Nonlinear Fracture Mechanics Models
July 14, 2008 — July 18, 2008
Coordinator:
- Alberto Carpinteri (Politecnico di Torino, Torino, Italy)
A survey of the most relevant nonlinear crack models is provided, with a view to analyzing the nonlinear mechanical effects occurring at the tips of macrocracks in quasi-brittle materials –like concrete, rocks, ceramics, polymers, high–strength metallic alloys– and in brittle matrix fibre-reinforced composites. Such local effects, as, for example, plastic deformation, yielding, strain-hardening, strain-softening, matrix micro-cracking, grain debonding, fibre bridging and slippage, crazing, and so on, should be properly described through different simplified models, representing the peculiarities of the phenomena involved.
In Linear Elastic Fracture Mechanics (LEFM) the energy dissipation due to the crack growth occurs in an infinitesimal zone at the crack tip, where the stress field is assumed to be singular. On the other hand, in real materials, the energy dissipation preceding the crack advancement takes place in a diffused damaged zone ahead of the crack tip, which is of finite size and in which the stress field is not singular. From a practical point of view, we could assert that the process zone is microscopic for linear cracks and macroscopic for nonlinear cracks, where the prefixes micro- and macro- usually refer to the size-scale of the material microstructure. This could be, for instance, the atomic scale for crystals, the molecular scale for polymers, the grain scale for ceramics and metals, the aggregate scale for concrete and rocks.
Excluding particularly ductile metallic alloys, for which the bulk behaviour cannot be modelled as linear elastic, the structural materials may generally be studied through nonlinear crack models, where the nonlinearity is concentrated only in the crack tip region. The selection of the most consistent and suitable model depends on the morphological and phenomenological characteristics of the process zone. When the mechanical damage is confined in a narrow band along the strain-softening crack line, the use of the Cohesive Crack Model is preferred. On the other hand, when the bridging and restraining reactions of the reinforcements are distributed in a more discrete way, the use of the Bridged Crack Model is more convenient.
All the most relevant nonlinear crack models represent simple theoretical approaches, since there are only a few paramters involved in each model. In LEFM, for instance, there is only one critical parameter (i.e., the fracture toughness), whereas in the cohesive crack model there are three significant parameters (i.e., the specific fracture energy, the tensile strength, and the shape of the cohesive law). In any case, a theory with a rather low number of parameters is simple, universal and testable. This means, that only a simple model may be applied to many different cases and positively assessed.
Particular attention will be paid to the stability conditions for nonlinear cracks. As in the limit case of LEFM the stability condition is represented by the well-known Griffith-Irwin equation, from the nonlinear crack models we obtain peaks and valleys in the load versus deflection structural response. After the peaks but before the valleys we can find softening branches with negative slope or even very unstable branches with positive slope. At each loading peak, we can observe snap-through or snap-back instabilities, according to the controlling parameter (load or displacement). In this way, it is possible to describe the crack instability as well as the crack arrest. In a composite material with different components, inclusions, phases, voids, etc., the response may be particularly irregular. This kind of mechanical behaviour may be revealed by crackling noise emitted through elastic waves.
The course is addressed to Ph.D. students, post-doctoral fellows, young researchers, specialists in fracture mechanics working in the industry.