Critical plane fatigue life models of materials and structures under multiaxial stationary random loading: The state-of-the-art in Opole Research Centre CESTI and directions of future activities Ewald Macha, Adam Niesłony ⇑ Opole University of Technology, Faculty of Mechanical Engineering, Department of Mechanics and Machine Design, Poland article info Article history: Received 24 November 2010 Received in revised form 23 February 2011 Accepted 1 March 2011 Available online 5 March 2011 Keywords: Multiaxial fatigue Random loading Critical plane criteria abstract In the paper some fatigue life assessment models for materials and structures under triaxial random stress state in time and frequency domains developed over 25 years in Opole University of Technology are presented. The first stress models were formulated by generalization of the known models applied under cyclic loadings. The main operation of the calculation algorithm is reduction of the triaxial stress state to the equivalent uniaxial one. Such a reduction is performed with a criterion of multiaxial fatigue failure, suitable for a given material. Then, the determined equivalent stress history is processed, like in the case of uniaxial random fatigue. Later, the strain and energy models were formulated and developed. Also, the spectral method was presented. At the end of the paper some desired directions in future researches are proposed. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction The first proposals of the models for fatigue life calculations un- der triaxial random stress state were formulated in the seventies of the 20th century. They were based on generalization of known models applied for the cyclic loadings [1,2]. These models included (i) assumption of the shear and normal stresses on the critical planes of fatigue fracture as the quantities strongly influencing the failure, (ii) changes of the principal stresses and strains, and (iii) their conformity (in special cases) with the known and verified models applied under cyclic loadings. Thus, they were the stress models for fatigue life calculations within a great number of load- ing cycles of the Wöhler curve (r a –N f ). The models were improved many times but the basic scheme of the algorithm for fatigue life calculations still remains valid [3–13]. Its main operation is reduc- tion of the triaxial stress state to the equivalent uniaxial state with a suitable criterion of multiaxial fatigue failure. Then, the deter- mined history of the equivalent stress is processed in order to cal- culate the fatigue life like in the case of the uniaxial random fatigue. There are also strain [14–16] and energy [17–22] models where – like in the stress models – the equivalent histories of strains and the energy parameter are calculated according to the assumed criterion of multiaxial fatigue failure. The above models for calculations of fatigue life of materials and structures belong to the group of algorithmic methods where loading histories are analyzed in the time domain with the numerical algorithms of cy- cle counting. Another group of models, co called spectral method, applied spectral analysis of stochastic processes while fatigue life assessment [23–26]. In 1988, the first paper [27] about the spectral method in multiaxial fatigue was published. In this method, the tri- axial stress state is described with the matrix of power spectral density functions, and the fatigue life is calculated in the frequency domain [28–36]. Models of the spectral method are still being developed, and their review and possibilities of application have been presented in [36]. Multiaxial fatigue is a continuously current issue. Many scientific centres and researchers are dealing with this issue, making new tests and developing models for description of the fatigue phenomena, however still a universal solution has not been found [37–44]. This paper presents a review of dominating and most often ver- ified models of fatigue life calculations for materials and structures subjected to multiaxial random loadings in Opole Research Centre CESTI. The authors discuss some directions of the further research work as well. 2. Random stress and strain states In the random stress state each component r ij (t), (i, j = x, y, z) is a random function of time. Such stress state in the generic point of the material is described by a random stress tensor which can be presented as the six-dimensional stochastic process 0142-1123/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2011.03.001 ⇑ Corresponding author. Address: ul. Mikołajczyka 5, 45-271 Opole, Poland. Tel.: +48 77 40 06 163; fax: +48 77 40 06 343. E-mail address: a.nieslony@po.opole.pl (A. Niesłony). International Journal of Fatigue 39 (2012) 95–102 Contents lists available at ScienceDirect International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue