International Journal of Fatigue 21 (1999) 83–88 Expected principal stress directions under multiaxial random loading. Part I: theoretical aspects of the weight function method Andrea Carpinteri a,* , Ewald Macha b , Roberto Brighenti a , Andrea Spagnoli a a Department of Civil Engineering, University of Parma, Viale delle Scienze, 43100 Parma, Italy b Department of Mechanics and Machine Design, Technical University of Opole, ul. Mikotajczyka 5, 45-233 Opole, Poland Received 5 January 1998; received in revised form 11 May 1998; accepted 15 May 1998 Abstract As has been observed experimentally by many authors, the position of the fatigue fracture plane appears to strongly depend on the directions of the principal stresses or strains. In Part I of the present work the expected principal stress directions under multiaxial random loading are theoretically obtained by averaging the instantaneous values of the three Euler angles through some suitable weight functions which are assumed to take into account the main factors influencing fatigue behaviour. Then, in Part II, it is examined how such theoretical principal directions determined by applying the proposed procedure are correlated to the position of the experimental fracture plane for some fatigue tests reported in the literature.  1998 Elsevier Science Ltd. All rights reserved. Keywords: Principal stress directions; Multiaxial random loading; Euler angles; Weight function method; Critical plane approach; Fatigue frac- ture plane Nomenclature A matrix of the principal direction cosines A(t k ) matrix of the principal direction cosines at time instant t k c constant coefficient, with 0  c  1 l n ,m n ,n n , principal direction cosines (eigenvectors n = 1,2,3 of the stress tensor) m  =- coefficient depending on the slope m of 1/m the S–N curve for uniaxial tension– compression with loading ratio R =- 1 N number of time instants being considered, i.e. number of the stress tensor realizations N a S–N fatigue life for stress amplitude  a N t k S–N fatigue life at time instant t k t 1 ,t 2 ,…, time instants t k ,…,t N W summation of the weights W(t k ), from t 1 to t N W(t k ) weight function at time instant t k X(t) = six-dimensional vectorial stochastic * Corresponding author. 0142-1123/99/$—see front matter  1998 Elsevier Science Ltd. All rights reserved. PII:S0142-1123(98)00046-2 [X 1 (t),$, process representing the components of X 6 (t)] the stress tensor ,,, Euler–Rodriguez parameters  a stress amplitude  af fatigue limit stress  n , principal stresses (eigenvalues of the n = 1,2,3 stress tensor), with  1   2   3  xx (t), yy (t), normal stresses  zz (t)  xy (t), xz (t), shear stresses  yz (t) ,, Euler angles (t k ),(t k ), Euler angles at time instant t k (t k )  ˆ , ˆ , ˆ expected values of the Euler angles 1. Introduction Several fatigue models have been proposed by many authors. For example, Mode I, Mode II and Mode III, introduced by Irwin [1], Case A and Case B, formulated by Brown and Miller [2] or Type A, Type B and Type C, distinguished by Socie [3], can be applied to the case of cyclic loading, while they are not applicable to the case of multiaxial random loading.