چكيده به لاتين
Nanomaterials are being widely used in different industries, such as in medicine, electronics, nanocomposites, biomaterials and energy production. Therefore, in recent years there has been a surge of research in mechanics of materials at micro/nano scales. According to experimental evidences, mechanical properties of the solids and structures predicted by the classical continuum mechanics is different from empirical results at small scales. This phenomenon is known as size effect or scale effect and it affects the material properties such as strength, elastic response and hardness. Therefore, it can be concluded that the size of specimens has a significant effect on the response of bodies. Classical continuum theory is size-free theory and this theory does not consider the size effects arising from the small size, and so, cannot correctly predict the mechanical behavior of micro/nanostructures. To overcome this problem, non-classical continuum theories that contain additional material constants, such as nonlocal elasticity, classical couple stress, modified couple stress and strain gradient have been developed to capture the size effect at micro/nano-scale. Constitutive equations of micro/nanostructures based on non-classical models are functions of the deformation gradient tensor and higher derivatives of the deformation gradient tensor. Analyzing the mechanical behavior of nano/microstructures by finite element software packages encounters some restrictions, because, finite element software packages cannot ensure C_1/C_2 continuity of deformation gradient tensor at nodes placed on common edges of adjacent elements. To overcome the mentioned problem, the components of the deformation gradient tensor should be obtained using the least square meshless method. For this purpose, the problem is simulated in ABAQUS and then the components of the deformation gradient tensor at all the nodes in the finite element model are extracted by user subroutine UMAT. Afterward, the nearest nodes to gauss point are identified by means of FORTRAN programming. Thereafter, the components of the deformation gradient tensor at gauss point are estimated based on quadratic polynomial by using nodal information of close nodes and the least square method. In the present thesis, the large deflection of a cantilever beam under a concentrated vertical force at the free end is intended as a problem. In order to validate the obtained results, the slope of the beam at the free end is determined by the analytical method, the finite element method and the presented method in this thesis. It is worth noting that the proposed method merely obviates the C_1/C_2 discontinuity of deformation gradient tensor in finite element software packages and in the future works, it can be used as a powerful method to solve the problems of nanostructures.
