calculate_displacements_with_symbolical_integrals

dislocation_tools.backend.integral.calculate_displacements_with_symbolical_integrals(radii, angles, b, m, n, xi, c)

Compute the displacements with integral formalism and symbolical math

Calculate the displacement using Barnett and Lothe’s integral formalism [1], see [2] and [3]. The variables of the elastic problem are given similar names as in [2] and [3]. The solution involves integration, see equ. 4.1.25 in [3]. The integrands are combinations of matrices, which, in turn are contractions of the elastic stiffness tensor with two perpendicular vectors. A naive, fully numerical solution is slow, see calculate_displacements_with_numerical_integrals. Here, the integrands are made callable functions of the angle omega, which is the integration variable. This allows fast evaluation. However, problems can arise when attempting to perform trigonomic simplifications on the symbolic matrices, as well as when symbolically inverting the matrix nn. This might be a sympy issue or an issue of number overflow. Currently, no simplifications are performed and nn is inverted using LU-decomposition, which seems to be stable.

Parameters

  • radii (array-like) – N×1 array of perpendicular distances of the atoms from the line direction

  • angles (array-like) – N×1 array of angles in the plane perpendicular to xi, and relative to m

  • b (array-like) – Burgers vector (in laboratory coordinate system)

  • m (array-like) – x-direction of the dislocation coordinate system

  • n (array-like) – y-direction of the dislocation coordinate system

  • xi (array-like) – dislocation line direction; z-direction of the dislocation coordinate system

  • c (array-like) – 3×3×3×3 array representation of elastic stiffness, given in laboratory coordinates

Returns

u – displacements in laboratory coordinate system

Return type

numpy.ndarray

References

  1. Barnett, D.M.; Lothe, J. Phys. Norvegica, 7 (1973)

  2. Hirth, J.P.; Lothe, J. Theory of Dislocations, 2nd Edition; John Wiley and Sons, 1982. pp 467

  3. Bacon, D. J.; Barnett, D. M.; Scattergood, R. O. Progress in Materials Science 1978, 23, 51-262.