Secant Approach to Cracking of Fixed Orientation

For application to problems with monotonic crack opening close to mode I, Bazant and Oh (1983a) proposed a crack band model in which the stress-strain relations have a secant form, with varying secant compliances. Here we give an enhanced version that explicitly considers the crack sliding (crack shearing) and that naturally leads to a damage formulation of the cracking problem.

Consider first proportional paths in which the microcrack opening and shearing increase monotonically. For these very particular paths we can assume the traction vector to be a function of the fracturing strain vector, i. e.,
where F(-) is a vector-valued function of two vector arguments. Note that a is made to depend on the orientation of the crack; this is essential to disclose the structure of F(-). The material containing the crack band is assumed isotropic. Therefore, we require that, if wc rotate simultaneously the crack and the crack strain vector, we must obtain a traction vector rotated by the same angle. This means that the function F(i’f ,n) must be isotropic, i. e., that

F(Qef, Qn) = QF{e*,n) (8.5.8)

for any orthogonal second-order tensor Q. A classical representation theorem (Spencer 1971) then requires the traction vector to have the form

3 = SN (є{,, Єт)є{,п + ST(efN, e! r)e! r (8.5.9)

where 5лг(є^,Єу) and S-r(eJN, e{-) are scalar functions that have the meaning of normal and tangent secant stiffnesses; is the normal component of є?, є/у is the vectorial component of in the plane of the cracks, and £y is the magnitude of that component. Algebraically,

efN=ef-n, e^ = ef~efNfi, Є}т = |4i -= Je{- ■ є’!ґ (8.5.10)

If we similarly define the normal and shear components of the traction vector, i. e.,

oyv = 5 ■ n, Зт = 3 — orjn, от " 3-r f 3-r ■ 3t (8.5.11)

the foregoing equations reduce to

vn = SN(efN, efr)efN and ffT = Ят(є{,,є(ґ)є? г (8.5.12)

which has a beautiful uncoupled form, with от parallel to efy. Certainly wc could have assumed this from the onset. But we now have proved that this is the most general possibility consistent with the initial assumption (8.5.7) and the condition of isotropy. This means that we need to specify two functions of two variables to determine the material behavior for proportional monotonic loading.

No doubt, many simplifications will be required to characterize limited experimental evidence. How­ever, before attempting such simplifications, let us find the general structure of the stress-strain relations. First we solve for the components from (8.5.12) and substitute them into (8.5.3) to get the expression of the fracturing strain tensor:

є? = Cjvcw n 0 n + Ct(3t ® n)s (8.5.13)

Подпись: CN = Подпись: (8.5.14)
Secant Approach to Cracking of Fixed Orientation Secant Approach to Cracking of Fixed Orientation

Cn and C-j – are the normal and shear compliances, defined as

where a dependence on and efr is implied (although hidden from now on). This expression is now substituted into (8.5.5) to get the total strain tensor as

є — cr •- ~tr <T 1 + CNoN n0 n + Cr{o-v 0 n)s (8.5.15)

Г/ Jj

Подпись: £nn £ss ett Подпись: 1 + ECN — V — V Подпись: -V —v 1 -v -V 1 Подпись: &nn aSit пи

For computational purposes this relation is best expressed in a component form relative to the base {n, S, i) in Fig. 8.5.1c, arranging the stresses in the six-dimensional column matrix (стпгг, ctss, c>tt, crns, crsj, сг(п)7 and the strains in the corresponding column matrix (єпп, є,,8,єи,7п!,,’їні,’Пп)Т where 7,Jt/ = 2єд„. It turns out that the equations for the normal and shear components arc mutually uncoupled and can be written as

and

 

1

0

0

 

0 0

GCr 0

0 1 GCT

 

& ns (7 st (7tn

 

Ins

1st

 

1

G

 

(8.5.17)

 

1 111

 

where G — Е/2(1 -1 u) is the shear modulus.

The foregoing secant equations are particularly simple: they depend only on the two functions Cm and C-r that appear in only three diagonal elements. Now we must specify how these functions evolve. In the early times of the smeared crack applications (see Rots 1988, for the basic references), before fracture mechanics concepts became widely accepted, the structural finite element codes used to set both the normal and tangential stiffnesses to zero just after cracking starts. This is equivalent to setting Cm — Ct — oo in the foregoing equations. This turned out to lead to numerical problems because of the sudden energy release implied by such approximation, and a certain amount of shear stiffness was retained, such that jsn = cr.,n/(/3aG), where A was called the shear retention factor (Suidan and Schnobrich 1973; Yuzugullu and Schnobrich 1973), whose value was of a few tenths, typically 0.2. This is equivalent to setting the shear compliance Ct equal to a constant of value

^ і – A

Г’ (8-5-18)

The introduction of the shear retention factor, however, is not satisfactory for four reasons: (1) it has no physical interpretation, (2) it is difficult to measure experimentally (if possible at all), (3) it leads to a behavior in which the material always has a stiffness in shear even if the crack is widely open, which is completely unrealistic, and (4) it seems a variable made to play with in numerical simulations. At any rate, for cases in which the cracking occurs close to mode I, the importance of the shear retention factor is not great. However, the results by Rots (1988) indicate that very low values of A —even zero— give better results in most cases.

For the normal compliance, Bazant and Oh (1983a) introduced a progressively degrading compliance as dictated by the uniaxial tension data. This is the equivalent to postulating that shearing the crack does not contribute to degradation in the normal direction. This is certainly a simplification, but can be realistic if the magnitude of shear is limited. This hypothesis allows a complete determination of the normal compliance Сдг from the uniaxial stress-strain curve. Then, for monotonic straining normal to the crack, we have

CN = (8.5.19)

«4%)

where </>(£jv) >s ^le function a — ф(є?) which is deduced from uniaxial tests and has been repeatedly analyzed in the two previous sections (only the name of the argument changes, since for uniaxial loading ef – efN).

f f

The foregoing hypothesis implies that the cracking process is controlled by eJN. For very small eJN very little damage exists and the shear compliance should be small; for very large с д the damage is large and the shear compliance should be correspondingly large. Thus it seems logical, as done by Rots (1988) in a slightly different formulation to be defined later, to take Cp as an increasing function of Cn. The simplest of all is to assume Ct as proportional to Cm, i. e.,

Подпись: (8.5.20)Ct = crCW Ад-)

where c-r is a constant that should be determined by experiment.

This is a very simplified model, devised for monotonic crack opening, that can be easily brought to a more general formulation involving unloading to the origin, i. e., to a damage model as described next.