We consider anti-plane motions of an elastic plate taking into account surface energy within the linear Gurtin–Murdoch surface elasticity. Two boundary-value problems are considered that describe complete shear dynamics of a plate with free faces or with free and clamped faces, respectively. These problems correspond to anti-plane dynamics of an elastic film perfectly or non-perfectly attached to a rigid substrate. Detailed analysis of dispersion relations is provided. In particular, the influence of the ratio of a plate thickness to characteristic length on the dispersion curves is analysed.
This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)’.
Buckling of short multi-walled carbon nanotubes (MWCNTs) under external radial pressure is studied on the base of a multiple-shell model. The modified Mushtari-Donell-Vlasov type equations taking into account the van der Waals (vdW) interaction forces between adjacent tubes are used as the governing ones. In contrast to a majority of available studies on buckling of MWCNTs, which consider only the simply supported boundary conditions, this paper based on the asymptotic approach allows for the study of the buckling behavior of MWCNTs with different variants of the boundary conditions at the tube edges. At first, the pre-buckling membrane hoop stress-resultants induced by radial pressure are determined for each wall. Then, introducing a small parameter defined as a thickness-to-radius ratio, the asymptotic solutions of the boundary value problem are constructed for different cases which depend on the outermost radius of a MWCNT. The relevance of the present approach is confirmed by good agreement between asymptotic estimates and exact values of the buckling radial pressure for simply supported double-and triple-walled nanotubes determined on the base of the accepted shell model. In addition, the validity of the asymptotic estimates is justified by comparing theirs with existing data obtained on the base of the available multiple-shell model taking into account the pressure dependence of the interlayer vdW forces. The influence of the outermost radius, aspect ratio and boundary conditions as well on the buckling radial pressure is analyzed in this study.
The objective of study is to determine the effective Young modulus before and after the completed osseointegration process using mathematical modelling of a titanium porous structure. A novel model is proposed in the form of 3D arrays of Gibson-Ashby cells with rigid clamping of horizontal beams resting on elastic foundation. Calculations made on the basis of the developed model are compared with known models and literature data. The assumption is proved that the osseointegration process due to the bone tissues ingrowth into the pores of titanium implant could affect the Young modulus increasing its value in proportion to porosity of a specimen.
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