Issue 33

Y. Wang et alii, Frattura ed Integrità Strutturale, 33 (2015) 345-356; DOI: 10.3221/IGF-ESIS.33.38

  1 sin θ sin 2φ cos α sin 2θ cos φ sin α 2 1 sin θ sin 2φ cos α sin 2θ sin φ sin α                  2 2

             

             

 

n q n q n q

      

        

1                 2 3 4 5 d d d d d d

x x

 

 

   

 

 

 

 

 









2

y y

1 2

  sin 2θ sin α   

z z



 



(6)

d

y x   

x y n q n q n q n q n q n q   x z y z

1

  sin 2θ sin 2φ s 2   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                                               

z x

6      

z y

and s ( t ) is a six-dimensional vector process depending on   [ε ] t and defined as:

1 2

1 2

1 2

 

  

s

( ) ε (t ) ε (t ) ε (t ) t  





(t )

(t )

yz γ (t )

(7)

x

y

z

xy

xz

According to the quantities defined above, the variance of the shear strain, q ( ) t 

, resolved along direction q can then be

calculated directly as:   t  

  

  

  

( )   k k d s t 

  C d d T

  s t s t

q

 

 





d d

Var

Var

Cov

, ( )

(8)

 

i j

i

j

2

k

i

j

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)

    , 

 

 

Var   

j  then

j  then

where, when i

, whereas when i

s t s t

s t

Cov

 





i

j

i

    ,   . Now [ C ] can be rewritten in explicit form by using both the variance and covariance terms:     , Cov Cov i j j i s t s t s t s t        

x V C C C C C C V C C C C C C V C C C C C C V C C C C C C V C C C C C C V x,y x,z x,xy x,xz x,y y y,z y,xy y,xz x,z y,z z z,xy z,xz x,xy y,xy z,xy xy xy,xz x,xz y,xz z,xz xy,xz xz

        

         

x,yz

y,yz

  C

z,yz

 

(10)

xy,yz

xz,yz

x,yz

y,yz

z,yz

xy,yz

xz,yz

yz

where

  t



i 

, , , , ,  i x y z xy xz yz 

V

Var

    for 

(11)

 

 

i

  t

  t

 

 



, , , , ,  i x y z xy xz yz 

i 

j 

(12)

C

CoVar

, 

    for 

,  i j

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.

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