  This paper presents a two-step Bayesian framework for the estimation of the
intrinsic single crystal elastic stiffness parameters from the measurements of
spherical indentation stress-strain responses in multiple individual grains of
a polycrystalline sample, whose crystal lattice orientations have been measured
using electron back-scattered diffraction technique. The first step requires
the establishment of the functional dependence of the indentation elastic
modulus given the lattice orientation and the intrinsic single crystal elastic
stiffness parameters. Previous efforts for this step required a large database
of computationally expensive finite element (FE) simulations in order to
establish this function with adequate accuracy. In this paper, it is shown that
the introduction of a Bayesian framework can greatly reduce the number of
simulations necessary to establish this function, while introducing practically
useful measures of uncertainty which can guide the selection of specific
additional simulations that are expected to best improve the predictive
accuracy of the function. The second step involves a Markov Chain Monte-Carlo
(MCMC) sampling of the distribution of possible values for the single crystal
elastic stiffness parameters based on a given set of experimentally measured
elastic indentation moduli in individual grains of different lattice
orientations. This second step is accomplished by calibrating the available
experimental data to the function established in the first step. This novel
framework is presented and demonstrated in this paper for an as-cast cubic
polycrystalline Fe-3% Si sample and a hexagonal polycrystalline commercially
pure (CP-Ti) titanium sample.
