Issue34

K. Nowak, Frattura ed Integrità Strutturale, 34 (2015) 34 (2015) 507-514; DOI: 10.3221/IGF-ESIS.34.56 508 If the structure works at elevated temperature, that is in creep conditions, the analysis using SIF is limited to a few cases. It can be applied if the stress redistribution from elastic state to steady creep state is small. For steady state creep condition the C*-integral (analogous to J-integral) can be used (cf. e.g. [14, 15, 17]). Thanks to this parameter different configurations of cracks can be compared and parameters of single equivalent crack can be determined. But in this method the steady state stress field is considered as constant and the stress redistribution due to damage development is not taken into account. Another method is that by means of Continuum Damage Mechanics. It allows for tracing stress and damage history, and to determine the most probable crack development. This methodology was successfully applied to creep growth of single crack by [7, 13] and is used by authors in this paper to determine the paths of interacting multiple cracks in creep conditions. C REEP VERSUS ELASTIC SAFETY OF STRUCTURES or perfect elastic structures its loading capacity can be evaluated basing on two limit cases: ultimate strength for uncracked members or critical load to propagate existing defects. The first one is standard procedure for newly projected structure, the second one - for structures which contain one or multiple cracks. In creep conditions, however, which has to be taken into account when structures is expected to work in a severe environmental conditions (e.g. high temperature or chemically aggressive media), the concept of critical loading must be replaced by the notion of critical time to failure. The latter is an effect of material structure deterioration which occurs at any level of mechanical load. This is similar to the effect of sloping of both part of Wöhler diagrams and fatigue limit absence in high temperature applications. In the case of appearance of multiple linear defects (cracks) a new approach has to be developed because of interaction between individual cracks. In elasticity it requires solving a troublesome individual BVP (Boundary Value Problem) for each initial crack configuration. This procedure can be neglected when cracks interaction is weak, and critical load can be calculated basing on the SIF value for most dangerous crack. The assumption of nonexistence of cracks interaction can not hold in creep conditions. The material deterioration will grow for any load, even for initially flawless structures, causing the slowly movement of cracks which will always grow - albeit not necessarily dangerous - and may coalescence with catastrophic result. In the present paper the methodology of dealing with such a situation will be developed and illustrated by an simple example of a plane rectangular specimen subjected to uniaxial tension with two symmetric edge cracks and/or a central one. C ONSTITUTIVE EQUATIONS USED AND NUMERICAL METHODOLOGY nalytical modelling of material deterioration in creep conditions became possible due to L.M. Kachanov works initiated in late 50-ties of last century [8]. When damage is coupled with stress-strain rate equation (cf. [2]) it yields a set of differential nonlinear equations: 1 c ij ijkl kl ij D       , (1a) 1 n c ij eff eff ij t                  , (1b)   max 1 1 1 eff A t                      , (1c) where ij  , c ij  are total and creep strain tensors, ij  is stress tensor, D ijkl is elastic constants matrix,  , n , A ,  ,  are steady-state creep and damage material constants,  max ,  eff - main positive principal stress and Huber-Mises effective stress,  - scalar damage parameter (0    1), t - time. This approach has been used in several investigations [1, 2, 3] and presented also at the series of conferences “Crack Paths”. The picture below demonstrates effectiveness of this approach used in analysis of crack paths in 3D bending plates [3]: F A