Issue 70

D. Kosov et alii, Frattura ed Integrità Strutturale, 70 (2024) 133-156; DOI: 10.3221/IGF-ESIS.70.08

Using Voigt-notation in a 2D space, the displacement field u={u, v} T and the phase field  in (5) can be discretized as

m

m

1    i

1    i

i u i u





and

(20)

N

N

u

i i

In (20) N - denotes the shape function associated with node i from all m nodes of element. The corresponding derivatives can be discretized as:

m

m

1    i

i i B  

u i u i B

    



and

(21)

i

1

where  = {  xx ,  yy ,  xy } T . The strain-displacement matrices are expressed as:

   

    

N

0

, i x

, i x N         , i y N

u

 

B

, i y N N N 0

B

and

(22)

i

i

, i y i x ,

where N i,x and N i,y are the derivatives of the corresponding shape function with respect to x and y , respectively. According to ANSYS documentation [24], the displacement vector in (19) must be present in the following form {u}={u 1 ,v 1 ,  1, … , u 4 ,v 4 ,  4 } , and the nodal load vector { r }={ r u1 , r v1 , r  1, … , r u4 , r v4 , r  4 } . Thus, for implementation it is necessary to arrange the components of the stiffness matrix according to:

           

           

1 K K K K 1 1 u ru u

v

4 K K K K 1 4 u ru u

v

1

4

0 ... 0 ...

0 0

ru

ru

1

1

v

v

1

4

rv

rv

rv

rv

1

1

1

1





1 

4 

K

K

0 0

... ...

0 0

r

r

1

1

..

...

...

...

...

...



[ ] K

(23)

1 K K K K 4 1 u ru u

v

4 K K K K 4 4 u ru u

v

1

4

0 ... 0 ...

0 0

ru

ru

4

4

v

v

1

4

rv

rv

rv

rv

4

4

4

4





1 

4 

K

K

0 0

...

0 0

r

r

4

4

The elements of stiffness matrix (23) and load vector can be calculated from:

 

 K g uu

u

u

T

 ( ) ( ) 

B C B dV

ij

i

j

0

(24)

 

u

 B dV T u



 ( ) ( ) 

r

g

i

i

for degrees of freedom u i and v i , and

 

 K g uu

u

u

T

 ( ) ( ) 

B C B dV

ij

i

j

0

(25)

 

u

 B dV T u



 ( ) ( ) 

r

g

i

i

139