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

316

4

where 0  L isthe wryness tensor corresponding to the transition from the natural to the prestressed reference configuration       . To derive the constitutive relations for the coating with respect to the prestressed reference configuration  , we use the formulae connecting the Piola-type stress and couple stress tensors in different reference configurations (Levin 2017)





W Y

W





1 T

1 T





0 0   D C D D 1 , J    

0 0 0   H H G C G G H H 1 1 0 , , J             

C

;

det





0 

J

(8)

 



1

0

L









1

1

Itfollowsfromexpressions (1) – (3), (5) – (8) thatthetensors ,  G G are equal to zero for the deformation (2) of the considered composite structure, and the Piola-type stress tensors ,  D D are (Sheydakov et al. 2020)         2 1 1 1 1 1 1 1 r R z Z r R z Z f s f s s r s c f s a c f s                                                                                              D e e e e e e D e e e e e e (9)

a c

a c

r



c









 

f

c f





3,

3,

2 ,              2

s c f 

a        

s f

    

r

r

The equilibrium equations for the micropolar rod with prestressed coatingin the absence of mass forces and moments are written as follows(Eremeyev andZubov 1994; Zubov 1997; Levin 2017):     T 0 T 0 = , = , 0 = , = , r r r r r                    D 0 G C D 0 D 0 G C D 0     (10)

Thesymbol  representsthe vector invariant of a second-order tensor. Theboundary conditions(Sheydakov et al. 2020)   -T , , pJ            e D e C e D e D

  0

 

(11)

,

0 0 

f r f r  

f

0

r

r

r

r

r r 

r r 

r r 

0

0



expressthe action of external hydrostatic pressure p (referred to the unit area of the deformed configuration X )on the surface of the coating   = r r  , the rigid coupling of the rod and the coating   0 = r r ,and the absence of radial displacement on the axis of the rod   0  r . Bysolvingthe boundary value problem (10), (11)while taking into account relations (2), (9), we findthe unknown functions     , f r f r  :



f

  f r f r 

  f r f r  

,

 

2

1

1

r

Here the constants 1 1 2 , , f f f boundary conditions (11).

  are determined from the system of three linear algebraic equations following from the