PSI - Issue 32

Denis N. Sheydakov et al. / Procedia Structural Integrity 32 (2021) 313–320 Denis N. Sheydakov, Irina B. Mikhailova / Structural Integrity Procedia 00 (2021) 000 – 000

315

3

,  H H are the radius-vectors, the deformation gradients and the microrotation tensors corresponding to the transition from the chosen reference to the actual configuration   X   of the micropolar rod with prestressed coating. Itfollowsfromrelations (2) that the wryness tensors L and  L for the rod and the coatingcorresponding to the transition X   are equal to zero ( I is the unit tensor) (Pietraszkiewicz and Eremeyev 2009; Zubov 2016):

   

   

   

  

T

T

T

T

1 2

1 2

  

  

  

  

  

  

  

  

tr  L H H I H H 0 L H H I H H 0     , tr                   

whilethestretchtensors ,  Y Y are expressed as follows (hereinafter the ' denotes the derivative with respect to r )



f

f

T





T Y C H e e = = f 

, e e e e Y C H e e = = f      

r z   e e e e      

(3)

r        z

r

z

r

z

r

r

We will assume that the elastic properties of the rod   0 r r r    are described by the model of a physically linear micropolar material. In this case, the specific strainenergy of the rod W is a quadratic form of the tensors Y I  and L (Lurie 1990; Eremeyev andZubov 1994; Lakes 1995)  0 r r  and its coating 





1 tr





 

 







    

  tr

  Y I Y I  







T

2

, Y L

Y I

Y I

T L L L L tr tr     

2

2

2

tr

tr

W



 



1 



 

 

2

3

2

(4)

0,

2   

0,

0,

0

 



  

 

  



  

2

1

2

3

and the constitutive relations for the Piola-type stress and couple stresstensors D and G at

0 r r  have the form:

W



  





       Y I    

  tr Y G H L I = tr   D H Y I I W        

T Y I H   



(5)

T L L H  

 

 

  

1 

2

3

L



1 2 3 , , , , ,       arethe micropolar elastic parameters of the rod material. Thespecificstrainenergyof the coating W  is written as follows:                 T 2 2 1 1 1 1 1 1 T 2 2 1 1 2 1 1 3 1 1 1 1 , tr tr tr 2 2 2 1 1 1 tr tr tr W                                      Y L Y I Y I Y I Y I L L L L

where

 

(6)

2

2

2

0,

2      

0,

0,

0

   



     

 

  





  

2

1

2

3

Here   Y L are the stretch tensor and the wryness tensor corresponding to the transition from the natural (stress-free) reference configuration to the actual (deformed) state of the coating   X    . The expressions for thetensors 1  Y and 1  L areobtainedby using the formulae for transforming the deformation gradient and the microrotation tensor when changing the reference configuration(Truesdell 1977; Eremeyev and Pietraszkiewicz 2012; Levin 2017): 1 2 3 , , , , ,             arethemicropolarelasticparametersof the coating material; 1 1 ,

   

   

T

T

1 2

 

  

  

  

0       H H I H H        0  0 

T

T

= Y C Y H L L C L H L , ,                

(7)

tr

0 



1

0

0

1

0

0

0

0