PSI - Issue 52

Akihide Saimoto et al. / Procedia Structural Integrity 52 (2024) 323–339 A. Saimoto et al. / Structural Integrity Procedia 00 (2023) 000–000

325

3

Fig. 1. An anisotropic infinite plate with multiple line cracks of inclination angle β k , ( k = 1 , 2 , · · · ) subjected to remote stresses.

physical coordinate system x , y is in the direction of the material principle axis x m , y m rotated clockwise by an angle α , as seen in Fig.1, the complex parameter µ mj , ( j = 1 , 2) are transformed to the o ff -axis physical coordinate system rotated by α counter-clockwise to give µ j as

µ mj cos α + sin α cos α − µ mj sin α

, ( j = 1 , 2)

(5)

µ j =

At the same time, the constitutive relation between stresses and strains in physical coordinate system becomes    ε x ε y γ xy    =    c 11 c 12 c 16 c 12 c 22 c 26 c 16 c 26 c 66       σ x σ y τ xy    or ε = C σ where c i j are the transformed compliance coe ffi cients in the physical coordinate system x , y . They are defined as follows. (6)

+  −

1 G 12 

2 ν 21 E 2

1 E 2

1 E 1

cos 4 α

sin 4 α

sin 2 α cos 2 α

c 11 =

(7)

+

+

c 12 =  c 16 = 

1 G 12 

1 E 1 2 E 1

1 E 2 −

ν 21 E 2

sin 2 α cos 2 α −

(cos 4 α

4 α )

+ sin

(8)

+

+  −

1 G 12  2 ν 21 E 2

1 G 12 

2 ν 21

2(1 + ν 21 ) E 2

sin α cos 3 α

sin 3 α cos α

(9)

E 2 −

+

+

+  −

1 G 12 

1 E 1

1 E 2

sin 4 α

sin 2 α cos 2 α

cos 4 α

c 22 =

(10)

+

+

1 G 12 

1 G 12 

c 26 =  − c 66 = 4 

+ 

2(1 + ν 21 ) E 2

2 E 1

2 ν 21

sin α cos 3 α

sin 3 α cos α

(11)

E 2 −

+

+

E 2 

1 E 1

1 + 2 ν 21

1 G 12

sin 2 α cos 2 α

(cos 2 α − sin 2 α ) 2

(12)

+

+

The infinitesimal strain theory follows the definition of strain components as

∂ u ∂ y

∂ u ∂ x

∂ v ∂ y

∂ v ∂ x

(13)

ε x =

,ε y =

,γ xy =

+

where u and v are the components of displacement in x and y directions. It is known that the material parameters in the physical coordinate system µ j , ( j = 1 , 2) can be obtained by solving the following quadratic equation without using the transformation formula in Eq.(5). c 11 µ 4 − 2 c 16 µ 3 + (2 c 12 + c 66 ) µ 2 − 2 c 26 µ + c 22 = 0 (14)