A numerical study of crack tip constraint in ductile single crystals

ARTICLE IN PRESS

Journal of the Mechanics and Physics of Solids 56 (2008) 2265–2286 www.elsevier.com/locate/jmps

A numerical study of crack tip constraint in ductile single crystals Swapnil D. Patila, R. Narasimhana,, R.K. Mishrab a

Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, India b General Motors Corporation, 30500 Mound Road, Warren, MI 48090, USA

Received 7 July 2007; received in revised form 31 December 2007; accepted 15 January 2008

Abstract In this work, the effect of crack tip constraint on near-tip stress and deformation fields in a ductile FCC single crystal is studied under mode I, plane strain conditions. To this end, modified boundary layer simulations within crystal plasticity framework are performed, neglecting elastic anisotropy. The first and second terms of the isotropic elastic crack tip field, which are governed by the stress intensity factor K and T-stress, are prescribed as remote boundary conditions and solutions pertaining to different levels of T-stress are generated. It is found that the near-tip deformation field, especially, the development of kink or slip shear bands, is sensitive to the constraint level. The stress distribution and the size and shape of the plastic zone near the crack tip are also strongly influenced by the level of T-stress, with progressive loss of crack tip constraint occurring as T-stress becomes more negative. A family of near-tip fields is obtained which are characterized by two terms (such as K and T or J and a constraint parameter Q) as in isotropic plastic solids. r 2008 Elsevier Ltd. All rights reserved. Keywords: Single crystal; Constraint effects; Crack tip fields; Modified boundary layer analysis; Finite elements

1. Introduction There have been rapid strides in modeling capabilities in recent years, which have enabled simulation of the behavior of individual grains in a polycrystalline aggregate through finite element computations employing crystal plasticity theory. These computations can provide an understanding of the role of texture and hardening considering dislocation interaction on fracture and formability of engineering alloys. As a first step in this direction, it is important to investigate plastic deformation at a crack tip in a ductile single crystal. In this context, it must be noted that when the crack opening displacement is much less than the grain size, the crack tip fields are contained in a single grain. The need to understand ductile–brittle transition also requires characterization of the slip patterns arising in the vicinity of crack tips in ductile single crystals. Further, some key structural components are being fabricated in single crystal form. For example, blades in high pressure

Corresponding author.

E-mail address: [email protected] (R. Narasimhan). 0022-5096/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2008.01.002

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turbines of jet engines are made of single crystal, nickel-based superalloys. Therefore, it is necessary to understand and characterize the stress and deformation fields present at the tip of a crack in a single crystal. Rice (1987) proposed an asymptotic solution for the crack tip stress field in ductile single crystals under mode I plane strain conditions within small strain, ideal plasticity framework. His analysis considered the ¯ direction, and a crack on the (1 0 1) plane with cases of a crack on the (0 1 0) plane with crack front along ½1 0 1 ¯ direction for FCC and BCC crystals, respectively. The motivation for considering crack front along ½1 0 1 these crack orientations is because they have been frequently observed to occur in experimental studies on fracture of ductile single crystals (see, Neumann, 1974a, b; Garrett and Knott, 1975). The solution of Rice (1987) consists of sectors of constant stress, with concentrated kink and slip shear deformation on the sector boundaries. Subsequently, Saeedvafa and Rice (1989) extended these results by assuming power law hardening and proposed HRR-type asymptotic solutions for crack tip singular fields. A similar solution was also derived by Cuitino and Ortiz (1996) for FCC crystals obeying both diagonal and isotropic power law hardening. Drugan (2001) proposed asymptotic solutions near a stationary crack tip in an elastic-ideally plastic ductile single crystal that do not contain kink-type plastic shearing bands. Rice et al. (1990) conducted preliminary finite element simulations under 2D plane strain, small scale yielding (SSY) conditions in ductile single crystals within small strain plasticity framework. They found that the near-tip fields are consistent with the asymptotic solution of Rice (1987). They also reported some preliminary computational results for the center cracked panel (CCP) configuration considering a planar double-slip model. Mohan et al. (1992) subsequently performed 2D plane strain finite element analysis of a stationary crack tip in FCC and BCC crystals subjected to mode I loading under SSY conditions accounting for finite deformation and lattice rotations. Their calculations were based on a saturation-type hardening rule. Their observations are in partial agreement with earlier analytical and numerical solutions (Rice, 1987; Rice et al., 1990). Cuitino and Ortiz (1996) conducted 3D finite element analysis of the four point bending configuration using a dislocation hardening model. Their results showed notable differences in slip activity at the free surface and in the interior of the specimen. In a recent study, Flouriot et al. (2003) carried out 3D finite element simulations of a compact tension (CT) specimen made from a single crystal of nickel-based superalloy. Their computations were performed for three different orientations by assuming elastic-ideally plastic behavior. A good agreement was found between numerical results and experimental data. Motivated by the predictions of the above noted analytical and finite element solutions, some researchers have experimentally investigated the near-tip deformation fields in a variety of single crystals. Shield and Kim (1994), Shield (1996) and Crone and Shield (2001) studied these fields in a four point bend single crystal specimen using Moire´ interferometry. Their experimental results follow the general structure of the analytical solution (Rice, 1987), namely existence of constant stress sectors with sharp boundaries. Kysar and Briant (2002) performed electron back-scattered diffraction (EBSD) analysis of an aluminum single crystal specimen subjected to mode I loading. Their results showed existence of the kink shear sector boundary consistent with Rice’s (1987) solution. Flouriot et al. (2003) also demonstrated the presence of kink bands near the crack tip at the free surface in CT specimens of nickel-based superalloy (FCC) single crystals through EBSD analysis. Although several studies on crack tip fields in ductile single crystals have been undertaken in the literature as discussed above, some important issues still need to be addressed. The influence of higher order terms in the asymptotic solution which have been shown to play an important role in isotropic plastic solids (see, for example, Sharma and Aravas, 1991; O’Dowd and Shih, 1991) has not been examined in the context of single crystals. In particular, O’Dowd and Shih (1991) have demonstrated that a two-parameter characterization of crack tip fields involving J and a triaxiality (or constraint) parameter Q is necessary to satisfactorily describe the configuration dependence of fracture response in isotropic plastic solids, especially under large scale yielding conditions. They found that in tension dominated geometries, such as CCP and single edge notch under tension (SENT), Q can attain a significantly negative value leading to loss of crack tip constraint or stress triaxiality. Rice (1987) noted that his asymptotic solution may not apply under large scale yielding conditions in low constraint (single crystal) fracture geometries such as those mentioned above wherein the stresses around the crack tip may be much lower. He also pointed out the possible strong configuration dependence of the near-tip deformation fields. Thus, considerable work is needed to understand these issues pertaining to fracture of ductile single crystals.

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The objective of the present work is to examine the effect of crack tip constraint on near-tip fields in a FCC single crystal having the orientation chosen by Rice (1987) under mode I plane strain condition. In order to simplify the analysis and interpretation, elastic anisotropy is ignored. A family of crack tip fields is generated by a two-parameter (K2T based) modified boundary layer approach. The finite element simulations are carried out within a continuum crystal plasticity framework and isotropic hardening response characterized by the Pierce–Asaro–Needleman (PAN) model (Peirce et al., 1983) is assumed. The results show that the near-tip deformation field, especially, the development of kink or slip shear bands, is sensitive to the constraint level. The stress distribution and the size and shape of plastic zone near the crack tip are also strongly influenced by the level of T-stress. The value of the constraint parameter Q becomes highly negative with increase in T-stress in the negative direction, which is akin to isotropic plastic solids. This will have profound implications on crack initiation by cleavage cracking as well as by micro-void growth and coalescence in ductile single crystals. 2. Background Rice (1987) performed an asymptotic analysis of the crack tip stress and deformation fields at the tip of a stationary crack in both FCC and BCC single crystals subjected to mode I loading under plane strain conditions. The behavior of the ductile single crystal was modeled as rigid-ideally plastic, and slip line solutions were sought within the small-strain framework. For the case of FCC single crystal, the plane of the crack was taken to coincide with the (0 1 0) plane, and the crack front was chosen to lie along ½1¯ 0 1 direction. Three combinations of slip systems which result in plane strain deformation were considered. The first combination involves systems of the type ð1 1 1Þ½1 1¯ 0 and ð1 1 1Þ½0 1¯ 1 which jointly give rise to effective inplane shearing along the ½1 2¯ 1 direction. The second pair comprises of the systems ð1 1¯ 1Þ½1 1 0 and ð1 1¯ 1Þ½0 1 1 which combine to produce effective in-plane slip along the [1 2 1] direction. The third pair of ¯ 0 1 and ð1¯ 1 1Þ½1 0 1, operating jointly also result in plane strain deformation along the [1 0 1] systems, ð1 1 1Þ½1 direction. The unit vectors along and normal to these three slip line traces are denoted as ðSð1Þ ; Nð1Þ Þ, ðSð2Þ ; Nð2Þ Þ and ðSð3Þ ; Nð3Þ Þ, respectively, as shown in Fig. 1. These three slip line traces are inclined at 54.71, 125.31 and 01, respectively, with respect to the positive X 1 axis as indicated in Fig. 1. Rice (1987) showed that for a stationary crack having the orientation depicted in Fig. 1, the yield condition can be met asymptotically in all angular sectors around the crack tip, thus requiring discontinuities in the stress state. X2

[010]

S(2)

S(1)

N(1)

N(2) r

e

θ

X1 [101] N(3) S(3)

Slip plane traces Fig. 1. Schematic diagram showing families of straight lines which are traces of slip plane intersections with the plane of deformation (X 3 ¼ constant). Here SðaÞ and NðaÞ are unit vectors along and normal to the slip line trace for system (a).

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The slip line sectors in the asymptotic solution of Rice (1987), which assumes active plastic flow all around the crack tip, are depicted in Fig. 2. The cartesian stress components prevailing in the four constant stress sectors labeled as A, B, C and D in Fig. 2 are summarized in Table 1. The sector boundaries are necessarily stress and displacement discontinuities emanating from crack tip. Further, the orientation of a sector boundary is constrained to lie either parallel or perpendicular to one of the effective slip line traces mentioned above (see Figs. 1 and 2). Thus, the boundaries between sectors A and B, C and D, which are aligned parallel to the slip traces Sð1Þ and Sð2Þ , respectively, produce slip shear bands. By contrast, the boundary between sectors B and C is along the X 2 axis, which is perpendicular to the slip trace Sð3Þ and hence, produces a kink shear band. The slip shear mode is accommodated by edge dislocations emanating from the crack tip. Thus, in this mode, the crack tip itself acts as a dislocation source. By contrast, the kink shear mode necessitates the formation of dislocation dipoles (Rice, 1987). The activation of this mode requires abundant internal sources of dislocation emission and involves extensive lattice rotation. The solution depicted in Fig. 2 and Table 1 may apply under SSY conditions as well as fully plastic conditions in high constraint geometries such as deep double edge cracks under tension (Rice, 1987). On the other hand, under fully plastic conditions in geometries such as CCP and shallow cracked SENT, lower stresses may arise ahead of the crack tip due to the presence of angular sectors that are stressed below yield levels (Rice, 1987). Also, as noted by Rice (1987), the limit state flow field for CCP is non-unique and can involve a velocity discontinuity along any line or set of lines within the limits formed by the two rays at 35:3 and 54:7 to the crack line. Drugan (2001) developed a framework for incorporating elastic sectors in the neartip field and obtained a family of solutions that display lower stress triaxiality than that given by the solution of Rice (1987). It must be noted that, the effects of strain hardening and finite geometry changes have not been included in the above asymptotic solutions. In particular, when the latter is taken into account, shearing along stress discontinuities

B

C

N(3) S(2) D

70.5°

S(1)

A 54.7°

Fig. 2. Slip line sectors in the asymptotic solution proposed by Rice (1987) for the mode I plane strain geometry depicted in Fig. 1 showing constant stress sectors A–D along with stress discontinuity lines.

Table 1 Cartesian stress components prevailing in different slip sectors in the asymptotic solution of Rice (1987) shown in Fig. 2 Sector

t22 =t0

t11 =t0

t12 =t0

A B C D

7.35 4.9 2.45 0

4.9 3.67 3.67 2.45

0 1.73 1.73 0

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kink bands is accompanied by lattice rotations which can induce geometrical hardening or softening (Rice, 1987; Mohan et al., 1992). 3. Constitutive model The single crystal plasticity theory, discussed in detail by Asaro (1983) and Miehe and Schotte (2004), is used in this work. The deformation gradient F is assumed to decompose multiplicatively, as, F ¼ Fe Fp ,

(1)

e

p

where F is the elastic deformation gradient caused by stretching and rotation of the lattice and F is the deformation gradient due to plastic shearing on crystallographic slip systems. The plastic deformation map Fp defines the cumulative effect of dislocation motion. The evolution equation for the plastic deformation map is written as (Miehe and Schotte, 2004) Lp ¼ F_ p Fp

1

¼

m X

g_ a ma ,

(2)

a¼1

where g_a is the plastic slip rate on slip system a. Also, ma ¼ Sa  Na

(3)

is the plastic flow (Schmid) tensor, where Sa and Na are unit vectors along the slip direction and normal to slip plane, respectively. Assuming small elastic strains, a quadratic form of macroscopic free energy function is postulated as (Miehe and Schotte, 2004) ^ e ¼ 1Ee : L: ¯ Ee , c 2

(4)

where L¯ is the fourth-order elasticity tensor and Ee ¼ 12ðCe  1Þ is the elastic Green–Lagrange strain. On using the above macroscopic free energy function, the nominal stress Pe based on the unloaded configuration is obtained as ^ e, Pe ¼ Fe Se ¼ qFe c

(5)

e

where S is the second P–K stress based on the unloaded configuration. The resolved shear stress (i.e., Schmid stress) on ath slip system is computed as ta ¼ Pe : Fe ma .

(6)

A non-linear viscous constitutive formulation is used in this study, which is of the power law form (Cuitino and Ortiz, 1992; Peirce et al., 1983):  1=m ta if ta 40, (7) g_ a ¼ g_ 0 p ta ¼ 0 otherwise. (8) Here, tpa characterizes the strength of the slip system a, whereas, g_ 0 is a reference strain rate. Since tpa represents the flow stress for system a, its evolution governs the hardening of the crystal. The rate of change of tpa is postulated as t_ pa ¼

m X

hab g_ b

with tpa ð0Þ ¼ t0 ,

(9)

a¼1

where t0 is the initial value of the critical resolved shear stress and hab is a hardening matrix that represents interaction of slip systems. In this work, the PAN model (Peirce et al., 1983) is used to describe the hardening

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which assumes the form hab ¼ hðgÞ½q þ ð1  qÞdab , (10) P where g ¼ a ga is the accumulated slip on all active slip systems. The parameter q characterizes the latent hardening behavior. The choice q ¼ 1, which corresponds to isotropic hardening, has been used in the current study. The form of hðgÞ is postulated by Peirce et al. (1983) as   2 h0  hs hðgÞ ¼ hs þ ðh0  hs Þ sech g . (11) ts  t0 Here, h0 is the initial hardening rate, hs the saturation hardening rate and ts the saturation strength. The above model has been implemented in the general purpose finite element code FEAP (Zienkiewicz and Taylor, 1989) by writing a user material sub-routine. An implicit (Backward Euler) procedure is used for updating stresses and plastic variables (Miehe and Schotte, 2004). 4. Computational aspects In mode I, modified boundary layer formulation employed in this work, a large semi-circular disk containing a notch along one of its radii is modeled with symmetry conditions imposed on the line ahead of the notch tip. It must be mentioned that although a notch is considered, the near-tip stress and strain variations after a certain level of deformation would be insensitive to the initial notch diameter b0 (i.e., they will be the same as for an initially sharp crack), when radial distance from the notch tip is normalized by J=t0 , where J is the energy release rate (see, for example, McMeeking and Parks, 1979; O’Dowd and Shih, 1991). This is because the notch opening is expected to scale with J=t0 . The ratio of the radius of the disk to the notch diameter, R0 =b0 , is chosen as 70 000 so that the plastic zone is well contained within the boundary. This semicircular disk is modeled using the finite element mesh shown in Fig. 3(a) which comprises of 4992 four-noded ¯ isoparametric quadrilateral elements based on the B-formulation (Hughes, 1980; Moran et al., 1990) and 5152 nodes. A detailed perspective of the mesh near the notch tip is displayed in Fig. 3(b). The displacement components based on the first and second terms of the elastic mode I, plane strain crack tip field (Williams, 1957) which are given by    1=2 R0 ð1 þ nÞ y y ð1  n2 Þ u1 ¼ K I cos k  1 þ 2 sin2 TR0 cos y, þ E 2 2 E 2p

X2

r

θ X1

Fig. 3. (a) Finite element mesh used in the boundary layer analysis. (b) Detailed view of the near-tip region.

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Table 2 Values of normalized material parameters used in the modified boundary layer simulations g_ 0

q

h0 =t0

hs =t0

ts =t0

E=t0

n

0.001

1

3

0

2.5

7000

0.3

 u2 ¼

R0 2p

1=2

  ð1 þ nÞ y y nð1 þ nÞ K I sin k þ 1  2 cos2 TR0 sin y,  E 2 2 E

are prescribed on the outer boundary of the semi-circular domain. In these equations, R and y are polar coordinates with origin at the center of curvature of the notch. Also, E and n are Young’s modulus and Poisson’s ratio and k ¼ 3  4n for plane strain. Here, elastic anisotropy is ignored, which is a reasonable assumption for aluminum single crystals, in order to simplify the analysis and interpretation of results. In this case, the stress intensity factor K and T-stress govern the first and second terms of the elastic crack tip field, respectively. The simulations are conducted by incrementing K and T, while keeping the ratio T=K fixed throughout a particular analysis pffiffiffiffiffi (O’Dowd and Shih, 1991). Thus, results pertaining to different T-stress levels at a fixed value of K=ðt0 b0 Þ are obtained by conducting analyses with various values of the ratio T=K. The values of initial hardening modulus h0 , saturation slip resistance ts and Young’s modulus E, normalized by the initial slip resistance t0 , assumed in the present analysis are summarized in Table 2 along with other material parameters. The resolved shear stress versus shear strain response generated from these parameters is representative of ductile aluminum single crystals (Hosford et al., 1960). In the current study, the orientation of the FCC single crystal is chosen as shown in Fig. 1. As mentioned in Section 2, the combinations of three pairs of slip systems which result in plane strain deformation in the X 1 –X 2 plane are considered. These three pairs give rise to effective slip along ½1 2¯ 1, [1 2 1] and [1 0 1] directions, respectively (see Fig. 1). The equivalent hardening parameters in the 2D plane strain analysis are derived by equating the effective plastic slip with that arising from the combination of the corresponding 3D slip systems. 5. Results and discussion The results obtained from the simulations are presented in this section in the form of contour plots and near-tip angular and radial variations of plastic slip and stresses. Attention is focused on the effect of T-stress on these field quantities. 5.1. Contour plots of plastic strain ¯ ¯ In Figs. 4(a)–(c), fringe contour plots of effective plastic slip, g, in systems pffiffiffiffiffi ð1 1 1Þ½1 1 0 and ð1 1 1Þ½0 1 1, corresponding to T=t0 ¼ 2, 0 and 2, respectively, and same level of K=ðt0 b0 Þ ¼ 80 are displayed. Here, as well as in other contour plots to be presented subsequently, the notch tip coordinates are normalized by the initial notch diameter b0 . Also, the same contour levels are displayed for the different T-stress cases in order to facilitate direct comparison. It can be observed that all three figures show a slip shear band emanating from the notch tip and aligned at around 55 to the notch line. On comparing Figs. 4(a)–(c), it can be seen that a much stronger and radially longer slip shear band develops for the case T=t0 ¼ 2 as compared to those with non-negative T-stress. For example, the contour corresponding to g ¼ 0:006 extends up to a radial distance of 70b0 from the notch tip for T=t0 ¼ 2 (Fig. 4(c)), whereas it spreads up to a radial distance of 11:5b0 and 7b0 from the notch tip for T=t0 ¼ 0 and 2, respectively. Thus, an imposition of negative T-stress causes enhanced plastic slip on the above-mentioned slip systems and this becomes pronounced at high negative T. On the other hand, it can be observed from Figs. 4(b) and (a) that with increase in positive T=t0 , plastic slip on this system decreases slightly. ¯ is The effect of T-stress on the development of effective plastic slip, g, in systems ð1 1 1Þ½1 1¯ 0 and ð1 1 1Þ½0 1 1 delineated in Fig. 5. Fringe contours of g for T=t0 ¼ 2, 0 and 2 are displayed at the same level of normalized

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12.0

8.0

4.50-002

4.50-002 3.50-002

6.0

3.50-002

9.0

2.50-002

2.00-002

4.0

1.60-002

X2/b0

X2/b0

2.50-002

2.00-002

6.0

1.60-002 1.20-002

1.20-002 1.00-002

2.0

1.00-002

3.0

8.00-003

8.00-003

6.00-003

6.00-003

0.0 -2.0

0.0 -3.0

0.0

0.0

2.0 X1/b0

4.0

6.0

0.0

0.0

3.0

6.0

9.0

X1/b0

72.0 4.50-002 3.50-002

54.0

X2/b0

2.50-002 2.00-002

36.0

1.60-002 1.20-002 1.00-002

18.0

8.00-003 6.00-003

0.0 -18.0

0.0

0.0

18.0

36.0

54.0

X1/b0

¯ ¯ Fig. 4. Fringe contour plots of effective plastic slip g on conjugate pair pffiffiffiffiffi of slip systems ð1 1 1Þ½1 1 0 and ð1 1 1Þ½0 1 1 corresponding to (a) T ¼ 2t0 , (b) T ¼ 0 and (c) T ¼ 2t0 and the same level of K=ðt0 b0 Þ ¼ 80.

stress intensity factor as in Fig. 4. Figs. 5(a) and (b) show a slip shear band emanating from the notch tip at around 125 to the notch line, whereas, Fig. 5(c) displays a kink shear band aligned at around 45 . On comparing Figs. 5(a) and (b), it can be noticed that an imposition of positive T-stress promotes the tendency to develop a stronger and radially longer slip shear band (at about 125 ), which is in contrast to plastic slip activity on the systems ð1 1¯ 1Þ½1 1 0 and ð1 1¯ 1Þ½0 1 1 discussed previously. Thus, Figs. 5(a) and (b) show that the contour corresponding to g ¼ 0:006 extends radially up to a distance of 7b0 from the notch tip for T=t0 ¼ 0, whereas, it spans a radial distance of 47b0 for T=t0 ¼ 2. By contrast, as the level of negative T-stress increases, the above slip shear band becomes weaker and completely vanishes for the case T=t0 ¼ 2 (see Fig. 5(c)). However, high level of negative T-stress leads to development of a kink shear band at about 45 , which is not present for cases with TX0. ¯ The fringe contours of effective plastic slip, g, on systems ð1¯ 1 1Þ½1 0 1 and pffiffiffiffið1 ffi 1 1Þ½1 0 1 are shown in Figs. 6(a)–(c), for T=t0 ¼ 2, 0 and 2, respectively, at the same level of K=ðt0 b0 Þ as Fig. 4. It can be seen from these figures that, irrespective of the level of T-stress, a kink shear band emanates from notch tip at an angle of about 90 , due to slip activity on the above systems. However, for non-zero T-stress (Figs. 6(a) and (c)), this band prevails over a smaller radial distance as compared to the case T ¼ 0 (see Fig. 6(b)). Also, on

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8.0

40.0 4.50-002

4.50-002

3.50-002

30.0

3.50-002

6.0

2.50-002

2.50-002

2.00-002

1.60-002

X2/b0

X2/b0

2.00-002

20.0

4.0

1.60-002 1.20-002

1.20-002 1.00-002

10.0

0.0 -30.0

8.00-003

6.00-003

6.00-003

0.0

0.0

-20.0

-10.0 X1/b0

0.0

1.00-002

2.0

8.00-003

0.0

-6.0

10.0

-4.0

-2.0 X1/b0

0.0

2.0

8.0 4.50-002 3.50-002

6.0

2.50-002

X2/b0

2.00-002

4.0

1.60-002 1.20-002 1.00-002

2.0 8.00-003 6.00-003

0.0

0.0

-2.0

0.0

2.0 X1/b0

4.0

6.0

¯ ¯ Fig. 5. Fringe contour plots of effective plastic slip g on conjugate pair pffiffiffiffiffi of slip systems ð1 1 1Þ½1 1 0 and ð1 1 1Þ½0 1 1 corresponding to (a) T ¼ 2t0 , (b) T ¼ 0 and (c) T ¼ 2t0 and the same level of K=ðt0 b0 Þ ¼ 80.

examining Figs. 6(a) and (c), it can be noted that negative T-stress results in higher magnitude of plastic slip at comparable distances from notch tip as opposed to positive T-stress of same magnitude. In summary, it can be observed from Figs. 4–6 that for T ¼ 0, the near-tip deformation field shows the development of a prominent kink shear band at an angle of 90 , as well as two slip shear bands occurring at 55 and 125 to notch line. As jTj increases, one of the above slip shear bands dominates over the other bands. ¯ and Thus, for high positive T, large plastic slip accumulates in the conjugate pair ð1 1 1Þ½1 1¯ 0 and ð1 1 1Þ½0 1 1  hence, results in a dominant slip shear band at about 125 . On the other hand, with an imposition of high negative T-stress, plastic slip in the conjugate pair ð1 1¯ 1Þ½1 1 0 and ð1 1¯ 1Þ½0 1 1 becomes preponderant and leads to a dominant slip shear band at about 55 . Interestingly, for high negative T-stress level, plastic slip in ¯ leads to a kink shear band at about 45 , whereas the occurrence the conjugate pair ð1 1 1Þ½1 1¯ 0 and ð1 1 1Þ½0 1 1  of the slip shear band at 125 is suppressed. Cuitino and Ortiz (1996) and Flouriot et al. (2003) have conducted finite element analyses of four point bend and CT specimens of FCC single crystals, having the same lattice orientation as that considered here. Indeed, the plastic slip activity on various pairs of conjugate slip systems reported by them in these high constraint geometries are quite similar to those presented above from the modified boundary layer analyses

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12.0

18.0 4.50-002

4.50-002

3.50-002

9.0

3.50-002

13.5

2.50-002

2.50-002

6.0

1.60-002

2.00-002

X2/b0

X2/b0

2.00-002

9.0

1.60-002

1.20-002

1.20-002

1.00-002

3.0

1.00-002

4.5

8.00-003

8.00-003

6.00-003

0.0 -6.0

6.00-003

0.0

-3.0

0.0 X1/b0

3.0

0.0 -9.0

6.0

0.0

-4.5

0.0

4.5

9.0

X1/b0

12.0 4.50-002 3.50-002

9.0

2.50-002

X2/b0

2.00-002

6.0

1.60-002 1.20-002 1.00-002

3.0 8.00-003 6.00-003

0.0 -6.0

0.0

-3.0

0.0

3.0

6.0

X1/b0

¯ ¯ Fig. 6. Fringe contour plots of effective plastic slip g on conjugate pair pffiffiffiffiffi of slip systems ð1 1 1Þ½1 0 1 and ð1 1 1Þ½1 0 1 corresponding to (a) T ¼ 2t0 , (b) T ¼ 0 and (c) T ¼ 2t0 and the same level of K=ðt0 b0 Þ ¼ 80.

with TX0. In a very recent work, Patil et al. (2007) have performed a combined numerical and experimental study of crack tip fields in an aluminum single crystal SENT specimen. Their computational results show that in this low constraint geometry, dominant slip shear bands develop at 55 , while kink shear bands form at 45 and 90 to the notch line as in the present modified boundary layer analysis with high negative T-stress. This corroborates with experimental observations based on scanning electron microscopy and EBSD (Patil et al., 2007). Indeed, the latter clearly shows lattice misorientation band emanating at 45 and 90 to the notch line which confirms the presence of the kink shear bands predicted by the computational analysis. This suggests that the near-tip fields corresponding to different single crystal fracture geometries are part of a family which can be characterized by two parameters (such as K–T or J–Q) similar to isotropic plastic solids (see, Betegon and Hancock, 1991; O’Dowd and Shih, 1991). This issue is further discussed in Section 5.3 after analysing the radial and angular distribution of stresses. The fringepcontours of maximum principal logarithmic plastic strain, logðlp1 Þ, obtained for T=t0 ¼ 2, 0 and ffiffiffiffiffi 2 at K=ðt0 b0 Þ ¼ 80 are displayed in Figs. 7(a)–(c), respectively. It can be observed from these figures that the T-stress has a significant effect on the plastic zone shape and size, which is akin to isotropic plastic solids (see, for example, Jayadevan et al., 2001). The shape of the plastic zone is governed by the shear bands that

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18.0

40.0

4.50-002

4.50-002 3.50-002

20.0

3.50-002

13.5

2.50-002

2.50-002

2.00-002

2.00-002

1.60-002

X2/b0

X2/b0

30.0

9.0

1.60-002 1.20-002

1.20-002 1.00-002

10.0

0.0 -30.0

8.00-003

8.00-003

6.00-003

6.00-003

0.0

-20.0

-10.0 X1/b0

0.0

1.00-002

4.5

0.0

0.0

-9.0

10.0

-4.5

0.0 X1/b0

4.5

9.0

60.0 4.50-002 3.50-002

45.0

X2/b0

2.50-002 2.00-002

30.0

1.60-002 1.20-002 1.00-002

15.0

8.00-003 6.00-003

0.0 -15.0

0.0

0.0

15.0 X1/b0

30.0

45.0

Fig. 7. Fringe contour plots of maximum logarithmic plastic strain logðlp1 Þ corresponding to (a) T ¼ 2t0 , (b) T ¼ 0 and pffiffiffiffiprincipal ffi (c) T ¼ 2t0 and the same level of K=ðt0 b0 Þ ¼ 80.

form due to plastic slip on various systems. For T=t0 ¼ 0, a distinct contribution from two slip shear bands (see Figs. 4(b) and 5(b)) and a kink shear band (see Fig. 6(b)) to the plastic zone shape can be perceived in Fig. 7(b). It can be seen from Fig. 7 that the plastic zone size is smallest for T=t0 ¼ 0 and increases with jTj40. Thus, it can be noted from Fig. 7(b) that for T=t0 ¼ 0, the contour corresponding to logðlp1 Þ ¼ 0:006 spreads up to a maximum radial distance of 9:5b0 , at an angle of y ¼ 55 to the notch line. By contrast, for T=t0 ¼ 2, it extends up to a maximum radial distance of 38b0 and is inclined at an angle of 125 to notch line (see Fig. 7(a)). This implies that, contribution to plastic strain is predominant due to the slip shear band that ¯ for T40 (see Fig. 5(a)). forms due to plastic slip on the conjugate systems ð1 1 1Þ½1 1¯ 0 and ð1 1 1Þ½0 1 1 Further, a small but noticeable contribution to the plastic strain distribution from the kink shear band occurring at 90 and slip shear band forming at 55 to the notch line can be perceived in Fig. 7(a). Finally, for T=t0 ¼ 2, Fig. 7(c) shows that the contour corresponding to logðlp1 Þ ¼ 0:006 spans a radial distance of 64b0 from the notch tip and is inclined at an angle of 55 . This implies that the slip shear band that forms due to plastic slip on the conjugate systems ð1 1¯ 1Þ½1 1 0 and ð1 1¯ 1Þ½0 1 1 predominantly contributes to the plastic strain distribution for To0. The absence of the slip shear band at 125 for high negative T-stress is also reflected in the plastic zone shape shown in Fig. 7(c).

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5.2. Radial variations of stresses The radial variations of the normalized Kirchhoff stress components t11 =t0 and t22 =t0 with normalized distance r=ðJ=t0 Þ ahead of the notch tip are presented in Fig. 8, corresponding to different levels of constraint which is characterized by the T-stress in the present modified boundary layer analysis. Here, r denotes the distance ahead of the notch tip in the undeformed configuration and J ¼ K 2 ð1  n2 Þ=E is the energy release rate computed from the pffiffiffiffiffiremote elastic field. The variations are shown for T=t0 ¼ 2, 0, 1 and 2, but at the same level of K=ðt0 b0 Þ ¼ 80. It was found that the stresses and plastic strains when plotted in radial pffiffiffiffi ffi coordinates normalized by J=t0 remain invariant beyond the above level of K=ðt0 b0 Þ. It can be seen from Figs. 8(a) and (b) that the stresses computed from the finite deformation finite element solution increase with distance ahead of the tip, reach a peak value between r=ðJ=t0 Þ ¼ 0:5 and 1 and then decrease. It should be noted from these figures that negative T causes a significant downward shift in the radial variation of both t11 and t22 , while, positive T tends to increase the stresses above the T ¼ 0 case. However, the latter effect is much less pronounced than the former. Thus, at r=ðJ=t0 Þ ¼ 4, the value of t22 =t0 ¼ 7 for T ¼ 0, whereas it is 7.4 and 3.7 for T=t0 ¼ 2 and 2, respectively. It is also evident from Figs. 8(a) and (b) that the curves pertaining to different values of T=t0 are roughly parallel to each other for r=ðJ=t0 Þ42. This suggests that the deviation of the direct stress components from the T ¼ 0 case is essentially independent of normalized distance, r=ðJ=t0 Þ, from the notch tip. In order to quantify the above observation, the radial variation of the difference stress field, which is defined as (O’Dowd and Shih, 1991) Q^tij ¼

tij  ðtij ÞT¼0 , t0

(12)

ahead of the notch tip is examined. Here, the reference field is taken to be the T ¼ 0 solution as suggested by O’Dowd and Shih (1991) for isotropic plastic solids. The radial variations of Q^tij are plotted against normalized distance r=ðJ=t pffiffiffiffiffi 0 Þ ahead of the notch tip in Fig. 9 for different levels of T-stress and at the same value of K=ðt0 b0 Þ ¼ 80. On examining this figure, several features of the difference stress field which are remarkably similar to isotropic plastic solids (O’Dowd and Shih, 1991) can be perceived. First, it can be seen that all components of the difference stress field remain almost constant over a wide range of normalized radial distance from r=ðJ=t0 Þ ¼ 2 to 10. Following Basu and Narasimhan (2000), a parameter Q0 is

9

5

8

4

7

3

τ22/τ0

τ11/τ0

6

T = 2τo T=0 T = −1τo T = −2τo

2

T = 2τo T=0 T = −1τo T = −2τo

6 5

1

4

0 0

2

4

6 r/ (J/τ0)

8

10

3 0

2

4

6

8

10

r/ (J/τ0)

Fig. 8.pffiffiffiffi Radial variations of stress components: (a) t11 =t0 and (b) t22 =t0 at y ¼ 0 , for T=t0 ¼ 2, 0, 1 and 2 and same level of ffi K=ðt0 b0 Þ ¼ 80.

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1 T/τ0 = 2

0

T/τ0 = -1

Qτ^ij

-1

-2 T/τ0 = -2

-3

Qτ^11 Qτ^22 Qτ^ 33

-4 2

4

6

8

10

12

14

r/ (J/τ0)

Fig. 9. Radialpvariations of the difference stress fields with respect to the T ¼ 0 solution at y ¼ 0 , for different T-stress values and same ffiffiffiffiffi level of K=ðt0 b0 Þ ¼ 80.

Table 3 Various parameters associated with the difference stress field corresponding to different T-stress levels T=t0

Q0

2.00 1.00 2.00

0.004 0.03 0.019

Q 0.33 1.08 2.9

Q^t11 =Q^t22

Q^t33 =Q^t22

1.31 0.97 1.01

0.69 0.59 0.6

defined as Q0 ¼

Q^t22 jr=ðJ=t0 Þ¼8  Q^t22 jr=ðJ=t0 Þ¼2 . 6

(13)

The values of jQ0 j, evaluated for different T-stress cases, are summarized in Table 3 and can be seen to be less than 0.03. This confirms that the difference stress field is slowly varying ahead of the notch tip akin to isotropic plastic solids (O’Dowd and Shih, 1991). Also included in Table 3 are the values of Q^tkk =3 evaluated at r=ðJ=t0 Þ ¼ 4 (which is referred to in the sequel as the constraint parameter Q) and the ratios Q^t11 =Q^t22 and Q^t33 =Q^t22 computed at the same r=ðJ=t0 Þ for different levels of T-stress. From this table, it is clear that the ratio Q^t11 =Q^t22 for all negative T-stress cases is close to unity (within 3%), whereas, the ratio Q^t33 =Q^t22 is close to 0.6 (within 1:6%). Thus, the difference stress field is not truly triaxial although the components Q^t11 and Q^t22 are remarkably close to each other. It must be noted that unlike in isotropic plastic solids obeying the von Mises yield condition, the ratio of t33 =ðt11 þ t22 Þ under plane strain conditions is not 0.5, instead is close to the elastic Poisson’s ratio, for the case of ductile single crystals (when elastic anisotropy is ignored). On examining the values of the constraint parameter Q summarized in Table 3, it can be noticed that it is marginally positive (around þ0:33) for T=t0 ¼ 2, whereas it becomes progressively more negative with increase in T-stress in the negative direction attaining a value of 2:9 at T=t0 ¼ 2. Thus, the present results clearly establish that ductile single crystal fracture geometries would progressively lose stress triaxiality (or crack tip constraint) with increase in negative T-stress as in the case of isotropic plastic solids (O’Dowd and Shih, 1991). However, stress triaxialities much higher than those displayed by the solution pertaining to T ¼ 0 are difficult to attain. As noted below, this level of stress triaxiality is exhibited by the asymptotic solution of Rice (1987) for perfectly plastic single crystals and is around 5:31t0 . The above discussion underscores the need for a two-parameter characterization of crack tip fields in terms of J and Q in order to describe the

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configuration dependence of fracture response in ductile single crystals. Moreover, the slip activity in ductile single crystals is also profoundly influenced by crack tip constraint in line with the far-sighted observations of Rice (1987) resulting in possible fascinating dependence of fracture configuration on the slip and kink shear band patterns near a crack tip. 5.3. Near-tip angular variation of field quantities The angular variations of normalized Kirchhoff stress components t11 =t0 , t22 =t0 and t12 =t0 at r=ðJ=t0 Þ ¼ 4 from notch tip are shown in Figs. 10(a)–(c) with y measured from the positive X 1 axis. The plots corresponding to T=t0 ¼ 2, 0, 1 and 2 are presented at the same level of normalized stress intensity factor pffiffiffiffiffi K=ðt0 b0 Þ ¼ 80. It should be first observed from Fig. 10 that the near-tip angular distribution for the case of T ¼ 0 is similar to the asymptotic solution of Rice (1987). Thus, four constant stress sectors, as predicted by Rice (1987) can be perceived, with t22 stress discontinuity at y  55 , 90 and 125 (see Fig. 10(b)). However, as expected, the discontinuity at y ¼ 90 is not present in the t11 and t12 components (see Figs. 10(a) and (c)). The magnitudes of the stresses in the four constant stress sectors are also similar to Rice’s solution. For example, in the angular sector ahead of the tip, t11 =t0 and t22 =t0 are about 4.9 and 7.1, respectively, which despite some hardening, are close to the values 4.9 and 7.35 (see Table 1), given by Rice (1987). Also, t12 =t0 in

8

6 T = 2τo T=0 T = −1τo T = −2τo

6

2

τ22/τ0

τ11/τ0

4

4

0

2

−2

0 0

30

60

90 θ (deg)

120

150

T = 2τo T=0 T = −1τo T = −2τo

180

0

30

60

90 θ (deg)

0.5

150

180

τ12 το

0

-1.5 A

-1

-0.5

0

0.5

1

D

1.5

τ11-τ22

−0.5 τ12/τ0

120

-0.5 −1

2το

-1

T/το = 0 T/το = −1

−1.5 -1.5

T = 2τo T=0 T = −1τo T = −2τo

−2

B

-2

C

−2.5 0

20

40

60

80 100 120 140 160 180 θ (deg)

-2.5

Fig. 10. Angular pffiffiffiffiffivariations of stress components: (a) t11 =t0 , (b) t22 =t0 , (c) t12 =t0 at r=ðJ=t0 Þ ¼ 4, for T=t0 ¼ 2, 0, 1 and 2 and same level of K=ðt0 b0 Þ ¼ 80. (d) Stress trajectories in stress plane for T=t0 ¼ 0 and 1.

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the angular range from y  55 to 125 is about 1:83, which compares well with the value 1:73 predicted by the asymptotic solution. It can be seen from Fig. 10 that the near-tip angular stress distribution is significantly affected by the constraint level, which is characterized by the T-stress in the present analysis. A marginal increase in t11 and t22 components can be noticed when T=t0 changes from 0 to 2, with the angular distributions retaining all other features present in the solution pertaining to T ¼ 0. On the other hand, an imposition of negative T-stress dramatically lowers the t11 and t22 stresses all around the notch tip. Thus, from the angular stress distribution corresponding to T=t0 ¼ 2, it can be noticed that t11 =t0 and t22 =t0 are 1.6 and 3.9 in the sector ahead of the notch tip. These values are much lower than those pertaining to the T ¼ 0 case mentioned above. Further, Fig. 10(a) shows that t11 is negative on the notch flank and approaches a value between 2 and 3t0 when negative T is imposed. By contrast, when TX0, t11 is positive on the notch flank with a value between 2.1 and 2:3t0 (which agrees well with the value 2:4t0 given by the asymptotic solution). Finally, it must be noted from Fig. 10(b) that for high negative T, the discontinuity in t22 that occurs at about 90 becomes inconspicuous, while that which occurs at 125 for TX0 shifts to an angle close to 145 . Fig. 10(d) shows the locus of the stress state encountered on traversing around the notch tip in stress space for T=t0 ¼ 0 and 1. It is observed that for T=t0 X0 the locus in stress space follows the yield surface ABCD which is hexagonal in shape. However, due to isotropic hardening considered in this work, it shows a small self-similar expansion from the initial yield surface given by Rice (1987). Thus, the stress state all around the notch tip is at yield for T=t0 ¼ 0, resulting in high stress triaxiality ahead of the notch tip. However, the locus of the stress state pertaining to T=t0 ¼ 1, as shown in Fig. 10(d), first follows the path ABC similar to T=t0 ¼ 0, but later traces an arc CA ending at the initial point A. This deviation of the trajectory in stress space for negative T from the case of T ¼ 0 implies that at least one elastic angular sector must exist in the near-tip region. Drugan (2001), through his asymptotic solution has shown that the presence of such an elastic sector can lower the stress triaxiality ahead of the crack tip. Following his approach, an analytical, asymptotic solution with an elastic sector in ideally plastic FCC single crystal is constructed in Appendix A. This solution shows that the elastic sector extends from y ¼ 141:6 to 175:1 . The stress fields obtained from the finite element simulation pertaining to T=t0 ¼ 1 are in good agreement with this asymptotic solution (see Figs. 10(d) and A1). Hence, negative T-stress promotes the development of an elastic sector and the presence of this elastic sector gives rise to lower stress triaxiality ahead of the notch tip. The angular variations of the difference stress field with respect to the T ¼ 0 solution, Q^tij (see Eq. (12) for definition), up to y ¼ 90 are presented in Fig. 11 at a normalized distance of r=ðJ=t0 Þ ¼ 4 from the notch tip corresponding to T=t0 ¼ 1. It can be seen from Fig. 11 that the difference stress field remains almost constant at least up to y ¼ 90 except at about y ¼ 55 where the stresses change discontinuously. Also, Q^t11  Q^t22 , whereas Q^t33  0:3ðQ^t11 þ Q^t22 Þ in the above angular range. It is found that all the difference 0 Qτ^11 Qτ^ 22

Qτ^33

Qτ^ij

-0.5

-1

-1.5

-2 0

30

60

90

θ (deg)

pffiffiffiffiffi Fig. 11. Angular variations of difference stress field at r=ðJ=t0 Þ ¼ 4, for T=t0 ¼ 1 and K=ðt0 b0 Þ ¼ 80.

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stress components become more negative in the forward sector as T stress increases in the negative direction, whereas they are slightly positive for T=t0 ¼ 2. These features are similar to those reported by O’Dowd and Shih (1991) for isotropic plastic solids and confirm that constraint loss will become pronounced in the region ahead of the tip (up to y ¼ 90 ) as T=t0 becomes more negative. In Fig. 12, angular variations of effective plastic slip, g, for all three conjugate slip system pairs at r=ðJ=t0 Þ ¼ 4 are presented corresponding to T=t0 ¼ 2, 0, 1 and 2. Fig. 12(a) shows that plastic slip on the pair of conjugate systems ð1 1¯ 1Þ½1 1 0 and ð1 1¯ 1Þ½0 1 1 attains a peak value at about y  55 , irrespective of the level of T-stress. Thus, the slip activity on this pair of systems results in a slip shear band as noted earlier from fringe contour plots (see Fig. 4). It can be observed from Fig. 12(a) that the peak value of g enhances strongly with increase in T-stress in the negative direction. Thus, the peak value increases by a factor of 2.5 as T changes from 0 to 2t0 . By contrast, positive T-stress slightly lowers the magnitude of g with the peak value decreasing by a factor of 0.6 as T changes from 0 to 2t0 . ¯ (Fig. 12(b)) shows more interesting The effective plastic slip on conjugate pair ð1 1 1Þ½1 1¯ 0 and ð1 1 1Þ½0 1 1 response to different T-stress levels. It can be seen from this figure that the occurrence of the peak value of g (magnitude as well as angular location) is sensitive to T-stress. For high positive T, the peak value occurs

0.04

0.025 T = 2τo T=0 T = −1τo T = −2τo

0.03

T = 2τo T=0 T = −1τo T = −2τo

0.02

0.02

γ

γ

0.015

0.01 0.01 0.005

0

0 0

30

60

90 θ (deg)

120

150

0

180

30

60

90 θ (deg)

120

150

180

0.04 T = 2τo T=0 T = −1τo T = −2τo

γ

0.03

0.02

0.01

0 0

30

60

90 θ (deg)

120

150

180

Fig. 12. Angular variations of effective plastic slip g on all three conjugate slip system pairs: (a) ð1 1¯ 1Þ½1 1 0 and ð1 1¯ 1Þ½0 1 1, ¯ ¯ ¯ ¯ (b) ð1p 1 1Þ½1 ffiffiffiffiffi 1 0 and ð1 1 1Þ½0 1 1 and (c) ð1 1 1Þ½1 0 1 and ð1 1 1Þ½1 0 1 at r=ðJ=t0 Þ ¼ 4 for T=t0 ¼ 2, 0, 1 and 2 and same level of K=ðt0 b0 Þ ¼ 80.

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at about y ¼ 125 and hence leads to the development of a dominant slip shear band at this angle (see also Fig. 5(a)). On the other hand, for high negative T, the peak value is observed at y ¼ 45 , resulting in the development of a kink shear band at this angle to the notch line (also see Fig. 5(c)). It is important to note that, positive T shows tendency to suppress the peak in the g variation at y ¼ 45 and enhance that at y ¼ 125 , whereas exactly opposite trend can be perceived for high negative T. Both peaks can be observed in the g distribution for T ¼ 0. With reference to this case, the peak at y ¼ 125 increases by a factor of 2.25 as T changes to 2t0 , whereas that at y ¼ 45 elevates by a factor of 6 when T becomes 2t0 . The slip activity on the ¯ 0 1,displayed in Fig. 12(c), shows the peak occurring at about y  90 , pair of systems ð1¯ 1 1Þ½1 0 1 and ð1 1 1Þ½1 irrespective of T-stress level and leads to a kink shear band at this angle (see Fig. 6). The peak drops with increase in positive T-stress and enhances with imposition of negative T-stress. The full field solutions of Cuitino and Ortiz (1996), for four point bend geometry show features similar to those obtained in p the modified boundary layer simulation with T  0. Indeed, the biaxiality ratio, b ffiffiffiffiffipresent ffi (defined as b ¼ T pa=K), for their configuration (with crack length to width ratio a=w  0:3), is around 0:15 leading to small negative values of T-stress. Cuitino and Ortiz (1996) found that plastic slip on the conjugate slip systems ð1 1¯ 1Þ½1 1 0 and ð1 1¯ 1Þ½0 1 1 attains a peak at around y ¼ 55–60 , whereas plastic slip ¯ displays two peaks, occurring at about on the pair of conjugate slip systems ð1 1 1Þ½1 1¯ 0 and ð1 1 1Þ½0 1 1   y ¼ 30–40 and y ¼ 125–135 . In addition, the peak plastic slip on the conjugate pair ð1¯ 1 1Þ½1 0 1 and ¯ 0 1 occurs at about 90 and is more than any other slip system. All these features show good ð1 1 1Þ½1 qualitative agreement with the modified boundary layer simulation (Fig. 12) if T is taken to have a value between t0 and 0. In Fig. 13, the near-tip angular variations (at r=ðJ=t0 Þ ¼ 4) of maximum principal logarithmic plastic strain, logðlp1 Þ, are displayed for different values of T=t0 . It can be seen from this figure that the plastic strain distribution around the notch tip is sensitive to the T-stress level. All curves show more than one distinct peak value and their magnitude as well as angular position depend on the level of the T-stress. For example, the plot corresponding to T ¼ 0 shows three distinct peaks. The plastic slip on the pair of systems ð1 1¯ 1Þ½1 1 0 and ð1 1¯ 1Þ½0 1 1 which develops a slip shear band at y ¼ 55260 governs the first peak that occurs at the same angular position. The second peak at y ¼ 90 observed in logðlp1 Þ distribution is due to plastic slip in the ¯ 0 1 which corresponds to a kink shear band. Finally, the slip conjugate system pair ð1¯ 1 1Þ½1 0 1 and ð1 1 1Þ½1 ¯ shear band that develops at about y ¼ 125 due to slip on the conjugate pair ð1 1 1Þ½1 1¯ 0 and ð1 1 1Þ½0 1 1, governs the occurrence of the third peak in the plastic strain distribution for the T ¼ 0 case. For high positive T-stress (T=t0 ¼ 2), as discussed in Section 5.1, the slip band at y ¼ 125 dominates and hence, a prominent peak in the plastic strain distribution can be observed at this angular position. However, an imposition of high negative T (say, T=t0 ¼ 2) suppresses the development of the above slip shear band

0.04 T = 2τo T=0 T = −1τo T = −2τo

p

log (λ1)

0.03

0.02

0.01

0 0

30

60

90

120

150

180

θ (deg) Fig. 13. Angular of maximum principal logarithmic plastic strain, logðlp1 Þ, at r=ðJ=t0 Þ ¼ 4 for T=t0 ¼ 2, 0, 1 and 2 and same pffiffiffiffivariations ffi level of K=ðt0 b0 Þ ¼ 80.

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and enhances the one that occurs at y ¼ 55260 (see Section 5.1). As a result, the plastic strain distribution for high negative T shows only two peaks, with the more prominent peak at y ¼ 55260 and the other at y  90 , respectively. Thus, a progressive transition in plastic strain distribution around the notch tip as the T-stress changes from positive to a high negative value can be noticed in Fig. 13. 6. Conclusions In this work, the effect of crack tip constraint on near-tip stress and deformation fields in FCC single crystals has been studied under mode I, plane strain condition and for the specific crystal orientation that has the notch surface lying on the (0 1 0) plane and the notch front along ½1¯ 0 1 direction. As mentioned in Section 1, this orientation has been frequently observed to occur in experimental studies. The main conclusions of this work are summarized below.

(1) The near-tip deformation field for T ¼ 0 shows the development of a prominent kink shear band at an angle of 90 , as well as, two slip shear bands occurring at 55 and 125 to notch line which agrees with the asymptotic solution of Rice (1987). As jTj increases, one of these two slip shear bands dominates over the other bands. Thus, for high positive T, the dominant slip shear band occurs at about 125 , whereas, with an imposition of high negative T-stress, it develops at about 55 . Also, for high negative T level, a kink shear band develops at 45 , whereas, the occurrence of the slip shear band at about 125 is suppressed. The kink shear band at 90 to the notch line forms irrespective of T-stress. (2) The size and shape of crack tip plastic zone is strongly influenced by the level of T-stress and is governed by the shear bands that form due to plastic slip on the various slip systems. (3) The T-stress significantly affects the stress distribution near the notch tip. The near-tip angular stress variation for the case of T ¼ 0 is similar to the asymptotic solution of Rice (1987) and shows the occurrence of four constant stress sectors separated by discontinuities. These features are retained for positive T-stress with slight elevation in the t11 and t22 components. By contrast, imposition of negative T-stress leads to significant drop in the above components in the forward sector (up to about y ¼ 90 ). This suggests loss of crack tip constraint with negative T-stress which is akin to isotropic plastic solids. The reason for the loss of crack tip constraint in the present context is traced to the occurrence of an elastic sector near the notch tip. Indeed, the angular stress distribution obtained from an asymptotic solution for ideally plastic FCC crystal in Appendix A matches well with that for T=t0 ¼ 1. (4) An examination of the difference stress field (with respect to the T ¼ 0 solution), Q^tij , shows features which are remarkably similar to isotropic plastic solids with a few minor exceptions. Thus, Q^t11  Q^t22 in the angular range from y ¼ 0 up to 90 and for normalized radial distance from r=ðJ=t0 Þ ¼ 2 to 10. Further, in the above region, the difference stress components vary slowly (except at the location of the stress discontinuity at y  55 ). The constraint or triaxiality parameter Q becomes highly negative with increase in T-stress in the negative direction attaining a value of 2:9 at T ¼ 2t0 . These features exhibited by the difference stress field were also found to occur for another orientation wherein the notch ¯ The constraint loss will have profound lies on the (1 0 1) plane and the notch front is aligned along ½1 0 1. implications on crack initiation by cleavage cracking as well as by micro-void growth and coalescence in ductile single crystals. (5) The results of the modified boundary layer analysis suggest that the near-tip fields corresponding to different single crystal fracture geometries are part of a family which can be characterized by two parameters (such as K–T or J–Q) as in isotropic plastic solids. Acknowledgments The authors would like to gratefully acknowledge General Motors Research and Development Center, Warren, MI, USA, for financial support through sponsored project GM/IISc/SID/PC20037.

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Appendix A. Asymptotic solution for crack tip fields in the presence of an elastic sector in ideally plastic FCC single crystal As mentioned in Section 2, Rice (1987) proposed an asymptotic solution for stress field near the tip of a crack lying on the (0 1 0) plane of an ideally plastic FCC single crystal with the crack front along the ½1¯ 0 1 direction. For a stationary tensile crack, he assumed active plastic flow at all angles about the crack tip and constructed a solution that consists of four constant stress sectors separated by stress and displacement discontinuities as indicated in Fig. 2. Drugan (2001) extended Rice’s analysis to derive asymptotic solutions without kink-type shear bands by introducing sub-yield near-tip sectors. The general framework developed by Rice (1987) and Drugan (2001) is adopted here to obtain an asymptotic solution with low stress triaxiality ahead of the tip which corroborates with the results of the numerical simulation presented in Fig. 10. A.1. Asymptotic structure of the stress field in currently plastic sectors at stationary crack tip Rice (1987) showed that the cartesian stress components are constant in the near-tip sectors depicted in Fig. 2 which are stressed to yield and the stress state must change discontinuously from vertex to vertex of the yield surface (see Fig. A1) at the boundary between two adjacent sectors. He further demonstrated that the continuity of traction vector across the sector boundary requires 112ðt11 þ t22 ÞU ¼ 1LU,

(A.1)

where L is the arc length around the yield locus, having the units of stress and increasing in anticlockwise sense. A.2. Asymptotic structure of stress field in currently elastic sectors at stationary crack tip By assuming the elastic response as isotropic, Drugan (2001) showed that the most general stress field in a near-tip elastic sector that satisfies equilibrium along with compatibility has the form: t11  t22 ¼ C 2 sin 2y þ C 4 , (A.2a) 2t0 t11 þ t22 ¼ C 1 þ 2C 2 y, 2t0

(A.2b)

τ12 το -1.5 D

-1

0

-0.5

0.5

1

τ11 - τ22

A -0.2 -0.4 -0.6

D' 1.5 2το

elastic sector

-0.8 -1 -1.2 -1.4 -1.6 B

-1.8

C

Fig. A1. Asymptotic stress field in stress space for the solution structure shown in Fig. A2.

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t12 ¼ C 2 cos 2y þ C 3 , 2t0

(A.2c)

where C 1 –C 4 are undetermined constants. A.3. Construction of asymptotic solution for stationary crack with sub-yield sector and low stress triaxiality The near-tip stress distribution obtained from the modified boundary layer simulation corresponding to negative T-stress provides valuable information to construct a low triaxiality solution involving an elastic sector. The angular variation of stress components pertaining to negative T-stress (see Figs. 10(a)–(c)) clearly shows the existence of three constant stress sectors A, B and C. The stress discontinuity line between sectors A and B and sectors B and C is seen to be at around 55 and 90 , respectively. However, the sector C seems to extend beyond 125:3 which is the angle where sector C should end according to the asymptotic solution of Rice (1987). Also, the angular variation of maximum principal logarithmic plastic strain, logðlp1 Þ, for negative T-stress (see Fig. 13) suggests the existence of an elastic sector at about 145 . This also corroborates with the locus traversed by the near-tip stress state in stress space (see Fig. 10(d)) for the case of T=t0 ¼ 1. This elastic sector either extends all the way to the crack flank or connects to another sector D which is at yield. Guided by the above observations from the numerical simulations, an attempt is made to find a solution having the near-tip structure depicted in Fig. A2. An elastic sector is incorporated between sectors C and D so that the locus traced by the stress state on the stress plane is as shown in Fig. A1 which is similar to that found from the numerical simulation (Fig. 10(d)) for the case of T=t0 ¼ 1. It is important to note that this locus is different from that proposed by Drugan (2001) in which the stress state in the elastic sector varies along an arc commencing from point B and ending at point D0 . The constant stress states corresponding to plastic sectors A–D are obtained from yield locus shown in Fig. A1 and Eq. (A.1), whereas, the stress state in the elastic sector is derived by employing Eqs. (A.2) with full stress continuity across the radial boundaries y ¼ y1 and y2 . D For sector D, traction-free boundary conditions at crack flank requires tD 22 =t0 ¼ 0 and t12 =t0 ¼ 0. Thus, from the yield locus one gets pffiffiffi tD 11 ¼  6. t0 On enforcing full stress continuity across radial boundaries y ¼ y1 and y2 and employing Eqs. (A.2) and the yield locus, a set of six equations is obtained which after some manipulation assumes a form as follows: pffiffiffi sin 2y1  sin 2y2 3 2 , (A.3a) ¼ 4 cos 2y1  cos 2y2

stress discontinuities

54.7°

90° θ = θ1 C

B

elastic sector

θ = θ2

A D

Fig. A2. Schematic diagram showing near-tip solution structure for a stationary crack with elastic sector.

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Table A1 Cartesian stress components prevailing in the near-tip plastic sectors for tc ¼ 1:45 in the asymptotic solution with elastic sector shown in Fig. A2 Sector

t22 =t0

t11 =t0

t12 =t0

A B C D

5.737 3.287 0.838 0.0

3.287 2.062 2.062 2.450

0.0 1.732 1.732 0.0

pffiffiffi pffiffiffi 6 2 3ðy1  y2 Þ , ¼ tc þ 2 cos 2y1  cos 2y2 pffiffiffi 3 C2 ¼ , cos 2y1  cos 2y2 pffiffiffi 6  2C 2 y2 , C1 ¼  2

(A.3b)

(A.3c)

(A.3d)

C 3 ¼ C 2 cos 2y2 ,

(A.3e)

pffiffiffi 6  2C 2 sin y2 , C4 ¼  (A.3f) 2 C where tc is normalized biaxial stress in sector C, i.e., tc ¼ ðtC 11 þ t22 Þ=2t0 . To solve the two non-linear equations (Eqs. (A.3a) and (A.3b)) comprising of three unknown parameters (two elastic sector boundary angles y1 and y2 and tc ), some information is sought from the results of the numerical simulations. The modified boundary layer simulation corresponding to T ¼ t0 gives tc ¼ 1:45 (see Fig. 10). With this specific value of tc , a solution is obtained for Eqs. (A.3) which is as follows: y1 ¼ 141:6 ; C 2 ¼ 2:287;

y2 ¼ 175:1 ; C 3 ¼ 2:254

C 1 ¼ 12:753, and

C 4 ¼ 1:614.

The stress fields in plastic sectors determined with the help of Eq. (A.1) pffiffiffi C, B and A are then progressively pffiffiffi and noting that 1L=t0 UC2B ¼ 6=2 and 1L=t0 UB2A ¼ 3 6=4. The stress state in plastic sector obtained for the specific case of tc ¼ 1:45 is summarized in Table A1. The stress trajectory within the elastic sector governed by Eqs. (A.2) is plotted in stress space in Fig. A1. It is important to note that this trajectory is such that it does not violate yield anywhere including the end points C and D. The stress state given by the above asymptotic solution is summarized in Table A1. It can be seen that the stress triaxiality exhibited by this solution is significantly lower than that prevailing in the solution of Rice (1987). Further, the stress state given in Table A1 corroborates well with that obtained from numerical simulation for T=t0 ¼ 1 (see Fig. 10). Also, the elastic sector boundary angles y1 and y2 predicted by the asymptotic solution are in good agreement with those deduced from latter. Thus it is demonstrated in this Appendix and the work of Drugan (2001) that the presence of an elastic sector gives rise to lower stress triaxiality. Finally, it must be mentioned that a family of low stress triaxiality solutions can be generated along the line of this Appendix either by employing different values for tc or by commencing the elastic unloading at a point in between B and C in Fig. A1 (see also Drugan, 2001). References Asaro, R.J., 1983. Crystal plasticity. J. Appl. Mech. 50, 921–934. Basu, S., Narasimhan, R., 2000. A numerical investigation of loss of crack tip constraint in dynamically loaded ductile specimen. J. Mech. Phys. Solids 48, 1967–1985.

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