The bonds between atoms and molecules give an intrinsic lattice resistance, f_i, that give pure crystal structures an inherent strength: the covalent bonds in ceramics are naturally strong, whereas the bonds in metals are innately soft.

It is useful to increase this lattice resistance though various strengthening mechanisms.

Solid Solution Hardening

Impurities are inserted into a pure lattice to increase the resistance to dislocation movement. This can be substitutional or interstitial, depending on the size of the inserted atoms.


This is when solute atoms of a similar size replace the solvent ones.

substitutional solid solution hardening

The new dislocation yielf strength, \tau_Y increases:

    \[\tau_Y=\frac{f_i}{\vec b}+\frac{f_{ss}}{\vec b}\]

  • f_i is the intrinsic lattice resistance per unit length (N/m)
  • f_ss is the solute solution resistance per unit length (N/m)

As the concentration of the solute atom, C, increases, the roughness of the slip plane increases too. This, in turn, increases the critical shear stress to initiate slip as follows:

    \[\tau_Y\propto \sqrt{C}\]


This is when smaller or larger solute atoms fill the gaps in the crystal structure. Typically, they propagate at the dislocation core, relieving some of the tension in the area. They anchor the dislocation, preventing it from moving. Any applied shear stress must now be larger to induce movement.

Smaller solute atoms tend to diffuse and propagate faster, so even at low concentrations are excellent hardeners.

Precipitation Hardening

Various heat treatments cause precipitates to form in a crystal lattice, which act as obstacles to dislocation movement.

The precipitates do not allow the dislocation to move through them, so are pressed around them. This causes them to curve, which requires energy (as the line tension restoring force must be overcome).

The restoring force is maximum when the dislocation is semi-circular – the critical form above. Here, the applied shear force is equal to two times the line tension on either side of the bulge.

The dislocation can only be pushed through the gap when

    \[\tau \vec b L = 2T\]

  • \tau is the applied shear stress (MPa)
  • T is the line tension per unit length (J/m)
  • L is the gap between precipitates (m)
  • \vec b is the Burger’s vector (m)

The critical shear stress is therefore:

    \[\tau_Y = \frac{2T}{\vec b L}\]

The resistance against dislocation movement due to precipitates, f_0, is given as:


Work Hardening

Work strengthening forces dislocations to move and reform in a lattice. The dislocations are then impeded and will require more energy to move further. This is because

  • Like dislocations (tensile or compressive) will repel one another
  • Forcing dislocations to move irregularly creates dislocation jogs – they become intertwined networks and as such immobile.

A higher dislocation density means there is a shorter distance between dislocations, and so there are more obstacles to hinder further dislocation motion. The critical shear stress increases with the increased dislocation density:

    \[\tau_Y=\tau_i+\alpha G \vec b \sqrt{\rho}\]

  • \tau_Y is the critical shear stress (MPa)
  • \tau_i is the initial stress not due to work hardening
  • \alpha is a material constant
  • \rho is the dislocation density

Dislocation density increase proportionately with plastic strain.

Hall-Petch Effect (Grain Boundary Hardening)

Grain boundaries act as excellent barriers to dislocation movement for two reasons:

  1. Grains tend to be differently orientated, so the dislocation will have to change direction
  2. The crystal structure is disordered along a boundary, resulting in discontinuities of slip planes.

Grain size, therefore, has a dramatic effect on the yield shear stress of a material, \tau_Y. This is clearly shown in the Hall-Petch Equation:


  • \tau_i is the initial resistance shear stress due to other hardening mechanisms
  • k is the locking parameter, measuring the effect of hardening on grain boundaries
  • D is the mean average diameter

Common values are:

Mild Steel23.30.74
Nickel Aluminide1001.70

Recovery & Recrystallisation

This is a process in which fine crystals are produced.

  1. Plastic Deformation
    Strain hardening occurs, where the additional energy is stored in dislocations (the lattice has a higher dislocation density).
  2. Heat Treatment
    The microstructure and properties are refined in two stages
    1. Recovery
      Some of the stored strain energy is released through dislocation movement and annihilation through atomic diffusion. Dislocation density decreases slightly, softening the material. There is still significant strain energy in the grains.
    2. Recrystallisation
      The vast imbalance in strain energies between different grains is unstable, and as such a new set of strain-free and equally sized (equiaxial) grains forms. This happens initially trough nucleation and then growth.

Recrystallisation depends on time, temperature, and the initial plastic deformation. The recrystallisation temperature is defined as the temperature at which recrystallisation completes in one hour – generally about 1/3 to ½ of the melting temperature.

Increasing initial cold work reduces the recrystallisation temperature.

Hot working is when the recrystallisation is done above the required temperature, when the material is still relatively soft and ductile. This allows much larger deformations to be made, such as shaping the material, but it will settle strong.

  • Substitutional solid solution hardening is when solute atoms replace solvent ones:

        \[\tau_Y = \frac{f_i}{\vec b}+\frac{f_{ss}}{\vec b}\]

  • The concentration is related to the strength: \tau_Y\propto\sqrt{C}
  • Interstitial solid solution hardening is when smaller or larger solutes are added into gaps in the lattice to alleviate tensile energies
  • Precipitation hardening provides obstacles which dislocations must overcome – this requires a lot of energy
  • Work hardening strengthens the material by increasing the dislocation density:

        \[\tau_Y=\tau_i+\alpha G \vec b \sqrt{\rho}\]

  • The Hall-Petch equation links yield stress and grain size:

        \[\tau_Y = \tau_i + \frac{k}{\sqrt{D}}\]

  • Fine crystals are produced in the following steps:
    1. Plastic Deformation
    2. Heat Treatment
      1. Recovery
      2. Recrystallisation