Rotating Crack Model

When the smeared cracks of the primary direction start forming, which is the start of strain softening, there is actually only a system of discontinuous microcracks. If the maximum principal stress direction rotates, these microcracks partially close and microcracks of a new orientation begin to form (Fig. 8.5.4a). Eventually the secondary microcracks may become the major ones and the primary ones may get nearly closed. Although a precise description would be rather difficult (and would perhaps be best done in terms of the microplane model, Chapter 14), the fact that the previously formed microcracks may to a large extent close and microeracks of a new orientation may become dominant can be better described by assuming that the direction of smeared cracking rotates (Fig. 8.5.4b) and remains always normal to the maximum principal stress. In reality, of course, a crack, once formed, cannot actually rotate.

The notion of a rotating crack (also called swinging crack), originally proposed by Cope et al. (1980) and reformulated by Gupta and Akhbar (1984), and Crisfield and Wills (1987), is just a computational convenience. The reality is more complicated than the preceding discussion suggests. Even for a constant principal stress direction, the microcracks during the process of crack formation do not have the same orientation; due to heterogeneity of the microstructure, microcracks arise in all directions, and one can
only say that the microcracks that arc normal to the maximum principal stress direction are the statistically dominant ones.

The rotating crack model can be formulated in a way very similar to the fixed crack model, although the resulting equations are simpler. Consider first a single crack system. By the very definition of the model, the normal to the cracks n is now coincident with p„ the unit vector in the direction of the maximum principal strain, which coincides with the principal stress direction. Then, the crack displacements arc in pure opening and we can write, referring to Fig. 8.5.1b

Aw = Awpt =Ф – є* = £^p, (8.5.30)

Proceeding again as in Eqs. (8.5.3)-(8.5.5) with n replaced by p„ we get the total strain tensor as

Подпись: (8.5.31)1 + и и, f

є — <т – -=tr <7 1 + eJ p, ® p,

b b

in which the fracturing strain tensor now depends on only one scalar variable, єЛ

As for the equation governing microcracking, Eq. (8.5.6) now becomes a scalar equation since, by definition <7p, — <7,p„ where <7, is the principal stress with principal direction p,. Thus, we need only a relationship between eJ and <7/. This coincides with the uniaxial stress-fracturing strain relation (for the monotonic loading case), i. e.,

cr, – ф{е}) = SN(ef)ef (8.5.32)

in which we keep the nomenclature in the previous sections to keep the meaning clear. From the foregoing equation, wc can solve for e? (at constant secant stiffness) and get the secant formulation:

Подпись: 1 -Подпись: (8.5.33)V и _

-<7 – – Ftr f 1 + wv<7i p, ® p,


Подпись: є. Є.І Подпись: 1 -I- ECN -• v v — v 1 —v —v —u 1 Подпись: C|! Подпись: (8.5.34)

The component expression for this equation is particularly simple if the axes are taken along the principal stress and strain directions:

in which the similarity with (8.5.16) is blatant.

In the foregoing, Eq. (8.5.32) is valid for virgin loading. Unloading and reloading require further rules. The simplest is a damage model in which G’ is taken to be a function of єЛ the maximum fracturing strain ever reached. Then, if є? < єЛ the unloading-reloading proceeds at constant C. y and, for virgin loading, ef = and Cn increases.

The model can further be extended by considering three mutually orthogonal jointly rotating systems of cracks, normal to the three principal stress and strain directions. Each system is allowed to follow an independent cracking process, same as described before, characterized by fracturing strains el, with v — II, or III. The resulting equation, which incorporates the three fracturing strains, can he written as

Подпись: III + '£2CN{el)a„ Pi 8>р„ v—I Подпись: (8.5.35)1 I и v

Є — – -—<7—- – tr cr ]

b h

Подпись: E Подпись: 1 + ЕС к (if) -v —v -v 1 ECN(el) -v ~v -и 1 -I- ECN{eif,) . Подпись: 0 Chl СГщ Подпись: (8.5.36)
Rotating Crack Model

whose component form is

Note that although there are three different damage components, the behavior of the material is described with only one material function which can, in principle, be determined from the uniaxial test.