load which is applied to the specimen at a loading angle P that is between PII and Pc. As stated in subsection 2.2, such a loading angle results in mixed mode I/II loading on the notch under negative mode I conditions.
These contours show that the notch sustains two types of stresses. At the right side of the notch bisector line, tensile stresses exist while at the left side, the notch sustains com-pressive stresses. Because, the domain of variation of P in the entire experiments and also in the finite element analyses is between PII and Pc, the tangential stresses on the notch bisector line are always compressive, so the notch undergoes negative mode I and tends to close. It is worth mentioning that for the cases that P is between 0° and PII, the right side of the notch bisector line experiences tension and left side compression too, but the difference is that for such cases the notch bisector line sustains tensile stresses and hence the notch undergoes positive mode I and tends to open.
As mentioned above, the left side of the notch bisector line is affected by compressive stresses while the right side by
tensile stresses. Therefore, there is a competition between compressive and tensile stresses for fracturing the RV-BD specimen. To find out which kind of stress would be predominant in this competition, the values of the stresses are carefully analyzed. As shown in Fig. 7, two radial paths are defined at the right and left sides of the notch bisector line. The distributions of the tangential stresses along these paths are depicted in Fig. 8.
According to the RV-MTS criterion, the tangential stress at the critical distance should overcome the material bonding strength for the initiation of fracture. As computed in subsection 3.1, the critical distance under tension is equal to 0.1 mm for the present PMMA material. It has been recently reported in Refs. [58, 59] that the critical distance for the PMMA under compression is about 8 times greater than that under tension meaning that the compressive critical distance for the present PMMA material is about 0.8 mm. To estimate the fracture load of the specimen by tensile stresses, the load is increased till the stress at the critical distance of 0.1 mm attains the ten-
S, S22 (CSYS-1) (Avg: 75%)
75.63
60.09
44.54
29.00
13.46
-2.09
-17.63
-33.18
- 48.72
- 64.27 -79.81 -95.36 -110.90
Compressive stress path -.yj^i
. Compression side
Fig. 7. Sample contours of tangential stresses at the vicinity of the notch border corresponding to the load of4520 N (2a = 30°, P = 40°, p = = 2 mm)
Fig. 8. The distributions of the tangential stress along the radial paths of the (a) right side and (b) left side of the notch bisector line. Tensile (a) and compressive stress distribution (b)
7060- <3 PH S ® 50- <x> b 40300. R; \ a
.0 0.2 0.4 0.6 0.8 1.0 idial distance from the notch border, mm
sile strength of material (68.5 MPa in Table 1). Accordingly, the tensile fracture load is found to be equal to 4681 N. A similar procedure is followed to estimate the fracture load by compressive stresses (when the compressive stress at the critical distance 0.8 mm reaches the compressive strength 210 MPa) which is found to be equal to 15 820 N. Comparing these two potential fracture loads suggests that fracture takes place from the tension side of the notch bisector line which is well confirmed by the experimental observations too (see Sect. 2). Note that similar estimations could also be performed by means of RV-MS criterion. From both the experimental observations and the finite element analyses, it can be concluded that mixed mode I/II brittle fracture takes place by tensile stresses in the RV-BD specimens even under negative mode I conditions.
After determining that fracture takes place from the tensile side of the RV-BD specimens (i.e. the right side of the notch bisector line), one can simply predict the fracture load of the entire specimens tested under negative mode I conditions by means of the RV-MTS and RV-MS criteria. Dealing with the RV-MTS criterion, the critical distance rc and the critical stress are obtained to be equal to 0.1 mm and 68.5 MPa for the tested PMMA, respectively. To obtain the fracture load, it is required to apply a unit external load (i.e. 1 N) to the finite element model of the RV-BD specimen and record the value of the tensile tangential stress at the critical distance 0.1 mm along the tensile stress path in MPa (herein referred to as g1 ). Because the stress analysis is linear elastic, the fracture load would be simply equal to 68.5/ g1 N.
Similar method can be used to predict the fracture load in accordance with the RV-MS criterion. For this purpose, first a unit load (1 N) is applied to the finite element model of the RV-BD PMMA specimen and the average value of the tensile tangential stresses over the critical distance dc = 0.4 mm along the tensile stress path is computed in MPa (herein referred to as a1). Then, the fracture load can be simply estimated to be equal to 68.5/ a1 N.
As mentioned in Sect. 3, the critical distances rc and dc presented in Eqs. (1) and (2), respectively are for sharp cracks
and they are assumed to be the material properties and independent of the notch geometry. In the next section, two procedures are elaborated to achieve the experimental values of the critical distances rc and dc for the RV-BD specimens. The values are obtained by using the experimental fracture loads of the RV-BD specimens and used for predicting the fracture loads of the entire RV-BD PMMA specimens by means of the RV-MTS and RV-MS criteria.
4.1. Determination of the experimental critical distances for the RV-MTS and RV-MS criteria
For predicting the value of fracture load for a RV-BD specimen by means of the RV-MTS and RV-MS criteria, the critical distance of each criterion is essentially needed. In the past sections, these critical distances were computed by using Eqs. (1) and (2) which have been frequently used for predicting brittle fracture in cracked and notched components [11]. In this subsection, an experimental approach is proposed for determining these critical distances. Due to high stress gradient, the critical distance values are more important in failure prediction of notches with small radii. Therefore, for determining experimentally the critical distances of the RV-MTS and RV-MS criteria, V-notches with the notch radius of 0.5 mm are selected.
Table 4
The experimental values of the critical distances for selected RV-BD PMMA specimens
2a p, mm P rc, mm dc, mm
0.5 30° 0.12 0.30
30° 0.5 40° 0.10 0.25
0.5 50° 0.08 0.19
0.5 30° 0.09 0.23
60° 0.5 40° 0.12 0.30
0.5 45° - -
Average = 0.1 Average = 0.25
Fig. 9. The variations of the fracture load versus the loading angle for the notch angle 2a = 30° and various notch radii p = 0.5 (a), 1.0 (b), 2.0 (c), 4.0 mm (d) together with the experimental results of the RV-BD specimens
Fig. 10. The variations of the fracture load versus the loading angle for the notch angle 2 a = 60° and various notch radii p = 0.5 (a), 1.0 (b), 2.0 (c), 4.0 mm (d) together with the experimental results of the RV-BD specimens
Table 5
The experimental and theoretical fracture loads for the RV-BD specimens (Pf) together with the discrepancies
for the RV-MTS and RV-MS criteria
2a p, mm P Pf,N
Exp. RV-MTS Discrepancies, % RV-MS* Discrepancies, % RV-MS** Discrepancies, %
30° 0.5 30° 3717 3512 5.5 4042 8.7 3531 5.0
40° 4692 4679 0.3 5300 12.0 4653 0.8
50° 6972 7370 5.7 8220 17.9 7433 6.6
1.0 30° 3517 3659 4.0 4153 18.1 3824 8.7
40° 4248 4614 8.6 5189 22.2 4694 10.5
50° 7004 6822 2.6 7588 8.4 6922 1.2
2.0 30° 3869 3986 3.0 4361 12.7 4063 5.0
40° 4520 4681 3.6 5091 12.6 4743 4.9
50° 5960 6288 5.5 6839 14.8 6389 7.2
4.0 30° 4527 4339 4.1 4593 1.5 4396 2.9
40° 5496 4722 14 4960 9.7 4747 13.6
50° 6018 5686 5.6 6021 0.1 5760 4.3
60° 0.5 30° 4271 4376 2.5 4936 15.0 4380 2.6
40° 8368 8028 4.1 8771 4.8 8073 3.5
45° 11097 12522 12.8 13201 19.0 12487 12.5
1.0 30° 4194 4247 1.3 4756 13.0 4322 3.0
40° 7296 7003 4.0 7694 5.5 7118 2.4
45° 8871 10003 12.0 10894 22.0 10170 14.6
2.0 30° 4234 4250 0.4 4639 9.6 4328 2.2
40° 6634 6208 6.4 6746 1.7 6306 4.9
45° 7252 8207 13.1 8870 22.0 8320 14.7
4.0 30° 4961 4301 13.2 4531 8.7 4356 12.1
40° 5631 5471 2.8 5782 2.7 5623 0.1
45° 6774 6666 1.6 7011 3.5 6736 0.6
Average 5.7 Average 11.1 Average 6.0
* dc from Eq. (2),
** dc from experiment.
To determine the critical distance of the RV-MTS criterion for each notch angle 2a° and loading angle P°, the average experimental fracture load of the corresponding RV-BD specimen is first applied to the finite element model of the specimen. Then the tensile tangential stress versus the radial distance from the notch border is plotted (like that represented in Fig. 8, a). The distance associated with the stress value of 68.5 MPa (the critical stress of material) is regarded as the critical distance of the RV-MTS criterion (i.e. rc). Table 4 presents the values of rc for different notch angles 2a° and various loading angles P° together with their average value. As given in Table 4, the average value of rc is obtained to be equal to 0.1 mm which is exactly the same value as that obtained from Eq. (1). Therefore, no difference is expected be-
tween the theoretical predictions of the RV-MTS criterion by using the theoretical (Eq. (1)) and experimental critical distances. It is seen in the next section that a single theoretical curve can be plotted for the RV-MTS criterion due to the unique critical distance.
A similar procedure can also be followed for determining the experimental value of the critical distance for the RV-MS criterion dc. The plotted tensile tangential stress versus the radial distance from the notch border described in the above paragraph is again used but for the RV-MS criterion. To determine the experimental value of dc, a trial-and-error procedure is used as follows: (i) select an initial value for dc (for example 0.4 mm), (ii) compute the average stress over the selected dc and compare it with the material critical stress (68.5 MPa) and
8000
racture load, N G\ OO O O O O O O o o o
Ph 2000-
PH
2000
8000
-p = 0.5 mm
____p = 1.0 mm
------p = 2.0 mm
................ p = 4.0 mm
2000
8000
RV-MTS criterion, dc from Eq. (2)
30c
35c
40c
45c
£ 2000 -
RV-MS criterion, dc from experiment
RV-MS criterion, dc from experiment
30°
35°
40° 45c
50° (3
30°
35c
40°
45c
50c
Fig. 11. The variations of the theoretical fracture load versus the loading angle for different notch angles 2a = 30° (a-c), 60° (d-f) and various notch radii p = 0.5 (1), 1.0 (2), 2.0 (3), 4.0 mm (4)
-p = 0.5 mm
____p = 1.0 mm
------p = 2.0 mm
................ p = 4.0 mm
cS 12H
a
<D
bû 4
IE
-p = 0.5 mm
____p = 1.0 mm
-----p = 2.0 mm
............. p = 4.0 mm
0.0 0.1 0.2 0.3 0.4 Distance from notch border, mm
oJ
0.0 0.1 0.2 0.3 0.4 Distance from notch border, mm
Fig. 12. Two samples of the variations of the tangential stress versus the distance from the notch border for different notch radii and the load P of 1000 N. 2a = 30° (a) and 60° (b), B = 50° (a) and 40° (b)
(iii) change the value of dc till the value of the average stress attains 68.5 MPa. The dc value that satisfies these conditions is known as the experimental critical distance for the RV-MS criterion.
The experimental dc values for different notch angles and various loading angles are presented in the last column of Table 4. The average value of dc for the entire RV-BD specimens with 0.5 mm notch radius is obtained to be equal to
Table 6
The theoretical and experimental fracture initiation angles of the RV-BD specimens including the discrepancies
2a p, mm P e0xp 00x2p 60x3p 90- qfem 90 Discrepancy, %
30° 0.5 30° 62° 60° 62° 61.3° 57° 7.7
40° 73° 69° 72° 71.3° 63° 11.7
50° 78° 81° 79° 79.3° 72° 9.3
1.0 30° 63° 64° 62° 63.0° 57° 9.5
40° 66° 64° 68° 66.0° 63° 4.5
50° 76° 74° 75° 75.0° 69° 8.0
2.0 30° 59° 56° 62° 59.0° 57° 3.4
40° 61° 64° 66° 63.7° 63° 1.0
50° 68° 72° 72° 70.7° 69° 2.6
4.0 30° 54° 53° 52° 53.0° 51° 3.8
40° 61° 58° 62° 60.3° 60° 0.6
50° 64° 58° 65° 62.3° 66° 5.9
60° 0.5 30° 53° 55° 53° 53.6° 48° 10.6
40° 64° 62° 64° 63.3° 57° 10.0
45° 65° 63° - 64.0° 60° 6.3
1.0 30° 47° 48° 50° 48.3° 48° 0.7
40° 58° 55° 57° 56.7° 54° 4.7
45° 61° 61° 60° 60.7° 57° 6.0
2.0 30° 48° 52° 51° 50.3° 48° 4.6
40° 56° 55° 62° 57.7° 54° 6.4
45° 58° 62° - 60.0° 57° 5.0
4.0 30° 51° 50° 45° 48.7° 48° 1.4
40° 54° 59° 54° 55.7° 54° 3.0
45° 57° 56° - 56.5° 54° 4.4
Average 5.4
0.25 mm which is significantly different from the theoretical critical distance computed from Eq. (2). Consequently, a meaningful difference is expected between the theoretical predictions of the RV-MS criterion by using the experimental (0.25 mm) and theoretical (0.4 mm) critical distances. In the next section, two various theoretical curves are plotted for the RV-MS criterion; one of them corresponds to dc = 0.4 mm and the other to dc = 0.25 mm. Then, the fracture loads and the fracture initiation angles of the RV-BD PMMA specimens are predicted by means of the RV-MTS and RV-MS criteria.
5. Results and discussion
The predictions of the two brittle fracture criteria, namely RV-MTS and RV-MS criteria, for the fracture loads of the RV-BD specimens are represented in Figs. 9 and 10 for the notch angles of 30° and 60°, respectively, together with the related experimental results. As mentioned in the past sections, two distinct values of the critical distances are obtained for the
RV-MS criterion; one of which from Eq. (2) and the other from experiments (see Table 4). Therefore, two distinct curves are seen in Figs. 9 and 10 for the RV-MS criterion regarding the two critical distances. As expected, the predictions of the RV-MS criterion based on the experimental value of the critical distance are qualitatively seen in Figs. 9 and 10 to be more accurate than those based on the theoretical one. Moreover, the predictions of the RV-MS criterion based on the theoretical critical distance are also acceptable. It is useful to note that the curves of the RV-MTS criterion in the entire plots of Figs. 9 and 10 are almost lie on those of the RV-MS criterion with experimental critical distance implying that one can simply use the RV-MTS criterion in the theoretical predictions instead of the RV-MS criterion that uses the experimental critical distance. Note that this conclusion is obtained for the specific specimens studied in this research and it should be further studied for other test samples and brittle materials.
It is seen in Figs. 9 and 10 that as the loading angle P increases, the fracture load of the RV-BD PMMA specimens enhances significantly. The finite element stress analyses indicate that for a constant external load, as P increases from 30°, the angular position of the maximum tangential stress on the notch border increases (measured from the notch bisector line) while its value decreases considerably. Thus, larger values of the external load are required for breaking the RV-BD specimens when the angle B becomes larger.
For evaluating the theoretical results of the RV-MTS and RV-MS criteria quantitatively, the percent discrepancies between the theoretical and average experimental results are presented in Table 5. The average discrepancy for the RV-MTS criterion is found to be approximately 5.7%, while it is obtained for the RV-MS criterion (using the experimental critical distance) to be equal to 6%. The discrepancies clearly demonstrate the excellent effectiveness of the two criteria. It is also seen in Table 5 that the total average discrepancy of the RV-MS criterion with the theoretical critical distance is about 11% suggesting a satisfactory accuracy of the criterion.
Figure 11 represents the variations of the theoretical fracture loads versus the loading angle B for different notch angles and various notch radii. It is seen in Fig. 11 that for the notch angle of 30° and the loading angles larger than 40°, increasing the notch radius results in decreasing the fracture load. Similar behavior is seen for the notch angle of 60° over the entire domain of the loading angle. Such a behavior seems to be strange because greater fracture loads are usually expected for larger notch radii as a result of decreasing the stress concentration. To justify this behavior, additional finite element stress analyses are performed for the RV-BD specimens.
For the notch angle of 30°, a finite element model is created for each notch radius. Then, a constant load of 1000 N is applied to the models at P = 50°. The finite element stress analyses are performed and the variations of the tangential stress versus the distance from the notch border are plotted in Fig. 12, a for various notch radii. It can be seen in Fig. 12, a that the value of the tangential stress at the critical distance (rc = = 0.1 mm) increases as the notch radius increases meaning that lower loads are needed for fracturing the RV-BD specimens having larger notch radii. Similar behavior can be found in Fig. 12, b for 2a = 60°, P = 40° and different notch radii.
Table 6 summarizes the theoretical and experimental results regarding the fracture initiation angle 60 of the RV-BD specimens including their percent discrepancies. The average discrepancy of about 5% for the entire PMMA specimens demonstrates the success of the fracture criteria, which are supported by the finite element analyses. Note that in Table 6, 60i- (i = 1, 2, 3) denotes the fracture initiation angles for the three reported experiments and 60av denotes the average experimental value.
To better understand the fracture initiation angle definition, the result obtained from the finite element stress analysis is depicted in Fig. 13, a for the RV-BD specimen of 2a = 30°,
Fig. 13. The fracture initiation angles for the RV-BD specimen of 2a = 30°, p = 1 mm and P = 40°: the finite element analysis (a) and the experimental result (b)
p = 1 mm and P = 40° and is compared with the corresponding specimen broken in the experiments (Fig. 13, b). It is seen from Fig. 13, a that the angular position of the maximum tangential stress on the notch border recognized by the RV-MTS and RV-MS criteria as the fracture initiation angle (63°) coincides well with the corresponding experimental value (66° in Fig. 13, b).
6. Conclusions
Mixed mode I/II brittle fracture was assessed both experimentally and theoretically for blunt V-notches under negative mode I conditions. The fracture experiments were carried out by means ofthe round-tip V-notched Brazilian disk specimen made of PMMA. The experimental observations and the finite element analyses indicated that although the V-notch in the RV-BD specimens was loaded under negative mode I conditions, brittle fracture took place by tensile stresses from the right-side border of the notch. Two brittle fracture criteria, namely the RV-MTS and RV-MS criteria were used to predict the fracture loads. Two values of the critical distances were considered in the theoretical assessment by each criterion. The first value was considered to be equal to the critical distance for sharp cracks and the second one was experimentally determined by means of the fracture loads of the RV-BD specimens of 0.5 mm radius. It was found that both of the values were almost identical for the RV-MTS criterion, while they were significantly different for the RV-MS criterion. Therefore, just one theoretical curve was obtained for the RV-MTS criterion, while two distinct curves were extracted for the RV-MS criterion. It was found that the predictions of the RV-MTS criterion were almost identical to those of the RV-
MS criterion which utilized the experimental critical distance in its predictions. By approximately 94.5% accuracy, it was found that both the RV-MTS criterion and the RV-MS criterion with the experimental critical distance could provide excellent predictions to the experimentally recorded fracture loads. Moreover, the RV-MS criterion which utilized the critical distance of sharp cracks was found to be accurate enough showing an overall discrepancy of 11% with the experimental results. Moreover, the RV-MTS criterion supported by the finite element analysis could successfully predict the experimental fracture initiation angles for the RV-BD specimens loaded under mixed mode I/II with negative mode I conditions.
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nocTynana b peaaKUHro 30.12.2015 r.
CeedeHun 06 aemopax
Ali Reza Torabi, PhD, University of Tehran, Iran, a_torabi@ut.ac.ir
Bahador Bahrami, M. Sc. graduate, Iran University of Science and Technology, Iran, bahramibahador@ymail.com Majid R. Ayatollahi, PhD, Prof., Director, Iran University of Science and Technology, Iran, m.ayat@iust.ac.ir
