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

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5

3. Perturbed state of circular rod We assume that in addition to the above-described state of equilibrium X for the micropolar rod with prestressed coating, there is an infinitely close equilibrium state p X under the same externalloads, which is determined by the radius-vectors , p p  R R and the microrotation tensors , p p  H H : , , , = , = , = , = p p p p R R Z Z R R Z Z R R Z Z R R Z Z v v v v v v                                                    R R v R R v H H H ω H H H ω v e e e v e e e ω e e e ω e e e where  isasmallparameter; ,  v v are the vectors of additional displacements; ,  ω ω are the linear incremental rotation vectors, which characterizes the small rotation of themicropolar medium particles, measured from the deformed state X . The linearized equilibrium equationsfor the considered structure have the form(Eremeyev andZubov 1994):

  

  







= , D 0 

T      G v D C D 0 T = ,  

0

r r





 

0



(12)

  

  







T

T







= , D 0 

= , G v D C D 0          

r r r  





0





Here ,    G G are the linearized Piola-type stress and couple stress tensors for the rod and the coating. Bylinearizingtheconstitutiverelations (5), (8), the expressions for these tensors in the case of a physically linear micropolar material (4), (6) are obtained (Sheydakov et al. 2020). Linearizedboundaryconditionsonthesurfaceofthecoating   = r r  and at the interface between the coating and the rod   0 = r r are written as follows (Sheydakov2011):   -T T , = ; det r r r pJ J                        e D e C v I v e G 0 C    D D and ,

 

 

r r 

r r 





(13)





 e D e D   

 e G e G   

 v v

,

,

,









 ω ω

r

r

r

r

r r 

r r 

r r 

r r 

r r 

r r 

0

0

r r 

r r 

0

0

0

0

0

0

Here  isthe nabla-operator in Eulerian coordinates. We assume that there is no friction at the ends of the considered composite structure   l z = 0, , andconstant normal displacement is given. This leads to the following linearized endconditions(Sheydakovand Altenbach2016):

 e D e  

 e D e  

 e G e  

z e v 

= e ω e ω  

=

=

= 0,

=

= 0

z

R

z

z

Z

r





=0, z l

=0, z l

=0, z l

=0, z l

=0, z l

=0, z l

(14)







 e D e  

 e D e  

 e G e  

z e v 

= e ω e ω   

=

=

= 0,

=

= 0





z

R

z

z

Z

r





=0, z l

=0, z l

=0, z l

=0, z l

=0, z l

=0, z l

To solve the linearized boundaryvalue problem (12) – (14) for a system of twelve partial differential equations, the following substitution is used (Sheydakov 2011; Sheydakovand Altenbach2016)                       = cos cos , = sin cos , = cos sin , = cos cos , = sin cos , = cos sin , = sin sin , = cos sin , = sin cos , = sin sin , = cos sin , R R Z Z R R Z Z R R Z Z R R Z v V r n z v V r n z v V r n z v V r n z v V r n z v V r n z r n z r n z r n z r n z r n z                                                       = / , = 0,1, 2, ... = sin cos , Z m l m n r n z      