Local density approximation for the energy functional of three-dimensional dislocation systems

The elastic energy functional of a system of discrete dislocation lines is well known from dislocation theory. In this paper, we demonstrate how the discrete functional can be used to systematically derive approximations which express the elastic energy in terms of dislocation densitylike variables which average over the discrete dislocation configurations and represent the dislocation system on scales above the spacing of the individual dislocation lines. We study the simple case of two-dimensional systems of straight dislocation lines before we proceed to derive energy functionals for systems of three-dimensionally curved dislocation lines pertaining to a single as well as to multiple slip systems. We then illustrate several applications of the theory including Debye screening of dislocations in two and three dimensions, and the derivation of back stress and friction stress terms entering the stress balance from the free-energy functionals.

[1]  Victor L. Berdichevsky,et al.  Entropy of microstructure , 2008 .

[2]  U. F. Kocks,et al.  Elastic interactions between dislocations in a finite body , 1969 .

[3]  Robert A. Evarestov,et al.  Quantum chemistry of solids , 2007 .

[4]  Surachate Limkumnerd,et al.  Statistical approach to dislocation dynamics: From dislocation correlations to a multiple-slip continuum plasticity theory , 2007, 0710.5045.

[5]  M. Wilkens.  Das mittlere Spannungsquadrat 〈σ2〉 begrenzt regellos verteilter Versetzungen in einem zylinderförmigen Körper , 1969 .

[6]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[7]  M. Zaiser.  The energetics and interactions of random dislocation walls , 2013 .

[8]  M. Zaiser,et al.  Fluctuation phenomena in crystal plasticity—a continuum model , 2005, cond-mat/0505593.

[9]  P. Gumbsch,et al.  Analysis of dislocation pile-ups using a dislocation-based continuum theory , 2014 .

[10]  Andrew G. Glen,et al.  APPL , 2001 .

[11]  Amit Acharya,et al.  Size effects and idealized dislocation microstructure at small scales: Predictions of a Phenomenological model of Mesoscopic Field Dislocation Mechanics: Part I , 2006 .

[12]  R. Wiť.  The Continuum Theory of Stationary Dislocations , 1960 .

[13]  E. Giessen,et al.  A comparison of a statistical-mechanics based plasticity model with discrete dislocation plasticity calculations , 2004 .

[14]  V. Berdichevsky.  On thermodynamics of crystal plasticity , 2006 .

[15]  van der Erik Giessen,et al.  Aspects of boundary-value problem solutions with three-dimensional dislocation dynamics , 2002 .

[16]  P. D. Ispánovity,et al.  Scale-free phase field theory of dislocations. , 2014, Physical review letters.

[17]  Nasr M. Ghoniem,et al.  Fast-sum method for the elastic field of three-dimensional dislocation ensembles , 1999 .

[18]  István Groma,et al.  Link between the microscopic and mesoscopic length-scale description of the collective behavior of dislocations , 1997 .

[19]  Mark A. Kenamond,et al.  Coupling continuum dislocation transport with crystal plasticity for application to shock loading conditions , 2016 .

[20]  Michael Zaiser,et al.  Statistical dynamics of dislocation systems: The influence of dislocation-dislocation correlations , 2001 .

[21]  Christopher R. Weinberger,et al.  A non-singular continuum theory of dislocations , 2006 .

[22]  M. Hütter,et al.  Collective behaviour of dislocations in a finite medium , 2014 .

[23]  M. Lazar,et al.  Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity , 2005, cond-mat/0502023.

[24]  K. Le,et al.  Nonlinear continuum dislocation theory revisited , 2014 .

[25]  M. Peletier,et al.  Asymptotic Behaviour of a Pile-Up of Infinite Walls of Edge Dislocations , 2012, Archive for Rational Mechanics and Analysis.

[26]  Michael Zaiser,et al.  Spatial Correlations and Higher-Order Gradient Terms in a Continuum Description of Dislocation Dynamics , 2003 .

[27]  A. El-Azab,et al.  Statistical mechanics treatment of the evolution of dislocation distributions in single crystals , 2000 .

[28]  Rhj Ron Peerlings,et al.  Continuum modeling of dislocation interactions: why discreteness matters? , 2008 .

[29]  Amit Acharya,et al.  Continuum theory and methods for coarse-grained, mesoscopic plasticity , 2006 .

[30]  Michael Zaiser,et al.  A three-dimensional continuum theory of dislocation systems: kinematics and mean-field formulation , 2007 .

[31]  H. Mughrabi,et al.  X-ray line-broadening study of the dislocation cell structure in deformed [001]-orientated copper single crystals , 1984 .

[32]  Vasily V. Bulatov,et al.  On the evolution of crystallographic dislocation density in non-homogeneously deforming crystals , 2004 .

[33]  Mgd Marc Geers,et al.  A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity , 2006 .

[34]  Michael Zaiser,et al.  Continuum dislocation dynamics: Towards a physical theory of crystal plasticity , 2014 .

[35]  A. Wilkinson,et al.  Measurement of probability distributions for internal stresses in dislocated crystals , 2014 .

[36]  M. Zaiser,et al.  Scaling properties of dislocation simulations in the similitude regime , 2014, 1404.0230.

[37]  M. Hütter,et al.  Microscopically derived free energy of dislocations , 2015 .

[38]  D. Kuhlmann-wilsdorf,et al.  Low energy dislocation structures due to unidirectional deformation at low temperatures , 1986 .

[39]  Nasr M. Ghoniem,et al.  Parametric dislocation dynamics: A thermodynamics-based approach to investigations of mesoscopic plastic deformation , 2000 .

[40]  T. Hochrainer.  Multipole expansion of continuum dislocations dynamics in terms of alignment tensors , 2014, 1409.8461.