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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2

of composite bodies with internal stresses is the absence of a unified natural (stress-free) reference configuration. Therefore, when deriving the linearized equilibrium equations for the rod and the coating, the writing of the constitutive relations with respect to different reference configurations is used (Sheydakov et al. 2020). The considered composite rod is assumed to be made of materials with a complex microstructure. Examples of such materials are metal and polymer foams, which are widely used in the modern aerospace and automotive industries. The behavior of foams cannot be adequately described within the framework of the classical theory of elasticity due to the size effects (Lakes 1995). To introduce an internal length scale, one has to resort to generalized continuum models. In the present paper, we used the model of a micropolar body(Cosserat continuum), i.e., medium with couple stresses and rotational degrees of freedom (Cosserat and Cosserat 1909; Toupin 1964; Kafadar and Eringen1971; Eringen 1999), to take into account the effect of material microstructure on the loss of stability.Within the framework of this model, the microrotation can be viewed as the rotation of nodes between ribs in the foam, and the couple stress can be viewed as a spatial average of the bending and twisting moments transmitted by the foam ribs. 2. Circular rod with prestressed coating Consider a circular rod of radius 0 r and length l in the natural (stress-free) state within the framework of the nonlinear Cosserat continuum.The rod has a coatingin the form of a hollow circular cylinder.We will assume that this cylinder is attached to the rod afterinitial deformation of axial extension-compression, described by the following relations (hereinafter, the superscript "  " will denote the quantities related to the coating) (Zubov 1997): where , ,       arethecylindricalcoordinatesinthe naturalreferenceconfiguration   ofthecoating; , , i e l      are the internal and external radii, and the length of the undeformed hollow cylinder acting as the coating; r z , ,  arethecylindrical coordinates in the prestressed state  ;   , ,       e e e and   z r e ,e ,e  are the orthonormal vector bases of the corresponding cylindrical coordinates; a  is the given coefficient of axial extension-compression for coating; c  is some constant characterizing the radial deformation of the cylindrical coating, which is determined from the no-load conditions on its lateral surfaces;    is the nabla-operator in the reference configuration   ; 0 0 0 , ,    R С H are the radius-vector, the deformation gradient and the microrotation tensor corresponding to the transition from the natural reference configuration to the prestressed state of the coating       . The geometric parameters of the rod and thecoating are assumed to be given quantities that satisfy the contact conditions 0 i c r     and a l l    . Let us now study the deformation of the considered composite structure under axial compression and external hydrostatic pressure, choosing  as the reference configuration, which is natural for the rod, but prestressed for the coating.This deformation is described by the relations(Sheydakov et al. 2020)         0 0 , 0 , 0 2 , , , , , 0 , , , e f r r r R r c f r r r r Z z z l f r z f r z                                            R e e R e e H H e e e e e e (2) 0  R e c     0  0  0  ,      , ; , 0 2 ,    0 , , e e H e e e e e e                         i e r z r z r z r c  z a  l a c С R e e e e c a                                 e  (1)

R

Z

R

Z

r

R

z

Z

 





df

f

df

f





r z   e e e e          C R e e R

 C R e e    

R z   e e e e      

,

Z

r

Z

dr

r

dr

r

; 

 Z

Here R e ,e ,e  istheorthonormalvectorbasisofthecorrespondingcylindricalcoordinates;  is the given compression ratio along the axis of the considered composite structure;     , f r f r  are some unknown functions characterizing the radial deformationof the rod and the coating;   isthenabla-operatorinthereferenceconfiguration  ; ,  R R , ,  С С and R Z , ,  arethecylindricalcoordinatesintheactual (deformed) configuration X