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If an external force acting on a body or if one part of the body applies
force on another part, it is said that the body is in the state of stress.
Stress is defined in units of force per unit area
and can be characterized in general case by the stress tensor.
Strain describes the state of deformation of a solid;
there is dilatational strain which changes the volume,
but not the shape and deviatoric strain
which in contrast changes the shape, but not the volume.
For a field of deformations ,
we can define a symmetric strain tensor in the following way:

(10.1) 
Hook discovered that the strain is proportional to the stress :

(10.2) 
These relation between the stress and the strain
could be generalized for an anisotropic solid,
in the terms of tensors:

(10.3) 
or

(10.4) 
The fourthrank tensor
has components,
but thanks to cubic symmetry (of simple cubic, fcc and bcc lattices)
there are only three independent components of the elasticity tensor
.
It is customary to reduce the number of subscripts by means of
an abbreviated notation for pairs of coordinate directions, as follows [67]:
in the full tensor notation the indices of
: 11, 22 , 33, 23 32, 13 31, 12 21,
while in the abbreviated notation they are
: 1, 2, 3, 4, 5, 6
and there is a correspondence:

(10.5) 

(10.6) 

(10.7) 

(10.8) 

(10.9) 

(10.10) 
In these notation the three independent components of the elasticity tensor [67]:
Of course, one can use linear combinations of these elastic constants,
to express other physical quantities such as:
bulk modulus:
and
shear modulus:
The free energy of a distortion can be expressed in powers of the strain tensor components.
Assuming that the distortion of a crystal lattice too small
to break the fourfold cubic symmetry and neglecting high order terms we can write the free energy as:

(10.11) 
On the base of elasticity theory, Born [68]
derived the general conditions for stability of a crystal lattice:

(10.12) 
and showed that the melting temperature could be found from the condition
.
Therefore, calculation of these elastic coefficients
is very important in studying the melting transition.
The elastic constants
are calculated by means of fluctuation formulas
obtained by Ray and Rahman [57,58].
It was shown that the elastic coefficients
are related
to fluctuations of the stress tensor

(10.13) 
and the ensemble average of the Born term
.
The elastic coefficients are calculated in the following way:

(10.14) 
where
is the stress tensor,
is Kroneker delta,
and is ensemble averaging, are the volume,
the number of atoms and the temperature, respectively.
The Born term is defined as:

(10.15) 
here
are Cartesian indices
and
numerate the neighbor atoms, and
is distance between the atoms and .
Potential energy is represented by the Finnis  Sinclair potential:

(10.16) 
and therefore the Born term is given by the formula:

(10.17) 
where:

(10.18) 

(10.19) 
here

(10.20) 
and finally:

(10.21) 
while the function
defined as:

(10.22) 
The main advantage of this method is the fast convergence of
the stress fluctuation term to its equilibrium value.
In order to calculate the shear elastic modulus
using the above mentioned formula two steps are required.
The first step is to find the zerostress reference matrix
for the computational cell
and
the second step is to run the NVT
MD simulation to calculate the elastic coefficients,
using stress tensor fluctuations and average value
of the
term.
Next: Isothermalisotension ensemble
Up: POINT DEFECTS, LATTICE STRUCTURE
Previous: NoseHoover algorithm
20030115