Scale Dependent Critical External Pressure for Buckling of Spherical Shell Based on Nonlocal Strain Gradient Theory

Abstract

Instabilities in nanosized, externally pressurized spherical shells are important for their applications in nano and biotechnology. Mechanics at such length scale is described by nonlocal and Strain Gradient (SG) field theories. However, analysis of shell buckling is involved and becomes even more complicated in presence of nonlocal and SG interactions. This paper demonstrates that such analysis can be largely simplified by a shallow segment representation of the shell by assuming short wave lengths for the incipient buckling modes. The governing equations are derived and linearized equations are solved to obtain a closed form solution for the critical external pressure causing buckling for a pressurized nonlocal shell. Nonlocal interactions are shown to reduce, whereas the SG interaction increases the critical pressure. The relative reduction/increase becomes more prominent for higher modes of buckling and for increasingly thinner shell. A constricting relationship between the two set of wave numbers expressing the buckling modes is also shown to be modified by the nonlocal and SG scale parameters. Consequent wave numbers increase/decrease, accompanied by decreasing/increasing number of wavelengths, thereby further justifying the shallow segment representation employed herein.

A nonlocal strain gradient shell model with the surface effect for buckling analysis of a magneto-electro-thermo-elastic cylindrical nanoshell subjected to axial load

“…18 Malikan et al (2020) discussed the stiffness softening and stiffness hardening phenomena of nanoshells under the action of three-dimensional magnetic fields, based on the NSGT. 19 Alam et al (2021) discussed the effects of nonlocal interactions and strain gradient interactions on the critical pressure of nanoshells using NSGT. 20 NSGT considers not only the nonlocal stress field but also the strain gradient stress field, which can describe both stiffness softening and stiffness hardening effects, 21 and is widely used to investigate nanostructures such as carbon nanoshells and graphene.…”

“…19 Alam et al (2021) discussed the effects of nonlocal interactions and strain gradient interactions on the critical pressure of nanoshells using NSGT. 20 NSGT considers not only the nonlocal stress field but also the strain gradient stress field, which can describe both stiffness softening and stiffness hardening effects, 21 and is widely used to investigate nanostructures such as carbon nanoshells and graphene. However, the surface effect has not been taken into consideration when analyzing the buckling of nanoshells using NSGT.…”

Stochastic thermo-elastic vibration characteristics of functionally graded porous nano-beams using first-order perturbation-based nonlocal finite element model

“…After that, plenty of researches are accomplished to analyse the bending, buckling, and vibration response of distinct nano-structures such as nano sized beams, plates, and shells using this theory. [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33] Researchers have also used the nonlocal theory to study the cracks in the macro-structures. 34 Recently, Van et al 35 presented a size-dependent refined plate theory and studied the bending and vibrations of graphenereinforced composite nano-plates with different types of gradation of graphene platelets.…”