- Email: [email protected]
Stress intensity factors for large aspect ratio surface and corner cracks at a semi-circular notch in a tension specimen
Engineering Fracture Mechanics Great Britain.
Vol. 38, No. 6, pp. 461473, 1991
0013-7944/91 $3.00+ 0.00 Pergamon Press plc.
Printed in
STRESS INTENSITY FACTORS FOR LARGE ASPECT RATIO SURFACE AND CORNER CRACKS AT A SEMI-CIRCULAR NOTCH IN A TENSION SPECIMEN K. N. SHIVAKUMARt
and J. C. NEWMAN, Jr
NASA Langley Research Center Hampton, VA 23665, U.S.A. Abstract-Stress intensity factor solutions for semi-elliptic surface and quarter-elliptic corner cracks emanating from a semi-circular notch in a tension specimen are presented. A three-dimensional finite-element analysis in conjunction with the equivalent domain integral was used to calculate stress intensity factors @IF). SIF solutions for surface or comer crack (crack length to depth ratio of 2) at a notch are presented for a wide range of crack sizes and notch radii. Results showed that the SIF are larger for larger crack lengths and for larger notch radii. The SIF are nearly constant all along the crack front for deep surface cracks and for all corner cracks analysed.
INTRODUCTION SURFACE and corner cracks frequently initiate and grow at notches, holes, and lugs in structural components. A large percentage of the useful life of these components is consumed while the cracks are small compared to the component thickness. Hence, understanding and quantifying the severity of surface cracks and corner cracks are important parts of the development of life prediction methodologies. Current methodologies use the stress intensity factor (SIF) concept to quantify the severity of cracks. Development of SIF solutions for surface and corner cracks in structural components using analytical, numerical (finite-element method), and semi-analytical methods has continued for the last two decades (for example, see refs [l-S]). But limited results have been reported for surface or corner cracks emanating from edge notches[5]. Several numerical methods, like finite-element_alternating[3,4,6], weight function[7], and dislocation density[9], are being developed to analyse surface cracks in a stress-gradient field. However, conventional displacement-based finite-element methods (FEM) continue to give accurate and reliable results. Other methods have been compared with FEM solutions to confirm their accuracy and convergence. Reference[S] clearly demonstrates that even in the FEM, the experience and insight of the analyst are needed to accurately model complex configurations, for example, the intersection of the crack face with a notch boundary. In ref. [5], several surface-crack configurations at a semi-circular notch in a tension specimen were analysed. Finite-element solution were generated using an eight-noded hexahedron element[;?] in conjunction with the force method to calculate SIF. Reported solutions were only for a/c 6 1.O, where a and c are the semi-elliptic crack lengths in the thickness and width directions, respectively. Fatigue crack growth experiments of small cracks[lO, 111, however, indicate that most surface or corner cracks at a hole or notch grow at an aspect ratio (a/c) between 1 and 2. Although, cracks sometimes initiate with low-aspect ratios (a/c < l), they quickly become semi-circular and then grow into a semi-elliptic shape with a/c > 1.O. The purpose of this paper is to generate SIF solutions for practical configurations (aspect ratio a/c = 2) of surface and corner cracks in a single-edgenotched tension specimen. The edge notch is semi-circular. Although the SIF solutions are developed for the single-edge-notched tension specimen, they are generic and can be used to predict the life of aircraft structural components. The general configurations of a singIe-edge-notched specimen subjected to a uniform remote tension and the nomenclature used are shown in Fig. 1. A three-dimensional finite-element program, ZIP3D[12], was used in conjunction with the equivalent domain integral method[l3, 141 to calculate the SIF along the crack front. The specific configurations analysed were surface cracks
tAnalytica1 Services & Materials Inc., Hampton, VA 23666. This work was performed at the Langley Research Center as a part of NASA Contract NASl-18599. 467
468
K. N. SHIVAKUMAR
(a) Surface crack (A-A)
and J. C. NEWMAN, Jr
(b) Corner crack (B-B)
Fig. 1. Specimen configuration and description of surface and corner cracks.
(a/c = 2) with ~mbinations of a/t = 0.2, 0.5, and 0.8 for r/t = 1, 2, and 3; and corner cracks (a/c = 2) with a/t = 0.2, 0.5, and 0.8 for r/t = 1.5. THREE-DIMENSIONAL
FINITE-ELEMENT
ANALYSIS
A single-edge-notched tension specimen subjected to remote implane tensile stress, S, was considered. A semi-elliptical surface or quarter-elliptical corner crack emanating from the notch root (see Fig. 1) was assumed. The specimen width (w = 49 + r) and height (h 2 2%) selected were large enough for these parameters to have negligible effect on the SIF solutions. The material Poisson’s ratio was assumed to be 0.3. Due to symmetry of the specimen configuration and loading, only one-quarter for the surface crack specimen and one-half for the corner crack specimen were modeiled. Figure 2 shows a typical finite-element model for a/c = 2, a/t = 0.5, and r/t = 3 specimen, Eight-node hexahedron elements were used with reduced-shear integration[l5] in the model. The Crack front
-x
(b) I~llx8tton
(I) Flnitwlement ldullzatlon
at the cmck plrne
tz (c) Typlcal aectlon aor
the cmck front
Fig. 2. Finite-element idealization of a single-edge notched tension specimen.
SIFs for large aspects ratio cracks
469
finite-element program ZIP3D[ 121 in conjunction with an equivalent domain integral (EDI) method[ 13, 141was used to calculate the SIF along the crack front. Figure 2(a) shows the three-dimensional view of the finite-element mesh at the intersection of notch and crack plane. Following a procedure similar to that explained in ref. [5], illconditioned elements at the intersection of notch and the crack plane were avoided. The mesh pattern at the crack plane is shown in Fig. 2(b). The model had thirteen wedges in the 4-direction, with the smaller wedges near the intersection of the crack front and the notch boundary. A typical model had 4375 elements and 5380 nodes with 16,140 degrees-of-freedom. The SIF along the crack front were obtained using the ED1 method[l3, 141.The ED1 method is an extension of the well known two-dimensional J-integral method to three-dimensional problems. The resulting surface integral is converted into a volume integral through an application of the divergence theorem and the use of an s-function suggested by Lorenzi[l6]. An advantage of the ED1 method is that the finite-element mesh need not be orthogonal at the crack front[lrl]. Plane-strain conditions were used to calculate SIF from the computed J. Typical domain selected for the analysis is shown in the Fig. 2(c) (shaded region).
three-dimensional
RESULTS
AND DISCUSSIONS
First, the present finite-element model is evaluated by comparing the results for a semi-circular surface crack at a semi-circular notch in the tension specimen with those reported in ref. [5]. This case was selected because a considerable effort was made to establish the accuracy of the results in ref. [5]. Second, results for several semi-elliptic surface-crack configurations (a/c = 2) for various a/t and r/t ratios are presented. Finally, SIF solutions for three corner-crack configurations (a/c = 2, r/t = 1.5, and a/t = 0.2, 0.5, and 0.8) are presented. As in the previous studies[2,5,6,8], the SIF along the crack front is normalized by ,S,/(na/Q), where Q is the shape factor of an ellipse and is givn by the square of the complete elliptic integral of second kind[8]. An empirical equation for Q is given by Q = 1 + 1.464(a/c)‘,‘j5 for a < c and Q = 1 + 1.464(c/a)‘,‘j5 for c d a. Figure 3 shows the distribution of the normalized SIF along the crack front of a semi-circular surface crack (a/c = 1.0, a/t = 0.8, r/t = 3) by various methods. The solid curve through cross symbols is from the ED1 method, the circle symbols are from the VCCT method[l7] and the square symbols are from ref. [5]. The finite-element mesh used in ref. [5] and in the present analysis has orthogonal mesh patterns along the crack front. Therefore, the VCCT method[l7] and force
a/c=1 a/t = .8 r/t
q
3
Fig. 3. Comparison of normalized stress intensity factors from various methods for a surface crack emanating from the center of a semi-circular notch in a tension specimen. EF”
386-H
470
K. N. SHIVAKUMAR 3
and J. C. NEWMAN, Jr
EDI method a/c=2
c
rn ~2 2
0
30
60 Cp
60
Wed
Fig. 4. Effect of n/t ratio on stress intensity factor for semi-elliptic surface cracks at the semi-circular notch root in a tension specimen.
method[5] should give accurate solutions. Results from all three methods agreed well with each other (within 3 percent) all along the crack front except very near 4 = 90”. The disagreement among the three methods near 4 = 90” is mainly due to a free-surface boundary layer effect[l8]. A part of this difference is also due to the plane-strain approximations used in calculating SIF from J or G (VCCT method). Figure 4 shows the distribution of normalized SIF along the crack front of semi-elliptical surface cracks (a/c = 2) for various a/t ratios. The finite-element mesh for all three cases were nonorthogonal (see Fig. 2), hence SIF was calculated by only the ED1 method. Comparison of the SIF for various a/t ratios shows that the normalized SIF are lower for larger a/t ratios all along the crack front except near 4 = 90”. Note that the absolute SIF, however, are larger for larger crack lengths. The previously mentioned trend is an artifact of normalization by da. For a/t = 0.2, the SIF at Q = 0 (deeper section) is larger than near the free surface (4 = 90 degrees), which indicates that an initial crack with a/c = 2 should tend towards a/c < 2 as it grows. However, for deeper cracks (u/t 2 0.5) the SIF near the free surface is greater than or equal to that at 4 = 0 (deeper section). Furthermore the SIF are nearly constant along the crack front except near the free surface. Hence an initial crack with a/c = 2 should tend towards a/c > 2. These results agree with the experimental results presented in refs [lo] and [ll]. Normalized stress intensity factors for r/t = 1, 2, and 3 with a/t = 0.2, 0.5, and 0.8 at various locations (4) along the crack front are given in Table 1. 3
EDI method a/c=2 P
0.0
0.2
0.6
0.4
0.6
1.0
alt
Fig. 5. Variation of normalized stress-intensity-factor with a/r ratio at q = 0” and cp = 80” for semi-elliptic surface cracks at the semi-circular notch root in a tension specimen.
SIFs for large aspects ratio cracks
471
Table 1. Normalized stress intensity factor K/[S,/(na/Q)] for surface crack at center of semi-circular notch in a tension specimen (a/c = 2) alt r/t 1.0
b, deg. 3 9 18 27 36 45 54 63 72 78 82 ii.3
2.0
3 9 18 27 36 45 54 63 72 78 82 E.3
3.0
3 9 :; 36 45 54 63 72 78 82 88 89.3
0.2
0.5
0.8
2.007 2.003 1.997 1.970 1.920 1.839 1.830 1.776 1.744 1.739 1.751 1.854 1.900
1.647 1.640 1.627 1.605 1.582 1.561 1.558 I.563 1.576 1.596 1.625 1.737 1.827
1.474 1.471 1.466
2.189 2.180 2.152 2.124 2.074 2.026 I .986 1.908 1.835 1.830 1.865 1.953 2.000
1.943 I .926 1.914 1.905 1.871 1.816 1.791 1.764 1.746 1.747 1.770 1.866 1.946
1.817 1.811 1.799 1.784 1.761 1.746 I.726 1.711 I.721 1.746 1.792 1.904 1.979
2.300 2.295 2.263 2.207 2.143 2.086 1.967 1.913 1.866 1.847 1.872 1.949 2.023
2.125 2.121 2.104 2.072 2.023 1.973 1.924 1.864 1.836 1.815 1.822 1.881 1.964
2.028 2.021 2.006 i ,983 1.958 I.927 1.902 1.865 1.849 1.852 1.873 1.962 2.026
1.457 1A49 1.445
1.452 1.475 1.503 1.546 1.602 1.736 1.808
Figure 5 shows the variation of normalized SIF with a/t ratio at Q, = 0 and 80” for various r/t ratios. The (b = 80” location was selected to avoid the boundary-layer effect near the free surface (90”). Results for a/t = 0 (for example, t is very large compared to a) at C#J= 0 and 80” were obtained from the product of the SIF for a surface crack in an infinite body]81 and the stress concentration factor of 3.17[1 I]. Results for aft = 1 at # = 0 were estimated from the two-dimensional solution for a through crack in an edge notched tension specimen[l9]. The limiting case solutions are represented by solid symbols in Fig. 5. For d, = 0, the normalized SIF is lower for larger a/t ratios in the range 0
K. N. SHIVAKUMAR
412
“r
and J. C. NEWMAN, Jr
EDI method
arc=2 (I_
piTi
4_
2_
I 0.2
01
0.0
I
I
I
0.4
0.6
0.3
I
alt
I
I 1.0
Fig. 6. Variation of stress intensity factor for a typical single-edge notched tension specimen (I = 2.3 mm).
Figure 7 shows the distribution of normalized SIF along the crack front for the three corner-crack configurations (a/t = 0.2, 0.5, and 0.8 with r/t = 1.5). Numerical values are also presented in Table 2. Again, as observed for surface cracks in Fig. 4, the absolute values of the SIF are larger for larger a/t ratios and the SIF are nearly constant all along the crack front for all three a/t ratios.
CONCLUDING
REMARKS
Three-dimensional finite-element analyses of large aspect ratio (a/c = 2) semi-elliptical surface or quarter-elliptical corner cracks in a single-edge notched tension specimen were performed. The width and height of the specimens were large compared to their thicknesses and notch radii. The three-dimensional equivalent domain integral was used to evaluate the stress intensity factor (SIF) distribution along the crack front. The analysis was verified for a reference case by comparing with results from the literature. SIF solutions for practical configurations of surface and corner cracks are presented. Results show that the SIF are larger for larger crack lengths and for larger notch radii. The SIF are nearly constant all along the crack front for deep surface cracks (a/t 2 0.5) and for all corner cracks analysed (0.2 < a/t < 0.8). 3
a/c=2
EDI method
r/i ~1.5
km
-
0 0
I
I
30
60
I 90
cp Ww) Fig. 7. Effect of a/r ratio on stress intensity factor for quarter-elliptic corner cracks at the semi-circular notch root in a tension specimen.
473
SIFs for large aspects ratio cracks Table 2. Normalized stress intensity factor, K/[S,/(na/Q)] for corner crack at semi-circular notch in a tension specimen (a/c = 2 and r/t = 1.5)
9
deg. 3 9 18 27 36 45 54 63 72 78 82 88 89.3
0.2
0.5
0.8
2.128 2.080 2.037 1.993 1.956 1.906 1.851 1.811 1.795 1.772 1.804 1.879 1.945
1.932 1.896 1.866 1.819 1.783 1.744 1.720 1.696 1.686 1.693 1.708 1.824 1.910
1.852 1.812 1.764 1.710 1.672 1.645 1.613 1.605 1.611 1.643 1.689 1.804 1.876
REFERENCES [l] F. W. Smith and T. E. Kullgren, Theoretical and experimental analysis of surface cracks emanating from fastener holes. AFFDL-TR-76-104, Air Force Flight Dynamics Laboratory (1977). [2] I. S. Raju, and J. C. Newman, Jr., Stress-intensity factors for two symmetric corner cracks in Fracture Mechanics, ASTM STP 677, (Edited by C. W. Smith) pp. 411430. Am. Sot. Testing Mater. Philadelphia (1979). [3] T. Nishioka and S. N. Atluri, Analytical solution for embedded elliptical cracks, and finite element-alternating method for elliptical surface cracks, subjected to arbitrary loadings. Engng Fracture Mech. 17, 247-268 (1983). [4] T. Nishioka and S. N. Atluri, An alternating method for analysis of surface flawed aircraft structural components. AIAA J. 21, 749-757 (1983). [5] P. W. Tan, I. S. Raju, K. N. Shivakumar,
and J. C. Newman, Jr. An evaluation of finite-element models and stress-intensity factors for surface cracks emanating from stress concentrations (to appear as ASTM Special Technical Publication). [6] I. S. Raju, S. N. Atluri and J. C. Newman, Jr. Stress-intensity factors for small surface and corner cracks in plates. Fracture Mechanics: Perspectives and Directions, ASTM STP 1020 (Edited by R. P. Wei and R. P. Ganglok) pp. 297-316. Am. Sot. Testing Mater., Philadelphia (1989). [7] X. R. Wu, Stress intensitifactors for half-elliptical surface cracks subjected to complex crack face loadings. Engng Fracture Mech. 19, 387405 (1984). [8] J. C. Newman, Jr and I. S. Raju, Stress-intensity factor equations for cracks in three-dimensional finite bodies. Fracture Mechanics: Fourteenth Symposium Volume I: Theory and Analysis, ASTM STP 791 (Edited by J. C. Lewis and G. Sines) pp. 1-238-I-265. Am. Sot. Testing Mater., (1983). [9] W. D. Keat, B. S. Annigeri and M. P. Cleary, Surface integral and finite element hybrid method for two- and three-dimensional fracture mechanics analysis. Inr. J. Fracture, 36, 35-53 (1988). [lo] J. C. Newman, Jr. and P. R. Edwards, Short-crack growth behavior in an aluminum alloy-an AGARD cooperative test program. AGARD Report No. 732, (1988). [l l] M. H. Swain and J. C. Newman, Jr., On the use of marker loads and replicas for measuring growth rates for small cracks. in Fatigue Crack Topography, pp. 12.1-12.17. AGARD-CP-376 (1984). [12] R. G. Chermahini, K. N. Shivakumar, and J. C. Newman Jr. Three-dimensional finite-element simulation of fatigue-crack growth closure, in Mechanics of Fatigue Crack Closure (Edited by J. C. Newman, Jr., and W. Elber), ASTM STP 982, Philadelphia, PA (1988). [13] G. P. Nikishkov, and S. N. Atluri, Calculation of fracture mechanics parameters for an arbitrary three-dimensional crack, by the equivalent domain integral method. Int. J. numer. Methodr Engng 24, 1801-1821 (1987). [14] K. N. Shivakumar and I. S. Raju, Three-dimensional equivalent domain integral for mixed mode fracture problems. NASA CR-182021 (1990). [15] R. H. MacNeal, (Ed.); The MSCjNASTRAN Theoretical Manual. The MacNeal-Schwendler Corp., Los Angeles, Calif., (August 1987). [16] H. G. de Lorenzi, Energy release rate calculations by the finite-element methods. Engng Fracture Mech. 21, 129-143 (1985). [17] K. N. Shivakumar, P. W. Tan and J. C. Newman, Jr. A virtual crack-closure technique for calculating stress-intensity factors for cracked three-dimensional bodies. Int. J. Fracture, 36, R43-RSO, (1988). [18] K. N. Shivakumar and I. S. Raju, Treatment of singularities in cracked bodies. Int. J. Fracture 45, 159-178 (1990). [19] P. W. Tan, The boundary force method for stress analysis of arbitrarily shaped plates with notches and cracks. Ph.D. Thesis, George Washington University (1986). (Received 27 February 1990)
Vol. 38, No. 6, pp. 461473, 1991
0013-7944/91 $3.00+ 0.00 Pergamon Press plc.
Printed in
STRESS INTENSITY FACTORS FOR LARGE ASPECT RATIO SURFACE AND CORNER CRACKS AT A SEMI-CIRCULAR NOTCH IN A TENSION SPECIMEN K. N. SHIVAKUMARt
and J. C. NEWMAN, Jr
NASA Langley Research Center Hampton, VA 23665, U.S.A. Abstract-Stress intensity factor solutions for semi-elliptic surface and quarter-elliptic corner cracks emanating from a semi-circular notch in a tension specimen are presented. A three-dimensional finite-element analysis in conjunction with the equivalent domain integral was used to calculate stress intensity factors @IF). SIF solutions for surface or comer crack (crack length to depth ratio of 2) at a notch are presented for a wide range of crack sizes and notch radii. Results showed that the SIF are larger for larger crack lengths and for larger notch radii. The SIF are nearly constant all along the crack front for deep surface cracks and for all corner cracks analysed.
INTRODUCTION SURFACE and corner cracks frequently initiate and grow at notches, holes, and lugs in structural components. A large percentage of the useful life of these components is consumed while the cracks are small compared to the component thickness. Hence, understanding and quantifying the severity of surface cracks and corner cracks are important parts of the development of life prediction methodologies. Current methodologies use the stress intensity factor (SIF) concept to quantify the severity of cracks. Development of SIF solutions for surface and corner cracks in structural components using analytical, numerical (finite-element method), and semi-analytical methods has continued for the last two decades (for example, see refs [l-S]). But limited results have been reported for surface or corner cracks emanating from edge notches[5]. Several numerical methods, like finite-element_alternating[3,4,6], weight function[7], and dislocation density[9], are being developed to analyse surface cracks in a stress-gradient field. However, conventional displacement-based finite-element methods (FEM) continue to give accurate and reliable results. Other methods have been compared with FEM solutions to confirm their accuracy and convergence. Reference[S] clearly demonstrates that even in the FEM, the experience and insight of the analyst are needed to accurately model complex configurations, for example, the intersection of the crack face with a notch boundary. In ref. [5], several surface-crack configurations at a semi-circular notch in a tension specimen were analysed. Finite-element solution were generated using an eight-noded hexahedron element[;?] in conjunction with the force method to calculate SIF. Reported solutions were only for a/c 6 1.O, where a and c are the semi-elliptic crack lengths in the thickness and width directions, respectively. Fatigue crack growth experiments of small cracks[lO, 111, however, indicate that most surface or corner cracks at a hole or notch grow at an aspect ratio (a/c) between 1 and 2. Although, cracks sometimes initiate with low-aspect ratios (a/c < l), they quickly become semi-circular and then grow into a semi-elliptic shape with a/c > 1.O. The purpose of this paper is to generate SIF solutions for practical configurations (aspect ratio a/c = 2) of surface and corner cracks in a single-edgenotched tension specimen. The edge notch is semi-circular. Although the SIF solutions are developed for the single-edge-notched tension specimen, they are generic and can be used to predict the life of aircraft structural components. The general configurations of a singIe-edge-notched specimen subjected to a uniform remote tension and the nomenclature used are shown in Fig. 1. A three-dimensional finite-element program, ZIP3D[12], was used in conjunction with the equivalent domain integral method[l3, 141 to calculate the SIF along the crack front. The specific configurations analysed were surface cracks
tAnalytica1 Services & Materials Inc., Hampton, VA 23666. This work was performed at the Langley Research Center as a part of NASA Contract NASl-18599. 467
468
K. N. SHIVAKUMAR
(a) Surface crack (A-A)
and J. C. NEWMAN, Jr
(b) Corner crack (B-B)
Fig. 1. Specimen configuration and description of surface and corner cracks.
(a/c = 2) with ~mbinations of a/t = 0.2, 0.5, and 0.8 for r/t = 1, 2, and 3; and corner cracks (a/c = 2) with a/t = 0.2, 0.5, and 0.8 for r/t = 1.5. THREE-DIMENSIONAL
FINITE-ELEMENT
ANALYSIS
A single-edge-notched tension specimen subjected to remote implane tensile stress, S, was considered. A semi-elliptical surface or quarter-elliptical corner crack emanating from the notch root (see Fig. 1) was assumed. The specimen width (w = 49 + r) and height (h 2 2%) selected were large enough for these parameters to have negligible effect on the SIF solutions. The material Poisson’s ratio was assumed to be 0.3. Due to symmetry of the specimen configuration and loading, only one-quarter for the surface crack specimen and one-half for the corner crack specimen were modeiled. Figure 2 shows a typical finite-element model for a/c = 2, a/t = 0.5, and r/t = 3 specimen, Eight-node hexahedron elements were used with reduced-shear integration[l5] in the model. The Crack front
-x
(b) I~llx8tton
(I) Flnitwlement ldullzatlon
at the cmck plrne
tz (c) Typlcal aectlon aor
the cmck front
Fig. 2. Finite-element idealization of a single-edge notched tension specimen.
SIFs for large aspects ratio cracks
469
finite-element program ZIP3D[ 121 in conjunction with an equivalent domain integral (EDI) method[ 13, 141was used to calculate the SIF along the crack front. Figure 2(a) shows the three-dimensional view of the finite-element mesh at the intersection of notch and crack plane. Following a procedure similar to that explained in ref. [5], illconditioned elements at the intersection of notch and the crack plane were avoided. The mesh pattern at the crack plane is shown in Fig. 2(b). The model had thirteen wedges in the 4-direction, with the smaller wedges near the intersection of the crack front and the notch boundary. A typical model had 4375 elements and 5380 nodes with 16,140 degrees-of-freedom. The SIF along the crack front were obtained using the ED1 method[l3, 141.The ED1 method is an extension of the well known two-dimensional J-integral method to three-dimensional problems. The resulting surface integral is converted into a volume integral through an application of the divergence theorem and the use of an s-function suggested by Lorenzi[l6]. An advantage of the ED1 method is that the finite-element mesh need not be orthogonal at the crack front[lrl]. Plane-strain conditions were used to calculate SIF from the computed J. Typical domain selected for the analysis is shown in the Fig. 2(c) (shaded region).
three-dimensional
RESULTS
AND DISCUSSIONS
First, the present finite-element model is evaluated by comparing the results for a semi-circular surface crack at a semi-circular notch in the tension specimen with those reported in ref. [5]. This case was selected because a considerable effort was made to establish the accuracy of the results in ref. [5]. Second, results for several semi-elliptic surface-crack configurations (a/c = 2) for various a/t and r/t ratios are presented. Finally, SIF solutions for three corner-crack configurations (a/c = 2, r/t = 1.5, and a/t = 0.2, 0.5, and 0.8) are presented. As in the previous studies[2,5,6,8], the SIF along the crack front is normalized by ,S,/(na/Q), where Q is the shape factor of an ellipse and is givn by the square of the complete elliptic integral of second kind[8]. An empirical equation for Q is given by Q = 1 + 1.464(a/c)‘,‘j5 for a < c and Q = 1 + 1.464(c/a)‘,‘j5 for c d a. Figure 3 shows the distribution of the normalized SIF along the crack front of a semi-circular surface crack (a/c = 1.0, a/t = 0.8, r/t = 3) by various methods. The solid curve through cross symbols is from the ED1 method, the circle symbols are from the VCCT method[l7] and the square symbols are from ref. [5]. The finite-element mesh used in ref. [5] and in the present analysis has orthogonal mesh patterns along the crack front. Therefore, the VCCT method[l7] and force
a/c=1 a/t = .8 r/t
q
3
Fig. 3. Comparison of normalized stress intensity factors from various methods for a surface crack emanating from the center of a semi-circular notch in a tension specimen. EF”
386-H
470
K. N. SHIVAKUMAR 3
and J. C. NEWMAN, Jr
EDI method a/c=2
c
rn ~2 2
0
30
60 Cp
60
Wed
Fig. 4. Effect of n/t ratio on stress intensity factor for semi-elliptic surface cracks at the semi-circular notch root in a tension specimen.
method[5] should give accurate solutions. Results from all three methods agreed well with each other (within 3 percent) all along the crack front except very near 4 = 90”. The disagreement among the three methods near 4 = 90” is mainly due to a free-surface boundary layer effect[l8]. A part of this difference is also due to the plane-strain approximations used in calculating SIF from J or G (VCCT method). Figure 4 shows the distribution of normalized SIF along the crack front of semi-elliptical surface cracks (a/c = 2) for various a/t ratios. The finite-element mesh for all three cases were nonorthogonal (see Fig. 2), hence SIF was calculated by only the ED1 method. Comparison of the SIF for various a/t ratios shows that the normalized SIF are lower for larger a/t ratios all along the crack front except near 4 = 90”. Note that the absolute SIF, however, are larger for larger crack lengths. The previously mentioned trend is an artifact of normalization by da. For a/t = 0.2, the SIF at Q = 0 (deeper section) is larger than near the free surface (4 = 90 degrees), which indicates that an initial crack with a/c = 2 should tend towards a/c < 2 as it grows. However, for deeper cracks (u/t 2 0.5) the SIF near the free surface is greater than or equal to that at 4 = 0 (deeper section). Furthermore the SIF are nearly constant along the crack front except near the free surface. Hence an initial crack with a/c = 2 should tend towards a/c > 2. These results agree with the experimental results presented in refs [lo] and [ll]. Normalized stress intensity factors for r/t = 1, 2, and 3 with a/t = 0.2, 0.5, and 0.8 at various locations (4) along the crack front are given in Table 1. 3
EDI method a/c=2 P
0.0
0.2
0.6
0.4
0.6
1.0
alt
Fig. 5. Variation of normalized stress-intensity-factor with a/r ratio at q = 0” and cp = 80” for semi-elliptic surface cracks at the semi-circular notch root in a tension specimen.
SIFs for large aspects ratio cracks
471
Table 1. Normalized stress intensity factor K/[S,/(na/Q)] for surface crack at center of semi-circular notch in a tension specimen (a/c = 2) alt r/t 1.0
b, deg. 3 9 18 27 36 45 54 63 72 78 82 ii.3
2.0
3 9 18 27 36 45 54 63 72 78 82 E.3
3.0
3 9 :; 36 45 54 63 72 78 82 88 89.3
0.2
0.5
0.8
2.007 2.003 1.997 1.970 1.920 1.839 1.830 1.776 1.744 1.739 1.751 1.854 1.900
1.647 1.640 1.627 1.605 1.582 1.561 1.558 I.563 1.576 1.596 1.625 1.737 1.827
1.474 1.471 1.466
2.189 2.180 2.152 2.124 2.074 2.026 I .986 1.908 1.835 1.830 1.865 1.953 2.000
1.943 I .926 1.914 1.905 1.871 1.816 1.791 1.764 1.746 1.747 1.770 1.866 1.946
1.817 1.811 1.799 1.784 1.761 1.746 I.726 1.711 I.721 1.746 1.792 1.904 1.979
2.300 2.295 2.263 2.207 2.143 2.086 1.967 1.913 1.866 1.847 1.872 1.949 2.023
2.125 2.121 2.104 2.072 2.023 1.973 1.924 1.864 1.836 1.815 1.822 1.881 1.964
2.028 2.021 2.006 i ,983 1.958 I.927 1.902 1.865 1.849 1.852 1.873 1.962 2.026
1.457 1A49 1.445
1.452 1.475 1.503 1.546 1.602 1.736 1.808
Figure 5 shows the variation of normalized SIF with a/t ratio at Q, = 0 and 80” for various r/t ratios. The (b = 80” location was selected to avoid the boundary-layer effect near the free surface (90”). Results for a/t = 0 (for example, t is very large compared to a) at C#J= 0 and 80” were obtained from the product of the SIF for a surface crack in an infinite body]81 and the stress concentration factor of 3.17[1 I]. Results for aft = 1 at # = 0 were estimated from the two-dimensional solution for a through crack in an edge notched tension specimen[l9]. The limiting case solutions are represented by solid symbols in Fig. 5. For d, = 0, the normalized SIF is lower for larger a/t ratios in the range 0
K. N. SHIVAKUMAR
412
“r
and J. C. NEWMAN, Jr
EDI method
arc=2 (I_
piTi
4_
2_
I 0.2
01
0.0
I
I
I
0.4
0.6
0.3
I
alt
I
I 1.0
Fig. 6. Variation of stress intensity factor for a typical single-edge notched tension specimen (I = 2.3 mm).
Figure 7 shows the distribution of normalized SIF along the crack front for the three corner-crack configurations (a/t = 0.2, 0.5, and 0.8 with r/t = 1.5). Numerical values are also presented in Table 2. Again, as observed for surface cracks in Fig. 4, the absolute values of the SIF are larger for larger a/t ratios and the SIF are nearly constant all along the crack front for all three a/t ratios.
CONCLUDING
REMARKS
Three-dimensional finite-element analyses of large aspect ratio (a/c = 2) semi-elliptical surface or quarter-elliptical corner cracks in a single-edge notched tension specimen were performed. The width and height of the specimens were large compared to their thicknesses and notch radii. The three-dimensional equivalent domain integral was used to evaluate the stress intensity factor (SIF) distribution along the crack front. The analysis was verified for a reference case by comparing with results from the literature. SIF solutions for practical configurations of surface and corner cracks are presented. Results show that the SIF are larger for larger crack lengths and for larger notch radii. The SIF are nearly constant all along the crack front for deep surface cracks (a/t 2 0.5) and for all corner cracks analysed (0.2 < a/t < 0.8). 3
a/c=2
EDI method
r/i ~1.5
km
-
0 0
I
I
30
60
I 90
cp Ww) Fig. 7. Effect of a/r ratio on stress intensity factor for quarter-elliptic corner cracks at the semi-circular notch root in a tension specimen.
473
SIFs for large aspects ratio cracks Table 2. Normalized stress intensity factor, K/[S,/(na/Q)] for corner crack at semi-circular notch in a tension specimen (a/c = 2 and r/t = 1.5)
9
deg. 3 9 18 27 36 45 54 63 72 78 82 88 89.3
0.2
0.5
0.8
2.128 2.080 2.037 1.993 1.956 1.906 1.851 1.811 1.795 1.772 1.804 1.879 1.945
1.932 1.896 1.866 1.819 1.783 1.744 1.720 1.696 1.686 1.693 1.708 1.824 1.910
1.852 1.812 1.764 1.710 1.672 1.645 1.613 1.605 1.611 1.643 1.689 1.804 1.876
REFERENCES [l] F. W. Smith and T. E. Kullgren, Theoretical and experimental analysis of surface cracks emanating from fastener holes. AFFDL-TR-76-104, Air Force Flight Dynamics Laboratory (1977). [2] I. S. Raju, and J. C. Newman, Jr., Stress-intensity factors for two symmetric corner cracks in Fracture Mechanics, ASTM STP 677, (Edited by C. W. Smith) pp. 411430. Am. Sot. Testing Mater. Philadelphia (1979). [3] T. Nishioka and S. N. Atluri, Analytical solution for embedded elliptical cracks, and finite element-alternating method for elliptical surface cracks, subjected to arbitrary loadings. Engng Fracture Mech. 17, 247-268 (1983). [4] T. Nishioka and S. N. Atluri, An alternating method for analysis of surface flawed aircraft structural components. AIAA J. 21, 749-757 (1983). [5] P. W. Tan, I. S. Raju, K. N. Shivakumar,
and J. C. Newman, Jr. An evaluation of finite-element models and stress-intensity factors for surface cracks emanating from stress concentrations (to appear as ASTM Special Technical Publication). [6] I. S. Raju, S. N. Atluri and J. C. Newman, Jr. Stress-intensity factors for small surface and corner cracks in plates. Fracture Mechanics: Perspectives and Directions, ASTM STP 1020 (Edited by R. P. Wei and R. P. Ganglok) pp. 297-316. Am. Sot. Testing Mater., Philadelphia (1989). [7] X. R. Wu, Stress intensitifactors for half-elliptical surface cracks subjected to complex crack face loadings. Engng Fracture Mech. 19, 387405 (1984). [8] J. C. Newman, Jr and I. S. Raju, Stress-intensity factor equations for cracks in three-dimensional finite bodies. Fracture Mechanics: Fourteenth Symposium Volume I: Theory and Analysis, ASTM STP 791 (Edited by J. C. Lewis and G. Sines) pp. 1-238-I-265. Am. Sot. Testing Mater., (1983). [9] W. D. Keat, B. S. Annigeri and M. P. Cleary, Surface integral and finite element hybrid method for two- and three-dimensional fracture mechanics analysis. Inr. J. Fracture, 36, 35-53 (1988). [lo] J. C. Newman, Jr. and P. R. Edwards, Short-crack growth behavior in an aluminum alloy-an AGARD cooperative test program. AGARD Report No. 732, (1988). [l l] M. H. Swain and J. C. Newman, Jr., On the use of marker loads and replicas for measuring growth rates for small cracks. in Fatigue Crack Topography, pp. 12.1-12.17. AGARD-CP-376 (1984). [12] R. G. Chermahini, K. N. Shivakumar, and J. C. Newman Jr. Three-dimensional finite-element simulation of fatigue-crack growth closure, in Mechanics of Fatigue Crack Closure (Edited by J. C. Newman, Jr., and W. Elber), ASTM STP 982, Philadelphia, PA (1988). [13] G. P. Nikishkov, and S. N. Atluri, Calculation of fracture mechanics parameters for an arbitrary three-dimensional crack, by the equivalent domain integral method. Int. J. numer. Methodr Engng 24, 1801-1821 (1987). [14] K. N. Shivakumar and I. S. Raju, Three-dimensional equivalent domain integral for mixed mode fracture problems. NASA CR-182021 (1990). [15] R. H. MacNeal, (Ed.); The MSCjNASTRAN Theoretical Manual. The MacNeal-Schwendler Corp., Los Angeles, Calif., (August 1987). [16] H. G. de Lorenzi, Energy release rate calculations by the finite-element methods. Engng Fracture Mech. 21, 129-143 (1985). [17] K. N. Shivakumar, P. W. Tan and J. C. Newman, Jr. A virtual crack-closure technique for calculating stress-intensity factors for cracked three-dimensional bodies. Int. J. Fracture, 36, R43-RSO, (1988). [18] K. N. Shivakumar and I. S. Raju, Treatment of singularities in cracked bodies. Int. J. Fracture 45, 159-178 (1990). [19] P. W. Tan, The boundary force method for stress analysis of arbitrarily shaped plates with notches and cracks. Ph.D. Thesis, George Washington University (1986). (Received 27 February 1990)