On the Quantification of the Constraint Effect Along a Three-Dimensional Crack Front

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1 Internatinal Jurnal f Mechanical Engineering and Applicatins 206; 4(6): di: 0.648/j.ijmea ISSN: X (Print); ISSN: (Online) On the Quantificatin f the Cnstraint Effect Alng a Three-Dimensinal Crack Frnt F. Labbe Mechanical Engineering Department, Universidad Tecnica Federic Santa Maria, Casilla 0-V, Valparaís, Chile address: fernand.labbe@usm.cl T cite this article: F. Labbe. On the Quantificatin f the Cnstraint Effect Alng a Three-Dimensinal Crack Frnt. Internatinal Jurnal f Mechanical Engineering and Applicatins. Vl. 4, N. 6, 206, pp di: 0.648/j.ijmea Received: Octber 25, 206; Accepted: Nvember 7, 206; Published: December 2, 206 Abstract: Elbws with a shallw surface cracks in nuclear pressure pipes have been recgnized as a majr rigin f ptential catastrphic failures. Crack assessment is nrmally perfrmed by using the J-integral apprach. Althugh this neparameter-based apprach is useful t predict the ductile crack nset, it depends strngly n specimen gemetry r cnstraint level. When a shallw crack exists (depth crack-t-thickness wall rati less than 0.2) and/r a fully plastic cnditin develps arund the crack, the J-integral alne des nt describe cmpletely the crack tip stress field. In this paper, we reprt n the use f a three-term asympttic expansin, referred t as the J-A 2 methdlgy, fr mdeling the elastic-plastic stress field arund a three-dimensinal shallw surface crack in an elbw subject t internal pressure and utf-plane bending. The material, an A 56 Gr. 70 steel, used in the nuclear industry, was mdeled with a Ramberg-Osgd pwer law and flw thery f plasticity. A finite defrmatin thery was included t accunt fr the highly nnlinear behavir arund the crack tip. Numerical finite element results were used t calculate a secnd fracture parameter A 2 fr the J-A 2 methdlgy. We fund that the used three-term asympttic expansin accurately describes the stress field arund the cnsidered three-dimensinal shallw surface crack. Keywrds: Nnlinear Fracture Mechanics, Cnstraint, Nuclear Pressure Pipe, Asympttic Stress Fields. Intrductin Flaws in pwer plants cmpnent under nrmal perating cnditins typically initiate n the inner surface f the elbw walls. They grws by fatigue and/ r stress crrsin frming a surface crack. As a cnsequence, shallw surface cracks (depth crack-t-thickness wall rati less than 0.2) are ften encuntered and hence have been recgnized as a majr rigin f ptential catastrphic failure. In rder t prevent these failures, fracture mechanics methdlgies are nrmally applied t evaluate the structural integrity f pressure pipes. The applicatin f cnventinal fracture mechanics techniques relies n the ntin that a single parameter, such as J-integral, characterizes the resistance f the material t fracture [, 2]. The J-integral is used t measure the intensity f the stress and defrmatin fields n the singular ne-term crack-tip slutin fr a nnlinear material. It is referred t as the HRR singularity field since Hutchinsn and Rice [3] and Rsengren [4]. Hwever, the applicability f this neparameter-based apprach is limited t high cnstraint crack gemetry such as a deep crack in bending lading. When shallw surface crack exists and/r extended plastic cnditins arund the crack are meet, J-integral parameter alne des nt describe cmpletely the field arund the crack tip. The limited ability f a single parameter J t fully characterize crack-tip cnditins fr certain gemetry and lad cnditins is still an pen prblem. T slve it, an apprach recently prpsed is establishing a multi-term asympttic crack-tip stress field expansin [5]. A tw-term asympttic stress field expansin in fracture mechanics has been carried ut by O Dwd et al. Ref [6, 8] and Betegn and Hancck [9]. In the tw-parameter apprach f O Dwd and Shih, dented as J-Q methdlgy, a secnd parameter Q is used which measures the degree f triaxiality and cnstraint f the stress field. Applicatins f J- Q methdlgy t cracked cylindrical specimen has been dne by Labbe and Dns [0] and, in the case f flawed pressure vessels by Labbe and Dns []. Bth wrks have

2 Internatinal Jurnal f Mechanical Engineering and Applicatins 206; 4(6): shwn that J-Q methdlgy can nly render high-quality predictins fr deep crack [0]. The three-term asympttic stress field expansin [2-5] referred t as the J-A 2 methdlgy, is cntrlled by tw amplitude parameters that describe the stress field in the vicinity f the crack tip. This paper presents an applicatin f the J-A 2 methdlgy t analyze the stress field in the area clse t a three-dimensinal shallw crack in a pressurized pipe. The cnsidered shallw surface crack was represented as a semi-elliptic inner surface flaw. In rder t slve with high accuracy the stresses, a high-density mesh with threedimensinal higher rder finite elements was lcated arund the crack-tip. Large nnlinear strain effects arund the cracktip were als included n the finite element mdel and we used stress values in the regin clse t the crack-tip t evaluate a secnd parameter A 2 fr the J-A 2 methdlgy. We fund that the used three-term asympttic expansin accurately describes the stress field in the area which is significant fr fracture. 2. Theretical Backgrund 2.. Tw-Parameter Fracture Mechanics: J-Q Methdlgy Accrding t Yang et al [2] the tw-term asympttic expansin that represents clse t the crack tip, but still utside f the zne f finite strain, the actual stress field is: / n+ J r σ ij σ φij σ ψ ij α σεinr J/ σ = ( θ, n) + Q ( θ, n) q () where r and θ are the lcal cylindrical crdinates centered at the crack tip. The nrmalizing factr I n and the angular distributin f the dimensinless cnstants φ ij (θ, n) and ψ ij (θ, n) are functins f the strain-hardening expnent n. The first term in the abve expansin crrespnd t the HRR field [3, 4], after Hutchinsn [3] and Rice and Rsengren [4], with the J-integral parameter as amplitude. The dimensinless parameter Q measures the amplitude f the secnd term. Numerical results f O Dwd et al. [6, 8] shw that the secnd term is apprximately independent f the nrmalized radial distance, r = J / σ <<, and the crrespnding ψ ij (θ,n) functins d nt depend n θ. This means that the secnd term f the abve expansin acts as a hydrstatic unifrm stress. Therefre, relatin () may be written as: ( ) σ = σ + Q σ δ (2) ij ij HRR ij where (σ ij ) HRR is the HRR field and Q represents a triaxiality parameter. A negative Q means that the hydrstatic stress is reduced (lw stress triaxiality) in cmparisn with the reference state (HRR field), and therefre the J neparameter apprach is n lnger valid. On the ther hand, structures with Q 0 exhibit high stress triaxiality, gd agreement with the HRR fields, and they are, cnsequently, highly cnstrained. Frm Eq (2) we can evaluate Q as the difference between the actual full-stress field, and the HRR field: ( σθθ ) HRR σθθ Q = (3) σ fr θ = 0 and nrmalized distance r = 2J/σ. Unfrtunately, this apprach represents well the stress nly fr a deep cracks Three-Term Asympttic Expansin: J-A 2 Methdlgy The general elastic-plastic behavir f hardening material is described by the Ramberg-Osgd pwer-law stress-strain relatin. Generalizing this equatin t a multidimensinal stress by the J 2 defrmatin thery and using small defrmatin thery [2, 5], we can btain the three-term asympttic crack-tip stress field which is cntrlled by nly tw amplitude parameters, the J-integral parameter and the A 2 parameter: σ σ ij s 2 2s2 + r ( n+ ) ( ) () r (2) 2 r n+ (3) = A σij ( θ, n) + A2 σij ( θ, n) + A2 σij ( θ, n) L L L (4) where A is given by A = J α σεinl /( n+ ) In equatin (4), r and θ are the lcal cylindrical crdinates centered at the crack tip, and (5) σ ( θ, n), σ ( θ, n), σ ( θ, n) are the angular distributin f () (2) (3) ij ij ij the dimensinless cnstants. These are functins f the strain-hardening expnent n, the stress pwer expnent s 2 and the factr I n. The latter depends nly n the hardening expnent n. L is a characteristic length parameter. A critical part in the evaluatin f the parameters J and Q, is the evaluatin f the crack-tip stress field arund the defect. Defects such as a surface cracks prduce cmplex stress fields within the area clse t the crack-tip due t

3 228 F. Labbe: On the Quantificatin f the Cnstraint Effect Alng a Three-Dimensinal Crack Frnt three-dimensinal effects, large defrmatins and micrstructural fracture prcess. Therefre, the evaluatin f these parameters requires a detailed nnlinear threedimensinal stress analysis. The first term in the expansin (4) crrespnds t the HRR field with the J-integral parameter as amplitude. The dimensinless parameter A 2 is an undetermined parameter that can be related t the lading and gemetry cnfiguratin. Frm Eq (4), the A 2 can be calculated by applying a least square fitting t the values f the stress in the area clse t the crack-tip. A negative value f A 2 means that the hydrstatic stress is reduced (lw stress triaxiality) in cmparisn with the reference state (HRR field), and therefre the J-integral apprach is n lnger valid. On the ther hand, structures with A 2 0 exhibit high stress triaxiality, gd agreement with the HRR fields, and cnsequently the J-integral apprach is valid. 3. Example 3.. Numerical Calculatins As an example, cnsider an elbws with a shallw surface cracks in a nuclear pressure pipe. A typical material in the nuclear industry; a high strength steel A 56 Gr. 70 steel. In elastic-plastic fracture-mechanics, the true stress strain is usually apprximated by the Ramberg-Osgd pwer-law relatin: ε σ = ε α σ We tk in the Ramberg-Osgd pwer law relatin, the reference stress σ 0 as equal t the 0.2 % ffset yield stress and the reference strain ε as equal t σ / E, being E the Yung s mdulus, α is a hardening parameter and n is the strain-hardening expnent. The values used in this mdel were σ = 360 MPa., α =.07 and n = 3. The material was mdeled with incremental thery f plasticity, Vn Mises criterin, and istrpic hardening rule. In the finite element mdel (FEM), we als cnsidered finite strain plasticity and large strain/rtatin effects arund the crack tip. The finite element cde ABAQUS [6] was used in the analysis f the elbw using a ttal number f,200 higherrder 20-ndes, isparametric elements. The FEM meshing f the elbw are presented in Figs., 2, 3. n Fig.. Outer Elbw Finite Element view mdel, inner radius R i = 52.4 mm (6 ), thickness t = 0 mm. Fig. 2. Inner view f the Elbw Finite Element Mdel.

4 Internatinal Jurnal f Mechanical Engineering and Applicatins 206; 4(6): Fig. 3. Finite Element Crack Detailed. We cnsidered and inner radius R i = 52 mm (6 in) and wall thickness t = 0 mm fr the pressurized elbw with a shallw surface crack. The crack was mdeled as a threedimensinal semi elliptic inner surface flaw with a depth-tlength rati f a/2c = 0.. The assumed shallw crack had a depth-t-thickness wall rati f a/t = The finite element mdel f the half-length pipe was sufficiently lng (2,000 mm) t avid end effects n lading. Due t lad and gemetry symmetry, by using apprpriate bundary cnditins impsed n the planes f symmetry, it was necessary t cnsider nly ne-furth f the cmplete elbwpipe. The applied pressure lading in the pipe was 8 MPa and we cnsidered an ut-f-plane bending f 300 N-m. We evaluated the numerical stresses at the critical psitin, which ccurred n the minr semi-axis f the crack. The J- integral values were calculated by a virtual crack extensin methd implemented by using ABAQUS [6] Numerical Results The A 2 parameter fr the three-term asympttic slutin was btained frm Eq. 4 by fitting the whle-field stress values cmputed by finite element arund the crack-tip. The value calculated by a least square adjustment was A 2 = The negative value f A 2 allw us t cnclude that the hydrstatic stress is reduced (lw stress triaxiality) in cmparisn with with the reference state (HRR field). This means that the J-integral apprach shuld nt be apprpriate fr the assessment f the cnsidered crack. Figure 4 shws a finite element methd (FEM) results, the HRR singularity stress field and a three-term apprximatin (J-A 2 field) fr the nrmalized pening stress arund the shallw crack, versus the crack-tip distance nrmalized by using J/ σ. The nrmalizing factr J/ σ fr the crack-tip distance is useful t set the lcal size scale n which large strains and stresses develp and the prcesses f micrscpic ductile fracture ccur. It can be bserved in Fig. 4 that, as expected, the HRRfield is far frm the finite element slutins. Frm these results, we cncluded that the J-apprach based nly n the HRR asympttic ne-term slutin cannt characterized three-dimensinal shallw cracks in elbws subjected t internal pressure and ut-f-plane bending. On the ther hand, the data pltted in Fig. 4 allwed us t verify the quality f the near-tip stress fields rendered by the three-term elastic-plastic asympttic expansin. As shwn in Fig 4, the J-A 2 three-term asympttic slutin is essentially similar t the finite element slutins ver the interval < r /(J/σ ) < 0; J-A 2 fields match very well with the FEM results. This allw us t cnclude that, fr failure assessment f a shallw surface cracks in pressure pipes, it is cnvenient t use the Fracture Mechanics apprach based n three-term asympttic expansin, with parameters J and A 2. Mrever as bserved in Fig. 4, fr the very near crack-tip regin, r < /(J/σ ), there are a drastic breakdwn in the pening stress behavir. This phenmenn is cnsistent with the existence f large nnlinear effects n that regin that invalidate the HRR asympttic stress field based n the small strain thery.

5 230 F. Labbe: On the Quantificatin f the Cnstraint Effect Alng a Three-Dimensinal Crack Frnt Fig. 4. Nrmalized FEM, J-A 2 and HRR pening stress and Plane stress and strain versus nrmalized crack-tip distance fr varius a/t values; applied pressure lading p = 8 MPa and Out-f-Plane lad f 300 N-m. 4. Summary and Cnclusins The cnventinal fracture mechanics techniques use the J- integral as a measure f the intensity f the stress and defrmatin fields n the singular ne-term crack-tip slutin fr nnlinear materials. This apprach, based n a single parameter, yields the referred t as HRR asympttic stress field. Hwever, its applicatin is limited t deep cracks in bending lading. The J-integral parameter alne des nt describe cmpletely the field arund a shallw surface crack with r withut surrunding plastic cnditins. Althugh the tw-term asympttic stress field expansin, dented as the J-Q methdlgy, allws als including in the analysis the degree f triaxiality and cnstraint f the stress field, this methdlgy can nly render high-quality predictins fr deep cracks. Shallw surface cracks (depth crack-t-thickness wall rati less than 0.2) in pwer plant cmpnents initiate at the inner surface f elbws in pressurized pipes. As pinted ut abve, fr this gemetry and lad cnditins, the J-integral and the J-Q methdlgy have limited ability t characterize the crack-tip behavir. Therefre, in rder t evaluate the whlefield stress fr a shallw surface crack in a pressure pipe, we applied the three-term asympttic crack-tip stress field expansin, referred t as the J-A 2 methdlgy. The applied J-A 2 methdlgy is cntrlled by tw amplitude parameters that allwed us t describe the stress field in the vicinity f the crack tip. We represented the shallw surface crack as a semi-elliptic inner surface flaw. In rder t btain high accuracy in the stresses within the area clse t the crack-tip, a high high-density mesh with threedimensinal higher rder finite elements was lcated arund the crack-tip. Large nnlinear strain effects arund the cracktip were als included n the finite element mdel and we evaluated a secnd parameter A 2 fr the J-A 2 methdlgy by fitting the whle-fiel stress values cmputed by finite element arund the crack tip. We fund that the three-term elastic-plastic asympttic expansin accurately describes the stress field in the regin clse t the crack-tip which is significant fr fracture. In this regin, the J-A 2 three-term asympttic slutin and the finite element slutins cincided. Fr the very near crack-tip area, we bserved a drastic breakdwn in the pening stress behavir that shuld be linked with the existence f large nnlinear effects in that regin. We argue that the nnlinear effects, bserved in the regin clse t the crack-tip, invalidate the HRR asympttic stress field based n the small strain thery. This means that the J- apprach, based nly n the HRR asympttic ne-term slutin, cannt be used t analyze the stress field in the area clse t a three-dimensinal shallw crack in a pressurized pipe. We cnclude that the failure assessment, f the threedimensinal shallw cracks in elbws subjected t internal pressure and ut-f plane bending, requires an apprach bases n three-term asympttic expansin with parameters, J and A 2. Althugh in the example shwn abve, we applied a threeparameter fracture mechanical apprach t describe the stress field ahead f the crack-tip, mre analysis shuld be perfrmed t validate the prpsed research. Specifically, in the shrt term, we are planning t examine: a range f pressure lading; results fr tensin/bend lading; small strain vs. large strain; use f SSY reference field vs. use f the HRR [7]; a range f different materials; the mesh sensitivity f the result.

6 Internatinal Jurnal f Mechanical Engineering and Applicatins 206; 4(6): Acknwledgement The authr acknwledge the financial supprt f CONICYT (Chile) Cnsej Nacinal de Ciencia y Tecnlgía, Natinal Tender Prcess fr Team Research Prjects (Anills) in Science and Technlgy, Preis ACT40, References [] J. W. Hutchinsn, Fundamentals f the phenmenlgical thery f nnlinear fracture mechanics, Jurnal f Applied Mechanics, 983, Vl. 50, pp [2] J. D. Landes, Elastic-plastic fracture mechanics: where has it been? where is it ging, Fatigue and Fracture Mechanics, 30th Vlume, ASTM STP 360, Editres: Paris, P.C., Jerina, K. L. 999, pp [3] J. W. Hutchinsn, Singular behavir at the end f a tensile crack in a hardening material, Jurnal f the Mechanics and Physics f Slids, 968,Vl. 6, pp [4] J. R. Rice, G. F. Rsengren, Plane strain defrmatin near a crack tip in a pwer law material, Jurnal f the Mechanics and Physics f Slids, 968, Vl. 6, pp. -2. [5] S. M. Sharma and N Aravas, N., Determinatin f higherrder terms in asympttic elastplastic crack tip slutins, Jurnal f the Mechanics and Physics f Slids, 99, 39, pp [6] N. P. O Dwd and C. F. Shih, Family f crack-tip fields characterized by a triaxiality parameter-i Structure f fields, Jurnal f the Mechanics and Physics f Slids, 99, Vl.39, pp [7] N. P. O Dwd and C. F. Shih, Family f crack-tip fields characterized by a triaxiality parameter-ii Fracture applicatins, Jurnal f the Mechanics and Physics f Slids, 992, Vl.40, pp Mechanics; Thery and Applicatins: Twenty-Furth Vlume, ASTM STP 207, American Sciety fr Testing and Materials, 994, [9] C. Betegn, and J. W. Hancck, Tw-parameter characterizatin f elastic-plastic crack-tip fields, Jurnal f Applied Mechanics,99, Vl. 68, pp [0] F. Labbé and J. R Dns, Q stresses and cnstraint behavir f the ntched cylindrical tensile specimen, Engineering Fracture Mechanics 202, 68, pp [] F. Labbé and J. R., Dns, Nnlinear finite element mdeling f surface cracks in a nuclear pressure pipe, Prceedings f the First MIT Cnference n Cmputatinal Fluid and Slids Mechanics, Massachussetts Institute f Technlgy, Cambridge, MA. Editr J. K. Bathe, June 200, pp [2] S. Yang, Y. J. Cha, and M. A. Suttn, Higher Order Asympttic Crack Fields in a Pwer- Law Hardening Material, Eng. Frac. Mech., 993, 45, -20. [3] Y. J. Cha, S. Yang and M. A. Suttn, On the Fracture f Slids Characterized by ne r Tw Parameters: Thery and Practice, J. Mech Phys. Slids, 994, 42, [4] Y. J. Cha and X. K. Zhu, J-A 2 Characterizatin f Crack-Tip Fields: Extent f J-A 2 Dminance and Size Requirements, Int J. Frac., 998, 89, [5] Y. J. Cha, X. K. Zhu, P. S. Lam, M. R. Luthen and N. C. Iyer, Applicatin f the Tw Parameter J-A 2 Descriptin t Ductile Crack Grwth, Fatigue and Fracture Mechanics, 3th Vlume, G. R. Halfrd and J. P. Gallagher, Eds., American Sciety fr Testing and Materials, ASTM 389, 200, [6] ABAQUS Thery Manual, Hibbit, Karlssn and Srensen, Inc., Prvidence, R. I [7] N. P. O Dwd, Applicatins f Tw Parameter Appraches in Elastic-Plastic Fracture Mechanics, Eng. Frac. Mech., 2005, 52, [8] N. P. O Dwd and C. F. Shih, Tw-Parameter Fracture