A review of higher order strain gradient theories of plasticity: Origins, thermodynamics and connections with dislocation mechanics

In this paper we review developments in higher order strain gradient theories. Several variants of these theories have been proposed in order to explain the effects of size on plastic properties that are manifest in several experiments with micron sized metallic structures. It is generally appreciated that the size effect arises from the storage of geometrically necessary dislocations (GNDs) over and above the statistically stored dislocations (SSDs) required for homogeneous deformations. We review developments that show that the GNDs result from the non-homogeneous nature of the deformation field. Though the connection between GNDs and strain gradients are established in the framework of single crystal plasticity, generalisations to polycrystal plasticity has been made. Strain gradient plasticity inherently involves an intrinsic length scale. In our review, we show, through a few illustrative problems, that conventional plasticity solutions can always be reduced to a scale independent form. The same problems are solved with a simple higher order strain gradient formulation to capture the experimentally observed size effects. However, higher order theories need to be thermodynamically consistent. It has recently been shown that only a few of the existing theories pass this test. We review a few that do. Higher order theories require higher order boundary conditions that enable us to model effects of dislocation storage at impermeable boundaries. But these additional boundary conditions also lead to unique conceptual issues that are not encountered in conventional theories. We review attempts at resolving these issues pertaining to higher order boundary conditions. Finally, we review the future of such theories, their relevance and experimental validation.