Constraint effects on crack-tip fields in elasticperfectly

Transcription

1 Journal of the Mechanics and Physics of Solids 49 (2001) Constraint effects on crack-tip fields in elasticperfectly plastic materials X.K. Zhu, Yuh J. Chao * Department of Mechanical Engineering, University of South Carolina, Columbia, SC 29208, USA Received 8 September 1999; received in revised form 28 March 2000 Abstract Asymptotic crack-tip stress fields accounting for constraint effects are developed for a stationary plane strain crack under mode-i, mode-ii or mixed-mode I/II loading. The mixedmode loading is considered only within small-scale yielding. Materials are taken into account in incompressible, elastic-perfectly plastic materials, and plastic deformation of materials obeys von Mises yield criterion. This investigation is an extension of the solution obtained by Li and Hancock [Li, J., Hancock, J.W., Mode I and mixed mode fields with incomplete crack tip plasticity. International Journal of Solids and Structures 36 (5), ] with special attention on what constraint parameters existed in the elastic-plastic crack-tip fields. Results indicate that the asymptotic crack-tip field is a 4-sector solution for mode-i cracks and a 6-sector solution for mixed-mode cracks, and is comprised of plastic sectors and elastic sector(s), and contain two undetermined parameters T p and T π which are hydrostatic stresses ahead of the crack tip and on the crack flank, respectively. When T p and T π vanish, the present elastic-plastic crack-tip field reduces to the fully plastic Prandtl slip-line field. Comparison shows that the asymptotic crack-tip stress fields can precisely match with elastic-plastic finite element results over all angles around a crack tip for various fracture specimens with constraint levels from high to low. The magnitudes of T p and T π determine the level of crack-tip constraint in plastic sectors and in elastic sector, respectively, due to geometric and loading configurations or mode mixity. Thus the parameters T p and T π can be used as constraint parameters to effectively characterize the entire crack-tip field in elastic-perfectly plastic materials under the plane strain conditions Published by Elsevier Science Ltd. Keywords: Crack-tip field; C. Asymptotic analysis; Constraint effect; Plane strain; Elastic-perfectly plastic material * Corresponding author. Fax: address: chao@sc.edu (Y.J. Chao) /01/$ - see front matter 2001 Published by Elsevier Science Ltd. PII: S (00)

2 364 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) Introduction Accurate characterization of asymptotic crack-tip fields is imperative in the constraint analysis and choice of fracture parameters. There are many published works on elastic-plastic deformation of near-tip field and constraint effect at a crack tip for power-law hardening materials. Since Hutchinson (1968a), Rice and Rosengren (1968) developed the HRR dominant singularity field for plane strain mode-i cracks based on a single parameter J-integral proposed by Rice (1968), the J-integralbased single parameter fracture criterion is gradually applied to fracture assessment of engineering structures. In practice, however, it is found that specimen geometry and loading configurations play a significant role in crack-tip fields. As such, the J- integral fracture criterion and the HRR field only have limited application to real flawed structures. Recently, more attention has been paid to constraint analysis and multi-parameter crack-tip fields. Typical representatives are the two-parameter crack-tip fields governed, respectively, by J T of Betegon and Hancock (1991), J Q of O Dowd and Shih (1991) and J A 2 of Chao et al. (1994) through extension of the HRR field using higher-order terms for mode-i cracks. These parameters are identified to be able to quantify the constraint effects on crack-tip fields for different cracked specimens. Further comments can be found in Chao and Zhu (1998). Constraint effects on mixed mode crack-tip fields in power-law hardening materials are presented in Du et al. (1991) and Roy and Narasimhan (1997). At complete yielding, analyses based on perfect plasticity could provide meaningful insights and reference values for low-hardening materials, and possibly for moderate-hardening materials. In fact, the earliest study of specimen geometry effect on crack-tip fields is for perfectly plastic materials using the slip-line theory. Based on the assumption that the plastic sectors entirely surround the crack tip, Prandtl (1920) first developed the well-known Prandtl slip-line field for a semi-infinite plane strain mode-i crack. Green and Hundy (1956), Green (1956), Ewing and Hill (1967) and Ewing (1968) reported the slip-line fields for different crack depths of Single Edged Notched plate under pure Bending (SENB) and Double Edged Crack Plate in tension (DECP). It is found that the Prandtl field can exist only in deeply cracked DECP specimens. Hutchinson (1968b) presented the slip-line field for plane strain mode- II cracks and observed that the HHR field approaches the Prandtl field in the limit of non-hardening. McClintock (1971) introduced the slip-line field of Center Crack Plate in tension (CCP), and summarized the results of slip-line fields for different notch or crack specimens. He found that the Prandtl field and CCP slip-line field are two extreme cases, all other slip-line fields of cracked specimens are in between the two extremes. As such, the stress and velocity slip-line fields around a crack tip are generally different for cracked specimens with different geometry or loading configurations. Wu et al. (1990) reviewed the slip-line field solutions for several conventional fracture specimens including both deep and shallow cracks. They concluded that the fully plastic crack-tip fields are similar to the Prandtl field for specimens with high triaxiality such as deeply cracked SENB and DECP, and Compact

3 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) Tension (CT) specimens, but considerably different for specimens with low triaxiality such as CCP specimens. Detailed investigation of constraint effects on near-tip fields in non-hardening materials was presented first by Du and Hancock (1991). Based on the modified boundary layer formulation, they numerically examined the effects of elastic T-stress on the near-tip field of a plane strain mode-i crack under contained yielding conditions. They found that a positive T-stress can cause plasticity to envelop the crack tip and yields the Prandtl field, while a negative T-stress reduces the triaxiality of the stress state at the crack tip and forms an incomplete Prandtl field. Under largescale yielding, Lee and Parks (1993) carried out the fully plastic analyses of single edge cracked specimens subject to different combined tension and bending for a sufficiently deep crack. Kim et al. (1996) and Zhu and Chao (2000) performed detailed finite element analyses (FEA) for SENB, CCP and DECP specimens, respectively, to study the effects of crack depth and loading level on crack-tip constraint for elastic-perfectly plastic materials under plane strain conditions. Both results indicate that only for deeply cracked SENB and DECP specimens, the values of crack-tip constraint remain almost constant for all range of deformation levels, and the crack-tip fields are very close to the Prandtl field. Under other cases, with the decrease of constraint levels, the hydrostatic stress ahead of the crack tip decreased from the Prandtl field and an elastic sector occurred on the crack flanks. Zhu and Chao (1999) showed that the two-parameter asymptotic field such as the J A 2 three-term solution can capture the essential features of elastic-plastic fields in the plastic sector ahead of the crack tip, but not in the elastic sector near the crack flanks. To explore the general structure of crack-tip fields, Ibragimov and Tarasyuk (1976) and Nemat-Nasser and Obata (1984) first discussed the possibility of near-tip fields containing elastic sectors for plane strain mode-i stationary cracks in elastic-perfectly plastic materials. More recently, Li and Hancock (1999) assembled crack-tip fields with slip-line plastic sectors and elastic sectors under contained yielding to match with the FEA results with the incomplete plasticity obtained by Du and Hancock (1991). The incomplete plasticity leads to a loss of crack-tip constraint and the appearance of an elastic sector on the crack flank. For mixed-mode-i/ii loading, using the theoretical methods of Hutchinson (1968a,b), Shih (1974) constructed an asymptotic crack-tip field under plane strain and small-scale yielding conditions based on the assumption that plasticity entirely surrounds the crack tip at all angles. With the exception of near mode-ii fields, these require a discontinuity of radial stress component in a sector trailing the crack front. In contrast, by use of FEA under the small-scale yielding, Dong and Pan (1990) and Hancock et al. (1997) calculated mixed-mode near-tip fields. These near-tip fields differ from those constructed by Shih (1974) in that plasticity does not entirely surround the crack tip and stress components contain no discontinuities. And with the exception of fields close to mode-ii cracks, an elastic sector appears on one crack flank. Dong and Pan (1990), Li and Hancock (1999) and Sham et al. (1999) assembled mixed mode crack-tip fields with slip-line plastic sectors and elastic sectors directly using the numerical border angles obtained in the FEA.

4 366 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) The deviation from the HRR fields at a crack tip in power-law hardening materials due to specimen geometry and loading configuration is often attributed to the loss of constraint. Similarly, the deviation from the Prandtl fields at a crack tip in elasticperfectly plastic materials can also be attributed to the effect of constraint. A key issue in studying the constraint effect is to identify the proper parameter(s) that can quantify the constraint levels. The present paper examines in detail the elastic-plastic crack-tip fields under mode-i, -II or mixed-mode-i/ii loading. Our investigation is an extension of the solution by Li and Hancock (1999) for incompressible, elastic-perfectly plastic materials under the plane strain conditions. Special attention is focused on what constraint parameters existed in the elastic-plastic crack-tip fields. Closed-form asymptotic solutions of crack-tip fields are developed. Results indicate that asymptotic crack-tip fields are comprised of plastic sectors and elastic sector(s), containing two undetermined parameters T p, T π and no discontinuities of stresses. To verify these asymptotic crack-tip fields, comparisons with numerical results are performed for various fracture specimens with constraint levels from high to low. The level of crack-tip constraint quantified by the parameters T p and T π is addressed. 2. Governing equations and asymptotic analysis Consider a stationary crack in an elastic-perfectly plastic material under plane strain conditions. Applied loading is accounted as mode-i, mode-ii or mixed-mode- I/II loading, respectively. Since both FEA of Dong and Pan (1990) and Du and Hancock (1991) showed that the effect of compressibility on the crack-tip stress distributions is very small for elastic-perfectly plastic materials, the material considered here is assumed to be incompressible, and deform plastically according to von Mises yield criterion and the associated flow rule Equilibrium equations Both Cartesian coordinates (x 1, x 2 ) and polar coordinates (r, q) (q=0 corresponding to the positive x 1 -direction) are introduced and centered at the crack tip. With reference to the polar coordinates, the equilibrium equations can be written as s rr r 1 s rq r q s rr s qq 0 (1) r 1 s qq r q s rq r 2s rq r 0 For elastic-perfectly plastic materials, the numerical calculations of Dong and Pan (1990) indicated that the stresses near a crack tip are nonsingular within both the plastic and elastic sectors around the crack tip. Accordingly, all stress components near the crack tip are only functions of the polar angle q, but not the distance r from

5 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) the crack tip (i.e. s ij =s ij (q), as r 0). Thus, the partial differential equations in (1) reduce to ordinary differential equations ds rq dq s rr s qq 0 (2) ds qq dq 2s rq Plane strain conditions For the incompressible, elastic-perfectly plastic material, the elastic-plastic constitutive relations of materials are the Prandtl-Reuss equations ė ij 2m ṡ 1 ij 1 3 ṡ kkd ij ls ij (3) where ė ij and ṡ ij are strain rate and stress rate components, respectively. m is the shear modulus, d ij is the Kronecker delta, s ij =s ij s kk /3 are the deviatoric stress components and l is the plastic flow factor. l 0 in elastic regions. Latin indices i, j, k have a range of 1 3. The plane strain conditions require e 33 =0. From (3), we have s (s 11 s 22 ) 0, in elastic region (4) C 0 e 2ml, in plastic region From the continuity of stress components at the border between an elastic region and a plastic region, the integration constant C 0 is equal to zero. For the incompressible, elastic-perfectly plastic material, therefore, the plane strain conditions are equivalent to s (s 11 s 22 )ors (s rr s qq ) (5) both in elastic regions and in plastic regions Yield criterion Under the plane strain condition (5), the von Mises yield criterion can be expressed as 1 4 (s rr s qq ) 2 s 2 rq k 2 (6) where k is the yield strength in shear and k=s 0 / 3, s 0 is the tensile yield stress. The yield criterion (6) is satisfied automatically if the in-plane stress components

6 368 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) can be expressed by using a stress function y(q) (cf Zhu et al., 1997) in the following form s rr (q)=s m (q) k cos y(q) s qq (q)=s m (q)+k cos y(q) (7) s rq (q)=k sin y(q) where s m =(s rr +s qq )/2 is a mean stress or hydrostatic stress. Comparing with (5), it holds that s m =s Asymptotic solution in plastic sector Without loss of generality, it is assumed that a near-tip field is comprised of plastic sectors and elastic sectors around a crack tip. Substituting (7) into the equilibrium Eq. (2), one obtains the governing field equations in plastic sectors as follows cos y(q) dy(q) dq 2 =0 ds m (q) dq =k sin y(q) dy(q) (8) dq 2 Solving the ordinary differential Eq. (8), we obtain two different sets of solutions which correspond to two different sets of plastic sectors at a crack tip. (i) Plastic constant stress sector (s ij =constant) y(q)=2q+y 0 (9) s m (q)=c 1 (ii) Plastic non-constant stress or fan sector (s rr =s qq ) y(q)= p 2 +np s m (q)= 2kq cos(np)+c 2 (10) The constants y 0, C 1, C 2 and the integer n in (9) and (10) can be determined by boundary and continuity conditions, and then the stress components in plastic sectors are given by (7) Asymptotic solution in elastic sector In an elastic sector, in addition to satisfying the equilibrium equations, the deformation field has to be compatible. Under the incompressible and plane strain conditions, the compatibility equation in terms of stress components gives

7 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) r 2 1 r r 1 2 r 2 q 2 (s rr s qq ) (11a) For the bounded stresses at a crack tip, Eq. (11a) becomes an ordinary differential equation d 2 dq 2(s rr s qq ) 0 (11b) Solving (2) and (11b), one can obtain an asymptotic stress field in the elastic sector as follows s rr (q)= A cos 2q B sin 2q+2Cq+D s qq (q)=a cos 2q+B sin 2q+2Cq+D (12) s rq (q)=a sin 2q B cos 2q C where the integration constants A, B, C and D can be determined from the boundary and continuity conditions Boundary conditions The equilibrium considerations alone require that traction σ qq and s rq along the border between any two sectors (either two plastic sectors or a plastic and an elastic sector) must be continuous, but radial stress s rr is not. To seek a fully continuous solution, it is assumed that s rr is also continuous along the border. And thus complete continuity of all stress components is required, or it is expressed mathematically as s ab (q i ) s ab (q + i ) (13) where q i and q + i represent the angle just before and after the border-delimitation angle q i, respectively. The traction-free conditions on the crack flanks are s qq ( p) 0; s rq ( p) 0 (14) 3. Crack-tip stress fields under mode-i loading For a mode-i opening crack, only the upper plane (0 q p) needs to be considered. The traction-free conditions on the crack flank and the deformation symmetric condition require s rq (p) 0, s qq (p) 0, s rq (0) 0 (15) Using boundary condition (15) and continuity condition (13), a fully plastic or elastic-plastic crack-tip stress field can be constructed.

8 370 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) Fully plastic crack-tip stress field This section assumes that plasticity entirely surrounds the crack tip at all angles. In terms of the stress function y(q) in (7), the boundary condition (15) becomes sin y(0) 0, sin y(p) 0, s m (p) k cos y(p) (16) Solving the ordinary differential Eq. (8) with the boundary condition (16), two general solutions of this boundary-value problem are obtained. One of the general solutions is and p 0 q 4 p y(q) 2q+np, 2 +np, p 4 q 3p 4 2q p+np, 3p 4 q p p cos(np), 0 q 4 s m (q) (1+p)k (1+ 3p 2 2q)k cos(np), p 4 q 3p 4 k cos(np), 3p 4 q p (17a) (17b) This solution shows that the fully plastic stress field is comprised of three plastic sectors: a constant stress sector ahead of the crack tip, followed by a centered-fan sector, and terminating with another constant stress sector adjacent to the crack surface. The other general solution is y(q) 2q np, s m (q) k cos(np) (18) The above equations hold for all angles in 0 q p. This solution shows that the full plastic stress field consists of a single uniform field around the entire crack tip. Notice that both in (17) and (18), the parameter n is an arbitrary integer, i.e. n =0, ±1, ±2,... Due to the periodic characteristics of the trigonometric function, one obtains only four sets of plastic solutions for a plane strain mode-i crack when (17) and (18) is substituted into (7), respectively First set of plastic solutions As n is taken as the even integers in (17), e.g. n=0, substitution of (17) into (7) obtains the first set of fully plastic crack-tip stress field that is precisely the Prandtl field under far-field tensile loading perpendicular to the crack line as follows

9 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) Sector 1 (0 q p/4) s rr =k(1+p cos 2q) s qq =k(1+p+cos 2q) s rq =k sin 2q 371 (19a) Sector 2 (π/4 q 3p/4) s rr =s qq =k(1+3p/2 2q) s rq =k (19b) Sector 3 (3π/4 θ π) s rr =k(1+cos 2q) s qq =k(1 cos 2q) s rq = k sin 2q (19c) Second set of plastic solutions As n is taken as the odd integers in (17), e.g. n= 1, substitution of (17) into (7) obtains the stress fields that are (19) with a negative sign added, i.e. the negative Prandtl field. Thus, this second set of plastic solutions corresponds to a mode-i crack under far-field compressive loading perpendicular to the crack line Third set of plastic solutions As n is taken as the even integers in (18), e.g. n=0, substitution of (18) into (7) has s rr = k(1+cos 2q) s qq = k(1 cos 2q), 0 q p (20) s rq =k sin 2q Under the rectangular coordinates, (20) becomes s xx (q)= 2k, s yy (q)=0, s xy (q)=0. Thus the uniform stress field in (20) appears to represent a crack under a far-field compressive load, 2k, parallel to the crack line Fourth set of plastic solutions As n is taken as the odd integers in (18), e.g. n=1, substitution of (18) into (7) obtains the stress field that is (20) with a negative sign. Thus the fourth set of fully plastic solutions corresponds to a crack subjected to a far-field tensile load, 2k, parallel to the crack line. In conclusion, except for the positive or negative sign of the stress fields, there

10 372 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) Fig. 1. Angular distributions of fully plastic crack-tip stress fields for mode-i cracks (a) the Prandtl field; (b) the uniform stress field subject to tension of far-field parallel to crack line. are only two independent fully plastic crack-tip fields as shown in Fig. 1 (i.e. the Prandtl field (19) and the uniform stress field (20)). It is noted that both stress fields in Fig. 1 meet the condition s qq (0) s rr (0) for mode-i tensile cracks. The following sections will show that these two fully plastic solutions are the two extremes of general elastic-plastic fields. In other words, the elastic-plastic crack-tip fields for a common specimen fall in between the two extreme cases shown in Fig Elastic-plastic crack-tip stress field Without loss of generality, it is assumed that the crack-tip field is comprised of plastic and elastic sectors. Based on the FEA results available for the materials considered, an elastic-plastic crack-tip stress field can be constructed by four sectors or three sectors as shown in Fig. 2. It is referred to as a 4-sector solution or 3-sector solution in the present work. In the 4-sector solution, the crack-tip field consists of Structures of elastic-plastic crack-tip field for mode-i cracks (a) 4-sector solution; (b) 3-sector sol- Fig. 2. ution.

11 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) three plastic sectors and an elastic sector: a plastic constant sector ahead of the crack tip over 0 q q 1, followed by a plastic fan sector in q 1 q q 2 and another plastic constant stress sector in q 2 q q 3, terminating with an elastic sector adjacent to the crack flank in q 3 q p. Here q 1,q 2,q 3 are the border angles delimitating sectors. Using the continuity condition (13) and boundary condition (15), from (9), (10) and (12), one can obtain the stress function in plastic sectors p 0 q q1= 4 p y(q) 2q, 2, q 1 q q 2 p 2 +2(q q 2), q 2 q q 3 and the hydrostatic stress around all angles p 0 q q1= 4 k 1+ s m (q) k(1+p)+tp, 3p 2 2q +T p, q 1 q q 2 k 1+ 3p 2 2q 2 +T p, q 2 q q 3 k cos(q 3 2q 2 ) sin q 3 2k(p q) cos 2q 2 1 cos 2q 3, q 3 q p (21a) (21b) where T p is an undetermined constant related to the border angles q 2 and q 3, and is given in (26). And thus, the stress components of the 4-sector solution are obtained as follows Plastic constant stress sector 1 (0 q p/4) s rr =k(1+p cos 2q)+T p s qq =k(1+p+cos 2q)+T p s rq =k sin 2q (22a) Plastic fan sector 2 (p/4 q q 2 ) s rr =s qq =k 1+ 3p 2 2q +T p s rq =k (22b) Plastic constant stress sector 3 (q 2 q q 3 )

12 374 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) s rr =k 1+ 3p 2 2q 2 +k sin 2(q q 2 )+T p s qq =k 1+ 3p 2 2q 2 k sin 2(q q 2 )+T p (22c) s rq =k cos 2(q q 2 ) Elastic sector 4 (q 3 q p) s rr = k cos(q 3 2q 2 ) sin q 3 [1+cos 2q] k cos 2q 2 1 cos 2q 3 [2(p q) sin 2q] s qq = k cos(q 3 2q 2 ) sin q 3 [1 cos 2q] k cos 2q 2 1 cos 2q 3 [2(p q)+sin 2q] s rq =k cos(q 3 2q 2 ) sin q 3 sin 2q k cos 2q 2 1 cos 2q 3 [1 cos 2q] (22d) Comparing (22a) and (22b) with (19a) and (19b), it is found that the stress field of the 4-sector solution in the plastic sectors 1 and 2 is related to the Prandtl field by s ij s Prandtl ij T p d ij (23) where s Prandtl ij denotes the components of stresses in the Prandtl field. It is obvious that the parameter T p in the solution of (22) is a hydrostatic stress and can be determined at q=0 by T p s app qq (0) s Prandtl qq (0) (24) Similarly, we can define another hydrostatic stress parameter T p on the crack surface as 2T p s app rr (p) s Prandtl rr (p) (25) where s app qq (0) and s app rr (p) stand for the known stress components at q=0 and q=p, respectively, from the slip-line field or a FEA. Using the two parameters T p and T p, the delimitation angles q 2 and q 3 in the 4- sector solution (22) can be determined by solving the following equation T p k = cos(q 3 2q 2 ) 1 sin q 3 T p k = 2q 2 3p 2 1 (p q 3 ) cos 2q 2 cos(q 3 2q 2 ) sin 2 q 3 sin q 3 (26)

13 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) Discussions about the 4-sector solution In terms of the yield condition (6), the radial stress s app rr (p) on the crack surface has the range of s app rr (p) 2k. Accordingly, from (25) and (26), there exist several special cases in the 4-sector solution as below: (a) When s app rr (p)=2k, we have T p 0, T p 0; q 2 3p/4, q 3 p Under this case, the elastic sector 4 disappears and the 4-sector solution (22) reduces to the fully plastic Prandtl field (19). (b) When s app rr (p)= 2k, we have T p 2k, T p (2 p)k, q 3 q 2 q 1 p/4 Under this case, the plastic sectors 2 and 3 vanish and the elastic sector 4 becomes a plastic sector under constant pressure 2k. And the 4-sector solution (22) reduces to the fully plastic uniform stress field (20). (c) When 2k s app rr (p) 2k, an elastic sector 4 appears near the crack surface. The angular span f(=p q 3 ) of the elastic sector 4 increases with the decrease of parameters T p and T p, thus stress components s app rr (p) and s app qq (0). The Prandtl field (19) and the plastic uniform stress field (20) are the two extremes of the general elastic-plastic stress field (22). Ranges of all unknown parameters in (26) are (2 p)k T P 0, 2k T p 0; p/4 q 2 3p/4, p/4 q 3 p With these ranges, from (21b), angular variation of the hydrostatic stress s m with different values of the parameters T p and T p can be illustrated in Fig. 3. It is seen Fig. 3. Sketch of angular variation of the hydrostatic stress s m with different values of the parameters T p and T p for mode-i cracks.

14 376 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) that parameters T p and T p can characterize constraint levels in the plastic sectors ahead of crack tip and in the elastic sector on the crack surface, respectively. The two parameters are generally independent to each other and so both can be used as constraint parameters. (d) When q 3 =q 2, the 4-sector solution reduces to a 3-sector solution. From (26), we have T p =k(cot f 1) T p =k p 1+cot f f (27) 2 sin f 2 where f=p q 2 is the angular span of the elastic sector 3 in 3-sector solution. Eliminate the unknown f from (27), the two constraint parameters T p and T p are related by T p k p 2 T p k 1 (T p+k) 2 k arctan k (28) 2 T p +k Notice that if the trigonometric function arctan[k/(t p +k)] is less than zero, it must be substituted by p arctan[k/(t p +k)]. Under the condition (28), only one of the two constraint parameters T p and T p is independent, the 4-sector solution degenerates to 3-sector solution as shown in Fig. 2(b). Eq. (28) is thus the existence condition of the 3-sector solution. From (22), the stress field of the 3-sector solution is simplified as Plastic constant stress sector 1 (0 q p/4) Stress components are the same as those in (22a) Plastic fan sector 2 (p/4 q q 2 ) Stress components are the same as those in (22b) Elastic sector 3 (q 2 q p) s rr = k cot q 2 [1+cos 2q]+k(1 cot 2 q 2 ) (p q) 1 2 sin 2q s qq = k cot q 2 [1 cos 2q]+k(1 cot 2 q 2 ) (p q)+ 1 2 sin 2q (29) s rq =k cot q 2 sin 2q+k(1 cot 2 q 2 )sin 2 q where q 2 =p f is determined by (27), and p/4 q 2 3p/4. It should be noted that the 3-sector solution is invalid for the elastic angular span 0 f p/4 in which the yield criterion is violated in the postulated elastic sector. Comparison shows that our 3- sector solution is the same as the solution developed by Li and Hancock (1999). Fig. 4 shows that the angular span f, thus the size of the elastic sector, indeed increases with the decreasing of constraint parameters T p and T p in the 3-sector solution. When f=p/4, T p =T p =0 and q 2 =3p/4, stress components of (29) in the elastic

15 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) Fig. 4. Variation of constraint parameters T p and T p in the 3-sector solution with the elastic angular span f=p q 2 for mode-i cracks. sector reduce to (19c) in the plastic sector, and so the 3-sector solution reduces to the fully plastic Prandtl field (19). When f=3p/4, T p = (2+p)k, T p = 2k and q 2 =p/4, (29) reduces to (20) in the plastic sector, and so the 3-sector solution reduces to the fully plastic uniform field (20). 4. Crack-tip stress fields under mode-ii loading For mode-ii crack problems under the plane strain conditions, the equilibrium equations and yield criterion are the same as (2) and (6), respectively. If the stress components are still represented by (7), then the governing equation of the plastic stress field remains the same as (8). The difference between mode-ii and mode-i cracks is only the boundary condition. For a mode-ii shear crack, from the skewsymmetric deformation along the remaining ligament and traction free on the crack face, the boundary conditions are s rr (0) 0, s qq (0) 0; s rq (p) 0, s qq (p) 0 (30) 4.1. Fully plastic crack-tip stress field Again, it is assumed that plasticity entirely surrounds the crack tip. Using the stress function y and the hydrostatic stress s m, the boundary conditions (30) can be rewritten as

16 378 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) cos y(0) 0, sin y(p) 0; s m (0) 0, s m (p) k cos y(p) (31) Solving the ordinary differential field Eq. (8) with the boundary conditions (31), the general solution of this boundary-value problem is obtained as and 2 y(q) p +np, 0 q q 1 2(q q 1 )+ p 2 +np, q 1 q q 1 + p 2 3p 2 +np, q 1+ p 2 q 3p 4 2q+np, 3p 4 q p cos(np), 0 q q1 1 cos(np), q 1 q q 1 + s m (q) 2kq p 2 2kq 4k q 1 + p q 4 cos(np), 1 + p 2 q 3p 4 4k q 1 p 8 cos(np), 3p 4 q p (32a) (32b) where the parameter n is an arbitrary integer, i.e. n=0, ±1, ±2,... q 1 is the delimitation angle between the first sector and the second sector and q 1 = 1 4 +p. This solution shows 8 that the fully plastic crack-tip stress field is comprised of four plastic sectors: a fan sector ahead of the crack tip, followed by a constant sector and another fan sector, and terminating with another constant sector adjacent to the crack surface. Due to the periodic characteristics of the trigonometric functions, substituting (32) into (7), one obtains only two sets of plastic solutions for a plane strain mode- II crack First set of plastic solutions As n is taken as the even integers in (32), e.g. n=0, substitution of (32) into (7) yields fully plastic crack-tip stress fields as follows Sector 1 (0 q q 1 ) s rr =s qq = 2kq s rq =k (33a)

17 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) Sector 2 (0 1 q q 1 +p/2) s rr = 2kq 1 +k sin 2(q q 1 ) s qq = 2kq 1 k sin 2(q q 1 ) s rq =k cos 2(q q 1 ) (33b) Sector 3 (q 1 +p/2 q 3p/4) s rr =s qq =2kq 4k q 1 + p 4 s rq =k (33c) Sector 4 (3p/4 q p) s rr = k k cos 2q s qq = k+k cos 2q s rq =k sin 2q (33d) The angular variations of stress components in (33) are shown as in Fig. 5(a). Through simple comparison, one can find that the present plastic solution (33) is exactly the slip-line field of plane strain mode-ii crack given by Hutchinson (1968b). Accordingly, this set of plastic solutions corresponds to a plane strain mode-ii crack under far-field clockwise shear forces Second set of plastic solutions As n is taken as the odd integers in (32), e.g. n=1, substitution of (32) into (7) obtains the stress field (33) with a negative sign added or the negative stress field of (33) which corresponds to plane strain mode-ii crack under counter-clockwise applied shear forces. Therefore, the crack-tip stress field (33) for plane strain mode- II crack is indeed unique Elastic-plastic crack-tip stress field It is possible that an elastic sector occurs on one crack surface in the mode-ii crack-tip stress field under action of the far-field T-stress. As such, without loss of generality, this section assumes that the crack-tip field is fully plastic on the lower plane, but on the upper plane it is comprised of three plastic sectors and an elastic sector: a plastic fan sector ahead of the crack tip over 0 q q 1, followed by a plastic constant sector in q 1 q q 2 and another plastic fan stress sector in q 2 q q 3, terminating with an elastic sector adjacent to the crack surface in q 3 q p. Here q 1,q 2,q 3

18 380 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) Fig. 5. Angular distributions of crack-tip stress fields for mode-ii cracks (a) fully plastic stress field; (b) elastic-plastic stress field. are the border angles delimitating sectors. Using the continuity condition (13) and boundary condition (30), from (9), (10) and (12), one can obtain the elastic-plastic stress field as follows Plastic fan sector 1 (0 q q 1 ) s rr =s qq = 2kq s rq =k (34a) Plastic constant stress sector 2 (q 1 q q 1 +p/2) s rr = 2kq 1 +k sin 2(q q 1 ) s qq = 2kq 1 k sin 2(q q 1 ) s rq =k cos 2(q q 1 ) (34b)

19 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) Plastic fan sector (q 1 +p/2 q q 3 ) s rr =s qq =2kq 4k q 1 + p 4 s rq =k Elastic sector 4 (q 3 q p) s rr =k cot q 3 [1+cos 2q]+k cot q 3 cot 2q 3 [2(p q) sin 2q] s qq =k cot q 3 [1 cos 2q]+k cot q 3 cot 2q 3 [2(p q)+sin 2q] s rq = k cot q 3 sin 2q+k cot q 3 cot 2q 3 (1 cos 2q) 381 (34c) (34d) where q 3 q 2 =q 1 +p/2. q 3 and q 1 are dependent on the radial stress s rr (p) and determined by q3=arctan 2k s rr (p) q 1 = 1 4 [q 3 cot q 3 (p q 3 )cot 2 q 3 ] (35) Fig. 5(b) plots the stress distribution of the elastic-plastic crack-tip field (34) with s rr (p)= 1.5k, q 1 =35 and q 3 =127. This is equivalent to the results of mode-ii cracks under a positive far-field T-stress. Comparing (34) with (33) and Fig. 5(b) with Fig. 5(a), one can find that the stress components ahead of the crack tip and on the lower plane remain invariant due to the occurrence of elastic sector on the upper crack surface. Therefore, it can be concluded that constraint has no effect on the stress states ahead of the crack tip for a mode-ii crack in elastic-perfectly plastic materials. A similar conclusion is obtained by Chao and Yang (1996) for power-law hardening materials. 5. Crack-tip stress fields under mixed-mode-i/ii loading For mixed-mode-i/ii crack problems, our attention is concentrated within the small-scale yielding (SSY). The SSY field is one field subjected to very small applied load levels for all geometries, which corresponds to the T=0 field. FEA results of Dong and Pan (1990), Hancock et al. (1997) and Li and Hancock (1999) showed that under the SSY conditions the mixed-mode crack-tip fields are (a) the Prandtl slip-line field for pure mode-i loading,

20 382 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) (b) the Hutchinson slip-line field for pure mode-ii loading, (c) fully plastic crack-tip fields for near mode-ii loading, (d) elastic-plastic crack-tip fields for other loading cases. Therefore, only case (d) above will be addressed in the following sections Elastic-plastic crack-tip stress field For mixed-mode crack-tip fields, it is observed from elastic-plastic FEA that the constant stress sector 1 ahead of the crack tip under mode-i loading rotates clockwise as the mode-ii loading is applied, and one elastic sector on the upper crack flank expands. It seems that this kind of mixed-mode crack-tip field is the perturbation of border angles in the mode-i crack-tip field. Under SSY, thus, it can be assumed that the elastic-plastic mixed-mode crack-tip field is constructed generally by 6-sector solution or 5-sector solution as shown in Fig. 6. These two solutions are the extension of the 4-sector solution and the 3-sector solution of mode-i cracks. The 6-sector solution is comprised of an elastic sector over q 5 q p and five plastic sectors: the first constant stress sector in p q q 1, the first fan sector in q 1 q q 2, the second constant stress sector in q 2 q q 3, the second fan sector in q 3 q q 4 and the third constant sector in q 4 q q 5. Here q 1,q 2,,q 5 are the border angles delimitating sectors. The 5-sector solution is a special case of the 6-sector solution when q 5 =q 4. Using the continuity condition (13) and boundary condition (14), from (9), (10) and (12), we obtain stress fields in the 6-sector solution as follows Plastic constant stress sector 1 ( p q q 1 ) s rr =k(1+cos 2q) s qq =k(1 cos 2q) s rq = k sin 2q (36a) Fig. 6. Structures of elastic-plastic crack-tip field for mixed-mode-i/ii cracks (a) 6-sector solution; (b) 5-sector soution.

21 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) Plastic fan sector 2 (q 1 q q 2 ) s rr =s qq =k 1+ 3p 2 +2q s rq = k (36b) Plastic constant stress sector 3 (q 2 q q 3 ) s rr =k 1+ p 2 +2q 3 +k sin 2(q q 3 ) s qq =k 1+ p 2 +2q 3 k sin 2(q q 3 ) (36c) s rq =k cos 2(q q 3 ) Plastic fan stress sector 4 (q 3 q q 4 ) s rr =s qq =k 1+ p 2 +4q 3 2q s rq =k (36d) Plastic constant sector 5 (q 4 q q 5 ) s rr =k 1+ p 2 +4q 3 2q 4 +k sin 2(q q 4 ) s qq =k 1+ p 2 +4q 3 2q 4 k sin 2(q q 4 ) (36e) s rq =k cos 2(q q 4 ) Elastic sector 6 (q 5 q p) s rr = k cos(q 5 2q 4 ) sin q 5 [1+cos 2q] k cos 2q 4 1 cos 2q 5 [2(p q) sin 2q] s qq = k cos(q 5 2q 4 ) sin q 5 [1 cos 2q] k cos 2q 4 1 cos 2q 5 [2(p q)+sin 2q] s rq =k cos(q 5 2q 4 ) sin q 5 sin 2q k cos 2q 4 1 cos 2q 5 [1 cos 2q] and the hydrostatic stress components over all angles are (36f)

22 384 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) k, p q q1 k 1+ 3p 2 +2q, q 1 q q 2 k 1+ p s m 2 +2q 3, q 2 q q 3 k 1+ p 2 +4q 3 2q, q 3 q q 4 k cos(q 5 2q 4 ) 2k(p q) cos 2q 4, q sin q 5 1 cos 2q 5 q p 5 (37) where the delimitation angles q 1 = 3p 4, q 2=q 3 p 2 ; q 3, q 4 and q 5 can be determined to solve the following equation system T p k cos(q 5 2q 4 ) sin q 5 1 (38a) 4 q 3 p 4 2q 4 3p 2 1 (p q 5 ) cos 2q 2 sin 2 q 5 cos(q 5 2q 4 ) sin q 5 (38b) T p k p 2q 3 1 2q 2 +sin 3 ; s rq (0) k 4q 3 p; s rq (0)=k (38c) in which the parameters T p and T p is defined as hydrostatic stresses ahead of crack tip and on the crack flank, respectively, which is similar to the mode-i case. They can be determined at q=0 and q=p by and T p sapp qq (0) s Prandtl s app qq (0); s rq (0) k qq (0) (1+3p/2)k; s rq (0)=k (39) 2T p s app rr (p) s Prandtl rr (p) (40) where s app qq (0) and s app rr (p) stand for the known stress components at q=0 and q=p, respectively, from the slip-line field or FEA. Once the parameters T p and T p are determined from (39) and (40), q 3, q 4 and q 5 can be solved from (38). Then the elastic-plastic stress field over all angles are given by (36).

23 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) Discussions about the 6-sector solution Due to s app rr (p) 2k for elastic-perfectly plastic materials, from (38) and (40), several special solutions exist in the 6-sector solution of mixed-mode I/II cracks: (a) When s app rr (p)=2k, we have T p 0, T p 0; q 2 p 4, q 3 p 4, q 4 3p 4, q 5 p Under this case, the elastic sector 6 disappears and the 6-sector solution (36) reduces to the fully plastic Prandtl field (19). (b) When s app rr (p)= 2k, we have T p 2k, T p (2 p)k; q p 2,q 3 1 2, q 4 q 5 p 4 Under this case, the plastic constant stress sector 5 vanishes and the elastic sector 6 becomes a plastic sector under constant pressure 2k. And the 6-sector solution (36) degenerates to a fully plastic near mode-ii stress field. (c) When 2k s app rr (p) 2k, an elastic sector 6 appears near the crack surface. The angular span f(=p q 5 ) of the elastic sector 6 increases as the parameters T p and T p, thus stress components s app rr (p) and s app qq (0), decrease. Ranges of all unknown parameters in (36) are (2 p)k T P 0, 2k T p 0; 1 2 p 2 q 2 p 4, 1 2 q 3 p 4, p 4 q 4 3p 4, p 4 q 5 p With these ranges, from (37), angular variation of the hydrostatic stress s m with different values of the parameters T p and T p can be illustrated in Fig. 7. It is seen again that parameters T p and T p can characterize constraint levels in the plastic sectors ahead of crack tip and in the elastic sector on the crack surface, respectively. The Prandtl field (19) and the fully plastic near mode-ii stress field are the two extremes of the elastic-plastic stress field (36). Notice that the pure mode-ii slipline field cannot be obtained from (36) because the elastic-plastic solution (36) is based on the extension from the pure mode-i solution [see Fig. 6(a)]. (d) When q 5 =q 4, the constraint parameters T p and T p depend on each other, and the 6-sector solution above reduces to the 5-sector solution, as shown in Fig. 6(b), which is the same as the solution developed by Li and Hancock (1999). The stress field of 5-sector solution can be obtained from (36) by letting q 5 =q Plasticity mixity for mixed-mode cracks Based on the current solution and the well-accepted concept of mode mixity, we determine in this section under what mode mixity an elastic sector will occur on the crack flank. The near-tip plasticity mixity parameter, M p, is defined by Shih (1974) in terms of opening and shear stresses ahead of the crack tip as

24 386 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) Fig. 7. Sketch of angular variation of the hydrostatic stress s m with different values of the parameters T p and T p for mixed-mode-i/ii cracks. M p 2 p arctan s qq lim (41) r 0s rq (r,q=0) with M p =1 for the pure mode-i and M p =0 for the pure mode-ii case. From the present elastic-plastic solution (42), M p is related to q 3 as p M p 2 arctan 1+p/2+2q 3+sin 2q 3 cos 2q, 0 q 3 p p arctan 1+ p 2 +4q 3, 1 2 q 3 0 (42) The analysis above gives the range of the delimitation angle q 3. Therefore from (42), the range of M p for the elastic-plastic 6-sector solution (36) can be determined as (a)when q 3 = p 4, M p=1; the 6-sector solution reduces to the Prandtl slip-line field. (b)when, q 3 = 1 2,M p=0.3302; the 6-sector solution reduces to the fully plastic near mode-ii stress field. (c)when 1 2 q 3 p 4, M p 1; an elastic sector occurs on the crack flank and

25 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) increases in size with the decrease of M p value. The crack-tip field is determined by the general 6-sector solution. Moreover, as 0 q 3 p 4, M p 1, the constant stress sector 3 crosses the line q=0; and as 1 2 q 3 0, M p , the fan sector 4 crosses the line q=0. (d)when 0 M p , the crack-tip field is for fully plastic near pure mode II cracks. 6. Constraint analysis of crack-tip field with elastic sector As displayed in previous sections, except for the pure or near mode-ii cracks, the crack-tip field is influenced by the constraint of specimen geometry and loading configurations. As a result, the opening stress ahead of the crack tip decreases and an elastic sector occurs on the crack flank with decreasing crack-tip constraint. This kind of mechanics behavior and constraint level can be theoretically described by the two hydrostatic stress parameters T p and T p. To verify the present asymptotic crack-tip fields, comparisons with numerical results are performed for various fracture specimens. Quantification of constraint levels using the parameters T p and T p is also addressed in this section. The constraint effect is discussed both for the SSY case and for finite-sized specimens. For finite-sized specimens, it is generally acknowledged that a deeply cracked DECP or SENB specimen is the representative for high constraint specimen geometry. And a CCP specimen is the representative for low constraint specimen geometry. The constraints of all other commonly used specimens generally fall in between these two extreme cases. As such, our study for finite-sized specimens includes these three specimens to cover a wide range of constraints. Particular emphasis is placed on how well the parameters T p and T p can be used to quantify the constraint level over the wide range Mode-I cracks under small-scale yielding For mode-i cracks in elastic-perfectly plastic materials under small scale yielding, Shih and German (1981) and Zhu and Chao (1999, 2000) showed that since the applied loading is small, the plastic zone at the crack tip is very small, but indeed encompasses the entire crack tip in practical structures. In this circumstance, the boundaries of a finite-sized specimen hardly affect the crack-tip fields. Their FEA results for SENB, CCP and DECP specimens indicate that as the crack tip is approached the tensile stress ahead of the crack tip attains the limiting value of 2.97s 0 given by the Prandtl field. For the far-field stress T=0, Dong and Pan (1990), Betegon and Hancock (1991) and Du and Hancock (1991) presented the crack-tip stress field which is close to the Prandtl field for the non-hardening materials by

26 388 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) using a boundary layer FEA approach. Consequently, it can be concluded that the mode-i crack-tip field under SSY is almost identical to the Prandtl field, and so the constraint parameters T p =T p =0. In other words, the mode-i crack-tip field is nearly unaffected by the constraints of specimen geometry and loading under SSY SENB specimens in pure bending Consider a SENB specimen in pure bending with the specimen width W, the crack depth a and the ligament length b. For elastic-perfectly plastic materials, Kim et al. (1996) presented the numerical crack-tip fields for SENB specimens with various crack depths through detailed FEA. Using their numerical results (see Fig. 5 in Kim et al., 1996), we can determine the values of constraint parameters T p and T p from (24) and (25). Then the angular crack-tip stress distributions for a specific case can then be obtained from (22). Figs shows the angular distributions of the stress components s rr, s qq and s rq for the SENB specimens with a/w=0.1, 0.2, 0.3 determined from the 3-sector solution, the 4-sector solution and from the FEA by Kim et al. (1996) at the limit load. Fig. 11 is the angular variation of the hydrostatic stress or mean stress s m for the same specimen. Notice that the crack becomes enough deep, i.e. a/w 0.5, both the FEA results and the 4-sector or 3-sector solutions approach to the Prandtl field as shown in Fig. 1(a). The results in Figs 8(b) 11(b) show that the 4-sector solution can match exactly with the numerical results both for the stress distributions and the delimitation angles. However, the 3-sector solution can only match with the numerical results within the plastic sector ahead of the crack tip [see Figs 8(a) 11(a)], but not in elastic sectors. In the 3-sector solution the border angle q EP delimitating the elastic sector and the plastic sector is quite different from the FEA result. In general, Fig. 8. Angular distributions of stresses in SENB specimens with a/w=0.1. Notice that the symbol denotes the FEA solution from Kim et al. (1996) and the solid line denotes the present asymptotic solution (a) the numerical results and the 3-sector solution with T p = 0.969s 0 and q 2 75 ; (b) the numerical results and the 4-sector solution with T p = 0.969s 0, T p = 0.827s 0, q and q

27 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) Fig. 9. Angular distributions of stresses in SENB specimens with a/w=0.2. Notice that the symbol denotes the FEA solution from Kim et al. (1996) and the solid line denotes the present asymptotic solution (a) the numerical results and the 3-sector solution with T p = 0.588s 0 and q ; (b) the numerical results and the 4-sector solution with T p = 0.588s 0, T p = 0.877s 0, q and q Fig. 10. Angular distributions of stresses in SENB specimens with a/w=0.3. Notice that the symbol denotes the FEA solution from Kim et al. (1996) and the solid line denotes the present asymptotic solution (a) the numerical results and the 3-sector solution with T p = 0.119s 0 and q ; (b) the numerical results and the 4-sector solution with T p = 0.119s 0, T p = 0.352s 0, q and q therefore, it is necessary to use the two independent parameters T p and T p to characterize the constraint effects on the crack-tip field both in the plastic sector ahead of the crack tip and in the elastic sector on the crack flank, respectively. It is seen that the values of constraint parameter T p and T p increase with increasing a/w and approaches zero as coming closer to the limiting case, i.e. the Prandtl field. The values of T p /s 0 in the 4-sector solution are 0.969, 0.588, 0.119, 0.0, respectively for SENB specimens corresponding to a/w=0.1, 0.2, 0.3, 0.5 (or 0.7),

28 390 X.K. Zhu, Y.J. Chao / Journal of the Mechanics and Physics of Solids 49 (2001) Fig. 11. Angular distributions of hydrostatic stress of SENB specimens for a/w=0.1, 0.2, 0.3, 0.5 (or 0.7). Notice that the symbol denotes the FEA solution from Kim et al. (1996) and the solid line denotes the present asymptotic solution (a) the numerical results and the 3-sector solution; (b) the numerical results and the 4-sector solution. and the values of T p /s 0 are 0.827, 0.877, and 0.0. These values are plotted in Fig. 12 and are curve-fitted to yield T p 7.715(a/W) (a/W) for a/w 0.5 (43a) 0.0 for a/w 0.5 and Fig. 12. Variation of the parameters T p and T p with the crack depth a/w for SENB specimens at the limit load.