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

318

6

which leads to the separation of variables  and z in this problem and allows to satisfythe linearized end conditions (14).As a result,the stability analysis of a micropolar rod with prestressed coatingisreduced to solving a linear homogeneous boundaryvalue problem for asystem of twelve ordinary differential equations (Sheydakov et al. 2020). For its solvability, it is necessary toadditionally formulate six conditions at 0  r , which can be obtained by requiringthe boundedness of unknown functions Z Z R R V V V      , , , , , and their derivativeson the rod axis (Sheydakov 2011):

= = = 0 

V V V  

= = = 0 

V V V 

   

   

R

Z

R

Z





(15)

= 0 :

,

= 1:

n

n

= = = 0 

R 

= = = 0 

  

  

R

Z

Z





4. Results and Discussion In this paper, we have carried out the stability analysis in the case when the rodand the coating are made of dense polyurethane foam.The micropolar elastic parameters for this material have been determined by Lakes (1995):

7 79.73 10 Pa,

6 99.67 10 Pa,

5 86.67 10 Pa

    

     

  

  

  

26.65 N,

45.3 N,

34.65 N

   

  

  

 

 

1

1

2

2

3

3

For convenience, the following dimensionless parameters were introduced:the relative axial compression of the rod with coating    =1 ;the relative external pressure = / p p   ;the relative initial extension-compression of the coating = 1 a     ;the relative thickness of the coating   0 = h r r r     ;the length-to-radius ratio of the rod with coating = l l r   ;the relative radius of the rod with coating = b r r l    .Here b l is the characteristic length for bending. This is the engineering constant ofmicropolarmaterial(Lakes 1995), which is expressed through the elastic parameters and for the dense polyurethane foam -5 33 10 m b l   . 4.1. Simple loading case First, the case of simple compression was considered, when there is no external pressure   = 0 p .By numerical solution (Sheydakovand Altenbach2016) of the linearized boundaryvalue problem (12), (13), (15),wefound the spectra of critical values for the relative axial compression  , correspondingto the different bucklingmodes of the micropolar rod with prestressed coating. Based on these spectraanalysis, the critical axial compression was determined for rods of different sizes and coatings of various thicknesses and with different initial deformations.

h = 0.05

h = 0.10

0.050 

0.050 

r * = 1 r * = 2 r * = 5 r * = 10

r * = 1 r * = 2 r * = 5 r * = 10

0.025

0.025

-0.1

0

0.1

0.2

-0.1

0

0.1

0.2

 *

 *

Fig. 1.Influence of initial deformation of the coating on the micropolar rod stability in the case of simple compression   = 0 p .