Journal of Applied Mathematics and Mechanics

Energy supply into a semi-infinite one-dimensional Fermi-Pasta-Ulam-Tsingou (FPUT) crystal (chain) at a boundary subjected to sinusoidal kinematic loading is examined. It is demonstrated that, in the linear approximation, the energy input problem can be considered symmetric with respect to the boundary for all loading frequencies. Utilizing the renormalized dispersion relation for the chain, an asymptotic approximation for the input energy at large times is derived. It is shown that at low and moderate frequencies, the obtained estimate of the total energy aligns with the results of numerical simulations, whereas a divergence is observed at high loading frequencies.

Interest in graphene is due to a wide range of unique physical and mechanical properties: high Young's modulus, high shear modulus, high strength, etc., as well as high electrical and thermal conductivity. From this point of view, the study of the deformation properties of graphene is one of the most actual branches of modern nanomechanics of materials and structural members (nanodevices). The application of mechanics to the study of nanomaterials, in particular, two-dimensional nanomaterials (graphene, carbon nanotube) is aimed at creating and developing a continuum theory of deformation behavior and having based on this theory, studying a number of applied problems. The moment-membrane theory of elastic thin plates and shells gives an adequate continuum theory of the mechanical behavior of a graphene sheet and a single-layered carbon nanotube (which is constructed taking account of the natural modeling of interactions between atoms in their crystal lattices, i.e. considering this interaction as both force and moment). It is known that due to their unique electrical and mechanical properties, both graphene and carbon nanotube are can be treated as supersensitive elements in the creation of nanoelectromechanical systems. On this basis, it is actual to develop a magnetomechanical theory of the dynamic behavior of a graphene sheet (as well as a carbon nanotube) placed in a given homogeneous magnetic field. In this paper, based on the equations of three-dimensional magnetoelasticity as a moment theory of elasticity with independent fields of displacements and rotations, by using hypotheses concerning the characteristics of mechanical behavior and the characteristics of the behavior of the electromagnetic field in thin regions, a two-dimensional model of magnetoelasticity is proposed according to the moment-membrane theory of elastic plates, which then is applied to a modeling magnetoelastic dynamics of a graphene sheet. Based on the proposed model of magnetoelastic dynamics of a graphene sheet, a problem of free one-dimensional bending oscillations of a two-dimensional body placed in a given homogeneous magnetic field is considered. Analyzing the obtained numerical results, it is shown that magnetoelastic oscillations have a damping nature, the behavior of both the oscillation frequency and the oscillation damping parameter are established depending on the values of the induction of a given magnetic field. Based on these results, a possible range of application of a graphene sheet to a nanoelectromechanical resonator is discussed.

The paper concerns investigation of nonlinear moment shells theories equations application to solution of axisymmetric hyperelastic cylindrical shell static inflation problems. Elasticity equations used for calculation of shells deforming at arbitrary displacements and rotations and relations are obtained on the basis of a modified Kirchhoff–Love model. Results of calculations of linear elastic and neo-Hookean cylindrical shell based on moment theories relations and traditionally used for considered problems solution momentless theory are compared. Shell thickness is considered to be both constant and variable. It is shown that moment theories equations are suitable only when stress-strain state rapidly varying along a meridian. These equations possess a well-posedeness in comparison to equations of shells theory based on the modified Kirchhoff–Love model.

This paper briefly presents an overview of previous studies that are relevant to the current research. Background information is provided. This includes some basic knowledge of ground vibrations, ground vibration reduction techniques, the use of wave barrier for vibration isolation, responses of existing tunnels under near-burst, and mitigation measures to reduce tunnel vibration.