- Email: [email protected]
A new constraint based fracture criterion for surface cracks
Engineering Fracture Mechanics 74 (2007) 1233–1242 www.elsevier.com/locate/engfracmech
A new constraint based fracture criterion for surface cracks A.M. Leach a, S.R. Daniewicz a
b,*
, J.C. Newman Jr.
c
Department of Materials Science and Engineering, Georgia Institute of Technology, 771 Ferst Drive, N.W., Atlanta, GA 30332, USA b Department of Mechanical Engineering, Mississippi State University, 210 Carpenter Building, Mississippi State, MS 39762, USA c Department of Aerospace Engineering, Mississippi State University, 330 Walker Engineering Laboratory, Hardy Street, Mississippi State, MS 39762, USA Received 20 December 2005; received in revised form 17 July 2006; accepted 23 July 2006 Available online 22 September 2006
Abstract A new methodology for predicting the location of maximum crack extension along a surface crack front in ductile materials is presented. Three-dimensional elastic–plastic finite element analyses were used to determine the variations of a constraint parameter (ah) based on the average opening stress in the crack tip plastic zone and the J-integral distributions along the crack front for many surface crack configurations. Monotonic tension and bending loads are considered. The crack front constraint parameter is combined with the J-integral to characterize fracture, the critical fracture location being the location for which the product Jah is a maximum. The criterion is verified with test results from surface cracked specimens. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Surface Crack; Constraint; FEA
1. Introduction Accurate life assessment of structural components may require advanced life prediction methodologies and criteria. Surface cracks are among the most prevalent life inhibitors in structural components as these cracks commonly initiate at material and manufacturing discontinuities. Under applied monotonic loading, a surface crack may exhibit stable crack growth until the crack has reached a critical size, at which time it loses stability and fracture ensues. Fracture prediction is not well established for materials containing surface cracks. The traditional approach involves the use of a single-parameter fracture criterion, Irwin’s stress intensity factor K [1] for brittle materials or Rice’s J-integral [2] for ductile materials. It has been shown [3] that predicting the onset of fracture as the point where K is equal to the plane strain fracture toughness KIC is a conservative approach. More recently, two-parameter fracture criteria have been proposed [4–10], incorporating the influence of crack front *
Corresponding author. E-mail addresses: [email protected] (A.M. Leach), [email protected] (S.R. Daniewicz), j.c.newman.jr@ae. msstate.edu (J.C. Newman Jr.). 0013-7944/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2006.07.011
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Nomenclature a c E h M n S SU SY t w x,y,z ag ah e eY / j m r rzz
crack depth, mm crack half-width, mm Young’s Modulus, GPa specimen half-height, mm applied maximum bending stress, MPa hardening exponent applied axial stress, MPa ultimate strength, MPa yield strength, MPa specimen thickness, mm specimen half-width, mm Cartesian coordinates global constraint factor hyper-local constraint factor strain, mm/mm extensional strain, mm/mm crack parametric angle, rad Ramberg–Osgood fitting constant Poisson’s ratio stress, MPa crack opening stress, MPa
constraint. Constraint has been quantified using different techniques, such as the T-stress [5], the Q-stress [6], and more recently, the average opening stress in the crack tip plastic zone [7]. A two-parameter approach for ductile materials incorporating the average opening stress and the J-integral is presented herein. Finite Element Analysis (FEA) was performed in conjunction with an experimental test program to investigate the accuracy of using the proposed two-parameter criterion to predict the location of initial crack extension along a surface crack front under monotonic tension or bending. For comparison, similar predictions were made using a single-parameter J-integral approach. The two-parameter criterion more accurately predicted initial crack extension for surface crack configurations under monotonic tension or bending loads.
2. Experimental test program The Idaho National Engineering and Environmental Laboratory (INEEL) conducted a test program sponsored by NASA Langley Research Center, NASA Marshall Space Flight Center, and the FAA William J. Hughes Technical Center on surface crack specimens fabricated from D6AC steel. The specimens were heat-treated to give a yield strength SY = 1330 MPa and ultimate strength SU = 1434 MPa. Specimens were tested under remote monotonic tension or bending loads. The specimen geometry is illustrated in Fig. 1. The geometries tested and the applied stress that first induced crack extension are presented in Table 1. For all specimens, the width w and thickness t were 12.7 mm and 6.35 mm, respectively. Prior to application of the monotonic tensile or bending load, the surface crack specimens were treated in an identical fashion. Triangular electrical discharge machined (EDM) notches were placed in the center of each specimen to aid crack initiation, and the specimens were fatigue pre-cracked under remote bending to reach the desired crack configuration. Careful observation of the fracture surfaces revealed no obtrusive shear-lip formation, indicating that the pre-cracking loads were not excessive for the crack geometries considered. The fracture surfaces for selected tension and bending specimens discussed in Section 5 indicate the
A.M. Leach et al. / Engineering Fracture Mechanics 74 (2007) 1233–1242 S
M
1235
z y x 2w h t
A a
φ c
A
w
Section A-A
M
S
Fig. 1. Surface Crack Configuration.
Table 1 Surface crack configuration, crack extension load, and location of maximum crack extension on the crack front for each surface crack specimen Tension
Bending
Specimen
a/c
a/t
S (MPa)
/crit
Specimen
a/c
a/t
M (MPa)
/crit
AT-01 AT-04 AT-05 AT-06 AT-07 AT-08 AT-09 BT-01 CT-01 CT-02 CT-03 DT-02
0.76 0.72 0.70 0.72 0.61 0.61 0.63 0.47 0.54 0.53 0.52 0.27
0.19 0.50 0.52 0.52 0.65 0.67 0.67 0.73 0.56 0.57 0.54 0.27
1300.8 1152.3 1111.7 1140.1 1006.3 992.5 987.8 841.0 872.5 998.4 1044.2 1205.6
0.68 0.43 0.28 0.45 0.34 0.31 0.33 0.40 1.00 0.75 0.44 0.94
AB-01 AB-04 AB-07 BB-01 CB-01 DB-03 DB-04
0.62 0.70 0.83 0.47 0.64 0.36 0.24
0.68 0.50 0.24 0.73 0.27 0.36 0.61
1720.6 1574.5 1969.6 1369.8 2010.3 1772.2 1047.7
0.20 0.23 0.21 0.00 0.32 0.67 0.32
extent of pre-cracking with respect to the initial EDM notch. After pre-cracking, 12 tension and 7 bending specimens were tested at room temperature under laboratory air conditions. The specimens were loaded monotonically until a 5% potential drop was recorded, indicating that a small amount of crack extension had occurred. The load magnitude at the 5% potential drop was recorded. After the first occurrence of crack extension, the specimens were cyclically loaded at a reduced load level to mark the location and extent of stable crack growth. The specimens were subjected to three instances of crack extension and cyclic marking before loading to complete failure; however, only the first instance is considered here. 3. Finite element analysis Elastic–plastic FEA was performed using surface crack geometries and applied stresses identical to the INEEL test configurations (Table 1) to obtain the J-integral and the opening stresses in the plastic zone. FEA-Crack version 2.5.625 [11] was employed to generate finite element meshes and WARP3D release 15 [12] was used for the analyses. The J-integral distributions along the crack front were obtained using the domain integral method intrinsic to WARP3D.
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Fig. 2. Finite element mesh detail on the crack plane.
The finite element meshes consisted of three-dimensional l3disop 8-noded isoparametric elements. Each mesh contained a highly refined region of elements near the crack front and in the crack plane. A typical mesh detail on the crack plane is shown in Fig. 2. The D6AC steel material response was modeled using a Ramberg–Osgood [13] material model. The Ramberg–Osgood model is defined by Eq. (1), where SY is the 0.2% offset yield strength, eY is the corresponding extensional strain at yield, n is the hardening exponent, and j is a fitting constant. n e S S ¼ þj ð1Þ eY S Y SY Using this stress–strain relation, a curve fit was generated in agreement to tensile test data provided by the INEEL. The input parameters used were eY = SY/E, n = 50, and j = 0.315. The finite element model comprised one-quarter of the specimen shown in Fig. 1. The model was constrained in a manner to simulate the symmetry planes of the surface crack. Zero-displacement boundary conditions were applied to the y–z plane in the x-direction and the un-cracked ligament on the crack plane in the z-direction. To prevent model translation, a single node located at the farthest point from the model origin ((x, y, z) = (w, t, 0)) was constrained in the y-direction. The models were loaded monotonically in either tension or bending with an applied pressure on the z = h plane. The applied pressures used were consistent with the 5% potential drop load (load at which crack exten-
Fig. 3. Constraint path definition for a semi-elliptical surface crack.
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Fig. 4. Fracture surface image showing location of maximum crack extension.
Fig. 5. J and Jah distributions for selected surface crack models subjected to monotonic tension.
sion first occurred) recorded during testing and listed in Table 1. The elastic modulus E was 209.7 GPa and Poisson’s ratio m was 0.3. Within WARP3D, the von-Mises yield criterion and its associated flow rule were used. Linear-kinematic hardening was assumed.
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4. Constraint factor definition The hyper-local constraint factor ah [14,15] is similar to the global constraint factor ag developed by Newman et al. [7] for through-the-thickness cracks. The hyper-local constraint factor is a normalized, averaged crack opening stress acting along a ray perpendicular to and originating from the crack front and extending to the plastic zone boundary. The constraint factor is defined as Z Lð/Þ 1 rzz ah ð/Þ ¼ dl ð2Þ Lð/Þ 0 SY where (rzz/SY) is the normalized opening stress lying along a given ray L(/) as shown in Fig. 3. Nodal values of stress were used in the constraint calculations, and nodes were considered yielded if the von-Mises stress was greater than or equal to SY. The nodal stress values were calculated as a direct average using Gauss point values in the surrounding elements. Each ray of nodes perpendicular to the crack front (as evident in Fig. 2) was defined by the parametric angle / along the crack front. 5. Fracture prediction criteria Fracture was predicted to occur at the location along the crack front where the crack front constraint and the J-integral magnitudes are the highest. A first approximation to this fracture criterion considers the product Jah. Initial crack extension along the crack front is expected at the / location exhibiting the highest Jah. The ah and J-integral distributions along the crack front were calculated for each surface crack model from the finite element results. Using fracture surface images and digitizing software, the location of maximum crack extension was measured for each test specimen. In this work, we make the inherent assumption that the location of maximum crack extension corresponds to the location of initial crack instability. The location of maximum
Fig. 6. J and Jah distributions for selected surface crack models subjected to monotonic bending.
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crack extension on both sides of the crack was averaged to obtain a single location for comparison to the predicted value. The averaged / value is considered as the critical location /crit along the crack front. A typical fracture surface image detailing the crack extension measurements is presented in Fig. 4. To predict the location of maximum crack extension, both J and ah must be computed along the crack front. The J and Jah distributions were normalized by the product of the maximum applied stress S (tension) or M (bending) and the thickness t. Representative J,ah, and Jah distributions for selected tension and bending models are shown in Figs. 5 and 6, respectively. The corresponding fracture surface images are presented in Fig. 7 for tension and Fig. 8 for bending. The cases of tension and bending illustrate that the hyper-local constraint factor is near unity at the free surfaces, and is larger at the maximum depth location. The J-integral is higher near the free surfaces, but is lower at the maximum depth location. The bending configuration exhibits a higher constraint in the interior than the tension specimen. Note from these figures that the J-based and Jah-based criteria (maximum value) do not predict the same location of initial crack extension along the crack front. In some models, large applied loads resulted in plastic zones that exceeded the most refined region and spanned into the transition zone of the mesh. This resulted in reduced values of ah at these / locations. Under large bending loads, specimens with large a/t exhibited a loss of constraint at the deepest point of penetration
Fig. 7. Fracture surface of selected tension specimens. The fracture surfaces correspond to the J and Jah distributions presented in Fig. 5.
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with ah ! 1. These regions of numerical difficulty did not affect the results of interest wherein Jah was a maximum. For the tension case, the J-integral reaches a maximum and nearly constant value just below the free surface. The constraint factor reaches a constant value as the deepest point of penetration is approached, as seen in Fig. 5. Thus for tension, the product Jah reaches a maximum and nearly constant value just below the free surface, resulting in a range of / values for the potential critical location. This range is determined to be the / values that span a ±5% deviation from the constant value of Jah, as determined from the calculated J and ah distributions. The proposed criterion does not predict an exact critical location along the crack front for the tension cases (due to the product of crack drive and constraint being constant over a large range of /); however, the measured critical values lie within the predicted range of / values for all but one of the tension cases considered. The bending cases exhibit higher Jah variations along the crack front as shown in Fig. 6. The J-integral for a surface crack under bending reaches a distinct peak value below the free surface, while the constraint distribution behaves similarly to that for the tension case and approaches a constant maximum value towards the deepest point. The resulting product Jah demonstrates a definite peak value along the crack front between the free surface and the deepest point of crack penetration. The validity of Jah as a two-parameter fracture prediction criteria for surface cracks under tension or bending are presented in Table 2 and Fig. 9, respectively. The fracture criterion does not provide an exact critical location for the case of monotonic tensile loading; however, as mentioned previously, the measured location of maximum crack extension (and thus the assumed location of initial instability) falls within the range of predicted / values for all but one of the tension specimens. Observation of Table 2 also indicates that the Jah criterion predicts the critical location to occur over a smaller range of / values than a single-parameter J criterion predicts. For monotonic bending, the proposed criterion performs quite well, as presented in Fig. 9. The criterion predicts the critical location within 10 percent of the measured value for all but 2 of the bending spec-
Fig. 8. Fracture surface of selected bending specimens. The fracture surfaces correspond to the J and Jah distributions presented in Fig. 6.
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Table 2 Comparison of J and Jah for prediction of the location of maximum crack extension along the surface crack front for monotonic tension specimens Specimen
J /min/max
Jah /min/max
/crit
AT-01 AT-04 AT-05 AT-06 AT-07 AT-08 AT-09 BT-01 CT-01 CT-02 CT-03 DT-02
0.47–1.00 0.10–1.00 0.10–1.00 0.10–1.00 0.10–0.68 0.09–1.00 0.09–0.70 0.10–0.79 0.57–1.00 0.44–1.00 0.49–1.00 0.77–1.00
0.57–1.00 0.22–1.00 0.22–1.00 0.18–1.00 0.17–0.65 0.21–1.00 0.18–0.91 0.17–0.62 0.54–1.00 0.52–1.00 0.57–1.00 0.78–1.00
0.68 0.43 0.28 0.45 0.34 0.31 0.33 0.40 1.00 0.75 0.44 0.94
The critical locations for J and Jah are provided for a range of / values, from a minimum value /min to a maximum value /max. The /min and /max values are determined as the bounds of a 5% variation from the maximum J and Jah for each specimen.
Fig. 9. Comparison of Jah and J for prediction of the location of maximum crack extension along the surface crack front for monotonic bending specimens. The bending specimens correspond to those presented in Table 1.
imens (BB-01 and AB-07), which are predicted within 15% of the measured values. As in the case of monotonic tension, the two parameter Jah approach outperforms the single-parameter J-integral criterion for nearly all cases. 6. Summary and conclusions A two-parameter fracture criterion for prediction of the location of initial instability in surface cracks under monotonic loading was introduced and the validity was investigated. Using the product Jah to predict the
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critical location along the crack front was more accurate than a single-parameter J-integral approach for surface cracks under both tension and bending loads. The criteria accurately predicted the critical location for all bending cases within 15% error, and accurately predicted the range of critical locations for the tension cases considered. Specifically, our results indicate that a fracture criterion that includes the effects of crack front constraint quantified by the opening stress provides an accurate prediction of the critical location for specimens under tension or bending loading conditions. Since most components are subjected to complex stress states which may be considered as a combination of tension and bending, the proposed criterion shows great potential. Acknowledgements The authors express their appreciation to the NASA Marshall Space Flight Center, NASA Langley Research Center, and the FAA William J. Hughes Technical Center for supporting this work. The authors also thank the Idaho National Engineering and Environmental Laboratory for fabricating and conducting tests on the surface crack specimens. References [1] Irwin GR. Relation of stresses near a crack to the crack extension force. Proceeding of the 9th international congress on Appl Mech. Brussels, Belgium, 1956. [2] Rice JR. A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 1968;35:379–86. [3] Reuter WG, Newman Jr JC, Skinner JD, Mear ME, Lloyd WR. Use of KIC and constraint to predict load and location for initiation of crack growth in specimens containing part-through cracks. ASTM STP 2002;1406:353–80. [4] Hancock JW, Mackenzie AC. On the mechanisms of ductile failure in high-strength steels subjected to multi-axial stress-states. J Mech Phys Solids 1976;24:147–69. [5] Hancock JW, Reuter WG, Parks DM. Constraint and toughness parameterized by T. ASTM Symposium on Constraint Effects in Fracture, Indianapolis, IN, 1991. [6] O’Dowd NP, Shih CF. A family of crack tip fields characterized by a triaxiality parameter Part I – structure of field. J Mech Phys Solids 1991;39:989–1015. [7] Newman Jr JC, Bigelow CA, Shivakumar KN. Three-dimensional elastic–plastic finite-element analyses of constraint variations in cracked bodies. Engng Fract Mech 1993;46:1–13. [8] Rice JR, Tracey DM. On the ductile enlargement of voids in triaxial stress fields. J Mech Phys Solids 1969;17:201–17. [9] McClintock FA. A criterion for ductile fracture by the growth of holes. J Appl Mech 1968;35:363–71. [10] Sommer E, Aurich D. On the effect of constraint on ductile fracture. Defect assessment in components – fundamentals and applications, ESIS/EGF9 1991:141–74. [11] FEA-Crack ver 2.5.625. Structurable Reliability Technology. 2465 Central Avenue Suite 110. Boulder, Colorado 80301. Available from:. [12] Gullerud AS, Koppenhoefer KC, Roy A, RoyChowdhury S, Walters MC, Dodds Jr RH. WARP3D – Release 15 Manual. Civil Engineering, Report No. UILU-ENG-95-2012m, University of Illinois, 2004. [13] Ramberg W, Osgood WR. Description of stress–strain curves by three parameters. NACA Tech Note, 902, 1943. [14] Aveline Jr CR, Daniewicz SR. Variations of constraint and plastic zone size in surface-cracked plates under tension or bending loads. ASTM STP 2000;1389:206–20. [15] Newman Jr JC, Reuter WG, Aveline Jr CR. Stress and fracture analyses of semi-elliptical surface cracks. ASTM STP 1999;1360:403–23.
A new constraint based fracture criterion for surface cracks A.M. Leach a, S.R. Daniewicz a
b,*
, J.C. Newman Jr.
c
Department of Materials Science and Engineering, Georgia Institute of Technology, 771 Ferst Drive, N.W., Atlanta, GA 30332, USA b Department of Mechanical Engineering, Mississippi State University, 210 Carpenter Building, Mississippi State, MS 39762, USA c Department of Aerospace Engineering, Mississippi State University, 330 Walker Engineering Laboratory, Hardy Street, Mississippi State, MS 39762, USA Received 20 December 2005; received in revised form 17 July 2006; accepted 23 July 2006 Available online 22 September 2006
Abstract A new methodology for predicting the location of maximum crack extension along a surface crack front in ductile materials is presented. Three-dimensional elastic–plastic finite element analyses were used to determine the variations of a constraint parameter (ah) based on the average opening stress in the crack tip plastic zone and the J-integral distributions along the crack front for many surface crack configurations. Monotonic tension and bending loads are considered. The crack front constraint parameter is combined with the J-integral to characterize fracture, the critical fracture location being the location for which the product Jah is a maximum. The criterion is verified with test results from surface cracked specimens. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Surface Crack; Constraint; FEA
1. Introduction Accurate life assessment of structural components may require advanced life prediction methodologies and criteria. Surface cracks are among the most prevalent life inhibitors in structural components as these cracks commonly initiate at material and manufacturing discontinuities. Under applied monotonic loading, a surface crack may exhibit stable crack growth until the crack has reached a critical size, at which time it loses stability and fracture ensues. Fracture prediction is not well established for materials containing surface cracks. The traditional approach involves the use of a single-parameter fracture criterion, Irwin’s stress intensity factor K [1] for brittle materials or Rice’s J-integral [2] for ductile materials. It has been shown [3] that predicting the onset of fracture as the point where K is equal to the plane strain fracture toughness KIC is a conservative approach. More recently, two-parameter fracture criteria have been proposed [4–10], incorporating the influence of crack front *
Corresponding author. E-mail addresses: [email protected] (A.M. Leach), [email protected] (S.R. Daniewicz), j.c.newman.jr@ae. msstate.edu (J.C. Newman Jr.). 0013-7944/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2006.07.011
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Nomenclature a c E h M n S SU SY t w x,y,z ag ah e eY / j m r rzz
crack depth, mm crack half-width, mm Young’s Modulus, GPa specimen half-height, mm applied maximum bending stress, MPa hardening exponent applied axial stress, MPa ultimate strength, MPa yield strength, MPa specimen thickness, mm specimen half-width, mm Cartesian coordinates global constraint factor hyper-local constraint factor strain, mm/mm extensional strain, mm/mm crack parametric angle, rad Ramberg–Osgood fitting constant Poisson’s ratio stress, MPa crack opening stress, MPa
constraint. Constraint has been quantified using different techniques, such as the T-stress [5], the Q-stress [6], and more recently, the average opening stress in the crack tip plastic zone [7]. A two-parameter approach for ductile materials incorporating the average opening stress and the J-integral is presented herein. Finite Element Analysis (FEA) was performed in conjunction with an experimental test program to investigate the accuracy of using the proposed two-parameter criterion to predict the location of initial crack extension along a surface crack front under monotonic tension or bending. For comparison, similar predictions were made using a single-parameter J-integral approach. The two-parameter criterion more accurately predicted initial crack extension for surface crack configurations under monotonic tension or bending loads.
2. Experimental test program The Idaho National Engineering and Environmental Laboratory (INEEL) conducted a test program sponsored by NASA Langley Research Center, NASA Marshall Space Flight Center, and the FAA William J. Hughes Technical Center on surface crack specimens fabricated from D6AC steel. The specimens were heat-treated to give a yield strength SY = 1330 MPa and ultimate strength SU = 1434 MPa. Specimens were tested under remote monotonic tension or bending loads. The specimen geometry is illustrated in Fig. 1. The geometries tested and the applied stress that first induced crack extension are presented in Table 1. For all specimens, the width w and thickness t were 12.7 mm and 6.35 mm, respectively. Prior to application of the monotonic tensile or bending load, the surface crack specimens were treated in an identical fashion. Triangular electrical discharge machined (EDM) notches were placed in the center of each specimen to aid crack initiation, and the specimens were fatigue pre-cracked under remote bending to reach the desired crack configuration. Careful observation of the fracture surfaces revealed no obtrusive shear-lip formation, indicating that the pre-cracking loads were not excessive for the crack geometries considered. The fracture surfaces for selected tension and bending specimens discussed in Section 5 indicate the
A.M. Leach et al. / Engineering Fracture Mechanics 74 (2007) 1233–1242 S
M
1235
z y x 2w h t
A a
φ c
A
w
Section A-A
M
S
Fig. 1. Surface Crack Configuration.
Table 1 Surface crack configuration, crack extension load, and location of maximum crack extension on the crack front for each surface crack specimen Tension
Bending
Specimen
a/c
a/t
S (MPa)
/crit
Specimen
a/c
a/t
M (MPa)
/crit
AT-01 AT-04 AT-05 AT-06 AT-07 AT-08 AT-09 BT-01 CT-01 CT-02 CT-03 DT-02
0.76 0.72 0.70 0.72 0.61 0.61 0.63 0.47 0.54 0.53 0.52 0.27
0.19 0.50 0.52 0.52 0.65 0.67 0.67 0.73 0.56 0.57 0.54 0.27
1300.8 1152.3 1111.7 1140.1 1006.3 992.5 987.8 841.0 872.5 998.4 1044.2 1205.6
0.68 0.43 0.28 0.45 0.34 0.31 0.33 0.40 1.00 0.75 0.44 0.94
AB-01 AB-04 AB-07 BB-01 CB-01 DB-03 DB-04
0.62 0.70 0.83 0.47 0.64 0.36 0.24
0.68 0.50 0.24 0.73 0.27 0.36 0.61
1720.6 1574.5 1969.6 1369.8 2010.3 1772.2 1047.7
0.20 0.23 0.21 0.00 0.32 0.67 0.32
extent of pre-cracking with respect to the initial EDM notch. After pre-cracking, 12 tension and 7 bending specimens were tested at room temperature under laboratory air conditions. The specimens were loaded monotonically until a 5% potential drop was recorded, indicating that a small amount of crack extension had occurred. The load magnitude at the 5% potential drop was recorded. After the first occurrence of crack extension, the specimens were cyclically loaded at a reduced load level to mark the location and extent of stable crack growth. The specimens were subjected to three instances of crack extension and cyclic marking before loading to complete failure; however, only the first instance is considered here. 3. Finite element analysis Elastic–plastic FEA was performed using surface crack geometries and applied stresses identical to the INEEL test configurations (Table 1) to obtain the J-integral and the opening stresses in the plastic zone. FEA-Crack version 2.5.625 [11] was employed to generate finite element meshes and WARP3D release 15 [12] was used for the analyses. The J-integral distributions along the crack front were obtained using the domain integral method intrinsic to WARP3D.
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Fig. 2. Finite element mesh detail on the crack plane.
The finite element meshes consisted of three-dimensional l3disop 8-noded isoparametric elements. Each mesh contained a highly refined region of elements near the crack front and in the crack plane. A typical mesh detail on the crack plane is shown in Fig. 2. The D6AC steel material response was modeled using a Ramberg–Osgood [13] material model. The Ramberg–Osgood model is defined by Eq. (1), where SY is the 0.2% offset yield strength, eY is the corresponding extensional strain at yield, n is the hardening exponent, and j is a fitting constant. n e S S ¼ þj ð1Þ eY S Y SY Using this stress–strain relation, a curve fit was generated in agreement to tensile test data provided by the INEEL. The input parameters used were eY = SY/E, n = 50, and j = 0.315. The finite element model comprised one-quarter of the specimen shown in Fig. 1. The model was constrained in a manner to simulate the symmetry planes of the surface crack. Zero-displacement boundary conditions were applied to the y–z plane in the x-direction and the un-cracked ligament on the crack plane in the z-direction. To prevent model translation, a single node located at the farthest point from the model origin ((x, y, z) = (w, t, 0)) was constrained in the y-direction. The models were loaded monotonically in either tension or bending with an applied pressure on the z = h plane. The applied pressures used were consistent with the 5% potential drop load (load at which crack exten-
Fig. 3. Constraint path definition for a semi-elliptical surface crack.
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Fig. 4. Fracture surface image showing location of maximum crack extension.
Fig. 5. J and Jah distributions for selected surface crack models subjected to monotonic tension.
sion first occurred) recorded during testing and listed in Table 1. The elastic modulus E was 209.7 GPa and Poisson’s ratio m was 0.3. Within WARP3D, the von-Mises yield criterion and its associated flow rule were used. Linear-kinematic hardening was assumed.
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4. Constraint factor definition The hyper-local constraint factor ah [14,15] is similar to the global constraint factor ag developed by Newman et al. [7] for through-the-thickness cracks. The hyper-local constraint factor is a normalized, averaged crack opening stress acting along a ray perpendicular to and originating from the crack front and extending to the plastic zone boundary. The constraint factor is defined as Z Lð/Þ 1 rzz ah ð/Þ ¼ dl ð2Þ Lð/Þ 0 SY where (rzz/SY) is the normalized opening stress lying along a given ray L(/) as shown in Fig. 3. Nodal values of stress were used in the constraint calculations, and nodes were considered yielded if the von-Mises stress was greater than or equal to SY. The nodal stress values were calculated as a direct average using Gauss point values in the surrounding elements. Each ray of nodes perpendicular to the crack front (as evident in Fig. 2) was defined by the parametric angle / along the crack front. 5. Fracture prediction criteria Fracture was predicted to occur at the location along the crack front where the crack front constraint and the J-integral magnitudes are the highest. A first approximation to this fracture criterion considers the product Jah. Initial crack extension along the crack front is expected at the / location exhibiting the highest Jah. The ah and J-integral distributions along the crack front were calculated for each surface crack model from the finite element results. Using fracture surface images and digitizing software, the location of maximum crack extension was measured for each test specimen. In this work, we make the inherent assumption that the location of maximum crack extension corresponds to the location of initial crack instability. The location of maximum
Fig. 6. J and Jah distributions for selected surface crack models subjected to monotonic bending.
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crack extension on both sides of the crack was averaged to obtain a single location for comparison to the predicted value. The averaged / value is considered as the critical location /crit along the crack front. A typical fracture surface image detailing the crack extension measurements is presented in Fig. 4. To predict the location of maximum crack extension, both J and ah must be computed along the crack front. The J and Jah distributions were normalized by the product of the maximum applied stress S (tension) or M (bending) and the thickness t. Representative J,ah, and Jah distributions for selected tension and bending models are shown in Figs. 5 and 6, respectively. The corresponding fracture surface images are presented in Fig. 7 for tension and Fig. 8 for bending. The cases of tension and bending illustrate that the hyper-local constraint factor is near unity at the free surfaces, and is larger at the maximum depth location. The J-integral is higher near the free surfaces, but is lower at the maximum depth location. The bending configuration exhibits a higher constraint in the interior than the tension specimen. Note from these figures that the J-based and Jah-based criteria (maximum value) do not predict the same location of initial crack extension along the crack front. In some models, large applied loads resulted in plastic zones that exceeded the most refined region and spanned into the transition zone of the mesh. This resulted in reduced values of ah at these / locations. Under large bending loads, specimens with large a/t exhibited a loss of constraint at the deepest point of penetration
Fig. 7. Fracture surface of selected tension specimens. The fracture surfaces correspond to the J and Jah distributions presented in Fig. 5.
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with ah ! 1. These regions of numerical difficulty did not affect the results of interest wherein Jah was a maximum. For the tension case, the J-integral reaches a maximum and nearly constant value just below the free surface. The constraint factor reaches a constant value as the deepest point of penetration is approached, as seen in Fig. 5. Thus for tension, the product Jah reaches a maximum and nearly constant value just below the free surface, resulting in a range of / values for the potential critical location. This range is determined to be the / values that span a ±5% deviation from the constant value of Jah, as determined from the calculated J and ah distributions. The proposed criterion does not predict an exact critical location along the crack front for the tension cases (due to the product of crack drive and constraint being constant over a large range of /); however, the measured critical values lie within the predicted range of / values for all but one of the tension cases considered. The bending cases exhibit higher Jah variations along the crack front as shown in Fig. 6. The J-integral for a surface crack under bending reaches a distinct peak value below the free surface, while the constraint distribution behaves similarly to that for the tension case and approaches a constant maximum value towards the deepest point. The resulting product Jah demonstrates a definite peak value along the crack front between the free surface and the deepest point of crack penetration. The validity of Jah as a two-parameter fracture prediction criteria for surface cracks under tension or bending are presented in Table 2 and Fig. 9, respectively. The fracture criterion does not provide an exact critical location for the case of monotonic tensile loading; however, as mentioned previously, the measured location of maximum crack extension (and thus the assumed location of initial instability) falls within the range of predicted / values for all but one of the tension specimens. Observation of Table 2 also indicates that the Jah criterion predicts the critical location to occur over a smaller range of / values than a single-parameter J criterion predicts. For monotonic bending, the proposed criterion performs quite well, as presented in Fig. 9. The criterion predicts the critical location within 10 percent of the measured value for all but 2 of the bending spec-
Fig. 8. Fracture surface of selected bending specimens. The fracture surfaces correspond to the J and Jah distributions presented in Fig. 6.
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Table 2 Comparison of J and Jah for prediction of the location of maximum crack extension along the surface crack front for monotonic tension specimens Specimen
J /min/max
Jah /min/max
/crit
AT-01 AT-04 AT-05 AT-06 AT-07 AT-08 AT-09 BT-01 CT-01 CT-02 CT-03 DT-02
0.47–1.00 0.10–1.00 0.10–1.00 0.10–1.00 0.10–0.68 0.09–1.00 0.09–0.70 0.10–0.79 0.57–1.00 0.44–1.00 0.49–1.00 0.77–1.00
0.57–1.00 0.22–1.00 0.22–1.00 0.18–1.00 0.17–0.65 0.21–1.00 0.18–0.91 0.17–0.62 0.54–1.00 0.52–1.00 0.57–1.00 0.78–1.00
0.68 0.43 0.28 0.45 0.34 0.31 0.33 0.40 1.00 0.75 0.44 0.94
The critical locations for J and Jah are provided for a range of / values, from a minimum value /min to a maximum value /max. The /min and /max values are determined as the bounds of a 5% variation from the maximum J and Jah for each specimen.
Fig. 9. Comparison of Jah and J for prediction of the location of maximum crack extension along the surface crack front for monotonic bending specimens. The bending specimens correspond to those presented in Table 1.
imens (BB-01 and AB-07), which are predicted within 15% of the measured values. As in the case of monotonic tension, the two parameter Jah approach outperforms the single-parameter J-integral criterion for nearly all cases. 6. Summary and conclusions A two-parameter fracture criterion for prediction of the location of initial instability in surface cracks under monotonic loading was introduced and the validity was investigated. Using the product Jah to predict the
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critical location along the crack front was more accurate than a single-parameter J-integral approach for surface cracks under both tension and bending loads. The criteria accurately predicted the critical location for all bending cases within 15% error, and accurately predicted the range of critical locations for the tension cases considered. Specifically, our results indicate that a fracture criterion that includes the effects of crack front constraint quantified by the opening stress provides an accurate prediction of the critical location for specimens under tension or bending loading conditions. Since most components are subjected to complex stress states which may be considered as a combination of tension and bending, the proposed criterion shows great potential. Acknowledgements The authors express their appreciation to the NASA Marshall Space Flight Center, NASA Langley Research Center, and the FAA William J. Hughes Technical Center for supporting this work. The authors also thank the Idaho National Engineering and Environmental Laboratory for fabricating and conducting tests on the surface crack specimens. References [1] Irwin GR. Relation of stresses near a crack to the crack extension force. Proceeding of the 9th international congress on Appl Mech. Brussels, Belgium, 1956. [2] Rice JR. A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 1968;35:379–86. [3] Reuter WG, Newman Jr JC, Skinner JD, Mear ME, Lloyd WR. Use of KIC and constraint to predict load and location for initiation of crack growth in specimens containing part-through cracks. ASTM STP 2002;1406:353–80. [4] Hancock JW, Mackenzie AC. On the mechanisms of ductile failure in high-strength steels subjected to multi-axial stress-states. J Mech Phys Solids 1976;24:147–69. [5] Hancock JW, Reuter WG, Parks DM. Constraint and toughness parameterized by T. ASTM Symposium on Constraint Effects in Fracture, Indianapolis, IN, 1991. [6] O’Dowd NP, Shih CF. A family of crack tip fields characterized by a triaxiality parameter Part I – structure of field. J Mech Phys Solids 1991;39:989–1015. [7] Newman Jr JC, Bigelow CA, Shivakumar KN. Three-dimensional elastic–plastic finite-element analyses of constraint variations in cracked bodies. Engng Fract Mech 1993;46:1–13. [8] Rice JR, Tracey DM. On the ductile enlargement of voids in triaxial stress fields. J Mech Phys Solids 1969;17:201–17. [9] McClintock FA. A criterion for ductile fracture by the growth of holes. J Appl Mech 1968;35:363–71. [10] Sommer E, Aurich D. On the effect of constraint on ductile fracture. Defect assessment in components – fundamentals and applications, ESIS/EGF9 1991:141–74. [11] FEA-Crack ver 2.5.625. Structurable Reliability Technology. 2465 Central Avenue Suite 110. Boulder, Colorado 80301. Available from: