Evolution of Crack Tip Constraint in a Mode II Elastic-Plastic Crack Problem

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.


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 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. .or such cases, which very often happen for metallic alloys, the failure mechanism can be either brittle fracture or ductile failure. .or 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 c J 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 c J 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 PHYSICAL MESOMECHANICS Vol. 21 No. 2 2018 fracture criteria. However, some modifications are needed to account for the effect of crack tip plasticity. .or 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 c J 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 I K 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 [57].
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 II , Q they developed a QT diagram to estimate the crack tip constraint from the T-stress. However, it is important to study the range of validity of the JT formulation using practical crack specimens when the external shear load gradually increases in mode II problems. In this paper the constraint parameter II Q is predicted from the QT dia-gram for a mode II specimen. The parameter II Q is also determined directly from the finite element results. These two calculated values of II Q are compared for increasing values of external load and the related results are discussed.

QT 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 T term 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 [57].
The boundary layer model can also be used for quantifying the crack tip constraint [6,7]. The mode I constraint parameter I Q corresponding to brittle fracture is determined as where 0 σ is the yield stress, θθ σ is the tangential stress and RE.
( ) θθ σ 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 II Q 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 be studied along the direction of maximum tangential stress 0 θ around the crack tip. Therefore, the constraint parameter in brittle fracture for mode II loading II Q can be determined from Here RE.
( ) θθ σ is either the HRR solution for θθ σ in 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 be-tween Q and the T-stress for hardening coefficient n = 3, 8, 13. .igure 1 shows the QT diagram for n = 8 under mode II loading.

.INITE 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 QT diagram using the value of the T-stress corresponding to the load. In this paper, the variation of II Q with applied load is obtained for a mode II specimen. The specimen, as shown in .ig. 2, is subjected to positive shear for tensile loading and negative shear for compressive loading [14,16]. The finite element results for II Q are used to study the extent of validity of the results obtained from the QT diagram.
The mode II specimen is considered to be elasticplastic with n = 8, α = 1.2 in the RambergOsgood stressstrain relation and with Youngs modulus E = 214 GPa, Poissons ratio ν = 0.3 and yield stress 0 σ = 400 MPa. To calculate the T-stress, the specimen was first simulated by an elastic finite element analysis with the Youngs modulus and Poissons 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 I K is negligible relative to II . K Therefore, the specimen can be considered as a mode II crack specimen. The J-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.

VARIATION O. Q II WITH T .OR THE MODE II SPECIMEN
To study the evolution of crack tip constraint, two finite element analyses are carried out for tensile and compressive loading. In the first analysis, II Q 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 II Q is calculated at r = 0 2J σ along the direction of maximum tangential stress at different load increments throughout the analysis.
.or 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 s P can be written in general as s s , Y is a constant factor depending on the type of loading. The factor s Y is +5.6 and 5.6 MPa/kN for the positive shear and negative shear models, respectively. .or the second analysis, the T-stress is determined (using Eq. (3)) at the same loads used to calculate II Q in the first finite element analysis. These values of T are employed to determine II Q from .ig. 1 according to the QT diagram for n = 8.

RESULTS AND DISCUSSION
.igure 3a shows the results for II Q obtained from the full field solution compared with those determined from the QT diagram for the positive T specimen (tensile loading). Similar results are shown in .ig. 3b for the negative T specimen (compressive loading). It is .ig. 1. Q II T diagram for n = 8 [15]. 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. .or loads higher than those to cause full plasticity, the absolute value of II Q 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 .ig. 3 that the extent of agreement between the results of the full field solution and those of the QT diagram and also the onset of the drop in the results of the full field solution vary slightly for +T and T shear specimens.
.igure 4 displays the variation of the tangential stress θθ σ normalized with respect to the yield stress 0 σ obtained from the first finite element analysis with elastic-plastic behavior for the specimen. .igure 4a shows the variations of 0 θθ σ σ with the normalized distance 0 r J σ along the direction of maximum tangential stress 0 θ for the positive T specimen (tensile loading). .igure 4b 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 where a is the crack length.
.or the positive T specimen (.ig. 4a) the tangential stress initially increases until a load corresponding to 0 log( ( )) J aσ = 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 < 0 r J σ < 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 .ig. 3a. Since the T-stress is positive in tensile loading, the constraint parameter II Q calculated using the QT diagram increases as shown in .ig. 3a. However, the QT diagram, obtained for small to moderate scale yielding, does not give accurate results for large scale yielding. Therefore, the JT approach cannot be used beyond 0 log( ( )) J aσ = 3.4 where the tangential stress begins to reduce due to excessive plastic deformation  (b) (a) and loss of constraint. The stresses are still parallel up to full plasticity implying that the JQ 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 JQ approach is not suitable to describe the crack tip stresses.
.igure 4b shows that for the negative T specimen, the tangential stress is always below the HRR solution. The stresses are parallel between 1 < 0 r J σ < 5 up to 0 log( ( )) J aσ = 3.1 and diverge considerably beyond it. It is seen from .ig. 3b that again the JT approach is valid for small to moderate scale yielding, the JQ approach can be used up to full plasticity and that a two-parameter characterisation is no longer applicable beyond the full plasticity.

CONCLUSIONS
The mode II constraint parameter II Q was determined in terms of 0 T σ 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 II Q was calculated for two types of shear loading using both the full field solution and the QT diagram. The results of the QT 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 JT approach can be used for small to moderate scale yielding and the JQ approach can be used up to full plasticity.