Damage model for brittle elastic solids with unequal tensile and compressive strengths

The paper presents a rate-type constitutive analysis of damage, applicable to brittle materials whose elastic properties degrade during a deformation process. Different tensile and compressive material responses are modeled incorporating positive and negative projections of the stress or strain tensors. Proposed evolution laws for the rate of compliance tensors are consistent with some of the prominent features of brittle material response. A new structure of the damage surface is introduced for a more accurate account of the effects of the hydrostatic states of stress on the overall response. Derived rate constitutive equations provide the explicit representation of the tangent compliance tensor. The proposed model is applied to uniaxial tension and compression to illustrate nonlinear relationships between stress and longitudinal, lateral, and volumetric strains. The proposed model is compared with some of the existing theories. DEGRADATION of elastic properties reflecting accumulating damage in brittle materials is primarily a consequence of the evolution of internal microcrack structure. Depending on the material microstructure and the current state of stress and strain and their rates, instantaneous material response and further evolution of damage may involve activation of different microcrack mechanisms. The tensile hoop stress generated at the surface of relatively large pores represents a preferential crack nucleation mechanism in very porous rocks. In low porosity, compact rocks frictional sliding of crack surfaces can destabilize original shear cracks, causing them to kink and develop wing cracks. Other mechanisms of microcracking are also possible. Some of these mechanisms were implemented in the failure analysis of brittle solids by Ashby and Sammis (I), and others. Progressive degradation of mechanical properties is an inherent feature of brittle material behavior. Several analytical models were developed to estimate the effective elastic properties of a solid weakened by a given distribution of microcracks or other defects. A comprehensive review of these models can be found in a recent treatise by Nemat-Nasser and Hori (2). An important issue in the formulation of continuum damage models is related to the appropriate choice of the mathematical form for the damage variable. This has been recently studied by Lubarda and Krajcinovic (3), who considered several frequently encountered two and three dimensional distri- butions of microcracks. Their analysis demonstrated the shortcomings of the scalar and second- order tensor damage variables, and the accuracy gained by using the fourth-order tensor variable. The mode and stability of crack growth and, therefore, behavior of brittle material strongly depends on the sign and magnitude of applied stresses. For instance? the response of brittle material subjected to a compressive loading is strongly dependent on the magnitude of lateral confinement. An unconfined specimen fails by axial splitting, attributed to unstable growth of a single crack, at relatively small microcrack density. As the confinement is increased, axial splitting is suppressed and at large levels of confinement homogeneous microcracking prevails throughout the sample, resulting in a quasi-ductile overall response (Horii and Nemat-Nasser (4)). The analysis presented in this paper addresses the important issue of modeling different tensile and compressive responses of brittle materials. The concept of positive and negative projections of stress and strain tensors is used to account approximately for two basic, tensile and compressive, damage evolution modes. This idea was independently introduced by Ladeveze and Lemaitre (5), Ortiz (6) and Mazars (7), and was subsequently utilized by Simo and Ju (8), Yazdani and Schreyer (9), Ju (IO), Stevens and Liu (l 11, Yazdani (12), and others. The positive parts of the stress