PSI - Issue 2_A

Yingyu Wang et al. / Procedia Structural Integrity 2 (2016) 3233–3239 Author name / Structural Integrity Procedia 00 (2016) 000–000

3235

3

⎢ ⎢ ⎢ ⎣ ⎡

⎥ ⎥ ⎥ ⎦ ⎤

( ) ( ) ( ) t t t

( ) ( ) ( ) t t t

( ) ( ) ( ) t t t

σ

τ

τ τ

x

xy

xz

[ ] ( ) t σ

(1)

=

τ τ

σ

xy

y

yz

τ

σ

xz

yz

z

( ) t

γ

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤

( ) t

γ

xy

xz

( ) t

ε

x

2

2

( ) t

( ) t

γ

γ

[ ] ( ) t ε

xy

yz

(2)

( ) t

=

ε

y

2

2

( ) t

γ

( ) t

γ

yz

xz

( ) t

ε

z

2

2

Then the shear strain resolved along of a generic direction on a generic plane, γ q ( t ), can be obtained by tensor rotation of the above strain tensor to this direction. Alternatively, γ q ( t ) can also be expressed by the following scalar product:

( ) ( )

t q 2

γ

(3)

e d = • t

where, d and e can be expressed as follows: [

]

2 1

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤

2

[ 2 1 ) ) sin(2 ) cos( sin( ) ) sin(2 ) cos( sin( θ α α φ θ φ θ α α φ θ − ) ) cos( ) cos( cos( ) sin( ) cos(2 ) sin( ) ) cos( ) sin( cos( ) cos(2 ) ) cos( sin( ) ) cos(2 ) sin( cos( ) sin(2 ) sin(2 ) sin( 2 1 ) ) sin(2 ) sin( sin( ) ) sin(2 ) cos( sin( θ φ α θ φ α θ φ α θ φ α θ φ α θ φ α θ α φ + − − + − + 2 1 ) sin(2 ) sin(

d d d d d d

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤

]

6 5 4 3 2 1

2

d

=

=

(4)

( ) t

( ) t

( ) t

γ

γ

⎢ ⎣ ⎡

⎥ ⎦ ⎤

γ

( ) t

( ) t

( ) t

( ) t

xy

yz

xz

(5)

e

=

ε

ε

ε

x

y

z

2

2

2

Therefore, the variance of the shear strain resolved along a generic direction q can be expressed in the following form: ( ) ( ) ( ) ( ) [ ] ⎥ = ∑∑ ⎦ ⎤ ⎢⎣ ⎡ ⎥ = ∑ i j j i i j k k k q d d Cov e t e t Var d e t t Var , 2 γ (6)

⎢ ⎣ ⎡

⎦ ⎤

All the candidate critical planes can be obtained by determining the global maxima of the variance of the resolved shear strain, as discussed by Susmel (2010) and Wang and Susmel (2016).