Geometric damage tensor based on microplane model

An appealing approach to formulate constitutive models for characterizing distributed damage due to microcracks and voids is continuum damage mechanics with the concepts of effective stress and strain equivalence. In that approach, in which damage is imagined to characterize the reduction of the net stress-transmitting cross-section area of the material, the constitutive model is separated into two independent parts, one for damage and the other for elastic and inelastic behavior (rheology) other than damage, which, if combined appropriately, give the overall constitutive behavior. However, the existing multidimensional formulations for damage are quite complex, and practical implementations capable of fitting experimental data are hard to obtain. The microplane models, by contrast, provide conceptual simplicity and close fits of multiaxial test data for concrete, soils, etc., although, as formulated in the past, various kinds of physica phenomena were mixed in the definition of the microplane stress-strain curves. In this work the microplane theory is reformulated in a manner that separates damage from rheology and makes the formulation fit the basic framework of continuum damage mechanics. Aside from a kinematic constraint between macrostrains and microstrains, the model satisfies a static constraint such that the effective microstresses are the resolved components of the effectiveness macrostresses.