Strain and Strain Invariants

In continuum mechanics, the geometrical state of a deformed body may be characterized by a strain tensor; particularly, for small deformation, the strain tensor, epsilon, is defined as

where the u's are the components of the displacement vector field that links the deformed state with the reference state and the x's are cartesian coordinates.

The main output from simulation programs usually consists of the displacements from the perfect lattice positions for each atom in the core region. A finite difference scheme is applied to evaluate the derivatives of the equation above at the atom positions given such data. A FORTRAN program has been written by students in the Virginia Tech Atomistic Simulation Lab to facilitate this calculation for BCC- and FCC-based crystal structures.

The strain tensor, E, is calculated as a function of position and the invariants of this tensor are plotted in order to visualize the changes and movements that a dislocation core may undergo, even under simulated stress. The invariants usually plotted are, for a dislocation line along z,

These invariants may be plotted as colors and/or contours superimposed upon a relevant lattice plane giving a visual representation of the extended core.

An excellent reference on tensors, strain and strain invariants is Continuum Mechanics by Frederick and Chang, 1965, Scientific Publishers, Inc.

Return to Strain Invariants and Dislocation Core Structure of Simulated B2 NiAl .

Strain and Strain Invariants / VPI&SU /