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## Option: crack

#### Syntax

`-crack <I|II|III> <stress|strain> <Ks> <p1> <p2> <ξ> <n> <μ> <ν>`

#### Description

This option inserts a straight crack in the system. The equations used by this option can be found e.g. in L.B. Freund, Dynamic fracture mechanics.

The user has to provide the following parameters:

- I|II|III: mode of the crack, must be I (opening), II (in-plane shear) or III (out-of-plane shear).
- stress|strain: decides if the crack is built assuming plane stress or plane strain, respectively.
- Ks: stress intensity factor (in GPa.Å1/2).
- p1, p2: coordinates of the crack in the plane normal to the crack tip line (see ξ below), by order of permutation X, Y, Z: if ξ=Z then p1 is the position along X and p2 the position along Y; if ξ=Y then p1 is along Z and p2 along X; if ξ=X then p1 is along Y and p2 along Z. The positions <p1> and <p2> are usually given in Å. It is also possible to give them with respect to the box dimensions with the keyword BOX and an operation (see this page).
- ξ: direction of the line of the crack tip, must be x, y or z.
- n: direction normal to the plane of cut (also normal to the surfaces forming the crack lips), must be x, y or z, and must be different from ξ.
- μ: shear modulus of the material (in GPa).
- ν: Poisson ratio of the material.

If plane stress is used then `κ=(3-ν)/(1+ν)`; otherwise (i.e. assuming plane strain) `κ=3-4ν`.

For a mode I crack (opening crack) the displacements applied to any atom in the system are:

`ux1 = (Ks/2μ) √r/2π cos(θ/2) [κ - 1 + 2sin2(θ/2)]`

`ux2 = (Ks/2μ) √r/2π sin(θ/2) [κ + 1 - 2cos2(θ/2)]`

where x3 is the direction <ξ> of the crack line, and x2 the normal to the plane of cut (i.e. x2=<n>), `r` is the distance of the atom to the crack tip, and `θ` is the angle between the plane normal to <n> and the segment formed by the atom and the crack tip. In addition for a mode I crack the box is elongated along x2 by `(Ks/μ)√pos2/2π(κ+1)`.

For a mode II crack (in-plane shear crack) the displacements are:

`ux1 = (Ks/2μ) √r/2π sin(θ/2) [κ + 1 + 2cos2(θ/2)]`

`ux2 = -(Ks/2μ) √r/2π cos(θ/2) [κ - 1 - 2sin2(θ/2)]`

And finally for a mode III crack (out-of-plane shear crack):

`ux3 = (Ks/μ) √r/2π sin(θ/2)`

In addition to these displacements, Atomsk also computes the theoretical (continuum) stresses associated with the crack and saves the Voigt components σxx, σyy, σzz, σyz, σxz and σxy as auxiliary properties for each atom.

In linear elastic materials a mixed-mode problem can be treated by using this option three times to add up the three contributions:

`u = umodeI + umodeII + umodeIII`

If a selection was defined (with the option `-select`) then the displacements described above are applied only to selected atoms.

#### Default

By default no crack is introduced at all.

#### Examples

- `atomsk initial.xyz -crack I stress 20 30.0 0.5*BOX z y 90 0.3 crack.cfg`

  This will read `initial.xyz` and insert a mode I crack along Z, assuming plane stress, the crack lips being normal to ξ=Y, using a stress intensity factor Ks=20 GPa.Å1/2, a shear modulus μ=90 GPa and a Poisson ratio ν=0.3. The crack will be at X=30 Å and in the center of the box along Y. The result will be output to `crack.cfg`.

- `atomsk initial.xyz -crack I stress 40 30 0.5*BOX z y 90 0.3 -crack II stress 10 30 0.5*BOX z y 90 0.3 -crack III stress 10 30 0.5*BOX z y 90 0.3 crack.cfg`

  This will read `initial.xyz` and insert a crack by adding three contributions of modes I, II and III. The result will be output to `crack.cfg`.

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