Brittle and Ductile Damage of Stochastically Homogeneous Solids

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Abstract
The damage variable is interpreted as the probability of failure of micro- elements in a quasi-homogeneous solid; cleavage and slip-type failures are taken into con sideration. Elastic and plastic phases of the solid are defined with regard to the type of damage. The ductile damage of the plastic phase depends on the micro-stress according to an extended Huber-Mises condition. The brittle damage of the elastic phase depends on the principal micro-stresses, and it is characterized by a second-rank tensor. Micro- stresses are amplified so that they increase more than the local stresses of the solid. The instability effect of the stress field originates its rupture. Hooke's law degenerates in the presence of micro-cracks, and the local stress-strain relations of the elastic phase are nonlinear after an averaging procedure of the random micro-stress and strain. Similarity of the deviators and incompressibility conditions are assumed in the plastic phase. Fur thermore, the internal equilibrium, continuity, and compatibility conditions of the elastic and plastic phases are taken into account, and constitutive equations for the two-phase ag gregate are derived. The basic concepts come from some of my earlier works; however, they have been revised and developed in a more consistent way.

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