From bicrystals to spherical inclusions: A superposition method to derive analytical expressions of stress fields in presence of plastic strain gradients

• Stress solutions in infinite bicrystal with spatial variations of plastic strains.
• From bicrystal to spherical inclusion problems by a superposition method.
• Inclusion problems with spatially non-uniform plastic strain in the inclusion.
• Size-dependent response for inclusion problem with non-uniform plastic strain.

In the present paper, the stress field of an infinite bicrystal with a planar boundary that undergoes plastic distortion variations along the normal to its interface is first considered. It is shown that the stress field of the classic Eshelby–Kröner spherical inclusion problem can be retrieved by applying an appropriate superposition method to these bicrystal stress solutions. The methodology is explained for interior and exterior points (i.e., inside and outside the inclusion). Such a superposition method provides a convenient geometrical interpretation of Eshelby–Kröner results. Besides, this method makes it also possible to handle spherical inclusion (or grain) problems with spatially non-uniform plastic strain in the inclusion. In particular, it is suited to handle easily intra-crystalline polynomial plastic strains with even exponents or any plastic strain that can be written as a power series representation with even exponents like cosx,coshx,sinxx,sinhxx. The analytical expression of the interior stress tensor for the problem of a plastic strain in the inclusion that varies as a power law with a general even exponent is given. Internal stresses and stored energy are also derived analytically for the problem of a plastic strain in the inclusion that varies as sinhrlrl (r being the polar distance to the inclusion centre and l a characteristic length), chosen to describe realistically the accumulated plastic strain gradients within grains. Remarkably, a tanh  -shape is found for the evolution of the stored energy as a function of la in a log–log scale (a being the radius of the grain), resulting in very similar size effects as those derived from generalized continuum models.