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Carbon nanostructures are perhaps one of the greatest scientific successes which possess superior material properties.Continuous efforts, involving experimental and theoretical studies, have been made for good understanding of their mechanical performance. Experimental studies require advanced instruments and expensive equipments.Moreover, it is still quite difficult to conduct and control experiments at such nanometer scale which would unavoidably cause a broad discrepancy in the results.Theoretical modeling approaches can predict structural behaviors of these carbon nanostructures with good repeatability, reproducibility and controllability and thus they play an increasingly important role in the analysis of mechanical properties for carbon nanostructures. A high-performance computing technique is urgently required for the investigation of such carbon nanostructures in nanoscience and nanotechnology research and practical engineering.Atomistic simulations can precisely capture the delicate behaviors of carbon nanostructures but at a high cost of computational resource and are apparently limited to a small size.This research adopts an exquisite quasi-continuum method to investigate the mechanical properties of carbon nanostructures.It employs the higher-order gradient theory to establish the constitutive model.Unlike the traditional continuum models, the higher-order gradient theory is developed at the atomic level and provides a sound linkage of the deformation of crystal lattice structure to that of continuum displacement field.The distinct superiority of this quasi-continuum method is incorporating the structural information of crystal lattice which is described by introducing a representative cell.In the carbon nanostructures, the atomic structure that each carbon atom is connected to three neighboring carbon atoms by the covalent bonds is selected as the representative cell.The deformation of C-C bond vector is approximated by utilizing the higher-order Cauchy-Born rule whose involved second-order deformation gradient can accurately capture the bending effect and make the deformed C-C bond vector closer to the actual placement.Thus, the established constitutive model is more reasonable and accords extremely well with physical behaviors. A widely used multi-body potential, Brenner potential, is employed for the calculation of the energy stored in the C-C covalent bonds.As far as single-walled carbon nanotubes (SWCNTs) are concerned, the initial equilibrium configuration is determined by minimizing the potential energy of the representative cell, structural parameters and elastic properties, such as Youngs moduli and Poissons ratio, are thus obtained. However, there are some differences of the atomic structure between SWCNTs and single-walled carbon nanocones (SWCNCs) because the radius of SWCNCs increases in the longitudinal direction and this phenomenon gives rise to an effect on the mechanical properties. Subsequently, with the constructed constitutive relationship, a novel mesh-free numerically computational framework is proposed to study mechanical behaviors of these carbon nanostructures.The mesh-free shape function is one of the critical factors in the development of a mesh-free method.The newly moving Kriging interpolation possesses two distinct advantages, the higher order continuity and delta function property.The former fills the requirement of C1-continuous field function for the second-order deformation gradients involved in the higher-order deformation gradient constitutive model and the latter automatically satisfies the essential boundary conditions. Consequently, the moving Kriging interpolation is fme and suitable to be adopted to construct the mesh-free shape function.In addition, by introducing a semivariogram model, corresponding to the covariance, the constructed mesh-free shape function as well as its first-and second-order derivatives can be expressed simply and conveniently. This research is a systemic theoretical and numerical study of mechanical behaviors of carbon nanostructures.It gives a comprehensive study of carbon nanostructures with various loadings and boundary conditions for static and dynamic problems.A SWCNT is regarded as a seamless cylindrical hollow shape formed by rolling up a rectangle graphite sheet and similarly, a SWCNC is treated as a conical structure formed by mapping a tailored graphite sheet and connecting its two ends together.Several numerical examples are used to validate the present quasi-continuum approach.Computational results are compared with those obtained from atomistic simulation and existing data, and are found to be in good agreement.Since the free choice of nodes in the mesh-free computational framework, it can largely reduce the degrees of freedom of the system, and thus save a large amount of computational resources.A few nodes can ensure a high precision for homogenous deformation prior to buckling and an increasing number of nodes are needed to capture the buckling behavior.This makes this proposed method much attractive in engineering applications.Elastic properties, buckling and post-buckling behaviors, vibration characteristic and mass detection of carbon nanostructures with various constraints are further numerically simulated using the proposed mesh-free computational framework. Finally, this approach is extended to the study of multi-layer carbon nanostructures with considering the weak interlayer interaction which is contributed from van der Waals (vdW) force.In the present model, the vdW force between any two layers is considered and the interatomic interaction between different layers is treated as a tress rod, which is described by Lennard-Jones potential.This work provides a systemic and comprehensive understanding of mechanical performance of carbon nanostructures.It is noteworthy that the proposed approach is not limited to the study of carbon based structures but it also applicable for investigating any other materials by giving an appropriate potential function.