Tensorial characterisation of directional data in micromechanics

Information being dealt with in micro-mechanics is massive. Most of them are directional data. Macro-scale physical laws embedded with micro-scale fundamentals need to be developed in terms of the statistics of the micro-scale variable in a frame-indifferent form. Mathematical techniques and theories for characterizing the statistics of directional data with tensors are hence demanded. This is the main concern of the current paper. Starting with the general theory established in Kanatani (1984) of describing the directional distributions of orientations, mathematical formulations have been extended to address the directional distributions of vector-valued directional data, which is the most common data type being dealt with in micromechanical investigations. For vector-valued directional data, statistical analyses are required in regarding to both their directional probability density distribution and their representative values along each direction. The technique used here is to approximate these directional distributions by polynomials in unit directional vector n. The coefficients are in tensorial form and determined from observed directional data by applying the least square error criterion. These coefficient tensors serve as macro-scale variables representing the statistics of the micro-scale directional data, and are referred to as direction tensor. Orthogonal decompositions are addressed so that the coefficient tensor of different orders can be determined independently from each other. The coefficient tensors in the orthogonal decompositions are referred to as deviatoric direction tensor. The choice of sufficient approximation order is suggested. As an example, a general form of the stress–force–fabric relationship is derived for demonstrating the application of the proposed mathematical theory in the micro-mechanical investigation of the behaviour of granular materials.