PSI - Issue 39

Ilia Nikitin et al. / Procedia Structural Integrity 39 (2022) 599–607 Author name / Structural Integrity Procedia 00 (2019) 000–000

603

5

torsion loading. Here we present the way how to model such complex behavior. We introduce the example when normal and shear crack opening are in competition. 2.3.1 Smith–Watson–Topper criterion The criterion of multiaxial fatigue failure in the LCF-HCF mode with the development of normal-stress micro-cracks (stress-based SWT) corresponding to the left branch of the bimodal fatigue curve has the form:

max 1 1 / 2        u





L N

L

where 1  is the largest principal stress, 1 / 2   is its amplitude. According to the chosen criterion only tensile stresses lead to failure, so it has the value 1   is the spread of the largest principal stress per cycle,

max 1 1 ( ) H     . Let us put down the following designation: max max 1

max 1 n eq        Here the upper index n stands for denotation and should not be considered as a power. 2.3.2 Carpinteri–Spagnoli–Vantadori criterion The criterion of multiaxial fatigue failure in the LCF-HCF mode, including the concept of a critical plane (stress based CSV), corresponding to the left branch of the bimodal fatigue curve has the form: where / 2 n   is the amplitude of the tangential stress on the plane (critical), where it reaches its maximum value, / 2 n   is the amplitude of the normal (tensile) stress on the critical plane, max ( ) n n n H       . Here, the shear fatigue limit u  for a pulsating cycle is additionally introduced at a cycle asymmetry coefficient of 1 R   . In a simplified formulation, we can approximately accept / c u u k    and 3 c k  . This criterion includes the mechanism of fatigue fracture with the formation of shear micro-cracks. Let us put down the following designation:   Here the upper index  stands for denotation and should not be considered as a power. The degradation of a material is described by a decrease in its elastic Lamé moduli  and  to zero with an increase of damage  to the value 1   . The following nonlinear law of decreasing Lamé moduli is adopted: for *    0 ( ) (1 )       , 0 ( ) (1 )      , for * 1     0   , 0   , where the parameter 0 1 / 2    characterizes a relatively weak change in the moduli until the critical level * 1    of damage is reached, after which a state of complete fracture occurs with the loss of the ability of the elastic resistance of the material to any loads. All figures should be numbered with Arabic numerals (1,2,3,….). Every figure should have a caption. All photographs, schemas, graphs and diagrams are to be referred to as figures. Line drawings should be good quality scans or true electronic output. Low-quality scans are not acceptable. Figures must be embedded into the text and 1 / 2 2 2     2 ( / 2) ( / 2)  L n  c n u L k N      2 / 2) 3( / 2)  2 ( eq    n  n   