A non-singular continuum theory of point defects using gradient elasticity of bi-Helmholtz type

ABSTRACT

In this paper, we develop a non-singular continuum theory of point defects based on a second strain gradient elasticity theory, the so-called gradient elasticity of bi-Helmholtz type. Such a generalised continuum theory possesses a weak nonlocal character with two internal material lengths and provides a mechanics of defects without singularities. Gradient elasticity of bi-Helmholtz type gives a natural and physical regularisation of the classical singularities of defects, based on higher order partial differential equations. Point defects embedded in an isotropic solid are considered as eigenstrain problem in gradient elasticity of bi-Helmholtz type. Singularity-free fields of point defects are presented. The displacement field as well as the first, the second and the third gradients of the displacement are derived and it is shown that the classical singularities are regularised in this framework. This model delivers non-singular expressions for the displacement field, the first displacement gradient and the second displacement gradient. Moreover, the plastic distortion (eigendistortion) and the gradient of the plastic distortion of a dilatation centre are also non-singular and are given in terms of a form factor (shape function) of a point defect. Singularity-free expressions for the interaction energy and the interaction force between two dilatation centres and for the interaction energy and the interaction force of a dilatation centre in the stress field of an edge dislocation are given. The results are applied to calculate the finite self-energy of a dilatation centre.

KEYWORDS:

• Point defects
• strain gradient elasticity
• Green tensor
• regularisation
• characteristic lengths
• eigenstrain

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 The inelastic distortion comprises plastic and thermal effects, and is typically an incompatible field. When the inelastic distortion is absent the elastic distortion is compatible.

2 In classical elasticity, the double-divergence of the Green tensor of the Navier equation reads$G_{ij,ij}^0\left(\mathit{R}\right)=-\frac{1-2\nu }{2\mu (1-\nu )}\delta \left(\mathit{R}\right),$which is often erroneously neglected in the literature (e.g. [2,3,63]).

Additional information

Funding

The author gratefully acknowledges a grant from the Deutsche Forschungsgemeinschaft (Grant No. La1974/4-1).