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This page provides stress intensity factor solutions for common cases.

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Center Through Crack in Plate

Center Through Crack in Plate

Stress intensity: $$ K = (Y_{t}~\sigma_{t} + Y_{b}~\sigma_{b})~\sqrt{ \pi a } $$
Geometry factor, tension: $$ Y_{t} = \sqrt{ \sec{ \left( \pi a \over 2b \right) } } $$
Geometry factor, bending: $$ Y_{b} = { Y_{t} \over 2 } $$

Definitions:

σt tensile stress
σb bending stress
a crack half-length
b plate half-width

References:

  1. AFRL-VA-WP-TR-2003-3002, "USAF Damage Tolerant Design Handbook: Guidelines for the Analysis and Design of Damage Tolerant Aircraft Structures," 2002
  2. API 579-1 / ASME FFS-1, "Fitness-For-Service," The American Petroleum Institute and The American Society of Mechanical Engineers, 2007


Single Edge Through Crack in Plate

Single Edge Through Crack in Plate

Stress intensity: $$ K = (Y_{t}~\sigma_{t} + Y_{b}~\sigma_{b})~\sqrt{ \pi a } $$
Geometry factor, tension: $$ Y_{t} = 0.265 (1 - \alpha)^4 + { 0.857 + 0.265 \alpha \over (1 - \alpha)^{3/2} } $$
Geometry factor, bending: $$ Y_{b} = \sqrt{ {2 \over \pi \alpha} \tan{ \pi \alpha \over 2 } } \left[ { 0.923 + 0.199 \left( 1 - \sin{ \pi \alpha \over 2 } \right)^4 \over \cos{ \pi \alpha \over 2 } } \right] $$
Dimension ratio: $$ \alpha = { a \over b } $$

Definitions:

σt tensile stress
σb bending stress
a crack length
b plate width

References:

  1. Dowling, Norman E., "Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue," 3rd Edition


Elliptical Surface Crack in Plate

Elliptical Surface Crack in Plate

Stress intensity: $$ K = (Y_{t}~\sigma_{t} + Y_{b}~\sigma_{b})~\sqrt{ \pi a } $$
Geometry factor, tension: $$ Y_{t} = F \sqrt{ 1 \over Q } $$
Geometry factor, bending: $$ Y_{b} = H \cdot Y_{t} $$
 
$$ F = \left[ M_1 + M_2 \left({a \over t}\right)^2 + M_3 \left({a \over t}\right)^4 \right] f_{\phi} f_w g $$
$$ H = H_1 + (H_2 - H_1) (\sin \phi)^p $$
\(\phi\) = 0° at c-tip,  \(\phi\) = 90° at a-tip (full depth of crack)

$$ a <= c $$ $$ a > c $$
$$ Q = 1 + 1.464 \left({a \over c}\right)^{1.65} $$ $$ Q = 1 + 1.464 \left({c \over a}\right)^{1.65} $$
$$ M_1 = 1.13 - 0.09 {a \over c} $$ $$ M_1 = \sqrt{c \over a} (1 + 0.04 {c \over a}) $$
$$ M_2 = -0.54 + {0.89 \over 0.2 + {a \over c} } $$ $$ M_2 = 0.2 \left( { c \over a } \right)^4 $$
$$ M_3 = 0.5 - { 1 \over 0.65 + {a \over c} } + 14 \left( 1 - { a \over c } \right)^{24} $$ $$ M_3 = -0.11 \left( {c \over a} \right)^4 $$
$$ f_{\phi} = \left[ \left( {a \over c} \right)^2 \cos^2 \phi + \sin^2 \phi \right]^{1/4} $$ $$ f_{\phi} = \left[ \left( {c \over a} \right)^2 \sin^2 \phi + \cos^2 \phi \right]^{1/4} $$
$$ f_w = \left[ \sec \left( { \pi c \over 2b } \sqrt{a \over t} \right) \right]^{1/2} $$
$$ g = 1 + \left[ 0.1 + 0.35 \left({a \over t}\right)^2 \right] (1 - \sin \phi)^2 $$ $$ g = 1 + \left[ 0.1 + 0.35 \left({c \over a}\right) \left({a \over t}\right)^2 \right] (1 - \sin \phi)^2 $$
$$ p = 0.2 + {a \over c} + 0.6 {a \over t} $$ $$ p = 0.2 + {c \over a} + 0.6 {a \over t} $$
$$ H_1 = 1 - 0.34 {a \over t} - 0.11 {a \over c} {a \over t} $$ $$ H_1 = 1 + G_{11} {a \over t} + G_{12} \left({a \over t}\right)^2 $$
$$ H_2 = 1 + G_{11} {a \over t} + G_{21} \left({a \over t}\right)^2 $$ $$ H_2 = 1 + G_{21} {a \over t} + G_{22} \left({a \over t}\right)^2 $$
$$ G_{11} = -1.22 - 0.12 {a \over c} $$ $$ G_{11} = -0.04 - 0.41 {c \over a} $$
--- $$ G_{12} = 0.55 - 1.93 \left({c \over a}\right)^{3/4} + 1.38 \left({c \over a}\right)^{3/2} $$
$$ G_{21} = 0.55 - 1.05 \left({a \over c}\right)^{3/4} + 0.47 \left({a \over c}\right)^{3/2} $$ $$ G_{21} = -2.11 - 0.77 {c \over a} $$
--- $$ G_{22} = 0.55 - 0.72 \left({c \over a}\right)^{3/4} + 0.14 \left({c \over a}\right)^{3/2} $$

Definitions:

σt tensile stress
σb bending stress
a crack depth
c crack half-length
b plate half-width
t plate thickness

References:

  1. Anderson, T.L., "Fracture Mechanics: Fundamentals and Applications," 3rd Edition


Corner Surface Crack in Plate

Corner Surface Crack in Plate

Stress intensity: $$ K = (Y_{t}~\sigma_{t} + Y_{b}~\sigma_{b})~\sqrt{ \pi a } $$
Geometry factor, tension: $$ Y_{t} = F \sqrt{ 1 \over Q } $$
Geometry factor, bending: $$ Y_{b} = H \cdot Y_{t} $$
 
$$ F = \left[ M_1 + M_2 \left({a \over t}\right)^2 + M_3 \left({a \over t}\right)^4 \right] f_{\phi} f_w g_1 g_2 $$
$$ H = H_1 + (H_2 - H_1) (\sin \phi)^p $$
\(\phi\) = 0° at c-tip,  \(\phi\) = 90° at a-tip

$$ a <= c $$ $$ a > c $$
$$ Q = 1 + 1.464 \left({a \over c}\right)^{1.65} $$ $$ Q = 1 + 1.464 \left({c \over a}\right)^{1.65} $$
$$ M_1 = 1.08 - 0.03 {a \over c} $$ $$ M_1 = \sqrt{c \over a} (1.08 + 0.03 {c \over a}) $$
$$ M_2 = -0.44 + {1.06 \over 0.3 + {a \over c} } $$ $$ M_2 = 0.375 \left( { c \over a } \right)^2 $$
$$ M_3 = -0.5 - 0.25 {a \over c} + 14.8 \left( 1 - { a \over c } \right)^{15} $$ $$ M_3 = -0.25 \left( {c \over a} \right)^2 $$
$$ f_{\phi} = \left[ \left( {a \over c} \right)^2 \cos^2 \phi + \sin^2 \phi \right]^{1/4} $$ $$ f_{\phi} = \left[ \left( {c \over a} \right)^2 \sin^2 \phi + \cos^2 \phi \right]^{1/4} $$
$$ f_w = \left[ \sec \left( { \pi c \over 2b } \sqrt{a \over t} \right) \right]^{1/2} $$
$$ g_1 = 1 + \left[ 0.08 + 0.4 \left({a \over t}\right)^2 \right] (1 - \sin \phi)^3 $$ $$ g_1 = 1 + \left[ 0.08 + 0.4 \left({c \over t}\right)^2 \right] (1 - \sin \phi)^3 $$
$$ g_2 = 1 + \left[ 0.08 + 0.15 \left({a \over t}\right)^2 \right] (1 - \cos \phi)^3 $$ $$ g_2 = 1 + \left[ 0.08 + 0.15 \left({c \over t}\right)^2 \right] (1 - \cos \phi)^3 $$
$$ p = 0.2 + {a \over c} + 0.6 {a \over t} $$ $$ p = 0.2 + {c \over a} + 0.6 {a \over t} $$
$$ H_1 = 1 - 0.34 {a \over t} - 0.11 {a \over c} {a \over t} $$ $$ H_1 = 1 + G_{11} {a \over t} + G_{12} \left({a \over t}\right)^2 $$
$$ H_2 = 1 + G_{11} {a \over t} + G_{21} \left({a \over t}\right)^2 $$ $$ H_2 = 1 + G_{21} {a \over t} + G_{22} \left({a \over t}\right)^2 $$
$$ G_{11} = -1.22 - 0.12 {a \over c} $$ $$ G_{11} = -0.04 - 0.41 {c \over a} $$
--- $$ G_{12} = 0.55 - 1.93 \left({c \over a}\right)^{3/4} + 1.38 \left({c \over a}\right)^{3/2} $$
$$ G_{21} = 0.64 - 1.05 \left({a \over c}\right)^{3/4} + 0.47 \left({a \over c}\right)^{3/2} $$ $$ G_{21} = -2.11 - 0.77 {c \over a} $$
--- $$ G_{22} = 0.64 - 0.72 \left({c \over a}\right)^{3/4} + 0.14 \left({c \over a}\right)^{3/2} $$

Definitions:

σt tensile stress
σb bending stress
a crack depth
c crack length
b plate width

References:

  1. Anderson, T.L., "Fracture Mechanics: Fundamentals and Applications," 3rd Edition


Thumbnail Crack in Solid Cylinder

Thumbnail Crack in Solid Cylinder

Stress intensity: $$ K = (Y_{t}~\sigma_{t} + Y_{b}~\sigma_{b})~\sqrt{ \pi a } $$
Geometry factor, tension: $$ Y_{t} = G \left( 0.752 + 1.286 \beta + 0.37 H^3 \right) $$
Geometry factor, bending: $$ Y_{b} = G \left( 0.923 + 0.199 H^4 \right) $$
 
$$ G = 0.92 \left({2 \over \pi}\right) \sec{(\beta)}~\sqrt{ \tan \beta \over \beta } $$
$$ H = 1 - \sin \beta $$
$$ \beta = {\pi \over 2} {a \over b}$$

Definitions:

σt tensile stress
σb bending stress
a crack depth
b cylinder diameter

References:

  1. AFRL-VA-WP-TR-2003-3002, "USAF Damage Tolerant Design Handbook: Guidelines for the Analysis and Design of Damage Tolerant Aircraft Structures," 2002