University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications from the Deartment of Engineering Mechanics Mechanical & Materials Engineering, Deartment of May 2002 Closed-form solution for the size of lastic zone in an edge-cracked stri Xiangfa Wu Deartment of Engineering Mechanics, University of Nebraska-Lincoln, xfwu@unlserve.unl.edu Yuris A. Dzenis Deartment of Engineering Mechanics,University of Nebraska-Lincol, ydzenis@unl.edu Follow this and additional works at: htt://digitalcommons.unl.edu/engineeringmechanicsfacub Part of the Mechanical Engineering Commons Wu, Xiangfa and Dzenis, Yuris A., "Closed-form solution for the size of lastic zone in an edge-cracked stri" (2002). Faculty Publications from the Deartment of Engineering Mechanics. 34. htt://digitalcommons.unl.edu/engineeringmechanicsfacub/34 This Article is brought to you for free and oen access by the Mechanical & Materials Engineering, Deartment of at DigitalCommons@University of Nebraska - Lincoln. It has been acceted for inclusion in Faculty Publications from the Deartment of Engineering Mechanics by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln.
International Journal of Engineering Science 40 (2002) 1751 1759 www.elsevier.com/locate/ijengsci Closed-form solution for the size of lastic zone in an edge-cracked stri Xiang-Fa Wu *, Yuris A. Dzenis Deartment of Engineering Mechanics, Center for Materials Research & Analysis, University of Nebraska Lincoln, Lincoln, NE 68588-0526, USA Received 16 November 2001; received in revised form 20 February 2002; acceted 20 March 2002 Abstract This aer is concerned with the roblem of lastic zone at the ti of an edge crack in an isotroic elastolastic stri under anti-lane deformations. By means of comlex otential and Dugdale model, the stress intensity factor and the size of lastic zone are obtained in closed-form. Furthermore, the analytic solutions for an edge crack at the free boundary of a half-sace and a semi-infinite crack heading towards a free surface are determined as the limiting cases of the stri geometries. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Edge crack; Plastic zone size; Stri; Stress intensity factor 1. Introduction Crack ti oening dislacement and the size of lastic zone near crack tis are two imortant fundamental arameters utilized in nonlinear analysis of cracks in ductile materials. Based on the assumtion of constant cohesive stress at the leading edge of the crack, the Dugdale model was utilized extensively for investigating the lastic deformation near crack tis, and yielded reliable redictions at the ends of stationary slits in steel sheets [1]. The Dugdale model rovides a feasible method to estimate the size of the lastic zone near crack ti in the framework of linear fracture mechanics. A number of solutions for notches, cracks and satial enny-shaed cracks under anti-lane or in-lane deformations were obtained using the Dugdale hyothesis, for examle, see recent works * Corresonding author. Fax: +1-402-472-8292. E-mail addresses: xfwu@unlserve.unl.edu, wuxiangfa@yahoo.com (X.-F. Wu). 0020-7225/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S0020-7225(02)00031-9
1752 X.-F. Wu, Y.A. Dzenis / International Journal of Engineering Science 40 (2002) 1751 1759 by Singh and coworkers [2 10], Olesiak and Wnuk [11], Olesiak and Shadley [12], Tsai [13], Fan [14,15], Wang and Shen [16]. It is also imortant to mention the early work by Atkinson and Howard [17], Bilby et al. [18], Field [19], Koskinen [20], Rice [21], Smith [22]. A brief literature review on the Dugdale model and its related alications can be found in the recent aer by Vrbik et al. [10]. In the resent work, on the basis of comlex dislacement otential and Dugdale model, a method is roosed to find the closed-form solutions for the stress intensity factor (SIF) and the size of the lastic zone of an edge crack in an isotroic elastolastic stri under anti-lane deformations. The solution is obtained by introducing a conformal maing, which mas the edgecracked stri onto the whole lane with a semi-infinite cut. Comared with the methods develoed in literature, the current method rovides a more concise way to extract the closed-form solutions. As the limiting cases of the current roblem, the SIF and lastic zone size solutions for an edge crack at the free boundary of a half-lane as well as a semi-infinite crack heading towards a free surface can be determined directly without recourse to the method of dual integral equations or the Wiener Hof technique. 2. Formulation and solution rocedure Cracks in elastic media under deformations can be analysed as interactions between a dislocation and cracks. In the anti-lane case, the dislacement and stress comonents of an isotroic elastic body can be exressed in terms of an analytic function uðfþ as follows: u 3 ðn; gþ ¼ 2 l Im½uðfÞŠ; r 23ðn; gþ ¼ 2Re½u 0 ðfþš; r 13 ðn; gþ ¼2Re½iu 0 ðfþš; ð1þ where the rime ( 0 ) denotes the derivative with resect to f ¼ n þ ig, and l is the material shear modulus. Consider a semi-infinite crack interacting with a singularity (a screw dislocation or a line force) in an infinite lane as shown in Fig. 1(a). The singularity is located at f 0 ¼ n 0 þ ig 0 where n and g are the coordinates of the material oints, ðn; gþ with the origin located at the crack ti. Thomson [23] obtained the comlex dislacement otential for this anti-lane roblem as follows: Fig. 1. A semi-infinite crack in the whole sace under anti-lane deformations: (a) with an anti-lane singularity and (b) with a air of concentrated forces located at crack surfaces.
X.-F. Wu, Y.A. Dzenis / International Journal of Engineering Science 40 (2002) 1751 1759 1753 0 1 u 0 ðfþ ¼ q q f f 0 2 ffiffi B q q C @ ffiffi ffiffiffiffi þ ffiffi ffiffiffiffi A; ð2þ f f þ f 0 f þ f 0 where the over bar ðþ denotes the comlex conjugate. Here, the quantity q is defined as q ¼ bl 4 þ i 4 ; in which b is the Burgers vector of the screw dislocation and is the line force. The SIF (K III ) and the energy release rate (ERR) (G III ) are evaluated resectively as ffiffiffi K III ¼ lim 2 2n u 0 ðnþ; G III ¼ K2 III n!0 2l : ð3þ ð4þ Under the action of a singularity q at f 0, substitution of (2) and (3) into (4) yields 0 1 K III ¼ ffiffiffiffiffi B q 2@ ffiffiffiffi þ q C ffiffiffiffi A: ð5þ f 0 f 0 When the crack is oened by a air of self-equilibrated anti-lane forces P 0 acting on the crack surfaces a distance l behind the crack ti, the corresonding SIF and ERR may be extracted from (4) and (5) as K III ¼ 2P 0 ffiffiffiffiffiffiffi ; G III ¼ P 0 2 2l ll : ð6þ Let us now consider an edge crack at the free boundary of a stri with an anti-lane singularity (a screw dislocation or a line force) located at z 0 ¼ x 0 þ iy 0 as shown in Fig. 2(a). Here, a and W denote the crack length and the stri width, resectively. Consider the conformal maing, Fig. 2. Cracked-material with an anti-lane singularity: (a) an edge-cracked stri (z-lane) and (b) a semi-infinite crack in the whole lane (f-lane).
1754 X.-F. Wu, Y.A. Dzenis / International Journal of Engineering Science 40 (2002) 1751 1759 with 2 tgðz þ AÞ f ¼ 1; ð7þ tgðaþ Z ¼ z 2W ðx þ iyþ ¼ ; A ¼ a 2W 2W ; ð8þ which mas the edge-cracked stri onto the whole lane with a semi-infinite cut along its negative n-axis as shown in Fig. 2(b). Substituting (7) into (2), we obtain the comlex otential of the edgecracked stri in the z-lane as where ( ) u 0 ðzþ ¼ W tgðz þ AÞ qg 2 ðzþþqg 3 ðzþ sec2 ðz þ AÞ qg 1 ðzþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 tg 2 ðz þ AÞ tg 2 ðaþ ; ð9þ g 1 ðzþ ¼ 1 tg 2 ðz þ AÞ tg 2 ðz 0 þ AÞ ; ð10þ 1 g 2 ðzþ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tg 2 ðz þ AÞ tg 2 ðaþ þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð11þ tg 2 ðz 0 þ AÞ tg 2 ðaþ 1 g 3 ðzþ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð12þ tg 2 ðz þ AÞ tg 2 ðaþ þ tg 2 ðz 0 þ AÞ tg 2 ðaþ Z and A are defined in (8), and q is the singularity quantity defined in (3). Substitution of (9) (12) into (4) yields the SIF, which can be used as the Green s function to calculate the SIF and ERR of an edge-cracked stri under arbitrary anti-lane deformations. Now consider the SIF for the edge crack in an isotroic stri oened by a air of self-equilibrated anti-lane forces, P 0 at the crack surfaces a distance l behind the crack ti, as shown in Fig. 3(a). Substituting (9) into (4), we obtain the SIF as K III ¼ lim 2 ffiffiffiffiffiffiffi ffiffi 2x u 0 1ðxÞ ¼lim 4 2x u 0 ðxþ x!0 x!0 s ¼ 2P 0 a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a a sec tg tg a ða lþ 2 tg2 : ð13þ a 2W 2W 2W 2W 2W Furthermore, as shown in Fig. 3(b), using (13) as the Green s function, we may obtain the SIF for the stri with crack surfaces under the action of self-equilibrated forces ðxþ alied in the interval x 2½ l; 0Š as
X.-F. Wu, Y.A. Dzenis / International Journal of Engineering Science 40 (2002) 1751 1759 1755 Fig. 3. An edge-cracked stri under anti-lane deformations: (a) a air of concentrated forces located at crack surfaces and (b) uniform forces at crack surfaces. 2 sec a=2w K III ¼ ð ffiffiffiffiffi Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a a tg a 2W 2W Z l 0 ðxþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx: ð14þ tg 2 ða=2w Þ tg 2 ðða xþ=2w Þ When uniform forces are considered, say ðxþ ¼ 0, relation (14) reduces to rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2W a K III ¼ 0 a tg 1 2 a 2W arcsin sin½ða lþ=2w Š : ð15þ sinða=2w Þ Setting l ¼ a, relation (15) becomes rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2W a K III ¼ 0 a tg ; ð16þ a 2W which agrees with the result derived by means of Westergaard stress function [24], z r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ZðzÞ ¼ 0 sin sin 2 z sin 2 a : 2W 2W 2W Under anti-lane deformations, the discussed edge-cracked stri is mathematically equivalent to collinear eriodic cracks in an infinite sace. Now consider the lastic zone near the edge crack ti in the stri with uniform forces, 0 acting on the crack surfaces. The lastic zone is described by line 0 < x < d as shown in Fig. 3(b). On the basis of the Dugdale hyothesis [1], the shear stress in the yield zone is equal to a constant yield stress, Y, and the singularity of shear stress at the leading of the crack (x ¼ d) is removed. Utilizing (13) as the Green s function for the current roblem, we obtain the size of the lastic zone by solving the following integral equation:
1756 X.-F. Wu, Y.A. Dzenis / International Journal of Engineering Science 40 (2002) 1751 1759 Z dþl d ¼ 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx tg 2 ðða þ dþ=2w Þ tg 2 ðða þ d xþ=2w Þ Z d 0 Y ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx; ð17þ tg 2 ðða þ dþ=2w Þ tg 2 ðða þ d xþ=2w Þ which yields sinða=2w Þ arcsin ¼ sin½ða þ dþ=2w Š where 2ð1 þ kþ 1 þ 2k arcsin sin½ða lþ=2w Š ; ð18þ sin½ða þ dþ=2w Š k ¼ 0 Y : ð19þ Setting l ¼ a in (18), we obtain the lastic zone solution for uniform forces at the crack surfaces as sinða=2w Þ arcsin ¼ sin½ða þ dþ=2w Š 2ð1 þ kþ : ð20þ Consequently, letting d ¼ W a in (20), we obtain the limiting uniform force 0 at which the stri fully yields along the crack line across the stri as 0 ¼ W 1 Y : ð21þ a The exlicit relation (20) determines the lastic zone size near the edge crack ti in an isotroic elastolastic stri. Vrbik et al. [10] rovided a numerical solution of the same roblem with the hel of the Fredholm integrals of the second kind. The current results are comared with the results of [10] in Fig. 4. It is evident that the two methods redict similar trends. Since the resent Fig. 4. Variation of lastic zone size with k for different values of W =a ¼ 1:2, 1.5, 2, 3, 8.
X.-F. Wu, Y.A. Dzenis / International Journal of Engineering Science 40 (2002) 1751 1759 1757 solution (20) is an exlicit one, the deviation between the two aroaches may be caused by the numerical rocedure in [10]. 3. Examles and discussion Here we consider the SIF and lastic zone size solutions for an edge crack at the free boundary of a half-sace and a semi-infinite crack heading towards a free surface etc. As shown in Fig. 5(a) and (b), letting W!1in (13) and (15), we obtain the SIFs for the edge crack at the free boundary of a half-sace as K III ¼ 2P 0 a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a a 2 ða lþ 2 ð22þ and K III ¼ 0 a 1 2 arcsin 1 l : ð23þ a Setting l ¼ a, relation (23) reduces to ffiffiffiffiffi K III ¼ 0 a ; ð24þ which is the solution for uniform forces acting on the entire crack surfaces. Alternatively, we may derive relations (22) (24) by letting W!1in (7) and using the conformal maing function, f ¼ðz=a 1Þ 2 1. Furthermore, as shown in Fig. 6(a) and (b), relations (22) and (24) are exactly the SIFs for a Griffith crack in an infinite sace with two airs of self-equilibrated forces, P 0, and uniform anti-lane forces, 0 at the crack surfaces, resectively. Fig. 5. An edge-cracked half sace under anti-lane deformations: (a) a air of concentrated forces located at crack surfaces and (b) uniform forces at crack surfaces.
1758 X.-F. Wu, Y.A. Dzenis / International Journal of Engineering Science 40 (2002) 1751 1759 Fig. 6. A Griffith crack in the whole sace under anti-lane deformations: (a) two airs of concentrated forces located at crack surfaces and (b) uniform forces at crack surfaces. Letting W!1in (18), we obtain the corresonding lastic zone size as a arcsin ¼ 1 þ 2k a þ d 2ð1 þ kþ arcsin a l : ð25þ a þ d Furthermore, setting l ¼ a, relation (25) becomes arcsin a a þ d ¼ 2ð1 þ kþ ; which accords with the result derived by the method of dual integral equations [10]. As shown in Fig. 7(a) and (b), letting a!1and simultaneously keeing c ¼ðW aþ constant in (13) and (15), we obtain the SIFs for a semi-infinite crack heading towards a free surface as K III ¼ 2P 0 l þ c ffiffiffiffiffiffiffiffiffiffiffiffiffi ð27þ l cðl þ 2cÞ and rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lðl þ 2cÞ K III ¼ 2 0 : ð28þ c Consequently, letting c!1relation (27) returns to (6), and (28) becomes K III ¼ 2 ffiffi 2 ffiffiffiffiffi 0 l ; ð29þ ð26þ Fig. 7. A semi-infinite crack in a half sace under anti-lane deformations: (a) a air of concentrated forces located at crack surfaces and (b) uniform forces at crack surfaces.
X.-F. Wu, Y.A. Dzenis / International Journal of Engineering Science 40 (2002) 1751 1759 1759 which is the solution for a semi-infinite crack in the whole sace with uniform forces, 0 on the crack surfaces, 0 < x < l. Acknowledgements The suort of this work by the US Army Research Office is gratefully acknowledged. The authors would like to thank the anonymous reviewers for their helful suggestions to imrove this aer. References [1] D.S. Dugdale, J. Mech. Phys. Solids 8 (1960) 100 104. [2] B.M. Singh, H.T. Danyluk, J. Vrbik, Acta Mech. 55 (1985) 81 86. [3] H.T. Danyluk, B.M. Singh, Acta Mech. 56 (1985) 75 92. [4] B.M. Singh, H.T. Danyluk, J. Vrbik, A.P.S. Selvadurai, Eng. Fract. Mech. 24 (1986) 39 44. [5] B.M. Singh, A. Cardou, H.T. Danyluk, J. Vrbik, Theor. Al. Fract. Mech. 8 (1987) 193 197. [6] B.M. Singh, A. Cardou, M.C. Au, Eng. Fract. Mech. 29 (1988) 500 511. [7] H.T. Danyluk, B.M. Singh, J. Vrbik, Int. J. Fract. 51 (1991) 331 342. [8] H.T. Danyluk, B.M. Singh, J. Vrbik, Eng. Fract. Mech. 51 (1995) 735 740. [9] H.T. Danyluk, B.M. Singh, J. Vrbik, Int. J. Fract. 75 (1996) 307 322. [10] J. Vrbik, B.M. Singh, J. Rokne, R.S. Dhaliwal, Z. Angew. Math. Mech. (ZAMM) 81 (2001) 642 647. [11] Z. Olesiak, M. Wnuk, Int. J. Fract. Mech. 4 (1968) 383 385. [12] Z. Olesiak, J.R. Shadley, Int. J. Fract. Mech. 5 (1969) 305 313. [13] Y.M. Tsai, Int. J. Mech. Sci. 26 (1984) 245 252. [14] T.Y. Fan, Eng. Fract. Mech. 37 (1990) 1085 1087. [15] T.Y. Fan, Eng. Fract. Mech. 44 (1993) 243 246. [16] X.M. Wang, Y.P. Shen, Int. J. Fract. 59 (1993) R25 R32. [17] C. Atkinson, I.C. Howard, Int. J. Fract. Mech. 6 (1970) 96 97. [18] B.A. Bibly, A.H. Cottrell, K.H. Swinden, Proc. Roy. Soc.: Ser. A 272 (1963) 304 314. [19] F.A. Field, J. Al. Mech. 30 (1963) 622 623. [20] M.F. Koskinen, J. Basic Eng. Trans. ASME 85 (1963) 585 594. [21] J.R. Rice, in: Proceedings of 1st International Conference in Fracture, Sendi, Jaan, 1965,. 309. [22] E. Smith, Int. J. Eng. Sci. 5 (1967) 791 799. [23] R. Thomson, Solid State Phys. 39 (1986) 1 129. [24] H. Tada, P.C. Paris, G.R. Irwin, The Stress Analysis of Cracks Handbook, Del Research, Hellertown, PA, 1973.