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Slip :

Crystal | Metals | Slip Plane | Slip Direction | Number of Slip System |
---|---|---|---|---|

FCC | Cu, Al, Ni, Ag, Au | {111} | <110> | 12 |

BCC | α-Fe, W, Mo | {110} | <111> | 12 |

α-Fe, W | {211} | <111> | 12 | |

α-Fe, K | {321} | <111> | 24 | |

HCP | Cd, Zn, Mg, Ti, Be | {0001} | \( <11\bar{2}0> \) | 3 |

Ti, Mg, Zr | {\( {10\bar{1}0} \)} | \( <11\bar{2}0> \) | 3 | |

Ti, Mg | {\( {10\bar{1}1} \)} | \( <11\bar{2}0> \) | 6 |

• Means slip occures by breaking of all bonds simultaneously on imposing shear stress.

• As the slip occure by translation of one plane over another, and for this movement to occure in a perfect crystal a shere stress (\( \tau \)) is required.

• The distance betwwen atoms in the slip direction is b, and the spacing between adjecent lattice plan is a.

• The shear stress is initially zero when the two plans are in concidence. and it is zero when moved distance is multiple of b.

• The shear stress is also zero when the atoms of top plane are midways between those of bottome plane.

• And to break the bonds, the applied stress required to overcome the lattice resistance to shear is indicated by \( \tau_{max} \).

From Above fig. the first approximation of realtionship between shear stress and displacement can be expressed by sine function as - \[ \tau = \tau_{max} \big(sin \frac{2\pi x}{b} \big) \] For maimum shear stress x=a, and for small value of a/b the above equation can be written as - \[ \tau = \tau_{max} \Big(\frac{2\pi a}{b} \Big) \] As for rough approximation b = a, and also as \( \tau = G\gamma \) \[ \boxed{ \Rightarrow \tau_{max} = \frac{G}{2\pi} } \] Where \( \tau_{max} \) = Theoratical shear strength, and G = Shear Modulus.

• The movement of dislocation is related with the Caterpillar Movement

• In this model we only move planes bit by bit instead of whole plane at once. which reduces the shear stress required to slip.

• This model was introduced by peierls nabarro and the steess predicted by this model is called as ** peierls nabarro stress or lattice friction**.

• He also introduce a fact that dislocation has a width. The width of the dislocation is the distance on either side of dislocation upto with dislocation stess is applicable.

• Narrow dislocation are difficult to move.

\[ \boxed{ \tau_{PN} = \frac{2G}{1-\nu} exp{\Big(- \frac{2\pi w}{b} \Big)}} \]
Where w = Width of dislocation, and \(w = \frac{a}{1-\nu} \), a= inter-planer spacing

**Important Points : **

(1) For CPP \( a \uparrow \Rightarrow w \uparrow \Rightarrow \tau_{PN} \downarrow \Rightarrow \) Stress required for slip is less means ductile material.

(2) Slip will occure when \( \tau_{PN} \) is less , means slip will occure in Closed pack plane ( i.e., \( a \uparrow \) ) and closed pack direction \( b \downarrow \)

(3) In covalent crystal dislocation width(w) is very less => Difficult to move dislocation => brittle

(4) For ionic crystal, moderate value of \( \tau_{PN} \) is required. But brittle frature is observed due to large 'b'.

P = Applied Load

N = Slip plane normal

D = Slip DIrection

\( \lambda \) = Angle between P & N

\( \phi \) = Angle between P & D

\[ \tau_{RSS} = \frac{P}{A} = \frac{Fcos(\phi)}{\frac{A}{cos(\phi)}} \]
\[ \Rightarrow \tau_{RSS} = \sigma cos(\lambda) cos(\phi) \]
**Special Cases : **

(1) If \( \lambda = 90 \Rightarrow cos(\lambda) = 0\) => \( \tau_{RSS} = 0 \)

(2) If \( \phi = 90 \Rightarrow cos(\phi) = 0\) => \( \tau_{RSS} = 0 \)

Critical Resolved Shear Stress (\( \tau_{CRSS} \)) :

Slip in a material occures when the shearing stress on the slip plane in the slip direction reaches a threshold value called the *critical resolved shear stress. *

Where \( cos(\lambda) cos(\phi) \) is called as Schmid Factor and \( \sigma_{y} =\) Yield Strength

**Note : **

(1) If we change the direction of stress w.r.t. slip plane and slip direction, \( cos(\lambda) cos(\phi) \) will also change but \( \tau_{CRSS} \) will not change. Hance \( \sigma_{y} \) will change => \( \sigma_{y} \) is anisotropy in nature.

(2) \( \tau_{CRSS} \) is a material property.

(3) Slip system with higher Schmid factor will be active slip system.

**Condition For Dislocation Motion**
\[ \tau_{RSS} > \tau_{CRSS} \]