Minkowski functionals of the whole structure (also known as intrinsic volumes or quermassintegrals) play an important role in the geometrical characterization of spatial structures. Nevertheless, facing the complex spatial structure, the global functionals appear to be not discriminating enough. Searching for a finer geometrical characterization of the structure, we study the particular local extensions of the Minkowski functionals - the local Minkowski measures. In the present work, these local measures are considered in the Stochastic Geometry framework, i.e. applied to the random closed sets. Their behaviour for the simulated germ-grain models and relationship with the quantitative descriptor used in material science (particle elongation) are numerically studied.