# Stress-Strain Relations for Elements of Arbitrary Size

To be precise, we limit the analysis to elastic-softening materials and consider a fracturing model with one definite cracking orientation (fixed or rotating). Consider first a case in which the crack band evolves. with the cracks oriented parallel to one of the directions of the finite element mesh, as depicted in Fig. 8.6.1a-c. It is intuitively clear that the stress-strain relations for the direction normal to the band must be very close to the uniaxial formulation deduced in Section 8.3.5. However, in that uniaxial formulation, the transversal strains were ignored, while in the actual three – or two-dimensional elements, a mismatch of strains parallel to the cracks can occur between the crack band that softens and the remainder of the element that unloads. In view of the other simplifications involved, this might not cause a serious error, but it is not difficult to enforce the proper interface continuity requirements at this interface. In fact, the formulation in Section 8.4.2 ensures compatibility automatically.

Consider the simple case in Fig. 8.6.1 d. After the stress peak, the material inside the crack band softens, while the rest of the element unloads (for the elastic-softening case considered here, unloading means elastic behavior). We want to enforce that the strain components in the plane of the cracks be the same for the unloading and softening regions, which, with reference to the base vectors in Fig. 8.5.1c, is written as

where superscripts и and s refer to the unloading region and to the softening band. Now, according to the hypothesis of elastic-softening behavior, the strain tensor in the unloading region is related to the stress tensor by the elastic relations, while the strain tensor in the softening band is given by (8.5.5), and so we have:

 | (a) f (b) | (c)

Therefore, writing the components appearing in (8.6.1), and taking into account (8.5.4), we can reduce the strain conditions to identical conditions in stresses, i. e.,

a’il = atl. <s ^ <7Sss and aU ats (8-6-4)

If we now lake into account that the traction vectors on the interface of the softening and unloading parts must be equal, i. e.,

crun — <rsn (8.6.5)

it turns out that the remaining three components of the stress tensors in the two regions must also be mutually equal. Therefore, the compatibility and continuity equations are satisfied if the stress tensors in the softening band and in the unloading region are identical, so that we can write

<Tu — <ts = a (8.6.6)

Since the stress tensors are the same, the two regions are fully coupled in series. The average strain in the element can be obtained by stipulating that the virtual work of the mean fields is equal to the sum of the virtual works in the unloading and softening portions, that is

о ■ 5є(е)У{е) – ou ■ 5euVu + O’s ■ 6esVs (8.6.7)

where V^VU, and Vs are, respectively, the volumes of the element, the unloading region and the softening region. By virtue of (8.6.6), this condition is identically satisfied for all the stress states and virtual strain increments if

є(е) =(1 -/)eu + /es (8.6.8)

where / is the volume fraction of the crack band. Substituting (8.6.6) into (8.6.2) and (8.6.3) and the results in (8.6.8) we get the final expression

— —~~cr — -^tr cr 1 r f (e’f & n)S (8.6.9)

lit lh

which shows that the equation for the element has a structure identical to the original stress-strain model, except that the fracturing strain is affected by the factor /. This factor is trivially equal to hc/h for the simple cases shown in Fig. 8.6.1 in which the dements are rectangular and the crack band is perpendicular to one pair of sides.

This case occurs frequently in the analysis of test results for mode 1 crack growth, and its use was pioneered by Bazanl and Oh (1983a), who analyzed with success a tremendous amount of experimental data. Bazant and Oh used a crack band model with a finite strain slope in a finite element analysis with square meshes. In computations, small increments of the load-point displacement were prescribed, and the reaction, representing the load P, was calculated in each loading step, ‘flic same stress-strain relation was assumed to hold for all the finite dements, although only some of them entered nonlinear behavior.

A plane stress state was assumed for all the calculations. Although the width of the crack band (size of the square elements) was found to have very little effect, its optimum was approximately wc = 3da (da = maximum aggregate size), and this value was used throughout the computations.

The crack band theory was able to reproduce with accuracy the experimental results of Naus (1971), Walsh (1972), Kaplan (1961), Mindess, Lawrence and Kesler (1977), Huang (1981), Carpinteri (1980), Shah and McGarry (1971), Gjprv, Sorensen and Arnesen(1971), Hillerborg, Modder and Petersson( 1976), Sok, Baron and Frangois (1979), Wecharatana and Shah (1980), Brown (1972), and Entov and Yagust (1975), As an example, we plot in Fig. 8.6.2a-f the results for the six Walsh’s series (1972) described in Section 1.5 (secTables 1.5.1 and 1.5.1, Series A1-A6, and Figs. 1.5.1 and 1.5.2).