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This paper addresses some issues of the application of the effective stress tensor to coupled damage-plasticity from the point of view of a microplane. In this paper, damage and effective stress are defined on microplanes. The effective stress tensor of Murakami (1988) can be recovered as the second-order fabric tensor of the effective stress vector. The effective second invariantJ J 2 is defined as the integration of the square of the effective shear stress over the unit sphere. It is revealed that theJ J 2 possesses the form of the Hill (1950) anisotropic yield function based on the definition. Therefore, one can also use the Hill (1950) model to describe coupled anisotropic-damage-plasticity except for the conventional approach of effective stress.

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). : 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).

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