In the present paper, the stability of a nonlinear elastic cylindrical tube made of micropolar material is analyzed. It is assumed that the elastic properties of the tube vary through the wall thickness. The problem is studied for the case of axial compression of the tube under internal and external hydrostatic pressure. Applying linearization the neutral equilibrium equations have been derived, which describe the perturbed state of the tube. By solving these equations numerically the critical curves and corresponding buckling modes have been found, and the stability regions have been constructed in the planes of loading parameters (relative axial compression, relative internal or external pressure). Using these results, the influence of elastic properties, as well as the size of the tube, on the loss of stability is studied. Special attention has been paid to the analysis of how the pattern of change in elastic parameters affects the stability of a cylindrical tube made of micropolar material.
GibsonLJAshbyMF. Cellular solids: structure and properties, 2nd edn (Cambridge Solid State Science Series). Cambridge: Cambridge University Press, 1997.
9.
dell’IsolaFMadeoASeppecherP. Boundary conditions at fluid-permeable interfaces in porous media: A variational approach. Int J Solids Structures2009; 46(17): 3150–3164.
10.
KrönerE. Elasticity theory of materials with long range cohesive forces. Int J Solids Structures1967; 3(5): 731–742.
11.
EringenAC. Nonlocal Continuum Field Theories. New York: Springer, 2002.
12.
CosseratECosseratF. Theorie des Corps Deformables. Paris: Hermann et Fils, 1909.
13.
KafadarCBEringenAC. Micropolar media - I. The classical theory. Int J Eng Sci1971; 9: 271–305.
14.
MauginGA. On the structure of the theory of polar elasticity. Phil Tran R Soc Lond A1998; 356: 1367–1395.
15.
EringenAC. Microcontinuum Field Theory. I. Foundations and Solids. New York: Springer, 1999.
16.
AltenbachJAltenbachHEremeyevVA. On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Arch Appl Mech2010; 80: 73–92.
17.
ToupinRA. Theories of elasticity with couple-stress. Arch Rat Mech Anal1964; 17: 85–112.
18.
MindlinRD. Micro-structure in linear elasticity. Arch Rat Mech Anal1964; 16: 51–78.
19.
AlibertJSeppecherPdell’IsolaF. Truss modular beams with deformation energy depending on higher displacement gradients. Math Mech Solids2003; 8(1): 51–73.
20.
dell’IsolaFSeppecherPMadeoA. How contact interactions may depend on the shape of Cauchy cuts in Nth gradient continua: Approach”á la D’Alembert”. Z angew Math Phys2012; 63(6): 1119–1141.
21.
FerrettiMMadeoAdell’IsolaF. Modeling the onset of shear boundary layers in fibrous composite reinforcements by second-gradient theory. Z angew Math Phys2014; 65: 587–612.
22.
EremeyevVAZubovLM. On the stability of elastic bodies with couple-stresses. Mech Solids1994; 29(3): 172–181.
23.
ZubovLM. Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies. Berlin: Springer, 1997.
24.
NikitinEZubovLM. Conservation laws and conjugate solutions in the elasticity of simple materials and materials with couple stress. J Elasticity1998; 51: 1–22.
25.
PietraszkiewiczWEremeyevVA. On natural strain measures of the non-linear micropolar continuum. Int J Solids Structures2009; 46: 774–787.
26.
PietraszkiewiczWEremeyevVA. On vectorially parameterized natural strain measures of the non-linear Cosserat continuum. Int J Solids Structures2009; 46(11–12): 2477–2480.
27.
EremeyevVAPietraszkiewiczW. Material symmetry group of the non-linear polar-elastic continuum. Int J Solids Structures2012; 49(14): 1993–2005.
28.
LurieAI. Non-linear Theory of Elasticity. Amsterdam: North-Holland, 1990.
29.
SheydakovDN. Stability of elastic cylindrical micropolar shells subject to combined loads. Shell Structures: Theory and Applications, volume 2. Boca Raton, FL: CRC Press, 2010, pp. 141–144.
30.
LakesRS. Experimental methods for study of Cosserat elastic solids and other generalized elastic continua. Continuum models for materials with micro-structure. New York: Wiley, 1995, pp. 1–22.
31.
GreenAEAdkinsJE. Large Elastic Deformations and Non-Linear Continuum Mechanics. Oxford: Clarendon Press, 1960.
SheydakovDN. Buckling of elastic composite rod of micropolar material subject to combined loads. Mechanics of Generalized Continua – From Micromechanical Basics to Engineering Applications (Advanced Structured Materials, volume 7). Berlin: Springer-Verlag, 2011, pp. 255–271.
34.
LakesRS. Experimental microelasticity of two porous solids. Int J Solids Structures1986; 22: 55–63.
35.
RichtmyerRDMortonKW. Difference methods for initial-value problems. New York: Interscience Publishers, 1967.
36.
ZubovLMSheidakovDN. Instability of a hollow elastic cylinder under tension, torsion, and inflation. Transactions of ASME Journal of Applied Mechanics2008; 75(1): 011002.
37.
HaughtonDOgdenR. Bifurcation of inflated circular cylinders of elastic material under axial loading – II. Exact theory for thick-walled tubes. J Mech Phys Solids1979; 27(5–6): 489–512.
38.
LakesRS. Size effects and micromechanics of a porous solid. J Mater Sci1983; 18: 2572–2581.
39.
LakesRS. Experimental micro mechanics methods for conventional and negative Poisson’s ratio cellular solids as Cosserat continua. J Eng Mater Technol1991; 113: 148–155.
40.
AndersonWBLakesRS. Size effects due to Cosserat elasticity and surface damage in closed-cell polymethacrylimide foam. J Mater Sci1994; 29: 6413–6419.
41.
GauthierRDJahsmanWE. A quest for micropolar elastic constants. J Appl Mech1975; 42: 369–374.
42.
GauthierRDJahsmanWE. Bending of a curved bar of micropolar elastic material. J Appl Mech1976; 43: 502–503.
43.
Krishna-ReddyGVVenkatasubramanianNK. On the flexural rigidity of a micropolar elastic circular cylinder. J Appl Mech1978; 45: 429–431.
