Incremental Approach to Cracking of Fixed Orientation

The foregoing formulation can obviously be rewritten in incremental form, which is obtained by differ­entiating the equations. However, it is possible to directly formulate an incremental formulation that, in general, is not equivalent to a secant formulation because it does not satisfy the integrability conditions. This happens with the incremental approach proposed by Rots (1988) which is briefly outlined now.

The incremental form is obtained by establishing relationships between the rales of the variables. Eqs. (8.5.5) and (8.5.6) are replaced by

є ■= – -^trd – 1 T (e <g>n) (8.5.25)

a = &n = (8.5.26)

where the superimposed dot indicates the time rate, and S4 is a second-order tensor defining the tangent stiffnesses for the cracks. The structure of this tensor depends on the details of the model. The tensor need not be symmetric. It may have up to nine independent components, which are reduced to five if one assumes that the tensor depends only on the crack orientation n and the instantaneous cracking strain vector гЛ At any rate, much more information is required thaii is currently available from the experimental knowledge, and strong simplifications are introduced. Rots (4988) assumed that the normal and tangent components of stress and strain were mutually proportional, with no mixed stiffness terms. With this assumption, the equations are similar to those for the secant formulation. In particular, (8.5.12) is replaced by

aN – S^(£^,)e^ and от = S-r{e! N)£T (8.5.27)

where SlN and Sj, are incremental (tangent) stiffnesses. Rots (1988) further assumed that both SfN and S! p depended only on the normal cracking strain c(v,, but introduced independent functions to describe them. SlN is derived from the uniaxial tensile test and is related to the secant stiffness and the softening function ф(є^) by

of „ d(SNerN) _ дф(є{,)

N desN ■ dsfN

(a)

 

(b)

 

(c)

 

(d)

 

Figure 8.5.2 Multiple cracking: (a) primary cracks; (b) shear stress built up due to principal stress rotation; (c) secondary cracking formed; (d) tertiary cracking.

 

Incremental Approach to Cracking of Fixed Orientation

For the incremental shear stiffness Rots introduced a decreasing function that was infinite for /xro crack opening and decreased progressively to vanish for the normal cracking strain at which the normal stress drops to zero. He sjtowed that this is equivalent to using an incremental shear retention factor varying from 1 just after crack creation, down to 0 for a fully broken material.

At first glance, it may appear that this formulation is equivalent to the secant formulation. It is not; indeed, if we differentiate (8.5.12) to get the rate equations for the secant model (with the assumption that the stiffnesses depend only on the normal cracking strain), we find that the equation lor the normal component is equivalent, but the equation for the shear component is

Incremental Approach to Cracking of Fixed Orientation(8.5.29)

This is certainly not equivalent to (8.5.27) except for proportional straining (see exercises), and gives a lower tangential stiffness than (8.5.27) because Sr is decreasing. No comparative analysis of the two approaches has been performed to date. The difference is analogous to that between the incremental and the total-strain theories of plasticity. The latter is known to give better (softer) stiffness predictions for the first deviation from a proportional loading (predicting the so-called vertex effect), and the same probably applies here.