CC BY
129
УДК 539.42
Evolution of crack tip constraint in a mode II elastic-plastic crack problem
M.R. Ayatollahi1, F. Berto2
1 School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran, 16846, Iran 2 Department of Engineering Design and Materials, Norwegian University of Science and Technology, Trondheim, 7491, Norway
Numerous studies have shown that crack tip constraint has an important effect on the level of conservatism when crack extension is investigated in elastic-plastic fracture mechanics. Constraint effect has been explored extensively in the past but mainly for pure mode I problems. Very few researchers have dealt with the effects of crack tip constraint on mode II or mixed mode I/II fracture in metallic materials. In this paper, the evolution of mode II constraint parameter Q in terms of applied external load is determined numerically for a test specimen under pure mode II loading. The finite element method is utilized to model the specimen and to study the range of validity of mode II constraint parameter determined from a Q-T diagram. The parameter Q calculated from the finite element simulation (or from the full field solution) is compared with the values of Q determined from the Q-T diagram. For low levels of load, the results of full field solution are shown to be consistent well with the results obtained from the Q-T diagram. However, when the external load increases significantly, the results of Q-T diagram are no longer accurate and mode II constraint parameter Q should be calculated directly from finite element results.
Keywords: crack tip constraint, mode II, elastic-plastic fracture mechanics, finite element analysis
DOI 10.24411/1683-805X-2018-11011
Развитие стесненной деформации в вершине трещины при упругопластическом разрушении в условиях сдвиговой нагрузки
M.R. Ayatollahi1, F. Berto2
1 Иранский университет науки и технологии, Тегеран, 16846, Иран 2 Норвежский университет естественных и технических наук, Тронхейм, 7491, Норвегия
Многочисленные исследования показали, что стесненная деформация в вершине трещины имеет существенное значение при изучении распространения трещины в рамках механики упругопластического разрушения. Влияние стесненности деформации подробно изучено в основном для задач чистого отрыва. Очень мало работ посвящено рассмотрению влияния стесненности деформации в вершине трещины на разрушение типа II или смешанного типа I/II в металлических материалах. В настоящей работе проведено численное исследование параметра стесненности деформации при сдвиговом нагружении Q для образца в условиях чистого сдвига. С помощью метода конечных элементов проведено моделирование образца, а также изучен диапазон допустимых значений параметра стесненности сдвиговой деформации, определенный на основе диаграммы Q-T. Проведено сравнение параметра Q, рассчитанного путем моделирования конечных элементов (или на основе полного полевого решения), со значениями Q, определенными на основе диаграммы Q-T. Показано, что при низких уровнях нагрузки результаты полного полевого решения хорошо согласуются с результатами, полученными на основе диаграммы Q-T. Однако при значительном увеличении внешней нагрузки результаты на основе диаграммы Q-T становятся неточными и параметр стесненности деформации при сдвиговом нагружении Q должен рассчитываться на основе результатов, полученных методом конечных элементов.
Ключевые слова: стесненность деформации в вершине трещины, сдвиг, механика упругопластического разрушения, анализ конечных элементов
1. Introduction
The effects of geometry and loading conditions in brittle and ductile fracture have received much attention in recent decades, particularly in the power industry. The dependency of fracture toughness on the geometry and loading type of
cracked components is often attributed to the level of constraint around the crack tip. The traditional failure assessment methods based on high constraint fracture tests can lead to costly and unnecessary inspections and repairs. To avoid excessive conservatism in the safety assessment of
© Ayatollahi M.R., Berto F., 2018
components, the fracture toughness should be measured using specimens having the same level of constraint as that for the defective body. However, this requires appropriate parameters to quantify the crack tip constraint in mode I, pure mode II or mixed mode I/II crack problems.
Elastic-plastic fracture mechanics deals with cracked specimens in which a significant volume around the crack tip undergoes plastic deformation prior to the initiation of fracture. For such cases, which very often happen for metallic alloys, the failure mechanism can be either brittle fracture or ductile failure. For metallic specimens failing by the mechanism of cleavage fracture under static loading, the unstable crack growth takes place when the path independent integral J attains a critical value Jc which is a material property. Because the stress field inside the plastic zone near the crack tip is often described by J, the critical value Jc corresponds to the critical stress needed for initiation of crack extension in stress controlled models for brittle fracture.
Mixed mode specimens can also fail by the mechanism of brittle fracture even in the presence of significant plasticity around the crack tip. This has been shown for example through experiments carried out by Maccagno and Knott [1] for several steel alloys. The direction and the onset of crack growth for such cases can often be predicted by using the mixed mode fracture criteria. However, some modifications are needed to account for the effect of crack tip plasticity. For example, the maximum tangential stress (MTS) criterion [2] can be extended to solve the elastic-plastic crack problem in mixed mode loading. Maccagno and Knott [1, 3] showed for several steel alloys that the fracture load predicted using the elastic-plastic MTS criterion is in better agreement with the experimental results than that predicted by the linear elastic MTS criterion.
Meanwhile, the experimental studies for mode I crack show that the fracture toughness obtained from different conventional cracked specimens made of a given material are not the same. This indicates that the fracture toughness or the critical value of J for fracture initiation Jc is not merely a material property but depends also on the geometry and loading configurations. The geometry dependency of fracture toughness can be attributed to the effect of the crack tip constraint. Based on the classical theories of fracture mechanics, the stresses and strains around the tip of a mode I crack can be characterised by a single parameter such as Kj or J. This is true only when certain size restrictions are applied for each crack specimen [4]. However, the geometry dependency of the fracture toughness suggests that at least a second parameter like T or Q is required to predict the critical conditions for crack growth in different specimens [5-7].
Very little research has been carried out to study the effect of constraint in mixed mode I/II loading [8-12]. Those studies also deal only with quantification of constraint in
mixed mode loading under given loads and not its evolution with increasing loads. In particular, there is almost no paper concentrating on crack tip constraint and its evolution in pure mode II. This is partly because of a common assumption that the T-stress is always zero for mode II deformation. However, Ayatollahi et al. [13, 14] have shown that there are many real mode II loading conditions involving significant values of positive or negative T. As an exception, the effects of a far field T-stress on the near crack-tip elastic-plastic stresses have been investigated for mode II deformation by Ayatollahi et al [15]. Using a mode II constraint parameter Qjj, they developed a Q-T diagram to estimate the crack tip constraint from the T-stress. However, it is important to study the range of validity of the J-T formulation using practical crack specimens when the external shear load gradually increases in mode II problems. In this paper the constraint parameter Qjj is predicted from the Q-T diagram for a mode II specimen. The parameter Qjj is also determined directly from the finite element results. These two calculated values of Qjj are compared for increasing values of external load and the related results are discussed.
2. Q-T relation in mode II
It is common to use a so-called boundary layer model to study the effect of crack tip parameters on the stresses inside the plastic zone. In the boundary layer model a crack is considered in a circular region so that the crack tip is placed in the center of the region. The elastic stresses or displacements corresponding to the singular term and Tterm in the Williams' series expansions are applied to the boundary of the region. Material properties are considered to be elastic-plastic. To ensure the conditions necessary for contained yielding, the magnitudes of the boundary conditions should be limited to a level at which the maximum radius of plastic zone is small compared with the radius of the circular region. If the T term on the boundary is zero, the stresses inside the plastic zone are in good agreement with the stresses given by the HRR solution [5-7].
The boundary layer model can also be used for quantifying the crack tip constraint [6, 7]. The mode I constraint parameter QI corresponding to brittle fracture is determined as
Qj = ^99 - (^99 )ref along 0 = 0 f0r 1 < ^L <5, (1) a0 J where ct0 is the yield stress, ct99 is the tangential stress and (ct99 )ref is either the HRR solution for mode I or the boundary layer solution for small scale yielding with T = 0. A similar formulation can be used to determine QII for mode II cracks [15]. However, in this case, brittle fracture no longer takes place along the crack line. If the maximum tangential stress criterion [2] is adopted for predicting the direction of fracture initiation, the crack tip constraint should
Normalized' Fig. 1. Qn-T diagram for n = 8 [15]
be studied along the direction of maximum tangential stress 0O around the crack tip. Therefore, the constraint parameter in brittle fracture for mode II loading Qn can be determined from
Qii
i - (Gee )
'REF
r Gn
at e = e0 for 1 <-°- < 5. (2)
J
in
Here (gee )REF is either the HRR solution for gee mode II or the mode II boundary layer solution for small scale yielding (T = 0). Using a set of finite element analysis Ayatollahi et al [15] derived the relation between Q and the T-stress for hardening coefficient n = 3, 8, 13. Figure 1 shows the Q—T diagram for n = 8 under mode II loading.
3. Finite element modeling
In real specimens, the variation of the constraint parameter Q with load can be determined directly from the near crack tip stresses using finite element results. Alternatively, the Q parameter can be predicted from a Q-T diagram using the value of the T-stress corresponding to the load. In this paper, the variation of QII with applied load is obtained for a mode II specimen. The specimen, as shown in Fig. 2, is subjected to positive shear for tensile loading and negative shear for compressive loading [14, 16]. The finite element results for QII are used to study the extent of validity of the results obtained from the Q-T diagram.
The mode II specimen is considered to be elastic-plastic with n = 8, a = 1.2 in the Ramberg-Osgood stressstrain relation and with Young's modulus E = 214 GPa, Poisson's ratio v = 0.3 and yield stress g0 = 400 MPa. To
calculate the T-stress, the specimen was first simulated by an elastic finite element analysis with the Young's modulus and Poisson's ratio given above. The specimen was subjected to compressive and tensile reference loads of the same magnitude 5 kN. A comparison of the displacement components along the crack faces showed that the mode I stress intensity factor KI is negligible relative to Kn. Therefore, the specimen can be considered as a mode II crack specimen. The ./-integral was equal to 2027 N/m for both cases of tensile and compressive loading. The T-stress was determined by using the displacement method described in [14] for mixed mode loading. The value of T-stress for compressive loading was -28 MPa and for tensile loading was +28 MPa. With reference to the sign of the T-stress, in the present analysis the mode II specimen is called a positive T shear specimen for tensile loading and a negative T shear specimen for compressive loading.
4. Variation of Qn with T for the mode II specimen
To study the evolution of crack tip constraint, two finite element analyses are carried out for tensile and com-pressive loading. In the first analysis, QII is determined from a full field solution. The specimen is considered to be elastic-plastic and in the state of plane strain. The shear load is increased beyond the load at which full plasticity takes place in front of the crack tip. The constraint parameter QII is calculated at r = 2 J/ g0 along the direction of maximum tangential stress at different load increments throughout the analysis.
For the second analysis the relation between the T-stress and the applied load should be known. Using the reference elastic analysis described in the previous section, the relation between the T-stress and the applied load Ps can be written in general as
T = 7s Ps,
(3)
where Ys is a constant factor depending on the type of loading. The factor Ys is +5.6 and -5.6 MPa/kN for the positive shear and negative shear models, respectively. For the second analysis, the T-stress is determined (using Eq. (3)) at the same loads used to calculate QII in the first finite element analysis. These values of T are employed to determine QII from Fig. 1 according to the Q-T diagram for
Fig. 2. Mode II specimen. All dimensions are in mm. Thickness 20 mm
5. Results and discussion
Figure 3, a shows the results for QII obtained from the full field solution compared with those determined from the Q-T diagram for the positive T specimen (tensile loading). Similar results are shown in Fig. 3, b for the negative T specimen (compressive loading). It is seen that the results of the two approaches are in good agreement but only for lower load levels. As the load is increased, the difference between the results becomes significant. For loads higher than those to cause full plasticity, the absolute value
^0.4
• J-T formulation v J-Q formulation
0.2 0.4 0.6 0.5
Normalized T-stress 77g0
-0.8 -0.6 -0.4 -0.2 Normalized T-stress 77g0
Fig. 3. Variations of constraint with load for tensile (a) and compressive loading (b)
of Qn drops significantly by increasing load. This is mainly due to the excessive plastic deformation leading to the relief of constraint around the crack tip. It is observed from Fig. 3 that the extent of agreement between the results of the full field solution and those of the Q-T diagram and also the onset of the drop in the results of the full field solution vary slightly for +T and —T shear specimens.
Figure 4 displays the variation of the tangential stress g00 normalized with respect to the yield stress g0 obtained from the first finite element analysis with elastic-plastic behavior for the specimen. Figure 4, a shows the variations of g00/go with the normalized distance rG0/ J along the direction of maximum tangential stress 0o for the positive T specimen (tensile loading). Figure 4, b displays similar results but for the negative T specimen. In both figures, the tangential stress has been shown at different levels of load represented here by log(J/(aG0)) where a is the crack length.
For the positive T specimen (Fig. 4, a) the tangential stress initially increases until a load corresponding to log( JI (aG0)) = -3.4 and then decreases gradually below the small scale yielding solution with T = 0 (or the HRR
solution). The stresses are almost parallel for 1 < rG0/ J < < 5, although at higher levels of load the stress curves diverge slightly. The change in the stress curves can also be predicted by the results shown in Fig. 3, a. Since the T-stress is positive in tensile loading, the constraint parameter Qjj calculated using the Q-T diagram increases as shown in Fig. 3, a. However, the Q-T diagram, obtained for small to moderate scale yielding, does not give accurate results for large scale yielding. Therefore, the J-T approach cannot be used beyond log( J/(aG 0)) = -3.4 where the tangential stress begins to reduce due to excessive plastic deformation and loss of constraint. The stresses are still parallel up to full plasticity implying that the J-Q approach is valid for larger extents of plastic deformation. However, for loads higher than that corresponding to full plasticity, the stress curves diverge gradually and the J-Q approach is not suitable to describe the crack tip stresses.
Figure 4, b shows that for the negative T specimen, the tangential stress is always below the HRR solution. The stresses are parallel between 1 < rgo/ J < 5 up to log( JI (ag0)) = -3.1 and diverge considerably beyond it. It is seen from Fig. 3, b that again the J-T approach is valid
Fig. 4. Radial variations of tangential stress along 0O for tensile (a) and compressive loading (b)
for small to moderate scale yielding, the J-Q approach can be used up to full plasticity and that a two-parameter characterisation is no longer applicable beyond the full plasticity.
6. Conclusions
The mode II constraint parameter QII was determined in terms of T/g0 for small scale yielding. It can be expected that the mode II fracture toughness in brittle materials increases for shear specimens having a negative T-stress and decreases for those having a positive T-stress.
The parameter QII was calculated for two types of shear loading using both the full field solution and the Q-T diagram. The results of the Q-T diagram were in agreement with those of the full field solution for small to moderate scale yielding but not for large scale yielding.
Elastic-plastic finite element analysis of the shear specimen showed that the near crack tip tangential stresses can be predicted for constrained yielding using a two-parameter characterization approach. The J-T approach can be used for small to moderate scale yielding and the J-Q approach can be used up to full plasticity.
References
1. Maccagno T.M., Knott J.F. The low temperature brittle fracture behaviour of steel in mixed mode I and II // Eng. Fract. Mech. - 1991. -V. 38. - No. 2/3. - P. 111-128.
2. Erdogan F., Sih G.C. On the crack extension in plates under plane loading and transverse shear // J. Basic Eng. Trans. ASME. - 1963. -V. 85. - P. 519-525.
3. Maccagno T.M., Knott J.F. Brittle fracture under mixed mode I and II loading // Int. J. Fract. - 1985. - V. 29. - P. R49-R57.
4. Shih C.F., German M.D. Requirements for a one parameter characterization of crack tip fields by the HRR singularity // Int. J. Fract. -1981. - V. 17. - No. 1. - P. 27-43.
5. Betegon C., Hancock J. W. Two-parameter characterization of elastic-plastic crack-tip fields // J. Appl. Mech. - 1991. - V. 58. - P. 104110.
6. O'DowdN.P., Shih F.C. Family of crack-tip fields characterized by a triaxiality parameter. I. Structure of field // J. Mech. Phys. Solid. -
1991. - V. 39. - No. 8. - P. 989-1015.
7. O'DowdN.P., Shih F.C. Family of crack-tip fields characterized by a triaxiality parameter. II. Fracture applications // J. Mech. Phys. Solid. -
1992. - V. 40. - No. 5. - P. 939-963.
8. Du Z.Z., Betegon C., Hancock J. W. J dominance in mixed mode loading // Int. J. Fract. - 1991. - V. 52. - P. 191-206.
9. Shlyannikov V., Tumanov A. Characterization of crack tip stress fields in test specimens using mode mixity parameters // Int. J. Fract. -2014. - V. 185. - No. 1-2. - P. 49-76.
10. Qin H.B., Zhang X.P., Zhou M.B., Zeng J.B., Mai Y.-W. Size and constraint effects on mechanical and fracture behavior of micro-scale Ni/Sn3.0Ag0.5Cu/Ni solder joints // Mater. Sci. Eng. A. - 2014. -V. 617. - No. 1. - P. 14-23.
11. Matvienko Y. The effect of out-of-plane constraint in terms of the T-stress in connection with specimen thickness // Theor. Appl. Fract. Mech. - 2015. - V. 80. - P. 49-56.
12. Ding P., Wang X. Three-dimensional mixed-mode (I and II) crack-front fields in ductile thin plates—Effects of T-stress // Fatig. Fract. Eng. Mater. Struct. - 2017. - V. 40. - No. 3. - P. 349-363.
13. Ayatollahi M.R., Pavier M.J., Smith D.J. Determination of T-stress from finite element analysis for mode I and mixed mode I/II loading // Int. J. Fract. - 1998. - V. 91. - No. 3. - P. 283-298.
14. Ayatollahi M.R., Zakeri M. An improved definition for mode I and mode II crack problems // Eng. Fract. Mech. - 2017. - V. 175. - P. 235246.
15. Ayatollahi M.R., Smith D.J., Pavier M.J. Crack-tip constraint in mode II deformation // Int. J. Fract. - 2002. - V. 113. - No. 2. - P. 153-173.
16. Smith D.J., Ayatollahi M.R., Pavier M.J. On the consequences of T-stress in elastic brittle fracture // Proc. Roy. Soc. A. - 2006. -V. 462. - P. 2415-2437.
Поступила в редакцию 11.05.2017 г.
Сведения об авторах
Majid R. Ayatollahi, PhD, Prof., Director, Iran University of Science and Technology, m.ayat@iust.ac.ir Filippo Berto, Prof., Norwegian University of Science and Technology, Norway, filippo.berto@ntnu.no
