Issue 24

P.V. Makarov et alii, Frattura ed Integrità Strutturale, 24 (2013) 127-137; DOI: 10.3221/IGF-ESIS.24.14 128 in the present paper the main accent is made on studying the transition from the quasistationary phase of media evolution to the blow-up regime which is assumed as local failure at crack formation which applies to be the new vision of the failure process. In the present paper these problem also dares on the basis of ideas of the mathematical theory of evolution of loaded solids and media [2]. According to traditional ideas of fracture mechanics a local fracture in solids occurs when a maximum load is reached. All experience of application of this approach to the problem of limiting design has shown its comprehensible working capacity and correctness for many practical problems. However we can tell nothing about the failure process especially about its forecast. If a certain constant or changing load is enclosed to solid it is only possible to calculate the conforming stress-strain state and to answer a question whether a maximum load is reached or not somewhere. Such answers appear useful and sufficient in a number of important engineer cases but to tell something about the mechanisms and scenarios of failure locus formation is impossible. The fundamental law of fracture of any materials has been investigated in the 70s of the XX century: the final failure (not only fatigue but any) precedes more or less significant preparatory stage. For example for the silicate glasses which failure was considered as instant the speed of crack propagation in the beginning of failure has appeared in thousands times less than at the final stage [5] and this is with the fact that the whole failure process takes some ms. Rapid development of ideas and methods of nonlinear dynamics at the same years and the next decades have allowed the group of S.P. Kurdyumov to formulate the new concept of superfast catastrophic stages of evolution of nonlinear systems – the blow-up regimes [3] and both analytically and numerically to study the kinds and features of these regimes. These ideas and the qualitative results obtained on their basis are the key-ideas for understanding the failure process. In the considered case of brittle or quasibrittle failure (and also the failure of any materials and constructions, plastic metals, brittle concrete, rocks, geomedia etc.) the preparatory process of accumulation of inelastic deformations and damages in brittle media is localized in certain areas. This preparatory quasistationary stage because of the self-organized criticality of solid as nonlinear dynamic system passes sooner or later in the superfast catastrophic stage – blow-up regime by S.P. Kurdyumov [2, 4, 7]. It is clear that any failure forecast basically is not possible without studying the features of development of these stages and conditions of transferring of one stage of sustainable development of failure to unstable superfast regime. M ATHEMATICAL STATEMENT OF A PROBLEM . M ODEL OF QUASI - BRITTLE MEDIUM ccording to the evolutionary concept of the description of inelastic deforming and the subsequent failure processes [2, 4, 6-9] the full set of equations includes: 1. Fundamental conservation laws:  Mass      0 d div dt (1)  Impulse         ij i i j d F dt x (2)  Energy     1 ij ij d dE dt dt (3) where  is the material density,  i is the i -component of the speed vector, i F is the i -component of the massive force, is energy, t is time. 2. The evolutionary equations of the first group which have been written down in the relaxation form in which increments of stresses       ij t are proportional to increments of total deformations   T ij and relax proportionally to the development of inelastic deformation   P ij . The procedure of stresses reduction to the instant limiting surface means the instant stress relaxation on each time layer to some dynamic equilibrium state defined both by relaxation and the rate of strength and elastic parameters of medium degradation. At         0 P T ij ij ij , there is a relaxation, at         0 P T ij ij ij and stresses steepen:                      2 T P T P ij ij ij ij (4) A