# Brittleness Number

The concept of brittleness of structural failure, which is the opposite of ductility, is an old one, but for along time, the definition of brittleness has been fuzzy and has not stabilized. One of the fundamental reasons is that the apparent brittleness depends simultaneously on the material, the geometry of the structure and loading, and the size of the structure. Therefore, it is not easy to find a single figure properly incorporating all these influences.

The first idea in quantifying the brittleness is to look for a quantity that is vanishingly small for the perfect plasticity limit and infinitely large for the elastic-brittle limit. A number with these properties is the ratio D/l appearing in the general size effect law (10.1.1). Therefore, any variable proportional to it is a good candidate to be a brittleness number:

(10.1.2)

Over the years, there have been various proposals for brittleness numbers of these forms. Well known in the metals community is the brittleness characterization based on Irwin’s estimate of the nonlinear zone (see Section 5.2.2) which is at the basis of the AST. M E 399 condition for validity of the fracture toughness test. This brittleness number, say 0k, can be written as

Df’2

= (І0Л-3)

With this definition, the condition for valid fracture toughness measurements reduces to 0k > 2.5.

In the field of concrete, probably the first ratio used as a brittleness number was put forward by I lillerborg

and co-workers. It was defined as

D „ EGf

011 = 7-, (ch = —7Г (10.1.4)

«ch /(

in which G p refers to the fracture energy of the underlying cohesive crack model. Note that the foregoing two equations are essentially identical, because of Irwin’s relationship Kic — у/EGp.

The foregoing brittleness numbers arc useful to compare various materials and sizes for a given structural shape and loading, but they cannot be used directly to compare the brittleness of different structural geometries, because the dependence of brittleness on geometry is not included in their definition.

Pertaining to this category, but with a slightly different definition, is Carpinteri’s brittleness number sc (Carpinteri 1982):

*С = Щ (шл’5)

We notice that it is the inverse of a brittleness (the more brittle, the smaller so), and should better be called a ductility number. Note also that it is related to the Hillerborg brittleness by

This means that Carpinteri’s brittleness number can be used to compare brittleness of various materials only as long as they have the same f’JE ratio. It has the same limitations as ft/ in not giving comparable results for different geometries.

To get a brittleness number that embodies the influences of material, geometry, and shape, we may recall the concept of intrinsic size defined in Section 5.3.3 and use the brittleness number defined as

0n <x ^ (10.1.7)

in which D is given by (5.3.11) and (5.3.12). The first brittleness number of this category was introduced by Bazant (1987a; also Bazant and Pfeiffer 1987), although the concept of intrinsic size was still to come. Bazant’s brittleness number was defined as

where the second expression is the original definition, which is equivalent to the first because of (6.2.2).

Brittleness numbers similar in concept, but based on the cohesive crack model have also been extensively used. Planas and Elices (1989a, 1991a) introduced the obvious extension of Hillerborg’s brittleness number as

/3p = ~ (10.1.9)

which gives a unified representation of the size effect and brittleness properties in the medium and large range of sizes (ft> > 0.04) for most laboratory geometries (Llorca, Planas and Elices 1989; Guinea, Planas and Elices 1994a).

The foregoing definition of (ip is, however, sensitive to the shape of the softening curve. Although (as discussed in Chapters 7 and 9) there is not much variability of shapes for ordinary concrete, it turned out to be better to use a brittleness number that refers the intrinsic size to the properties of the initial portion of the softening curve, characterized by the tensile strength and the horizontal intercept Wi (Fig. 7.1.8); its definition is

This brittleness number adequately captures the fracture properties for small and medium sizes, including most practical situations (0.1 < ft < 1). Moreover, from the Planas-Elices correlation (7.2.14), we get
and it turns out that the two brittlenesses can be interchangeably used since the factor 5.3 is independent of the shape and size of the structure as well as of the material, as long as the material softening curve can be approximated by a straight line in its first part. This is usually the case for concrete. For granite, there is also evidence of this fact (Rocco et al. 1995). In the remainder of this chapter we will mainly use Bazant’s brittleness number /3.