Issue34

K. Nowak, Frattura ed Integrità Strutturale, 34 (2015) 507-514; DOI: 10.3221/IGF-ESIS.34.56 509 (a) (b) Figure 1 : Crack patterns on upper (a) and lower (b) surfaces of a 3D plate without initial damage. The above methodology allows for evaluation all three characteristic stages of damage growth terminated by: t 1 - time when first failure occurs in a point, t 2 - time when failure occurs across of structures characteristic length, t 3 - time when the cracks system makes structure unstable. It is worthwhile to notice the ‘Kachanov paradox’, which follows his approach, that for uniform stress distribution (uniaxial tension) the damage in all points grows simultaneously causing the specimen to ”evaporate” at time t 1 = t 2 = t 3 ! This is not the case when stress field in a structure is non-homogeneous one - like in plates, for example - and which is particularly strong in the case of initial discontinuities (cracks) existing in a structure. Because of its nonlinearity and complexity the problems considered below will be solved by means of numerical analysis. Time integration of Eq. (1) is done by Euler explicit method. Plane strain four- and three-node solid elements are used. A diversification of a FE mesh has been made in damage affected zones. The numerical solution of the problem is very strong mesh depended (cf. e.g. [14]), so the results can be considered as qualitative ones. The initial cracks are defined as the set of Gauss integration points in which initial value of damage parameter ω =1 (failure in a point). S PECIMEN GEOMETRY , LOADING AND MATERIAL CHARACTERISTICS rectangular plate of dimensions shown in Fig. 2a is subjected to uniformly distributed loading q on its end edges. The plane strain condition is assumed and the problem is solved as 2D one, the times t 2 and t 3 are equal. The following configurations of initial crack will be considered: two symmetric edge cracks (Fig. 2b), two symmetric central cracks (Fig. 2c), and combination of two previous configurations (Fig. 2d), where the distance denoted as H will be used as a problem free parameter. Only quarter part (right upper) of the plate is modelled due to symmetry. Unlikely to elastic considerations, for creep conditions an interaction between multiple cracks will always exists as damage field extends over the whole structure. The critical distance will be sought for which the interaction of edge and central cracks can be neglected (it is when they will span plate width independently). It is necessary to emphasise that this critical distance has to be understood as that corresponding to the situation when cracks which span specimen edges at time t 2 will be different form cases 2c and 2d by a small, arbitrarily defined amount. The material parameters used for simulation was that of type 316 stainless steel at 650°C according to [11]: E =0.144·10 6 MPa, ν =0.314,  =2.13·10 -13 (MPa) -n h -1 , n =3.5, A =3.42·10 -9 (MPa) -μ h -1 ,  =2.8,  = 1.0. The loading was assumed to be 50 MPa and is kept constant for all experiments. For the structure shown in Fig. 2b the length of initial edge crack is assumed to be l e =15 mm. SIF for this crack and assumed loading is 12.3 MPam 0.5 . This value is about one order less than critical value of SIF, so the structure is safe in elastic conditions. For the structure shown in Fig. 2c the length of initial central cracks are assumed to yield the same critical elastic load as in the case of central crack and equals to 18 mm (SIF is equal to 12.3 MPam 0.5 , too). For the initially non-cracked structure (Fig. 2a) the time to failure can be calculated independent of deformation since the stress σ x (the only nonzero- component of stress matrix) is - from equilibrium - equal to loading q . Simple integration of the Eq. (1c) in uniaxial stress state with initial condition ω =0 and failure at ω =1 yields: 0 1 5115 h ( 1) t Aq      . (2) A