Study On The Flexural Waves In Carbon Nanotubes Based On The Generalized Gradient Elastic Beam Theory Considering Surface Effects

The effective mechanical properties of materials and structures at the micro-nano scale are affected by surface effects.The development of continuous medium mechanical models and effective simulation methods for coupling surface effects,and the rational characterization of experimental mechanical behavior of nanomaterials under different conditions have become an indispensable way to study the mechanical behavior of materials and structures on the micro-nano scale.And it is also one of the hot spots of micro-nano mechanics research today.The research on the wave properties of carbon nanotubes(CNTs)is of great significance for understanding the micro-nano structure with carbon nanotubes as the core element and the mechanical properties of the composite materials with carbon nanotubes as the reinforcing material.In this paper,by introducing surface effects,the wave properties of CNTs in free space,elastic medium and initial stress state are studied under the continuous medium mechanics framework.Different from the past,the curvature form of the second derivative of deflection used in the theory of surface elasticity,and the first derivative of rotation angle is introduced in this paper,so that not only the bending deformation but also the shear deformation are considered in the surface effect.This paper mainly includes the following aspects:Firstly,based on the generalized gradient Timoshenko beam theory and generalized gradient high-order beam theory considering surface effects,the flexural wave behavior of CNTs in free space was studied.In the surface elastic model,the curvature of the beam adopts the first derivative of rotation angle,and the CNTs are simulated by Timoshenko beam and high-order beam,respectively.The corresponding governing equations of the generalized gradient beam model considering the surface effect are established,and the dispersion relationship of the flexural wave in the corresponding CNTs is derived.Based on the generalized gradient Timoshenko beam theory and the generalized gradient high-order beam theory,the influences of surface effects,scale effects,shear deformation and rotational inertia on the wave speed of CNTs in free space are discussed.Compared with the results of molecular dynamics,the theoretical results in this paper are in good agreement with the results of molecular dynamics,which verifies the validity and accuracy of the theoretical model.Secondly,based on the generalized gradient Timoshenko beam theory considering surface effects,the flexural wave behavior of CNTs in elastic medium is studied.In the surface elastic model,the curvature of the beam is calculated using the first derivative of the rotation angle,the CNTs are simulated using the generalized gradient Timoshenko beam,and the elastic medium is described using the two-parameter Pasternak basis.The governing equations of the generalized gradient Timoshenko beam considering surface effects are derived,and the dispersion relations of flexural waves propagating in single-walled and multi-walled CNTs in elastic medium are established.The influence of elastic medium,surface effects,scale factors,shear deformation,and rotational inertia on the wave speed of CNTs are discussed.For multi-walled CNTs,the influence of van der Waals force on the wave velocity of flexural waves propagation in CNTs is studied.Thirdly,based on the generalized gradient high-order beam theory considering surface effects,the flexural wave behavior of CNTs in elastic medium is studied.In the surface elastic model,the curvature of the beam adopts the first derivative of the rotation angle,the CNTs are simulated by the high-order beam,and the elastic medium is described by the two-parameter Pasternak basis.Compared with Timoshenko beam model,the high-order beam model not only satisfies that the shear stress on the circumference is zero,but also does not need to introduce additional shear correction coefficient.Based on the generalized gradient high-order beam theory considering surface effects,the wave governing equation is derived,and the dispersion relation of flexural waves in CNTs is established.The influence of elastic medium,surface effects,scale factors,shear deformation and rotational inertia on the wave speed of CNTs are discussed.For multi-walled CNTs,the influence of van der Waals force on the wave velocity of flexural waves propagation in CNTs is studied.Finally,based on the generalized gradient Euler-Bernoulli beam theory considering the surface effect,the influence of initial stress on the flexural wave behavior of CNTs is studied.Based on the generalized gradient Euler-Bernoulli beam theory,the governing equation and dispersion relations of Euler-Bernoulli beam under initial stress are derived.The influence of surface effects and initial stress on the wave speed of CNTs are discussed.For multi-walled CNTs,the effect of van der Waals force on the wave velocity of flexural wave propagation in CNTs under initial stress is studied.Through the research on the flexural wave of CNTs,the understanding of the relevant structural performance at the micro-nano scale has certain guiding significance,and lays a theoretical foundation for the safety design of the related micro-nano electromechanical system structure.