PSI - Issue 13

Milan Micunovic et al. / Procedia Structural Integrity 13 (2018) 2158–2163 Micunovic, Kudrjavceva/ Structural Integrity Procedia 00 (2018) 000 – 000

2161

4

3. Transversely isotropic materials 3.1. Transversely isotropic QRI materials When the material body possesses a single preferred anisotropy direction, say n , then the arguments of the evolution equation have to include the diadic n n   N . If n is unit vector then 1 n n tr    N . Therefore,   , , P S P     D T Ν with   , , P     T Ν . (6) We restrict our consideration here to a reduced set of invariants to be used as the source of tensor generators: (7) omitting eigen and mixed invariants of plastic strain tensor (cf. [12] for the complete set of invariants). Suppose now that  is a polynominal of third order in S . Then the loading function has the following simple form: 1 , s tr  T 2 2 , d s tr  T 3 3 , d s tr  T 1 { }, d tr   NT 2 2 { } d tr   NT

1 9

1 3

  

  

2

2   1 

1 2 2 3 1 2 a s a s s s b s 1 ( )

s

2



    



1

1 1

(8)

1 9

2 3

1 3

 

  

 

  

2

2 b s 

s

3 1 2 b s s    

s

2   





1

1 1

1 2

leading to the following evolution equation:

3 2

1 9

1 3

1 3

 

  

2 T S 1 s s

D S

s 1 N N 1   

a a

1 b s 

(

)   

1 

1 

 

P

d

d

d

1

2

1

2

1

1

(9)

1 1 1 9 2 

1 2

1 3

1 3

 

  

s   1 NT T N N 1 

2 b s 

1 



d

d

1

1 3

2 1 1 6 2 s    s

  

  

S

2 1  1 N S

b s

.

 

d

d

3

1

3.2. Classical J 2 theory of transversely isotropic materials In the classical theory of plasticity of transversely isotropic materials the evolution equation is based on the yield function and the modified equivalent stress i.e.

1

1 (

(10)

2    eq S

D

1  T N d R 

f   

),

S

P

2 ( ) Peq h 

2 ( ) Peq h 

2



1

2 3 1 2 2 s  

 

eq

2

2

f

R

with

1 and

.



1 





  

eq

3 ( ) Peq h 

 Comparing these expressions with the QRI evolution equation (9) we see that anisotropy coefficient R is proportional to 1 b . However, this classical theory does not contain either coefficients b 2 and b 3 or new invariant expressions present in (8). Indeed, analyzing experimental data in [15] we conclude that evolution equation (10) is not able to cover strain induced transverse anisotropy observed during straining of car body sheets. 3.3. Diffuse instability According to McClintock, (cf. [16]), a nonuniform strain field may develop twofold: (a) thinning during tension loads occurs very gradually in dimensions comparable with specimen dimensions and (b) it occurs in a region comparable with sheet (or specimen) thickness. The first is called diffuse instability, whereas for the second phenomenon the name localized instability is chosen. Due to above distinction the diffuse instability could appear mainly when cylindrical or some other 3Dspecimen are used. At this point Hill’s stability postulate (cf. [17]) is invoked stating in its local form that at a bifurcation point the following equality holds