Editing Fracture mechanics (section)

=== Stress intensity factor ===
{{main article|Stress intensity factor}}

Another significant achievement of Irwin and his colleagues was to find a method of calculating the amount of energy available for fracture in terms of the asymptotic stress and displacement fields around a crack front in a linear elastic solid.<ref name="Irwin57" /> This asymptotic expression for the stress field in mode I loading is related to the stress intensity factor <math>
   K_I
 </math> following:<ref name="notch">{{cite journal |last1= Liu |first1= M. |display-authors= etal |title= An improved semi-analytical solution for stress at round-tip notches |journal= Engineering Fracture Mechanics |year= 2015 |volume= 149 |pages= 134–143 |url= http://drgan.org/wp-content/uploads/2014/07/032_EFM_2015.pdf |doi= 10.1016/j.engfracmech.2015.10.004 |s2cid= 51902898 |access-date= 2017-11-01 |archive-date= 2018-07-13 |archive-url= https://web.archive.org/web/20180713213957/http://drgan.org/wp-content/uploads/2014/07/032_EFM_2015.pdf |url-status= live }}</ref>

:<math>\sigma_{ij} = \left(\cfrac{K_{I}}{\sqrt{2\pi r}}\right)~f_{ij}(\theta)</math>

where <math>
   \sigma_{ij}
 </math> are the [[Cauchy stress tensor|Cauchy stresses]], <math>
   r
 </math> is the distance from the crack tip, <math>
   \theta
 </math> is the angle with respect to the plane of the crack, and <math>
   f_{ij}
 </math> are functions that depend on the crack geometry and loading conditions.  Irwin called the quantity <math>
   K
 </math> the stress intensity factor.  Since the quantity <math>
   f_{ij}
 </math> is dimensionless, the stress intensity factor can be expressed in units of <math>\text{MPa}\sqrt{\text{m}}</math>.

Stress intensity replaced strain energy release rate and a term called [[fracture toughness]] replaced surface weakness energy. Both of these terms are simply related to the energy terms that Griffith used:

:<math>K_I = \sigma \sqrt{\pi a}\,</math>

and

<math>
   K_c = \begin{cases} \sqrt{EG_c} & \text{for plane stress} \\
                                                      \\
                     \sqrt{\cfrac{EG_c}{1-\nu^2}} & \text{for plane strain} \end{cases}
 </math>

where <math>K_I</math> is the mode <math>
   I
 </math> stress intensity, <math>K_c</math> the fracture toughness, and <math>\nu</math> is Poisson's ratio.

Fracture occurs when <math>K_I \geq K_c</math>. For the special case of plane strain deformation, <math>K_c</math> becomes <math>K_{Ic}</math> and is considered a material property. The subscript <math>
   I
 </math> arises because of the [[Stress intensity factor#Stress intensity factors for various modes|different ways of loading a material to enable a crack to propagate]]. It refers to so-called "mode <math>
   I
 </math>" loading as opposed to mode <math>
   II
 </math> or <math>
   III
 </math>:

The expression for <math>K_I</math> will be different for geometries other than the center-cracked infinite plate, as discussed in the article on the stress intensity factor. Consequently, it is necessary to introduce a [[dimensionless number|dimensionless correction factor]], <math>
   Y
 </math>, in order to characterize the geometry. This correction factor, also often referred to as the ''geometric shape factor'', is given by empirically determined series and accounts for the type and geometry of the crack or notch. We thus have:

:<math>K_I = Y \sigma \sqrt{\pi a}\,</math>

where <math>
   Y
 </math> is a function of the crack length and width of sheet given, for a sheet of finite width <math>
   W
 </math> containing a through-thickness crack of length <math>
   2a
 </math>, by:

:<math>Y \left ( \frac{a}{W} \right ) = \sqrt{\sec\left ( \frac{\pi a}{W} \right )}\,</math>