Issue 41

T. Morishita et alii, Frattura ed Integrità Strutturale, 41 (2017) 45-53; DOI: 10.3221/IGF-ESIS.41.07 47 These models correlate multiaxial data for certain materials under some loading conditions, but they cannot be considered generally applicable to life evaluation under non-proportional loading. Energy models based on the plastic work per cycle are proposed as a parameter for life to crack nucleation. Energy is a scalar quantity and does not address the result of the observation where cracks nucleate and propagate along specific planes. Critical plane models attempt to correlate fatigue damage with physical observations of the nucleation and growth of cracks. The concept of critical planes was first proposed by Brown and Miller. The approach defined a failure plane as a critical plane in terms of stage I (shear-type) or stage II (tensile-type) cracks. Firstly, the history of strain on the critical plane is analyzed, and then strain parameters are used to quantify the damage parameters on the critical plane. Various terms have been proposed for different materials and experimental results. The Brown and Miller critical plane approach is expressed as  max + S  n = C , where  max is the maximum shear strain amplitude,  n is the amplitude of the normal strain acting on the  max plane, and S is a constant. It is reasoned that  max governs crack growth and its direction, and that  n further assists crack propagation. Socie considered that the fatigue life criterion should be based on a physical mechanism. For a material with tensile-type failures, Socie modified the Smith–Watson–Topper (SWT) parameter by considering that crack growth was perpendicular to the maximum tensile stress. The parameters which control damage were the maximum principal strain amplitude   max /2 and the maximum principal stress  1 max on the maximum principal strain plane [16-21],     b cb N E N 2 f 2 f f f f max 1 max 1 2 2 2        (4) where  f  is the fatigue ductility coefficient and  f  is the fatigue strength coefficient, while b is the fatigue strength exponent and c is the fatigue ductility exponent. However, some disadvantages on these approaches can be listed as: i) Stress based critical plane approach advocated by Findley or McDiarmid is not available for short fatigue lives regime. ii) Strain based critical plane approach advocated by Brown-Miller has no reflection of material behavior, such as non- proportional cyclic hardening. iii) Strain-stress based critical plane approach advocated by Fetemi-Socie or Smith-Watson- Topper is complex on calculating the stresses from the multiaxial strain. Therefore, in order to avoid these disadvantages, Itoh et al. proposed one strain parameter related to material property and loading path, which are available in both of short-life and long-life fatigue regimes, takes response to material deformation and is expressed by simple calculation based on strain model. M ULTIAXIAL FATIGUE LIFE EVALUATION MODEL : IS- METHOD Definition of stress and strain n non-proportional multiaxial fatigue, principal directions of stresses and strains vary during a cycle. In such a case, strain and stress ranges and mean strain and stress cannot be easily determined. It is necessary to represent the principal stress/strain and the direction of principal axis as a function of the time. The principal stress/strain vectors at time t are set as S I (t), the index I equals to 1, 2 and 3, that is, maximum, median and minimum principal stress or strain vectors, respectively. In addition, S is the symbol denoting either stress (  ) or strain (  ). Fig. 1 illustrates three principal values, S I (t), applied to a cube at time t together with xyz-coordinates (spatial coordinates). S I ( t ) is the maximum amplitude of the principal stress or strain at the time t in the IS-method [2, 4], which is defined by the following equation.             t t t t tS 3 2 1 I I , , Max S S S S   (5) The maximum value of S I ( t ) during a cycle is taken as the maximum principal value S Imax , which is defined as,       tS t S I 0 I ax Im Max   S (6) That is, t 0 is the time when | S 1 ( t )| or | S 3 ( t )| takes the maximum value in one cycle. Fig. 2 illustrates two rotation angles,  (t)/2 and  (t), to express the direction change of principal value in XYZ- coordinates, where XYZ-coordinates are the material coordinates taking X-axis in the direction of SI max and the other two I