Research article

1. MATERIALS SCIENCE
&
ENGINEERING
Anandh Subramaniam & Kantesh Balani
Materials Science and Engineering (MSE)
Indian Institute of Technology, Kanpur- 208016
Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh
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A Learner’s Guide

2.  Near the dislocation line the stress fields and associated strains are so large that linear elasticity
theory breaks down → this region is known as the core of the dislocation.
 The minimum shear stress required to move a dislocation is the Peierls stress  can be
visualized as a lattice friction.
 For all practical purposes it is equivalent to the Critical Resolved Shear Stress (CRSS) (except
for the T aspect).
 The lattice friction stress* or the Peierls Stress is a sensitive function of the structure of the
core.
 The structure of the core is determined by the bonding in the crystal and the crystal structure.
 When the core is planar (lies on the slip plane) the Peierls stress can be described by an
exponential function. When the core is non-planar then atomistic calculations are required to
calculate the Peierls stress (e.g. screw dislocations in BCC materials).
 The original model is due to Peierls and Nabarro (PN) wherein they derived the lattice friction
stress as an exponential function of the ‘width of the dislocation’ and the Burgers vector (as
below).
◘ In their model the width (w) is the ‘effective extent; of the dislocation.
◘ ‘Wider dislocations’ have a lower PN stress
◘ Different slip systems have different values of Peierls stress
 Understanding the origin of the Peierls stress:
◘ The dislocation is in a local metastable equilibrium → sits in a Peierls valley
◘ Stress has to be applied to ‘pull’ the dislocation out of the valley (→ into the next valley)
Core of the dislocation and the Peierls Stress
* Stress required to move a dislocation on its slip plane

3.  There are two very similar quantities which we have seen:
 Peierls stress (or PN stress or Lattice friction stress)
 Critical Resolved Shear Stress (CRSS)
 Both of them are stresses required to cause plasticity
 Peierls stress is a quantity defined at the microscopic level (at the level of the slip plane)
 CRSS is more at the macroscopic level (level of the specimen).
 How are these quantities related? (Answer in diagram below)
 Peierls stress may corrected for an increase in temperature and hence the concept may be
extended to finite temperatures.
 Hence, often these two terms are interchangeably used  this is in some sense justified as they
are a measure of the same physical effect  inherent lattice resistance to the motion of
dislocations.
What is the connection between Peierls Stress and Critical Resolved Shear Stress?
Stress to cause microscopic plasticity
CRSS
Peierls stress At zero K, theoretically/computationally derived
At finite temperatures, experimentally determined
Funda Check

4. Stress to move a dislocation: Peierls – Nabarro (PN) stress
 We consider the original Peierls-Nabarro model (though this has been superseded by better models and computations).
 Width of the dislocation is considered as a basis for the ease of motion of a dislocation.
 Two extreme ‘widths’ are shown below for illustration.
 Unrelaxed condition- stiff
 Smaller width of displacement fields
  atomic adjustments required (for
any one atom) for dislocation motion
are large
Extreme
situations
 ‘Relaxed’ condition
 Large width of displacement fields
  atomic adjustments required for
dislocation motion are small

5. 






 b
w
PN e
G


2  G → shear modulus of the crystal
 w → width of the dislocation !!!
 b → |b|
Effect of w on PN
w 0 b 5b 10b
PN G G / 400 G / 1014 G / 1027
Hence,
► narrow dislocations are more difficult to move than wide ones
► dislocations with larger b are more difficult to move
Peierls – Nabarro stress (PN) → P-N stress → Lattice Friction
 The PN stress required to move a dislocation depends exponentially on the width of the
dislocation (w) & the modulus of the Burgers vector (b).
 Being an exponential function of both ‘w’ and ‘b’; PN stress is a sensitive function of these
factors:
 ‘w’ is determined by the bonding characteristics (metallic, ionic…)
 ‘b’ is determined by the crystal structure (superlattices have a large ‘b’; ordered structures
have a larger ‘b’ as compared to their disordered counterparts).

6.  Though the Peierls original formula has been superseded by more sophisticated theoretical
models and computational calculations; it worthwhile noting that if the core of the dislocation is
planar then the Peierls stress can be described by an exponential function similar to the one
originally conceived by Peierls.
 Additionally, a better feel can obtained for the PN stress by connecting the width of the
dislocation to the bonding characteristics of the material.
 Core splitting in BCC crystal is well studied by atomistic computational methods.







 b
w
PN e
G


2
Peierls stress is determined by
The nature of the bonding (which determines the core structure)
Crystal structure (which determines b)
 If the core is planar, then the width can be related to the interplanar spacing (d): w = d/(1)
 Equation (1) in conjunction with above implies that ‘d’ should be large  slip will occur on
widely spaced planes (usually close packed planes are widely spaced).
 Also, equation (1) implies that ‘b’ should be small this implies slip along close packed
direction.
 Hence, in metals like Cu slip occurs along close packed planes along close packed
directions.
(1)

7. Dependence of width of the dislocation on bonding of crystals
 Nature of chemical bonding in the crystal determines the → extent of relaxation & the width of the dislocation
Intermetallic compounds / complex crystal structures
 Intermetallic compounds and complex crystal structures (Fe3C, CuAl2) do not have good slip systems
→ favorable planes & directions → usually brittle Ordered compounds may have very large b
 In CuZn (an ordered compound) dislocations move in pairs to preserve the order during slip
 Quasicrystals have 4, 5 or 6 dimensional b and the 3D component is not sufficient to cause slip in the
usual sense
Covalent crystals
 Strong and directional bonds → small relaxation (low w) → high PN
 Usually fail by brittle fracture before PN is reached
Metallic crystals
 Weaker and non-directional bonds → large relaxation (high w) → low PN
 E.g. Cu can be cold worked to large strains
 Transition metals (e.g. Fe) have some covalent character due to ‘d’ orbital bonding → harder than Cu
Ionic crystals
 Moderate and non- directional bonds
 Surface cracks usually lead to brittle fracture
 Large b (NaCl: b = 3.95Å) → more difficult to move