Classical Nonlinear Theories

LEFM, which is expounded in Chapters 2-4, provides the basic tool today for the analysis of many structural problems dealing with crack growth, such as safety in presence of flaws, fatigue crack growth, stress corrosion cracking, and so on. However, soon after the introduction of the fracture mechanics concepts, it became evident that LEFM yielded good predictions only when fracture was very brittle, which meant that most of the structure had to remain clastic up to the initiation of fracture. This was not the case for many practical situations, in particular, for tough steels which were able to develop large plastic zones near the crack tip before tearing off. The studies of Irwin, Kies and Smith (1958) identified the size of the yielding zone at the crack tip as the source of the misfit. Then, various nonlinear fracture mechanics theories were developed, more or less in parallel. Apart from clastoplaslic fracture mechanics (essentially based on extensions of the J-flitegral concept, and outside the scope of this book), two major descriptions were developed: equivalent clastic crack models and cohesive crack models.

In the equivalent crack models, which will be presented in detail in Chapters 5 and6, the nonlinear zone is approximately simulated by staling that its effect is to decrease the stiffness of the body, which is approximately the same as increasing the crack length while keeping cveiything else elastic. This longer crack is called the effective or equivalent crack. Its treatment is similar to LEFM except that some rules have to be added to express how the equivalent crack extends under increased forces. In this context, Irwin (1958) in general terms, and more clearly Krafft, Sullivan and Boyle (1961), proposed the so-called R-curve (resistance curve) concept, in which the crack growth resistance li is not constant but varies with the crack length in a manner empirically determined in advance. This simple concept still remains a valuable tool provided that the shape of the R-curve is correctly estimated, taking the structure geometry into account.

For concrete, the equivalent crack models proposed by Jenq and Shah (1985a, b) and Bazant and co­workers are among the most extended and have led to test recommendations for fracture properties of concrete (see Chapters 5 and 6). The 1980s have also witnessed a rise of interest in the size effect, as one principal consequence of fracture mechanics. A simple approximate formula for the effect of structure size on the nominal strength of structures has been developed (Bazant 1984a) and later exploited, not only for the predictions of failures of structures, but also as the basis of test recommendations for the determination of nonlinear fracture properties, including the fracture energy, the length of the fracture process zone, and the R-curve. This R-curve concept has also been applied to ceramics and rocks with some success, although until recently it has not been recognized that the R-curves are not a true material property but depend on geometry.

The cohesive crack models, which are discussed in detail in Chapter 7, were developed to simulate the nonlinear material behavior near the crack tip. In these models, the crack is assumed to extend and to open while still transferring stress from one face to the other. The first cohesive model was proposed by Barenblatt (1959,1962) with the aim to relate the macroscopic crack growth resistance to the atomic binding energy, while relieving the stress singularity (infinite stress was hard to accept for many scientists). Barenblatt simulated the interatomic forces by introducing distributed cohesive stresses on the newly formed crack surfaces, depending on the separation between the crack faces. The distribution of these cohesive atomic forces was to be calculated so that the stress singularity would disappear and the stresses-would remain bounded everywhere. Barenblatt postulated that the cohesive forces were operative on only a small region near the crack tip, and assumed that the shape of the crack profile in this zone was independent of the body size and shape. Balancing the external work supplied to the crack tip zone —which he showed to coincide with Griffith’s Q— against the work of the cohesive forces —which was 2js by definition— he was able to recover the Griffith’s results while eliminating the uncomfortable stress singularity.

Dugdale (1960) formulated a model of a line crack with a cohesive zone with constant cohesive stress (yield stress). Although formally close to Barenblatt’s, this model was intended to represent a completely different physical situation: macroscopic plasticity rather than microscopic atomic interactions. Both models share a convenient feature: the stress singularity is removed. Although very simplified, Dugdale’s approach to plasticity gave a good description of ductile fracture for not too large plastic zone sizes. However, it was not intended to describe fracture itself and, in Dugdale’s formulation, the plastic zone extended forever without any actual crack extension.

More elaborate cohesive crack models have been proposed with various names (Dugdale-Barenblatt models, fictitious crack models, bridged crack models, cracks with closing pressures, etc.). Such models

include specific stress-crack opening relations simulating complete fracture (with a vanishing transferred stress for large enough crack openings) to simulate various physically different fracture mechanisms: crazing in polymers (which must take viscoelastic strains into account, see Chapter 11), fiber and crack bridging in ceramics, and frictional aggregate interlock and crack overlapping in concrete. All these models share common features; in particular, a generic model can be formulated such that all of them become particular cases, and the mathematical and numerical tools are the same (Elices and Planas 1989).

However, the fictitious crack model proposed by Hillerborg for concrete (Hillerborg, Modeer and Petersson 1976) merits special comment. In general, all the foregoing fracture mechanics theories require a preexisting crack to analyze the failure of a structure or component. If there is no crack, neither LEFM nor EFM, equivalent crack models or classical cohesive crack models, can be applied. This is not so with Hillerborg’s fictitious crack model. It is a cohesive crack in the classical sense described above, but it is more than that because it includes crack initiation rules for any situation (even if there is no precrack). This means that it can be applied to initially uncracked concrete structures and describe all the fracture processes from no crack at all to complete structural breakage. It provides a continuous link between the classical strength-based analysis of structures and the energy-based classical fracture mechanics: cohesive cracks start to open as dictated by a strength criterion that naturally and smoothly evolves towards an energetic criterion for large cracks. We will discuss this model in detail in Chapter 7.