Effective Stress and Vector-valued Orientational Distribution Functions

The original Kachanov—Rabotnov damage variable is inherently microplane-based and should be expressed by a scalar-valued orientation distribution function (ODF), and the corresponding effective stress is a vector-valued ODF. The analysis of vector-valued ODFs is established in this article in the spirit of the analysis of scalar-valued ODFs by Kanatani, K. (1984a). Distribution of Directional Data and Fabric Tensors, International Journal of Engineering Science, 22: 149—164 and Kanatani, K. (1984b). Stereological Determination of Structural Anisotropy, International Journal of Engineering Science, 22: 531—546. Explicit expansions of vector-valued ODFs up to the sixth-order have been developed, and the relationship of fabric tensors of different order is addressed. The fabric tensors and expansion of the Kachanov—Rabotnov effective stress vector can be fully determined by the scalar-valued ODF characterizing the microplane damage. The second-order effective stress and damage tensors of Murakami, S. (1988). Mechanical Modeling of Material Damage, ASME Journal of Engineering Materials and Technology, 55: 280—286 are related to the second-order expansion of the Kachanov—Rabotnov effective stress vector. Therefore, the analysis of vector-valued ODFs furnishes a rigorous and unified mathematical basis for the damage theory of Kachanov, L.M. (1958). Time of the Rupture Process under Creep Conditions, Izv. Akad. Nauk., USSR. Otd. Tekhn. Nauk. 8: 26—31, Rabotnov, I.N. (1963). On the Equations of State for Creep, Progress in Applied Mechanics, the Prager Anniversary, 8: 307—315, and Murakami, S. (1988). Mechanical Modeling of Material Damage, ASME Journal of Engineering Materials and Technology, 55: 280—286.