Frank network of dislocations within Mindlin's second strain gradient theory of elasticity

A special configuration of dislocation networks consisting of a honeycomb-shaped grid of dislocation lines, which is referred to as Frank network, has not thus far been investigated in the context of any higher-order theories of elasticity. Hence, the current paper is devoted to the determination of the elastic state of such a network of dislocations in the framework of Mindlin's second strain gradient theory of elasticity. To this end, the concepts of the plastic distortion and dislocation density tensors are utilized to represent a network of dislocations and, subsequently, a general solution will be derived for a spatially periodic network of dislocations contained in an infinitely extended isotropic body. Then, as a special case, a Frank network is considered and analytical expressions are obtained for the corresponding displacement and plastic distortion fields. The obtained results demonstrate that the employment of Mindlin's second strain gradient theory gives rise to the removal of all the classical singularities in the elastic fields, which manifest themselves especially at the plane of the dislocation network. It has also been shown that, by decreasing the edge length of the hexagons of the Frank network relative to the characteristic lengths of the constituent material of the body, the discrepancy between the strain-gradient and classical solutions becomes more pronounced or, in other words, the size effect plays a more significant role.