Mazars’ Scalar Damage Model
Mazars (1981, 1984, 1986) developed a scries of damage models, which aim at damage in tension and compression. When specialized for tension, the only primary internal variable is the scalar damage oj, varying form и) — 0 (for no damage), to to = 1 (for complete failure). The hardening-softening variable is /t-ё where є has the meaning of an equivalent uniaxial strain. The equations for this model are
Q(i) is a scalar function characterizing the material and (є) ’ is the positive (or tensile) part of the strain tensor, defined as the tensor possessing the same principal directions as є and having principal values that coincide with those of є when positive and are set to zero when negative. This model is restricted to tensile damage since, by its very definition, no damage is introduced if the principal strains are negative. Note also that, due to its simplicity, brought about by the scalar nature of the internal variable, the flow rule takes an integrated form.
The function Q(e) is uniquely determined from the uniaxial stress-strain curve. Indeed, taking the axis x] to lie along the specimen axis and the axes хг and xq to be normal to it, it turns out that in uniaxial tension (e)+if — £n5]i5ij, so that for monotonic straining є == є+ — £ц = £ (where є denotes the
The main problem with this model is that the prediction for the transverse strain is unrealistic. Indeed, it is easily verified that, for uniaxial tension in the direction of xq, we have £22 = £33 ~ — іоєц at all times. This means that, for full fracture, when £ 11 —> 00, we get £22 = £33 —з—oo, which is unrealistic. Therefore, a directional scalar damage model such as that given by (8.5.33) in which C, v is the scalar damage variable and £^ the hardening-softening variable, may often be more suitable to describe the fracture behavior.