# Size Effect Equations

With the foregoing definitions, the size effect consists in the variation of the nominal strength Onu with size D. There are various possible plots showing special aspects of the size effect, but the most widely used is the bilogarithmic plot already shown in Fig. 1.2.5 in which logc/vu is plotted vs. logD. As previously discussed in Section 1.2.5, the strength theory (based on yield or strength criteria) predicts no size effect (horizontal dashed line in Fig. 1.2.5b); this is the kind of response assumed in most engineering approaches and codes (see Chapter 10 for a detailed discussion about the need of including the size effect in the codes.) On the other extreme, we have the purely brittle behavior of structures that fail by crack instability at a fixed crack-to-size ratio (relative crack length). In the next chapter, after presenting the essentials of linear elastic fracture mechanics (LEFM) we will see that the size effect in such a case is shown in the plot of Fig. 1.2.5b as an inclined line of slope -1/2. The actual size effect behavior is best described by a transitional curve having the two straight lines as asymptotes, as sketched in Fig. 1.2.5b.

The simplest size effect law satisfying this condition was derived by Baz. ant (1984a) under very mild assumptions which apply, approximately, to a large number of practical cases. Bazant’s size effect equation brings into play the energy required for crack growth as shown in the next paragraph, where a short derivation is presented. However, the final expression can be written (without explicitly showing the fracture energy term) as a function depending on only two parameters as

where f{ is the tensile strength of the material, introduced only for dimensional purposes, D is a dimen­sionless constant, and Do is a constant with the dimension of length. Both В and Do depend on the fracture properties of the material and on the geometry (shape) of the structure, but not on the structure size. Simple derivations of this size effect law are given next.

1.4.2 Simple Explanation of Fracture Mechanics Size Effect

Consider a uniformly stressed panel as shown in Fig. 1.4.2. Imagine first that fracture proceeds as the formation of a crack band (or fracture band) of thickness h/ across the central section of the panel. Now, the extension of the crack band by a unit length will require a certain amount of energy that, per unit thickness of the specimen, is called fracture energy and is denoted as Gf. The value of G/ may be considered, for the present purposes, approximately a material constant. To determine the load required to propagate the band, an energy balance condition must be imposed by writing that the energy available is equal to the energy required for band extension.

To do so, one writes that the strain energy released from the structure at constant on (which is the condition of maximum load) is used to further propagate the crack band. As an approximation, we may assume that the presence of a crack band of thickness hf reduces the strain energy density in the band and cross-hatched area from crjv/2E (for the intact panel) to zero (E = elastic modulus of material). The cross-hatched area is limited by two lines of some empirical slope k. When the crack band extends by A a at no boundary displacements, the additional strain energy that is released comes from the densely cross – hatched strip of horizontal dimension A a (Fig. 1.4.2a). If the failure modes are geometrically similar, as

Figure 1.4.2 Sketches for explaining size effect: (a) blunt crack band, (b) slit-like process zone (adapted from ЛСІ Committee 446 1992).

is usually the case, then the larger the panel, the longer is the crack band at failure. Consequently, the area of the densely cross-hatched strip for a larger panel is also larger. Therefore, in a larger structure, more energy is released from the strip by the same extension of the crack band. This is the source of size effect.

Quantitatively, the energy released per unit panel thickness is given by the area of the densely cross – hatched region h f A a + 2/cao A a times the thickness, times the energy density of the intact panel аЦ2Е. Therefore, the release of energy from the aforementioned strip (at constant boundary displacement) is b(hfAa + 2/cao Aa)a2N/2Е, where b is the panel thickness. This must be equal to the energy required to create the fracture, which is G/b Aa. Thercfore,

b(hrAa + 2ka0 Aa)^-A = Gj 6 Да 2b

Solving for the nominal stress, one obtains the size effect law (1.4.10) in which

(1.4.12)

Note that Do depends on the structure shape through the constant к but is independent of the structure size if the structures are geometrically similar (D/ao = constant); f[ = tensile strength, introduced for convenience; and hf = width of the fracture band front, which is treated here approximately as a constant, independent of structure size.

Lest one might get the impression that this explanation of size effect works only for a crack band but not for a sharp line crack, consider the similar pane Is of differentsiz. es with line cracks as shown in Fig. 1.4.2b. In concrete, there is always a sizable fracture process zone ahead of the tip of a continuous crack, of some finite length which may, in the crudest approximation, be considered constant. Over the length of this zone, the transverse normal stress gradually drops from fl to 0. Because of the presence of this zone, the elastically equivalent crack length that causes the release of strain energy from the adjacent material is longer than the continuous crack length, ao, by a distance Cf which can be assumed to be approximately a material constant.

When the crack extends by length A a, the fracture process zone travels with the crack tip, and the area from which additional strain energy is released consists of the strips of horizontal dimension Да that are densely cross-hatched in Fig. 1.4.2b. Following the same procedure as before for the crack band, we see that the area of the zone from which energy is released is 2k(ao + Cf)Aa. So the total energy release is b2k(ao F Cf)Aaa,/2E, which must be equated to the energy required for crack extension, bG fAa, thus delivering the equation

(1.4.13)

Solving for cTjV, one again obtains the size effect law in (1.4.10) in which now

The foregoing equations are only approximate in their details, because of the simplifying assumptions in determining the structural energy release. I lowcvcr, their structure is correct. The same form is obtained using simplified theories for other geometries (e. g., bending). The fine-tuned equations require the use of more sophisticated fracture mechanics concepts, and their presentation will be deferred until Chapter 6.

As will be shown in Chapter 9, Fq. (1.4.10) can also be derived, in a completely general way, by dimensional analysis and similitude arguments (Bezant 1984a). This general derivation rests on two basic hypotheses: (1) the propagation of a fracture or crack band requires an approximately constant energy supply (the fracture energy, G/) per unit area of fracture plane, and (2) the energy released by the structure due to the propagation of the fracture or crack band is a function of both the fracture length and the size of the fracture process z. one at the fracture front.

Applications of F. q. (1.4.10) to brittle failures of concrete structures rest on two additional hypotheses: (3) the failure modes of geometrically similar structures of different sizes are also geometrically similar (e. g., a diagonal shear crack has at failure about the same slope and the same relative length), and (4) the structure does not fail at crack initiation (which is really a requirement of good design).

These hypotheses arc never perfectly fulfilled, soil must be kept in mind that F, q. (1.4.10) is approximate, valid only within a size range of about 1:2() for most structures (for a broader size range, a more complicated formula would be required). This size range is sufficient for most practical purposes, but for some structures the range of interest extends beyond the applicability range. This is so because a sufficiently large change of structure size may alter the failure mode and thus render Eq. (1.4.10) inapplicable beyond that size; this happens, for example, for the braz. ilian split-cylinder tests. The analysis of such ‘anomalous’ size effect will be deferred until Chapter 9.