material constants, or constitutive response, due to plastic deformation under pure compressive loading. This knowledge is critical in order to develop a predicting model capable to account for plastic strain reversal as for low cycle fatigue. Few attempts based on the cyclic accumulation of damage, or its associated variable, always resulted in predicted very short lives as a consequence of the fact that p usually accumulates quickly. Porosity models, such as the Gurson model, are incapable to predict material failure since porosity effects are fully recovered during compressive loading resulting in a unrealistic healing-material behavior, [7]. These premises clearly indicate that additional hypotheses must be formulated in order to describe properly material behavior under compressive stress states. If ductile damage can be imputed to the formation and growth of microcavities that have the effect to reduce the net resisting area, and consequently material stiffness, thus the following scenarios can be speculated. Scenario a). The material is initially stress-free and it is assumed that no strain history has modified its status from the one of “virgin material”. Let us assume to start to load a material reference volume element, RVE, under pure compressive uniaxial state of stress avoiding any buckling phenomena. In this configuration, microvoids cannot nucleate since the ductile matrix is compressed around the brittle inclusions. If the local stress in the particle overcomes the particle strength, the particle itself can eventually break. This kind of damage should not affect material stiffness since no reduction of the net resisting area is occurred. The only effect that we would expect is probably an anticipated microvoids nucleation, due to an early void opening since the particle is broken, when the stress state is reversed in tension, (i.e. a lower strain threshold value). Even though an irreversible process such as particle breaking will eventually occur under compressive loads, the stiffness should remain unaffected indicating no damage in compression. Scenario b). Let say that the virgin RVE is initially loaded in tension until some amount of damage. Then, the load is reversed in compression developing additional plastic strain. In this case, during the unloading from positive stress-state to zero, microvoids can close controlled by the large ductile matrix volume movement, (here, potential buckling of microcavities is neglected). Voids implode back to the particle from which they have nucleated. Void closure can eventually close to the zero displacement condition. During this phase the net resisting area is restored and the stiffness should be the same as for the virgin material. Continuing in the compressive ramp the stiffness, once again, should remain unaffected. Further compressive loads, will eventually breaks some particles, but no effects are expected on E. A new reload in tension would see both the opening of the previously grown voids plus the opening of the new ones nucleated at the broken particles. However, at this stage it can be assumed that compressive damage does not modify damage developed under positive stress states. It follows that ductile damage can accumulate under positive stress state only, while total plastic strain will accumulate under both. Consequently, the associate damaged variable has to be a redefined as an “active accumulated plastic strain” p+, i.e. the plastic strain that accumulates if and only if, the actual stress triaxiality is greater than zero. Similarly, the damage effect on material stiffness will also be activated if and only if the current stress triaxiality is positive. According to this, the damage model proposed by Bonora can be modified in terms of active damage D+ and active plastic strain p+ as follows: Y f D D ¶ ¶ l& & =- + = ( ) ( ) + + - + × × - ÷ ÷ ø ö ç ç è æ × - × p p D D D D cr eq H f th cr f & a a a s s e e a 1 1 0 / ) ln( (10) ( ) m eq H f D p s s l / ) (1 + + - = & & (11) ( ) m eq E E D H f s s/ ) (1 ~ + = - (12) where
RkJQdWJsaXNoZXIy MjM0NDE=