STRUCTURES IN FRACTURE PROCESSES

Fracture of materials, engineering structures, and natural objects is frequently accompanied by the appearance of ordered systems of defects which earlier were called fracture structures [1]. The fracture structures can form hierarchical systems. There are familiar observations in laboratory and in nature of these structures in quasi-brittle fracture of polymers and rocks, in the processes of plastic strain localization [2, 3,4], during the formation of slip planes in granular media [5], during the development of the structures in the process of energy dissipation at the strain wave front [6], in the spallation fracture [7], and in a number of other cases. In bodies subjected primarily to compressive loading (e.g., rocks subjected to tectonic loads), hierarchies of the fracture structures contain a number of self-similar blocks of the material. These blocks are inserted into one another, which gives ground to the assumption that the fracture process has a selective nature and that the fracture is quantized in a certain proportion [8-9]. On the other hand, the development of the fracture structures is accounted for by a fracture mechanism (fixed for each specific case) which starts acting on the attainment of the limit equilibrium state of the material. To observe the fracture structures and hierarchical systems of such structures in experiments, it is necessary to measure the characteristics of mechanical processes occurring on several conjugate geometric scales, which substantially differ in sizes in accordance with the rank of the structures in the hierarchy. Moreover, the development of the fracture structures is a boundary process occurring with small variations of the parameters in the vicinity of the limit equilibria of the examined media. One can separate several approaches which are used in the fracture mechanics for the analysis of the development of the fracture structures. We will mention here a number of approaches relying on the continuous description ofdeformable media (bodies). In the approach of [3, 11], the appearance of the fracture structures is accounted for by a strain instability on the scale of the entire body. According to this approach, the development of the fracture structure in the volume subjected to a uniform loading is the total (simultaneous) response of the material to this loading. An alternative approach has been suggested by the authors of the present paper [1, 12, 13]. It involves (i) the analysis of the development of the fracture mechanism eventually leading to the formation of isolated elements of the structure and (ii) the formation of a structure from these elements. In particular cases, the formation of this structure can have several stages. The development of the structural elements is investigated on the scale of these elements (internal scale of the problem), provided that the boundary conditions of the external problem (considered on the scale of the entire volume subjected to the loading) are satisfied on the matching boundary of the solutions of the internal and external problems. This permits one to take into account specific fracture mechanisms and consider multilevel scenarios of successive development of the hierarchy of the structures. This is convenient, in particular, if the scales of the internal and external problems differ substantially. In what follows, this approach is used for the analysis of a number of problems arising in the construction of the hierarchy of the fracture structures.