Issue 33

Y. Wang et alii, Frattura ed Integrità Strutturale, 33 (2015) 345-356; DOI: 10.3221/IGF-ESIS.33.38 348                                 2 1 2 2 3 4 5 6 1 sin θ sin 2φ cos α sin 2θ cos φ sin α 2 d 1 sin θ sin 2φ cos α sin 2θ sin φ sin α d 2 1 d sin 2θ sin α 2 d 1 d sin 2θ sin 2φ s 2 d x x y y z z x y y x x z z x y z z y n q n q n q n q n q n q n q n q n q                                                                      d                                 in α sin θ cos 2φ cos α cos 2θ cos φ sin α cos θ sin φ cos α cos 2θ sin φ sin α cos θ cos φ cos α                                          (6) and s ( t ) is a six-dimensional vector process depending on   [ε ] t and defined as: x y z xy xz yz 1 1 1 ( ) ε (t ) ε (t ) ε (t ) (t ) (t ) γ (t ) 2 2 2 t          s (7) According to the quantities defined above, the variance of the shear strain, q ( ) t  , resolved along direction q can then be calculated directly as:       Var Var ( ) Cov , ( ) 2 q T k k i j i j k i j t d s t d d s t s t C                       d d (8) where [ C ] is a symmetric square matrix of order six, and the terms of the covariace matrix are defined as     Cov ,  ij i j C s t s t      (9) where, when i j  then       Cov ,  Var i j i s t s t s t          , whereas when i j  then         Cov ,  Cov , i j j i s t s t s t s t          . Now [ C ] can be rewritten in explicit form by using both the variance and covariance terms:   x x,y x,z x,xy x,xz x,yz x,y y y,z y,xy y,xz y,yz x,z y,z z z,xy z,xz z,yz x,xy y,xy z,xy xy xy,xz xy,yz x,xz y,xz z,xz xy,xz xz xz,yz x,yz y,yz z,yz xy,yz xz,yz yz V C C C C C C V C C C C C C V C C C C C C C V C C C C C C V C C C C C C V                      (10) where   Var     for  , , , , ,  i i V t i x y z xy xz yz        (11)     ,  CoVar ,      for  , , , , ,  i j i j C t t i x y z xy xz yz         (12) Then Eq. (8) can be rewritten in the following simple form:     Var T q t C       d d (13) Eq. (13) makes it evident that the determination of the direction experiencing the maximum variance of the resolved shear strain is a conventional multi-variable optimization problem. It can be solved satisfactorily by simply using the so-called Gradient Ascent Method [19]. Fig. 2 reports the flowchart summarising the algorithm which was proposed in Ref. [9] to be used to determine the orientation of the critical plane.