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Abstract:
The instability of nanobeams rested on two-parameter elastic foundations is studied through the Bernoulli-Euler beam theory and the stress-driven nonlocal elasticity model. The size-dependency is incorporated into the formulation by defining the strain at each point as an integral convolution in terms of the stresses in all the points and a kernel. The nonlocal elasticity problem in a bounded domain is well-posed and inconsistencies within the Eringen nonlocal theory are overcome. Excellent agreement is found with the results in the literature, and new insightful results are presented for the buckling loads of nanobeams rested on the Winkler and Pasternak foundations.

Keywords:
buckling, closed form solution, nanobeam, nonlocal elasticity, Pasternak foundation, stress-driven

Abstract:
A nonlocal linear elastic fracture formulation is presented based on a discrete layer approach and an interface model to study cracked nanobeams. The formulation uses the stress-driven nonlocal theory of elasticity to account for the size-dependency in the constitutive equations, and the Bernoulli-Euler beam theory to define the kinematic field. Two fundamental mode I and mode II fracture nanospecimens with applications in Engineering Science are studied to reveal principal characteristics of the linear elastic fracture of beams at nanoscale. The domains are discretized both through the transverse and longitudinal directions and the field variables are derived by solving systems of the nonlocal equilibrium equations subjected to the variationally consistent and constitutive boundary and continuity conditions. The energy release rates of the fracture nanospecimens are calculated both from the global energy consideration and from the localized fields at the tip of the crack, i.e. the cohesive forces and the displacement jumps. The results are shown to be the same, proving the capability of the interface model to predict localized fields at the crack tip which are important for the cohesive fracture problems. It is found that the nanospecimens with higher nonlocality have higher fracture resistance and load bearing capacity due to higher energy absorptions and lower energy release rates. The crack propagation in the nanospecimens are also studied and load-displacement curves are presented. The nonlocality considerably increases the stiffness of the initial linear response of the nanospecimens. The fracture model is also able to capture the non-linear post-peak response and the unstable crack propagation, the snap-back instability, which is more intense for nanospecimens with higher nonlocality.

Keywords:
cracked nanobeams, nonlocal fracture, energy release rate, cohesive, crack propagation

Abstract:
The size-dependent buckling instability of shear deformable nanobeams rested on a two-parameter elastic foundation is studied through the stress-driven nonlocal theory of elasticity and the kinematic assumptions of the Timoshenko beam theory. The small-scale size effects are taken into account by nonlocal constitutive relationships, which define the strains at each point as integral convolutions in terms of the stresses in all the points and a kernel. In this manner, the nonlocal elasticity formulation is well-posed and does not include inconsistencies usually arising using other nonlocal models. The size-dependent governing differential equations in terms of the transverse displacement and the cross-sectional rotation are decoupled, and closed form solutions are presented for the displacement functions. Proper boundary conditions are imposed and the buckling problem is reduced to finding roots of a determinant of a matrix, whose elements are given explicitly for different classical edge conditions. The closed form treatment of the problem avoids the numerical instabilities usually occurring within numerical techniques, and allows to find also higher buckling loads and shape modes. Several nanobeams rested on the Winkler or Pasternak elastic foundations and characterized by different boundary conditions, shear deformability, and nonlocality are considered and the critical loads and shape modes are presented, including those for the higher modes of buckling. Excellent agreements are found with the available approximate numerical results in the literature and novel insightful findings are presented and discussed, which are in accordance with experimental observations.

Keywords:
nanobeam, buckling, elastic foundation, closed form solution, nonlocal elasticity, size effect

Abstract:
Interfacial tractions generated at the interface in two-layered nanobeams are studied through the stress-driven nonlocal theory of elasticity and an interface model. The model uses a layerwise description of the problem and satisfies the continuity conditions at the interface. The size-dependency are incorporated into formulation through a nonlocal constitutive law which defines the strain at each point as an integral convolution in terms of the stresses in all the points and a kernel. The Bernoulli-Euler beam theory is used separately for each layer to describe kinematic field, and to derive size-dependent system of coupled governing equations. The displacement components within the layers are derived and the interfacial tractions are obtained through the interfacial constitutive relations. Results are presented for the interfacial shear and normal tractions, exhibiting a different behavior at the nano-scale compared to those of the layered beams with large-scale dimensions including different maximum interfacial tractions and the location where maxima occur. A superior resistance of nanobeams against debondings and delaminations due to the interfacial normal tractions compared to that of the beams with large-scale dimensions is observed. The formulation and the understandings presented here are expected to stimulate further researches on multilayered nanobeams, including their interfacial fracture mechanics.

Keywords:
multilayered nanobeams, weak bonding, interfacial tractions, delamination, nonlocal elasticity

Abstract:
Free flexural vibrations of nanobeams with non-rigid edge supports are studied by means of the stress-driven nonlocal elasticity model and Euler-Bernoulli kinematics. The elastic deformations of the supports are modelled by transversal and flexural springs, so that, in the limit conditions when the springs stiffnesses tend to zero or infinity, the classical free, pinned, and clamped boundary conditions may be recovered. An analytical procedure is used to derive the closed form solution of the spatial differential equation. The problem of finding the natural frequencies is then reduced to find the roots of the determinant of a matrix, whose elements are explicitly given. The proposed technique, then, avoids the numerical instabilities usually arising when the numerical techniques are used to obtain the solution. The effects of both non-rigid supports elastic deformations and nonlocal parameter on the natural frequencies are studied also for higher vibrations modes. The comparison between the solutions of the proposed model and those available in the literature shows an excellent agreement, and new insightful results and discussions are presented.

Keywords:
elastically constrained beam, nanostructures, natural frequency, size effects, well-posed nonlocal formulation