Stability of inhomogeneous micropolar cylindrical tube subject to combined loads

Abstract

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.

Influence of internal stresses on stability of multilayer micropolar tubes

“…and the constitutive relations (2) for the Piola-type stress and couple stress tensors have the form [42,44]…”

“…In the case of a physically linear micropolar material (4), the tensors D • and G • satisfy the following relations [42,44]…”

“…Linearized boundary conditions on the inner ( r = r 0 ) and outer ( r = r N ) surfaces of the multilayer tube, and at the interfaces between its layers ( r = r k , k = 1 , . . . , N − 1 ) are written as follows [17,44]…”

“…This model has been successfully applied for describing the behavior of granular and powder-like materials [29,30], metal and polymer foams [31,32], polycrystalline and composite materials [33][34][35], bones [36,37], beam-lattice materials [38,39], masonry [40,41], and so on. Some stability problems for micropolar bodies were considered in the previous studies [17,39,[42][43][44][45]. In addition to the above, the Cosserat continuum is also used in the modeling of nanotubes [46][47][48].…”

A Lagrangian Hencky-type non-linear model suitable for metamaterials design of shearable and extensible slender deformable bodies alternative to Timoshenko theory

“…Open problems and challenges that could be tackled in the next future include: (i) a careful analysis of the stiffness parameters used to characterize the elastic response of the whole beam in the largedeformation regime, and in general when de Saint-Venant estimate of stiffness parameters does not apply; such parameters should be related to the material constitutive parameters of the material constituting the meso-beams such as the Young and tangential moduli, and to geometrical parameters of the beams cross-section, as the area, the shear correction factor, and the moment of inertia; (ii) the development of functionally graded materials, meaning those materials having stiffness parameters which are varying along the beam axis; an extended campaign of numerical simulations might unveil new and exotic mechanical behaviors, see [48][49][50][51]; (iii) the development of continuum models, as those developed and exploited, e.g., in [52][53][54][55][56][57][58][59][60], aimed at describing for large displacements systems with many discrete elements of the type presented here; besides being useful in unveiling so-called emerging phenomena, continuum models could help in identifying stiffness parameters; (iv) the exploitation of the presented approach to provide a validation and insight into new and existing approaches for the extension of stability theory in classical elastic media to micromorphic, strain-gradient [61,62], and Cosserat media, see, e.g., [12,[63][64][65][66][67][68][69][70]; (v) the extension of the presented approach to problems where dynamics effects are non-negligible, see, e.g., [71][72][73], like those studied in the active control of vibrations [74]; (vi) the validation of continuum approaches to the study of plane and curved structures moulded as, e.g., shells and tubes, see [75][76][77...…”