  A simple analytical model of intergranular normal stresses is proposed for a
general elastic polycrystalline material with arbitrary shaped and randomly
oriented grains under uniform loading. The model provides algebraic expressions
for the local grain-boundary-normal stress and the corresponding uncertainties,
as a function of the grain-boundary type, its inclination with respect to the
direction of external loading and material-elasticity parameters. The knowledge
of intergranular normal stresses is a necessary prerequisite in any local
damage modeling approach, e.g., to predict the intergranular stress-corrosion
cracking, grain-boundary sliding or fatigue-crack-initiation sites in
structural materials.
  The model is derived in a perturbative manner, starting with the exact
solution of a simple setup and later successively refining it to account for
higher order complexities of realistic polycrystalline materials. In the
simplest scenario, a bicrystal model is embedded in an isotropic elastic medium
and solved for uniaxial loading conditions, assuming 1D Reuss and Voigt
approximations on different length scales. In the final iteration, the grain
boundary becomes a part of a 3D structure consisting of five 1D chains with
arbitrary number of grains and surrounded by an anisotropic elastic medium.
Constitutive equations can be solved for arbitrary uniform loading, for any
grain-boundary type and choice of elastic polycrystalline material. At each
iteration, the algebraic expressions for the local grain-boundary-normal
stress, along with the corresponding statistical distributions, are derived and
their accuracy systematically verified and validated against the finite element
simulation results of different Voronoi microstructures.

Materials Science;Classical Physics