Grain boundary stresses in elastic materials

Dr. Samir El Shawish and dr. Timon Mede from the Reactor Engineering Division at Jožef Stefan Institute published an article “Grain boundary stresses in elastic materials” in European Journal of Mechanics / A Solids.

In the paper a simple analytical model is proposed for computing intergranular normal stresses induced in elastic polycrystalline materials by uniform macroscopic loading. The model is derived in a perturbative manner, where at each succeeding order additional contributions to grain-boundary-normal stresses are included – from more to less relevant. The most important effects come from grain-boundary orientation (with respect to external stress), its type and the neighbourhood of bicrystal pair of grains. In the final iteration, grain boundary becomes a part of a 3D structure consisting of five 1D chains with arbitrary number of grains, which is embedded in a homogeneous and isotropic elastic medium. Imposing 1D Reuss and Voigt approximations on different length scales, the constitutive equations can be solved in general, i.e., for arbitrary uniform loading, grain-boundary type and choice of elastic polycrystalline material, and local grain-boundary-normal stresses expressed algebraically as a function of grain-boundary type, its inclination with respect to loading direction and material-elasticity parameters.

A comparison with finite element simulations shows that while the model is not accurate enough on the local scale to predict crack-initiation sites, the corresponding statistical distribution of normal stresses on grain boundaries of a chosen type agree extremely well with those obtained numerically. Such statistical knowledge of intergranular normal stresses is a necessary prerequisite in any local damage modelling approach, e.g., to predict the probability for intergranular stress-corrosion cracking or fatigue-crack initiation in structural materials.

Figure 1: A 2D sketch of perturbative model for grain-boundary stresses, consisting of two anisotropic grains of unit size, enclosing the grain boundary, and several isotropic buffer grains of variable length, composing one axial and two (four in 3D) transverse chains.

Figure 2: Statistical stress distributions PDF(σ_nn) evaluated on three different grain-boundary types in Fe for uniaxial tensile loading. An excellent agreement between simulation results (solid lines) and model predictions (dashed lines) is shown for all three cases.