The role of strain gradients in the grain size effect for polycrystals

The role of grain size on the overall behaviour of polycrystals is investigated by using a strain gradient constitutive law for each slip system for a reference single crystal. Variational principles of Hashin-Shtrikman type are formulated for the rase where the strain energy density is a convex function of both strain and strain gradient. The variational principles are specialized to polycrystals with a general multi-slip strain gradient constitutive law. An extension of the Hashin-Shtrikman bounding methodology to general strain gradient composites is discussed in detail and then applied to derive bounds for arbitrary linear strain gradient composites or polycrystals. This is achieved by an extensive study of kernel operators related to the Green's function for a general ''strain-gradient'' linear isotropic incompressible comparison medium. As a simple illustrative example, upper and lower bounds are computed for linear face-centred cubic polycrystals: a size effect is noted whereby smaller grains are stiffer than large grains. The relation between the assumed form of the constitutive law for each slip system and the overall response is explored.