A Simple Continuum Damage Model

Let us now briefly review a very simple damage model. We base it on Dougili’s approach in which a material element is considered to be formed, ideally, by many infinitesimal rods connected in parallel (Fig. 8.4.3a). We assume that the rods are identical in all but strength, which is randomly distributed. Upon stretching, the weaker rods fail first. At a given strain level є, a fraction ui of the rods have failed. Then, the resulting stress is given by

a — E( 1 — иі)є (8.4.9)

Note that this is the average stress. The stress in the surviving rods is that for the virgin, undamaged material. Therefore, the relationship between the macrostress и anti the microstress r (also called the true-stress)is

r = ; (8.4.10)

1 — tо

This is the basic relation in continuum damage mechanics, initiated by Kachanov (see, e. g., Lemaitre and Chaboche 1985). This relation applies not only to brittle materials in which the relationship between the true stress and the strain is linear, as we have here, but to any other (true) stress-strain relationship.

The model must specify the evolution of damage. This is done on the basis of the uniaxial stress-strain curve as shown in Fig. 8.4.3b. Assuming that the data are in the form of the cr(i^) curve, we get the damage at each point by writing


A Simple Continuum Damage Model

= cr_ _ ф(є?)

~ Еє ~ ф{є!) + Eif

from which (8.4.11) follows. So, the two formulations are fully equivalent, even though the underlying micromodels seem to be completely different.


Подпись: Figure 8.4.4 Models with prepeak inelasticity: (a) stress-strain curve for monotonic loading; (b) stiffness degradation; (c) strength degradation; (d) mixed behavior.

Introducing Inelasticity Prior to the Peak

Although, for concrete in tension, the inelastic strain prior to the peak is relatively small, for some reinforced materials the prepeak nonlinearity can be important and must be taken into account. This can be done exactly as before, with the only assumption that the cracking strain (or damage) starts before the peak is reached. This is illustrated in Fig. 8.4.4a, which shows the full а (є) curve, and Figs. 8.4.4b-d, which show the three possibilities of unloading behavior.

Therefore, to get a model incorporating the prepeak nonlinearity, it suffices to use the adequate expres­sion for the function ф(ёf). Some candidates for such a model were given in Section 8.3.2.

We recall here that, when used in finite clement formulations in which the element width h is greater than the characteristic crack band width hc, the softening part of the curve must be sealed as indicated in Section 8.3.5.