On gradient field theories: gradient magnetostatics and gradient elasticity

Abstract

In this work, the fundamentals of gradient field theories are presented and reviewed. In particular, the theories of gradient magnetostatics and gradient elasticity are investigated and compared. For gradient magnetostatics, non-singular expressions for the magnetic vector gauge potential, the Biot–Savart law, the Lorentz force and the mutual interaction energy of two electric current loops are derived and discussed. For gradient elasticity, non-singular forms of all dislocation key formulas (Burgers equation, Mura equation, Peach–Koehler stress equation, Peach–Koehler force equation, and mutual interaction energy of two dislocation loops) are presented. In addition, similarities between an electric current loop and a dislocation loop are pointed out. The obtained fields for both gradient theories are non-singular due to a straightforward and self-consistent regularization.

Keywords:

• gradient theories
• gradient elasticity
• gradient magnetostatics
• dislocations
• Green functions
• size effects
• regularization

Acknowledgements

The author gratefully acknowledges Dr. Eleni Agiasofitou for many fruitful discussions and constructive remarks, which significantly influenced this work.

Notes

1 A more general constitutive relation than Equation (55 ) is , since . Using Equation (33 ), it does not change the Euler–Lagrange Equation (88 ), due to , and . Therefore, gradient magnetostatics possesses in a natural way only one internal length scale parameter, namely .

2 Due to an existing confusion in the literature, it is noted that is the elastic distortion tensor of gradient elasticity and it should not be confused with the elastic distortion tensor of classical elasticity.

3 In order to avoid the existing confusion and non-unique terminology in the literature of gradient elasticity (e.g. [50–53]), it has to be noted that and are the Cauchy stress tensor and the elastic strain tensor of gradient elasticity and they should not be confused with the Cauchy stress tensor and the elastic strain tensor of classical elasticity. On the other hand, Georgiadis et al. [32] used the notation of monopolar stress tensor for and dipolar stress tensor for . Georgiadis and Grentzelou [54] used the terminology: is the monopolar (or Cauchy in the nomenclature of Mindlin [14]) stress tensor and is the dipolar (or double) stress tensor.

4 Aifantis [70, 71] claimed that the gradient part, , of the elastic strain tensor of dislocations is determined from a homogeneous Helmholtz equation. This is obviously mistaken, since satisfies the inhomogeneous Helmholtz equation: , where is the classical elastic strain tensor.

5 Using an erroneous terminology in gradient elasticity, Polyzos et al. [50], Karlis et al. [52], Aravas and Giannakopoulos [72] and Aifantis [53] derived an inhomogeneous Helmholtz equation for the classical Cauchy stress tensor: , which is based on a physical misinterpretation of the Cauchy stress tensor in gradient elasticity.

The author acknowledges the grants from the Deutsche Forschungsgemeinschaft [grant number La1974/2-1], [grant number La1974/2-2], [grant number La1974/3-1].