This paper presents a bar-elastic substrate model to investigate the axial responses of nanowire-elastic substrate systems considering the effects of nonlocality and surface energy. The thermodynamics-based strain gradient model is adopted to capture the nonlocality of the bar-bulk material while the Gurtin-Murdoch surface theory is utilized to consider the surface energy. To characterize the bar-surrounding substrate interaction, the Winkler foundation model is employed. In a direct manner, system compatibility conditions are obtained while within the framework of the virtual displacement principle, the system equilibrium condition and the corresponding natural boundary conditions are consistently obtained. Three numerical simulations are conducted to investigate the characteristics and behaviors of the nanowire-elastic substrate system: the first is conducted to reveal the capability of the proposed model to eliminate the paradoxical behavior inherent to the Eringen nonlocal differential model; the second is employed to characterize responses of the nanowire-elastic substrate system; and the third is aimed at demonstrating the dependence of the system effective Young’s modulus on several system parameters.

Nonlocal and surface effects are incorporated into a bar-elastic substrate element to account for small-scale and size-dependent effects on axial responses of nanowires embedded in elastic substrate media. The virtual displacement principle, employed to consistently derive the governing differential equation as well as the boundary conditions, forms the core of the displacement-based finite element formulation of the nanowire-elastic substrate element. The element displacement shape functions, analytically derived based on homogeneous solution to the governing differential equilibrium equation of the problem, result in the exact element stiffness matrix and equivalent load vector. Two numerical simulations employing the proposed model are performed to study characteristics and behavior of the nanowire-substrate system. The first simulation involves investigation of responses of the wire embedded in elastic substrate. The second examines influences of several system parameters on the contact stiffness and reveals the size-dependent effect on the effective Young's modulus of the system.

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This paper proposes a beam model with inclusion of surface and nonlocal effects. Beam-bulk kinematics is formulated within the framework of Euler-Bernoulli beam theory; Eringen nonlocal elasticity theory is employed to account for long-range atomic interactions of a nanoscale beam; and Gurtin-Murdoch surface elasticity theory is used to represent the size-dependent effect inherent to a nanoscale beam.Surface-layer balance equation between the surface and bulk stresses is treated in a consistent manner. Virtual displacement principle is exploited to consistently derive the governing differential equilibrium equation as well as the natural boundary conditions of the problem. The general form of beam bending solution is presented. Two numerical simulations employing the proposed beam model are conducted to study characteristics and behaviors of the nanobeam with various model parameters. The first simulation investigates the coupled effects of nonlocality and surface energy on transverse displacement and bending moment characteristics of nanowires under various support conditions. The second examines the influences of several model parameters as well as the size-dependent effect on the effective bulk Young's modulus of the nanobeam. K E Y W O R D S analytical solution, effective bulk Young's modulus, Euler-Bernoulli beam, nanobeam, nonlocal elasticity, size effect, surface elasticity

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