Analysis Based on Stress Redistribution and Complementary Energy
The truss model also allows an easy alternative calculation of the energy release on the basis of complementary energy U*. For the sake of simplicity, we now consider the residual strength v,- — 0, although a generalization to finite vr would be feasible.
In the truss model, we isolate the representative cell limited by the shaded zone in Fig. 10.3.4. This cell must alone be capable to resist the applied shear force V. The compression failure of concrete in the band 12341 (Fig. 10.3.4) is considered to completely relieve the stress from the inclined strip 12561. If the applied shear force V is kept constant, the stress in the cell must redistribute such that all of the compression force in the inclined strut is carried by the remaining strips, shaded in Fig. 10.3.4. After that, all of the complementary energy in concrete in the cell is contained in the shaded strips. According to Bazant (1996b), the energy density is given by the shaded area in Fig. 10.3.6, and so the complementary energy density may be expressed as W* = (o2/2Ec} V in which V — b(D cos 0 – csin 0)D/ sin 9 — volume of the shaded strips (Fig. 10.3.4), ac = Fc/b(D cos 0 — csind) — average normal stress in the direction of the strut, and Fc — V/ sin 9 — vubD/ sin 0 = compression force transmitted by the stmt. This yields for the complementary energy, after the stress redistribution at constant shear force V, the expression:
According to (2.1.21), the energy release rate is obtained by differentiation of the complementary energy at constant load (or constant shear force V):
This must be equal to the energy dissipation rate, which is given by the following equations, same as
7Z—-—G,, h = (10.3.24)
There is now one difference from the previous approach. In (10.3.18), the energy release rate was constant, while in (10.3.23) it increases with c. This difference should not surprise since both solutions are approximate. In the case of variable Q, which is a typical case in fracture mechanics, the crack length at maximum load, that is, at a loss of stability, need not be considered as empirical, as done in our previous calculation based on the strain energy change, but can be calculated from the stability criterion. It is well known that, at the limit of stability, the curve of energy release rate at constant load must be tangent to the Я-curve (see Section 5.6.3):
dG __ dTZ dc dc
(This stability criterion could not be applied to the previous case with (10.3.18), because in that case, due to the approximations made, we had dQ/Oc = 0 and thus c was indeterminate.) Because Q = TZ, an equivalent condition is
1 3Q 1 dTZ
Q dc ‘TZ dc
which is more convenient. We may now substitute here the expressions in (10.3.23) and (10.3.24), and carry out the differentiations. This leads to a quadratic equation for c/D, whose only real solution is
This represents a theoretical expression for the length of the crack band at maximum load (i. e., at stability loss).
It may now be observed that с/ D lends to zero as the size D ~> oo. In that limiting case, the stress relief region would become an infinitely narrow strip, which would not be a realistic model. Therefore, (10.3.27) is meaningful only for sufficiently small sizes. For this reason, and for the sake of simplicity, we consider the second term under the square root in (10.3.27) to be small compared to 1. Because s/Y+ 2x ~ 1 – p x when x <C 1, (10.3.27) for small D yields the approximation:
c _ cold D ~ ~3~
Substituting this into the fracture propagation criterion Q — TZ, along with (10.3.23) and (10.3.24), we obtain an equation whose solution furnishes the simple result:
in which we have introduced the notations:
Figure 10.3.7 (a) Stress redistribution zones for initial diagonal shear cracks, (b) Localization of the openings of diagonal cracks into one major diagonal crack in a beam with stirrups, (c) Tensile stress-displacement diagram for the opening of a cohesive crack, (d) Mohr circle of strains, (e) Localization of openings of diagonal cracks into one major crack in a beam without stirrups, (f) Mohr circle of stresses.
v„ — Kc sin2flVcot0 (10.3.31)
The result we have obtained has the same form as (10.3.19), although the expressions for the size effect constants Dq and vp are partly different. The differences reveal the degrees of uncertainty caused by the simplifications of analysis we made. The comparison of (10.3.19) and (10.3.29) indicates that the general form of the size effect we obtained ought to be realistic although the coefficients Do and vp cannot be fully predicted by the present theory but must be corroborated on the basis of experiments.