Multipole expansion of continuum dislocations dynamics in terms of alignment tensors

Abstract

The development of a dislocation-based continuum theory of plasticity remains one of the central challenges of applied physics and materials science. Developing a continuum theory of dislocations requires the solution of two long-standing problems: (i) to find a faithful representation of dislocation kinematics with a reasonable number of variables and (ii) to derive averaged descriptions of the dislocation dynamics (i.e. material laws) in terms of these variables. In the current paper, we solve the first problem, i.e. we develop tensorial conservation laws for distributions of oriented lines. This is achieved through a multipole expansion of the dislocation density in terms of so-called alignment tensors containing information on the directional distribution of dislocation density and dislocation curvature. A hierarchy of evolution equations of these tensors is derived from a higher dimensional dislocation density theory. Low-order closure approximations of this hierarchy lead to continuum dislocation dynamics models of plasticity with only few internal variables. Perspectives for more refined theories and current challenges in dislocation density modelling are discussed.

Keywords:

• dislocations
• alignment tensors
• multipole expansion
• crystal plasticity

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 An often employed notation for these bases in continuum mechanics (e.g. [40]) is and .

2 We note that the flat crystal connection also renders the tangent bundle trivial in the case of large deformations, see e.g. [34]

3 The dynamic theory will of course have to consider interactions between dislocations of different Burgers vector or on distinct glide planes. We will focus on the kinematic theory in most of the paper and only touch upon the huge challenge of the dynamic case in Section 8.

4 It seems worth noting that there appears a non-trivial in the two-dimensional case which has no counterpart in the three-dimensional vector spherical harmonics, where and . The reason is that a ‘constant’ vector field only exists on the unit circle, while for topological reasons there are no such fields on the sphere.

5 In [34] I mistakenly assumed Equation (73(73) ) to be valid for the 2D case by just changing to . This remained unrecognized, because it had no consequences for the rest of that paper.

Additional information

Funding

The author gratefully acknowledges funding by the German Science Foundation DFG under project HO 4227/3-1.