Dislocation Slip in Single Crystals

resolved shear stress, slip in single crystals, plastic deformation systems

The resolved shear stress in a slip direction is given as:

    \[\tau_R=\frac{F \cos \lambda \cos \phi}{A}\]

  • F is the applied force (N)
  • \lambda is the angle between the slip direction and the applied force
  • \phi is the angle between the slip plane and the applied force
  • A is the slip plane area (m²)

This can be written in terms of applied tensile stress, \sigma, and orientation factor (Schmidt Vector), M:

    \[\tau_R = \sigma \cos \lambda \cos \phi = \frac{\sigma}{M}\]

The resolved shear stress is maximum when λ = φ = 45°

The dislocation yield strength (critically resolved shear stress) occurs at yield:

    \[\tau_Y = \tau_{crss} = \sigma_Y \cos \lambda \cos{\phi} = \frac{\sigma_Y}{M} = \frac{f}{\vec b}\]

Where  is the friction force per unit length on the dislocation (N/m) and  is the Burger’s vector.

Vector Notation

  • Directions tend to be expressed in square brackets, [x,y,z]
  • Planes to be in normal brackets, (x,y,z)

The angle between two vectors is given as:

    \[\cos \phi = \frac{\vec a . \vec b}{|\vec a| |\vec b|}\]

Dislocation Slip in Polycrystals

Polycrystals are made up of many differently orientated grains. This makes them stronger, with superior fracture and fatigue resistance.

Not all grains deform as easily as each other, and so the deformation is not uniform. Along the grain boundaries, however, the grains must have the same dimensions as their neighbours. Therefore, even if one grain is at 45° (the most favourable to slip), it will not slip until its neighbours do.

Direct vs Shear Stress

Tensile stresses induce a maximum shear stress half their own value:

    \[\sigma = 2\tau\]

The gross total yielding of a polycrystalline sample occurs at a stress higher than the dislocation yield strength, \tau_Y, by an amount given by the Taylor factor. It is close to 1.5:


Therefore, hardening mechanisms that increase shear yield stress also increase direct yield stress.

Deformation by Twinning

Twinning is a form of deformation that produces a region that is a mirror image of the neighbouring undeformed region. Like slip, it occurs in set twinning directions and planes.


  • Crystal orientation does not change
  • Occurs in distinct steps
  • Atomic displacement = interatomic spacing


  • Orientation of lattice changes across the twinning plane
  • Atomic displacement is less than the interatomic spacing

Twinning only tends to occur when there are few slip systems (like HCP structures or at low temperatures and high loads), and the reorientation twinning causes can lea to more slip systems.

Twinning and temperature, twinning vs slip

Twinning is more or less independent of temperature – it is more favourable below room temperature as the critically resolved shear stress is lower than for slipping.

  • The resolved shear stress of a single crystal is given by
  •     \[\tau_R=\frac{F \cos \lambda \cos \phi}{A}\]

  • This gives a shear yield stress \tau_Y = \frac{\sigma_Y}{M}=\frac{f}{\vec b}
  • In polycrystals, tensile stresses induce a maximum shear stress half their own value: \sigma = 2\tau
  • The polycrystal yield stresses are linked: \sigma_Y = 3\tau_Y
  • Twinning produces a region that is a mirror image of it undeformed neighbour