   For concrete as well as for other quasibrittle materials, the basic inelastic deformation mechanism in tension is cracking. If the material is subjected to tensile stress producing a crack, then is unloaded and the stress reversed into compression, the crack closes and the stiffness in compression is recovered to a large extent. As already pointed out, a nonlinear unloading behavior such as that sketched in Fig. 8.4.1 d is observed (except at strong lateral confinement). However, the simpler models based on damage mechanics may be more convenient for computational purposes, and-then some mechanism must be devised to ensure that the compliance reduction due to damage, as shown in Fig. 8.4.5a, would not appear on the compression side. Then, the split form (8.4.1) together with (8.4.2) is most efficient in handling the problem. It suffices to write that, for a < 0, the crack opening must be zero; this may be compactly written as

where (tr)+ is the positive part of <7, defined as a for positive values and zero for negative values, or, in algebraic terms: ■ _л + cr + H

■ff> = ~1T

The behavior becomes elastic, characterized by the initial elastic modulus, as soon as the stress becomes negative (Fig. 8.4.5b).   For the case of pure strength degradation, no special precaution needs to be taken since, by definition, the crack opening is fully irrecoverable. For the case of mixed unloading behavior (Fig. 8.4.4d), the positive part must include the entire expression (8.4.7), and so the total strain may be written as: Figure 8.4.5 Reversing the stress sign: (a) invalid result with stiffness degradation also in compression; (b) model with crack closure; (c) stress-strain curve showing softening in compression as well as in tension.

In this way, the material recovers the undamaged behavior in compression as soon as the unloading branch reaches the initial elastic line, as shown in Fig. 8.4.5c.

Certainly, inelastic strain and cracking occur in compression, too. In practical analysis of concrete structures, especially in the analysis of beams and plates based on a uniaxial or biaxial stress-strain diagram, it is normally assumed that the stress-strain diagram of concrete in uniaxial compression also exhibits a peak followed by strain softening (Fig. 8.4.5d). As a consequence of this hypothesis, all localization phenomena described for tension occur for compression as well.

This means that one needs to also use fracture mechanics for compression behavior. Similar to tension, one needs to introduce either a softening band in compression (Bazant 1976) or one might postulate a compressive fictitious crack, as suggested by Hillerborg (1989). If, however, triaxial stress-strain relations are considered, such assumptions do not reflect realistically the actual mechanism of compression failure. Compression strain softening is not due to large strain in the direction of compression, unlike tensile strain softening, but is due to volume expansion of the material which causes large strains in the directions transverse to the direction of compression. So, compression softening is a strictly triaxial phenomenon, while tensile strain softening can, to a large extent, be treated as a uniaxial phenomenon. If volume expansion (transfer of strains) is prevented, e. g., by strong enough confining reinforcement or encasement of concrete in a strong enough pipe, then there is no compression softening and the stress-strain relation has no peak in compression.

A realistic triaxial stress-strain relation for compression strain softening must reflect these features. But many existing triaxial constitutive models do not, and the biaxial ones, in fact, cannot because they do not involve volume expansion as a variable. In any case, whether compression softening is modeled directly as a function of the compression strain or as a triaxial phenomenon associated with volume expansion, the fracture mechanics aspects associated with localization of compression softening need to be taken into account. Much research remains to be done in this direction.