Structures Failing at Very Small Cracks Whose Size is a Material Property
The foregoing size-effect analysis applies always to structures in which the crack length at maximum load is proportional to the size of the structure. This kind of size effect, however, differs from that found in normal kinds of metallic and other structures which become unstable and fail (or must be assumed to fail) before a small flaw, represented by a microcrack, can become a macrocrack of significant length compared to the structure size.
If the crack is small compared to the distances over which the stresses vary appreciably (let’s call them microcracks, for short), it is easy to show (see the superposition method to compute Kj in the next chapter) that the stress intensity factor always takes the form
К і = koas/a (2.3.19)
where <r is the stress normal to the microcrack plane at the inicrocrack location computed as if no microcrack existed in the structure; a is the half-length or radius (for a pennyshape) of the microcrack and ко a constant depending on the exact shape and location of the crack (but not on microcrack or structure size). Consider now two structures that are similar (which means macro-geomctrically similar) and contain identical microcracks in homologous positions. Since we have proved in Section 2.3.1 that the stresses at homologous positions are identical, it follows that the microcrack at a specific site starts to grow in both structures at the same nominal stress level. If one further assumes that global failure follows immediately after one of the cracks starts to grow, it turns out that the two structures will fail at the same nominal stress level. Hence, in this type of similitude no size effect is present because, in fact, it is equivalent to analysis based on allowable-stress or strength criterion (in which the allowable stress or strength varies from microcrack site to microcrack site).
However, getting similar structures with identical distribution of microcracks is practically impossible, so actual structures are macroscopicaliy similar but microscopically random. Then the strength of the structure can be defined only on statistical (probabilistic) grounds, and a size effect appears because the probability of getting larger flaws in the more highly stressed regions of the structure increases with the structure size. The analysis of this kind of size effect will be deferred until Chapter 12, where we prove that the statistical size effect vanishes asymptotically when macrocracks or notches exist at the start of failure in the body (see also the short discussion in Section 1.3).
With respect to quasibrittle materials, and particularly concrete, it is important to note that they contain plenty of microcracks, but failure does not happen as soon as one of these microcracks starts to grow. It only occurs after many microcracks have grown and coalesced to form a macroscopic fracture process zone. This is a feature that makes the classical statistical theory of strength inapplicable to these materials.
2.16 Show that the stress intensity factor for a penny shaped crack of radius a coaxial to a cylindrical bar of diameter D subjected to uniaxial stress a must take the form (2.3.11). What is the form if Ki is written in terms of the axial load P = ar. D1 /4?
2.17 Find the general forms for the energy release rate of a penny shaped crack of radius a coaxial to a cylindrical bar of diameter D subjected to uniaxial stress a.
2.IS In most handbooks on stress intensity factors, Ki is written in the form Ki — Ycr^/тта where a is the crack length, a a characteristic stress, and Y a dimensionless factor depending only on geometrical ratios, in
particular on the relative notch depth a/D where D is a characteristic linear dimension of the body. Rewrite this in the form (2.3.11) and find the relationship between Y and k(a).
2.J9 Write the stress intensity factor of the DCB specimen of Fig. 2.1.3b in terms of the relative displacement и of the load-points. Show that the general form of Kr for imposed displacement must be Ki — (Eu//0)k(a) where E is the elastic modulus, u the displacement, D a characteristic dimension of the body, a — a/D the relative crack depth, and k(a) a nondimensional function.
2.20 To analyze the behavior of a large structure with cracks, which is assumed to behave in an essentially linear elastic manner, a reduced scale model is built at a 1/10 scale. The model is tested so that the stresses at homologous points are identical in both model and reality, and we require the full scale and reduced models to break at the same stress level. Determine the scale factors for (a) loads, (b) toughness, and (c) fracture energy.