Nonlinear Longitudinal Vibrations in Accreting One-Dimensional Nanostructures

The investigation of an accreting one-dimensional chain of atoms was carried out in this paper. A governing partial differential equation that describes the accreting nanocontinuum containing an attachment of infinite atoms was derived and analyzed. We transformed the space variable $u(t,r)→v(τ,x)$ (for a governing partial differential equation formulated in a previous publication) and introduced a function of linear growth. The boundary conditions were also transformed in terms of the new variables. We assumed that the left end of the accreting nanostructure was free $u(t,r=0)=0$ while the right end was fixed $∂u∂r|r=l[1+εf(t)]=0$⁠. The method of lines was employed to obtain numerical approximate solutions for the nonlinear partial differential equations because their exact solutions proved difficult to obtain. The approximate solutions were expressed using graphical plots in Mathematica®. The viscous damping term, δ, was introduced into the nanocontinuum and analysis was carried out by considering the damped continuum as an accreting nanostructure. Numerical simulations for both cases illustrated linear growth, which was observed via numerical simulation plots.