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Verification of stress-intensity factors for various middle-crack tension test specimens
Engineering Fracture Mechanics 72 (2005) 1113–1118 www.elsevier.com/locate/engfracmech
Technical Note
Verification of stress-intensity factors for various middle-crack tension test specimens J.C. Newman Jr. *, M. Jordan Haines Department of Aerospace Engineering, Mississippi State University, 330 Walker Engineering Laboratory, Hardy Street, Mississippi State, MS 39762, USA Received 26 July 2004; accepted 29 July 2004 Available online 21 September 2004
Abstract Recently, the stress-intensity factor equation used in the ASTM Standard Test Method for Measurement of Fatigue Crack Growth Rates (E-647) for the middle-crack tension M(T) specimens (friction-gripped or pin-loaded) has been questioned due to the influence of the specimen ‘‘height’’ specified in the standard. A boundary-element code has been used to calculate the stress-intensity factors for a wide range of crack-length-to-width ratios and various height-to-width ratios for M(T) specimens under remote uniform stress, remote uniform displacement, or pin-loaded holes. Comparisons are made with some of the well-known stress-intensity factor solutions and equations in the literature. Recommended specimen heights and stress-intensity factor equations have been made. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction The ASTM Standard Test Method for Measurement of Fatigue Crack Growth Rates (ASTM E-647 [1]) is used by many organizations to determine fatigue-crack-growth-rate properties for metallic materials. The test specimens that are used to measure fatigue-crack-growth rates must have accurate stress analyses or equations, which give the crack-tip stress-intensity factor, K, as a function of crack length. The stress-intensity factor, K, is used to characterize the fatigue-crack-growth properties and to make life predictions under various load histories. Recently, it came to the attention of an ASTM Task Group, by Francis I. Baratta, that the current stress-intensity factor equation used for two middle-crack tension, M(T), specimens (see Fig. 1)
*
Corresponding author. Tel.: +1 662 325 1521; fax: +1 662 325 7730. E-mail address: [email protected] (J.C. Newman Jr.).
0013-7944/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2004.07.008
1114
J.C. Newman, M. Jordan Haines / Engineering Fracture Mechanics 72 (2005) 1113–1118
S or δ
P
+ W/3
2L
2L
2a
2a
W
W
+
S or δ (a)
P (b)
Fig. 1. Schematic of the two middle-crack tension specimens used in ASTM E-647. (a) Friction gripped M(T) specimen, (b) pin-loaded M(T) specimen.
may have errors (>3%) for small and large crack-length-to-width (2a/W) ratios for the specified heights, 2L, in the current standard [1]. The discrepancies were due to the influence of the specimen ‘‘height’’ on stressintensity factors. One specimen is friction gripped and the other specimen is pin loaded. The M(T) specimens were some of the first specimens used to generate fatigue-crack growth and fracture data using the stress-intensity factor. These solutions and equations are tabulated in the two well-known reference manuals [2,3]. The influence of the height-to-width (2L/W) ratio on M(T) specimens under remote uniform tension or clamped-displaced edges (Fig. 1(a)) was studied by Isida [4]. However, his solutions were limited to 2a/W ratios less than 0.6–0.8 depending upon the 2L/W ratio. The stress-intensity factor solution for a crack in a pin-loaded M(T) specimen (Fig. 1(b)) is not included in the reference manuals [2,3]. The objective of this note is to use a two-dimensional boundary-element code, FADD2D [5], to calculate the stress-intensity factors for various M(T) specimen heights for both the pin-loaded and friction-gripped specimens. These results will allow an accurate assessment on the influence of specimen ‘‘height’’ on stressintensity factors and give some accuracy statements on the current stress-intensity factor equation used in the ASTM standard. The equation used in the ASTM standard was developed by Feddersen [6] and the equation has been used for many years for both types of specimens (see Fig. 1). This equation is pffiffiffiffiffiffi K ¼ S paF ;
ð1Þ
where the boundary-correction factor, F, is given by F ¼ ½secðpa=W Þ
1=2
:
ð2Þ
The applied stress S is calculated from the applied loading, P, specimen thickness, B, and specimen width, W. The half-crack length is a. The famous ‘‘secant’’ equation has been used and studied for many years. The secant correction appears to be close to the exact solution for a crack in an ‘‘infinite’’ long strip. However, in the laboratory, short ‘‘height’’ specimens are often used to minimize the amount of material used in the specimens.
J.C. Newman, M. Jordan Haines / Engineering Fracture Mechanics 72 (2005) 1113–1118
1115
2. Analysis The FADD2D code is a boundary-element code developed by Chang and Mear [5]. Cracks are modeled with linear elements, whereas internal or external boundaries are modeled with quadratic elements. The material was linear elastic with a PoissonÕs ratio of 0.3. The friction-gripped specimen, Fig. 1(a), in the ASTM standard has fixtures, which would impose roughly uniform displacement boundary conditions along the grip lines with L/W of 0.72. The M(T) specimen was analyzed for various L/W ratios, but only the results for L/W values of 0.6, 0.7 and 1.0 for both uniform displacement and uniform stress boundary conditions will be presented over a wide range in cracklength-to-width (2a/W) ratios. The uniform-displacement case simulated a clamped condition (horizontal displacements were fixed and a uniform vertical displacement was applied along the grip line). The uniform-stress case had the horizontal displacement free all along the edge of the specimen. The pin-loaded specimen, Fig. 1(b), was analyzed with simulated pin-loaded holes, assuming contact normal (cosine type) stresses acting over a 90°-segment of the hole. Here the L/W ratio of 1.5, as stated in the ASTM standard, was analyzed. Analyses were preformed for both types of M(T) specimens for 2a/W ratios of 0.1–0.95. Fig. 2 shows a convergence study on a uniformly stressed and pin-loaded specimen with a very large crack (2a/W = 0.9). The external boundary was modeled with a maximum of 120 elements and the crack was modeled with a maximum of 60 elements for a total of 180 elements. (For the pin-loaded specimen, 24 elements were used to model each hole and were not changed during the convergence study. And these elements were not included in the total number of elements.) The stress-intensity factor for various numbers of elements along the external boundary and crack is normalized by the 180-element solutions. Fig. 2 is a highly expanded scale and shows how rapid the solution converges. Even with as low as 50 elements (boundary and crack), the K solutions are within 0.5% of the highly refined solution. The K solutions from the FADD2D code are expected to be within ±0.2% of the exact solution. Fig. 3 shows the stress-intensity factors for various M(T) specimens normalized by the results from FeddersenÕs equation. The pin-loaded specimen results show that the equation is within 2% of the numerical results for 2a/W ratios less than about 0.9. The Feddersen formula is not capturing the K solution as
1.01 2a/W = 0.9 +0.2% 1.00
K / K180
-0.2%
0.99
Uniform stress Pin loaded
0.98
0.97 0
50
100
150
200
Number of elements Fig. 2. Convergence of stress-intensity factors for two M(T) specimens with a very large crack.
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J.C. Newman, M. Jordan Haines / Engineering Fracture Mechanics 72 (2005) 1113–1118 1.10
1.08
K / KFeddersen
1.06
Pin-loaded specimen (L/W = 1.5) Tada (L/W > 2) Isida-Uniform displacement (L/W = 0.6) Uniform displacement (L/W = 0.6) Uniform displacement (L/W = 0.7) Uniform stress (L/W = 0.7) Uniform displacement (L/W = 1.0) Uniform stress (L/W = 1.0)
1.04
1.02 +2% 1.00 -1% 0.98 0.0
0.2
0.4
0.6
0.8
1.0
2a / W Fig. 3. Ratio of stress-intensity factors for various M(T) specimens to the Feddersen equation.
the crack approaches the free boundary, as noted by Tada [7]. For large crack-length-to-width (2a/W) ratios, Tada [7] developed a modified equation as F ¼ ½secðpa=2Þ1=2 ½1 0:025a2 þ 0:06a4 ;
ð3Þ
where a = 2a/W. The Tada–Feddersen equation applies over a wide range in crack-length-to-width ratios (0 < 2a/W < 0.95) and was within 0.2% on the numerical results. For friction-gripped M(T) specimens, the results under uniform-displacement boundary conditions are shown in Fig. 3 for L/W ratios of 0.6, 0.7 and 1.0. The results for L/W = 0.6 agreed well with Isida [4]
1.03 Pin-loaded specimen (L/W = 1.5) Uniform displacement (L/W = 1.0) Uniform stress (L/W = 1.0)
K / KTada-Feddersen
1.02
+1%
1.01
1.00
0.99 -1%
0.98 0.0
0.2
0.4
0.6
0.8
1.0
2a / W Fig. 4. Ratio of stress-intensity factors for various M(T) specimens to the Tada–Feddersen equation.
J.C. Newman, M. Jordan Haines / Engineering Fracture Mechanics 72 (2005) 1113–1118
1117
Table 1 Normalized stress-intensity factors for various M(T) specimens F(1 a)1/2
Feddersen [6]
Isida [4] L/W = 1
a 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.85 0.90 0.95 1.00 a
1.0000 0.9546 0.9172 0.8864 0.8612 0.8409 0.8249 0.8129 0.8045 0.8016 0.7995 0.7983 –
1.0000 0.9544 0.9164 0.8849 0.8593 0.8391 0.8243 0.8151 0.8121 – 0.8151 – 0.8255a
Present L/W = 1
Present L/W = 1.5
Uniform stress
Uniform displacement
Pin-loaded holes
1.0000 0.9546 0.9168 0.8855 0.8600 0.8398 0.8250 0.8157 0.8126 0.8138 0.8170 0.8237 0.8255a
– 0.9630 0.9239 0.8911 0.8640 0.8421 0.8255 0.8158 0.8138 0.8147 0.8178 0.8241 0.8255a
– 0.9534 0.9154 0.8842 0.8588 0.8389 0.8245 0.8156 0.8128 0.8141 0.8176 0.8233 0.8255a
KoiterÕs exact limit [8].
and showed that the small 2a/W ratios produced K solutions that were as much as 4% higher than the Feddersen equation. The results for L/W = 0.7 (close to the standard specified value of 0.72) was about 3% higher than the Feddersen equation for both small and very large crack lengths. Herein, the L/W ratio was increased until the K solution was within about 1% of the Feddersen equation for small crack lengths. In practice, the L/W = 1 value has been used for many years as a guideline for where uniformly displaced and uniformly stressed specimens would give nearly the same K solutions. Fig. 4 shows the stress-intensity factors for pin-loaded (L/W = 1.5) and uniformly stressed or displaced (L/W = 1) specimen normalized by the Tada–Feddersen equation (Eq. (3)). All results were within 1% over a very wide range in crack-length-to-width (0 < 2a/W < 0.95) ratios. The numerical values for these three cases (normalized by the Koiter [8] factor) are given in Table 1 along with comparisons with FeddersenÕs equation and IsidaÕs numerical results for L/W equal to infinity. In 1965, Koiter had found the exact limiting solution as the crack length approached the free boundary [8].
3. Conclusions (1) For the pin-loaded middle-crack tension specimens (L/W = 1.5) in the ASTM standard, the Feddersen ‘‘secant’’ formula was within 2% for crack-length-to-width (2a/W) ratios less than about 0.9. The Tada– Feddersen equation was within 0.5% over a wide range in crack-length-to-width ratios (0 < 2a/ W < 0.95). (2) For the friction-gripped middle-crack tension specimens with L/W = 0.72 in the ASTM standard, the Feddersen ‘‘secant’’ formula was within about 3% over a wide range in crack-length-to-width ratios (0 < 2a/W < 0.95). (3) For a friction-gripped middle-crack tension specimens with L/W = 1, the Feddersen ‘‘secant’’ formula is within 1% for crack-length-to-width (2a/W) ratios less or equal to 0.8; whereas, the Tada–Feddersen equation was within 0.5% for a wide range in crack-length-to-width (0 < 2a/W < 0.95) ratios for uniformly-stressed specimens, but was within 1% for uniformly displaced specimens.
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References [1] Standard Test Method for Measurement of Fatigue Crack Growth Rates. ASTM E-647-00, 2000. p. 616–8. [2] Murakami Y, editor. Stress intensity factors handbook. Tokyo, Japan: Pergamon Press; 1986. [3] Tada H, Paris PC, Irwin GR, editors. The stress analysis of cracks handbook. New York, NY, USA: The American Society of Mechanical Engineers; 2000. [4] Isida M. Effect of width and length on stress intensity factors of internally cracked plates under various boundary conditions. Int J Fract Mech 1971;7(3):301–16. [5] Chang C, Mear ME. A boundary element method for two dimensional linear elastic fracture analysis. Int J Fract 1996;74:219–51. [6] Feddersen CE. Discussion to plane strain crack toughness testing. ASTM STP, 1966;410:77. [7] Tada H. A note on the finite width corrections to the stress intensity factor. Engng Fract Mech 1971;3(3):345–7. [8] Koiter WT. Note on the stress intensity factor for sheet strips with cracks under tensile loads. University of Technology, Report No. 314, Delft, The Netherlands, 1965.
Technical Note
Verification of stress-intensity factors for various middle-crack tension test specimens J.C. Newman Jr. *, M. Jordan Haines Department of Aerospace Engineering, Mississippi State University, 330 Walker Engineering Laboratory, Hardy Street, Mississippi State, MS 39762, USA Received 26 July 2004; accepted 29 July 2004 Available online 21 September 2004
Abstract Recently, the stress-intensity factor equation used in the ASTM Standard Test Method for Measurement of Fatigue Crack Growth Rates (E-647) for the middle-crack tension M(T) specimens (friction-gripped or pin-loaded) has been questioned due to the influence of the specimen ‘‘height’’ specified in the standard. A boundary-element code has been used to calculate the stress-intensity factors for a wide range of crack-length-to-width ratios and various height-to-width ratios for M(T) specimens under remote uniform stress, remote uniform displacement, or pin-loaded holes. Comparisons are made with some of the well-known stress-intensity factor solutions and equations in the literature. Recommended specimen heights and stress-intensity factor equations have been made. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction The ASTM Standard Test Method for Measurement of Fatigue Crack Growth Rates (ASTM E-647 [1]) is used by many organizations to determine fatigue-crack-growth-rate properties for metallic materials. The test specimens that are used to measure fatigue-crack-growth rates must have accurate stress analyses or equations, which give the crack-tip stress-intensity factor, K, as a function of crack length. The stress-intensity factor, K, is used to characterize the fatigue-crack-growth properties and to make life predictions under various load histories. Recently, it came to the attention of an ASTM Task Group, by Francis I. Baratta, that the current stress-intensity factor equation used for two middle-crack tension, M(T), specimens (see Fig. 1)
*
Corresponding author. Tel.: +1 662 325 1521; fax: +1 662 325 7730. E-mail address: [email protected] (J.C. Newman Jr.).
0013-7944/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2004.07.008
1114
J.C. Newman, M. Jordan Haines / Engineering Fracture Mechanics 72 (2005) 1113–1118
S or δ
P
+ W/3
2L
2L
2a
2a
W
W
+
S or δ (a)
P (b)
Fig. 1. Schematic of the two middle-crack tension specimens used in ASTM E-647. (a) Friction gripped M(T) specimen, (b) pin-loaded M(T) specimen.
may have errors (>3%) for small and large crack-length-to-width (2a/W) ratios for the specified heights, 2L, in the current standard [1]. The discrepancies were due to the influence of the specimen ‘‘height’’ on stressintensity factors. One specimen is friction gripped and the other specimen is pin loaded. The M(T) specimens were some of the first specimens used to generate fatigue-crack growth and fracture data using the stress-intensity factor. These solutions and equations are tabulated in the two well-known reference manuals [2,3]. The influence of the height-to-width (2L/W) ratio on M(T) specimens under remote uniform tension or clamped-displaced edges (Fig. 1(a)) was studied by Isida [4]. However, his solutions were limited to 2a/W ratios less than 0.6–0.8 depending upon the 2L/W ratio. The stress-intensity factor solution for a crack in a pin-loaded M(T) specimen (Fig. 1(b)) is not included in the reference manuals [2,3]. The objective of this note is to use a two-dimensional boundary-element code, FADD2D [5], to calculate the stress-intensity factors for various M(T) specimen heights for both the pin-loaded and friction-gripped specimens. These results will allow an accurate assessment on the influence of specimen ‘‘height’’ on stressintensity factors and give some accuracy statements on the current stress-intensity factor equation used in the ASTM standard. The equation used in the ASTM standard was developed by Feddersen [6] and the equation has been used for many years for both types of specimens (see Fig. 1). This equation is pffiffiffiffiffiffi K ¼ S paF ;
ð1Þ
where the boundary-correction factor, F, is given by F ¼ ½secðpa=W Þ
1=2
:
ð2Þ
The applied stress S is calculated from the applied loading, P, specimen thickness, B, and specimen width, W. The half-crack length is a. The famous ‘‘secant’’ equation has been used and studied for many years. The secant correction appears to be close to the exact solution for a crack in an ‘‘infinite’’ long strip. However, in the laboratory, short ‘‘height’’ specimens are often used to minimize the amount of material used in the specimens.
J.C. Newman, M. Jordan Haines / Engineering Fracture Mechanics 72 (2005) 1113–1118
1115
2. Analysis The FADD2D code is a boundary-element code developed by Chang and Mear [5]. Cracks are modeled with linear elements, whereas internal or external boundaries are modeled with quadratic elements. The material was linear elastic with a PoissonÕs ratio of 0.3. The friction-gripped specimen, Fig. 1(a), in the ASTM standard has fixtures, which would impose roughly uniform displacement boundary conditions along the grip lines with L/W of 0.72. The M(T) specimen was analyzed for various L/W ratios, but only the results for L/W values of 0.6, 0.7 and 1.0 for both uniform displacement and uniform stress boundary conditions will be presented over a wide range in cracklength-to-width (2a/W) ratios. The uniform-displacement case simulated a clamped condition (horizontal displacements were fixed and a uniform vertical displacement was applied along the grip line). The uniform-stress case had the horizontal displacement free all along the edge of the specimen. The pin-loaded specimen, Fig. 1(b), was analyzed with simulated pin-loaded holes, assuming contact normal (cosine type) stresses acting over a 90°-segment of the hole. Here the L/W ratio of 1.5, as stated in the ASTM standard, was analyzed. Analyses were preformed for both types of M(T) specimens for 2a/W ratios of 0.1–0.95. Fig. 2 shows a convergence study on a uniformly stressed and pin-loaded specimen with a very large crack (2a/W = 0.9). The external boundary was modeled with a maximum of 120 elements and the crack was modeled with a maximum of 60 elements for a total of 180 elements. (For the pin-loaded specimen, 24 elements were used to model each hole and were not changed during the convergence study. And these elements were not included in the total number of elements.) The stress-intensity factor for various numbers of elements along the external boundary and crack is normalized by the 180-element solutions. Fig. 2 is a highly expanded scale and shows how rapid the solution converges. Even with as low as 50 elements (boundary and crack), the K solutions are within 0.5% of the highly refined solution. The K solutions from the FADD2D code are expected to be within ±0.2% of the exact solution. Fig. 3 shows the stress-intensity factors for various M(T) specimens normalized by the results from FeddersenÕs equation. The pin-loaded specimen results show that the equation is within 2% of the numerical results for 2a/W ratios less than about 0.9. The Feddersen formula is not capturing the K solution as
1.01 2a/W = 0.9 +0.2% 1.00
K / K180
-0.2%
0.99
Uniform stress Pin loaded
0.98
0.97 0
50
100
150
200
Number of elements Fig. 2. Convergence of stress-intensity factors for two M(T) specimens with a very large crack.
1116
J.C. Newman, M. Jordan Haines / Engineering Fracture Mechanics 72 (2005) 1113–1118 1.10
1.08
K / KFeddersen
1.06
Pin-loaded specimen (L/W = 1.5) Tada (L/W > 2) Isida-Uniform displacement (L/W = 0.6) Uniform displacement (L/W = 0.6) Uniform displacement (L/W = 0.7) Uniform stress (L/W = 0.7) Uniform displacement (L/W = 1.0) Uniform stress (L/W = 1.0)
1.04
1.02 +2% 1.00 -1% 0.98 0.0
0.2
0.4
0.6
0.8
1.0
2a / W Fig. 3. Ratio of stress-intensity factors for various M(T) specimens to the Feddersen equation.
the crack approaches the free boundary, as noted by Tada [7]. For large crack-length-to-width (2a/W) ratios, Tada [7] developed a modified equation as F ¼ ½secðpa=2Þ1=2 ½1 0:025a2 þ 0:06a4 ;
ð3Þ
where a = 2a/W. The Tada–Feddersen equation applies over a wide range in crack-length-to-width ratios (0 < 2a/W < 0.95) and was within 0.2% on the numerical results. For friction-gripped M(T) specimens, the results under uniform-displacement boundary conditions are shown in Fig. 3 for L/W ratios of 0.6, 0.7 and 1.0. The results for L/W = 0.6 agreed well with Isida [4]
1.03 Pin-loaded specimen (L/W = 1.5) Uniform displacement (L/W = 1.0) Uniform stress (L/W = 1.0)
K / KTada-Feddersen
1.02
+1%
1.01
1.00
0.99 -1%
0.98 0.0
0.2
0.4
0.6
0.8
1.0
2a / W Fig. 4. Ratio of stress-intensity factors for various M(T) specimens to the Tada–Feddersen equation.
J.C. Newman, M. Jordan Haines / Engineering Fracture Mechanics 72 (2005) 1113–1118
1117
Table 1 Normalized stress-intensity factors for various M(T) specimens F(1 a)1/2
Feddersen [6]
Isida [4] L/W = 1
a 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.85 0.90 0.95 1.00 a
1.0000 0.9546 0.9172 0.8864 0.8612 0.8409 0.8249 0.8129 0.8045 0.8016 0.7995 0.7983 –
1.0000 0.9544 0.9164 0.8849 0.8593 0.8391 0.8243 0.8151 0.8121 – 0.8151 – 0.8255a
Present L/W = 1
Present L/W = 1.5
Uniform stress
Uniform displacement
Pin-loaded holes
1.0000 0.9546 0.9168 0.8855 0.8600 0.8398 0.8250 0.8157 0.8126 0.8138 0.8170 0.8237 0.8255a
– 0.9630 0.9239 0.8911 0.8640 0.8421 0.8255 0.8158 0.8138 0.8147 0.8178 0.8241 0.8255a
– 0.9534 0.9154 0.8842 0.8588 0.8389 0.8245 0.8156 0.8128 0.8141 0.8176 0.8233 0.8255a
KoiterÕs exact limit [8].
and showed that the small 2a/W ratios produced K solutions that were as much as 4% higher than the Feddersen equation. The results for L/W = 0.7 (close to the standard specified value of 0.72) was about 3% higher than the Feddersen equation for both small and very large crack lengths. Herein, the L/W ratio was increased until the K solution was within about 1% of the Feddersen equation for small crack lengths. In practice, the L/W = 1 value has been used for many years as a guideline for where uniformly displaced and uniformly stressed specimens would give nearly the same K solutions. Fig. 4 shows the stress-intensity factors for pin-loaded (L/W = 1.5) and uniformly stressed or displaced (L/W = 1) specimen normalized by the Tada–Feddersen equation (Eq. (3)). All results were within 1% over a very wide range in crack-length-to-width (0 < 2a/W < 0.95) ratios. The numerical values for these three cases (normalized by the Koiter [8] factor) are given in Table 1 along with comparisons with FeddersenÕs equation and IsidaÕs numerical results for L/W equal to infinity. In 1965, Koiter had found the exact limiting solution as the crack length approached the free boundary [8].
3. Conclusions (1) For the pin-loaded middle-crack tension specimens (L/W = 1.5) in the ASTM standard, the Feddersen ‘‘secant’’ formula was within 2% for crack-length-to-width (2a/W) ratios less than about 0.9. The Tada– Feddersen equation was within 0.5% over a wide range in crack-length-to-width ratios (0 < 2a/ W < 0.95). (2) For the friction-gripped middle-crack tension specimens with L/W = 0.72 in the ASTM standard, the Feddersen ‘‘secant’’ formula was within about 3% over a wide range in crack-length-to-width ratios (0 < 2a/W < 0.95). (3) For a friction-gripped middle-crack tension specimens with L/W = 1, the Feddersen ‘‘secant’’ formula is within 1% for crack-length-to-width (2a/W) ratios less or equal to 0.8; whereas, the Tada–Feddersen equation was within 0.5% for a wide range in crack-length-to-width (0 < 2a/W < 0.95) ratios for uniformly-stressed specimens, but was within 1% for uniformly displaced specimens.
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J.C. Newman, M. Jordan Haines / Engineering Fracture Mechanics 72 (2005) 1113–1118
References [1] Standard Test Method for Measurement of Fatigue Crack Growth Rates. ASTM E-647-00, 2000. p. 616–8. [2] Murakami Y, editor. Stress intensity factors handbook. Tokyo, Japan: Pergamon Press; 1986. [3] Tada H, Paris PC, Irwin GR, editors. The stress analysis of cracks handbook. New York, NY, USA: The American Society of Mechanical Engineers; 2000. [4] Isida M. Effect of width and length on stress intensity factors of internally cracked plates under various boundary conditions. Int J Fract Mech 1971;7(3):301–16. [5] Chang C, Mear ME. A boundary element method for two dimensional linear elastic fracture analysis. Int J Fract 1996;74:219–51. [6] Feddersen CE. Discussion to plane strain crack toughness testing. ASTM STP, 1966;410:77. [7] Tada H. A note on the finite width corrections to the stress intensity factor. Engng Fract Mech 1971;3(3):345–7. [8] Koiter WT. Note on the stress intensity factor for sheet strips with cracks under tensile loads. University of Technology, Report No. 314, Delft, The Netherlands, 1965.