Issue34

B. Schramm et alii, Frattura ed Integrità Strutturale, 34 (2015) 280-289; DOI: 10.3221/IGF-ESIS.34.30 284 the initial area of the crack and the crack tip. Accordingly, the crack sees two fracture mechanical different materials, so that the material functions (threshold value curve  K I,th and cyclic fracture toughness curve  K IC ) vary in dependency of the polar-coordinate  and the gradation angle  M = 30° and show a jump for a sharp material transition (Fig. 5b). Below the threshold value curve  K I,th (  ) the crack is not able to propagate, whereas above the cyclic fracture toughness curve  K IC (  ) unstable crack growth occurs. The region of stable fatigue crack growth is situated between both curves. a) b) Figure 5: a) Fracture mechanical graded structure with the materials M1 and M2 and the gradation angle  M = 30°, b) threshold value curve and cyclic fracture toughness curve in polar coordinate system The TSSR-concept is a modification of the MTS-concept of Erdogan and Sih for homogeneous and isotropic materials [3] and compares stress values with material values as well. Due to the fact that the fracture mechanical material properties change in dependency of the existing gradation, a material function is considered instead of a constant material value. For the determination of the beginning and the direction of fatigue crack growth the threshold value curve  K I,th (  ) is used as material function and the cyclic tangential stress   (Eq. (1) with the Mixed Mode ratio V=K II /(K I +K II )) as stress function. 3 I V 3 Δσ 2π ΔK cos sin cos 2 1 V 2 2 r               (1) To determine the beginning of stable crack growth and the direction of propagation  TSSR the TSSR-concept looks for the cyclic stress function which has the first intersection point with the threshold value curve  K I,th (  ). For this the cyclic stress function    2  r is equalized with the material function  K I,th (  ) (Eq. (2)). 3 I I,th V 3 Δσ 2π ΔK cos sin cos ΔK ( ) 2 1 V 2 2 r                 (2) Transposition of this equation according to Eq. (3) and applying the potential kinking angles  0,MTS ,  M and  M ±180° lead to the cyclic stress intensity factors   th I 0,MTS Δ K    ,   th I M Δ K    and   th I M Δ 180 K      .   I,th th I 3 ΔK ( ) ΔK V 3 cos sin cos 2 1 V 2 2          (3)