Numerical implementation of static Field Dislocation Mechanics theory for periodic media

Abstract

This paper investigates the implementation of Field Dislocation Mechanics (FDM) theory for media with a periodic microstructure (i.e. the Nye dislocation tensor and the elastic moduli tensor are considered as spatially periodic continuous fields). In this context, the uniqueness of the stress and elastic distortion fields is established. This allows to propose an efficient numerical scheme based on Fourier transform to compute the internal stress field, for a given spatial distribution of dislocations and applied macroscopic stress. This numerical implementation is assessed by comparison with analytical solutions for homogeneous as well as heterogeneous elastic media. A particular insight is given to the critical case of stress-free dislocation microstructures which represent equilibrium solutions of the FDM theory.

Keywords:

• Field Dislocation Mechanics
• internal stresses
• periodic microstructure
• Fourier transform
• infinite bicrystal
• zero-stress dislocation distributions

Acknowledgments

It is a pleasure for AA to acknowledge the hospitality and support of the CNRS, Laboratoire de Mécanique et d’Acoustique, Marseille, during a sabbatical visit in 2008–2009 when this work was initiated. AB gladly acknowledges the support of Université Paris Nord during its scientific stay in Paris in 2012.

Notes

1 This can be shown by prescribing . In this case, we havewhich admits non-periodic solutions, for instance,

2 The functional spaces and are defined in Appendix 2.

PS acknowledges the financial support of Labex MEC and of A*Midex [grant numbers ANR-11-LABX-0092 and ANR-11-IDEX-0001-02].