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Streszczenie:
A unified approach is applied for determining both strain- and stress-driven differential formulations of Timoshenko nano-beams in presence of loading discontinuities. The consequent models can simulate small scale effects with different types of constitutive laws (such as pure nonlocal, mixture of local and nonlocal phases, and nonlocal gradient). A specific novel feature of the proposed models is the ability to consider loading discontinuities, i.e. points of discontinuities for generalized internal forces occurring in presence of external supports, forces, or couples concentrated at internal points of the nano-beam. To this end, novel constitutive continuity conditions (CCCs) are imposed at the beam interior points of loading discontinuities. CCCs contain integral convolutions of generalized forces or displacements over suitable parts of the nano-beam; they represent a valid alternative to Dirac delta function and are different from the well-known constitutive boundary conditions (CBCs) imposed at the end-points of the nano-beam. Finally, the proposed models are applied for finding closed-form solutions to cases of practical interest.

Słowa kluczowe:
nonlocal gradient elasticity, constitutive boundary conditions, constitutive continuity conditions, nano-beams, NEMS

Streszczenie:
A novel buckling model is formulated for the Bernoulli-Euler nanobeam resting on the Pasternak elastic foundation. The formulation is based on the local-nonlocal stress-driven gradient elasticity theory. In order to incorporate the size-dependency, the strain at each point is defined as the integral convolutions in terms of the stresses and their first-order gradients in all the points, accounting also for the local contribution. The differential form of the nonlocal constitutive equation, together with a set of constitutive boundary conditions, are used to define the buckling equation in terms of transverse displacement, which is solved in closed form. Both variationally consistent and the constitutive boundary conditions are imposed to calculate the buckling loads and the corresponding mode shapes. The predictions of the present model are in agreement with the results available in the literature for the carbon nanotubes based on the molecular dynamics simulations. Insightful results are presented for the first three buckling modes of local-nonlocal nanobeams considering the gradient effects. The distinctive feature of the present model is its capability to capture both stiffening and softening behaviors at the small-scales, which result in, respectively, higher and lower buckling loads of the nanobeams with respect to those of the large-scale beams.

Słowa kluczowe:
nanobeams, nonlocal elasticity, stress gradient, buckling, Pasternak foundation

Streszczenie:
The size-dependent bending of perfectly/imperfectly bonded multilayered/stepwise functionally graded nanobeams, e.g. multiwalled carbon nanotubes with weak van der Waals forces, with any arbitrary numbers of layers, exhibiting different material, geometrical, and length-scale properties, is studied through a layerwise formulation of the stress-driven nonlocal theory of elasticity and the Bernoulli-Euler beam theory. The formulation is also valid for the continuously graded nanobeams, where the through-the-thickness material gradation with any arbitrary distribution is approximated in a stepwise manner through many layers. The size-dependency of each layer is accounted for through nonlocal constitutive relationships, which define the strains at each point as the output of integral convolutions in terms of the stresses in all the points of the layer and a kernel. Linear elastic uncoupled interfacial laws are implemented to model the mechanical response of the interfaces. The size-dependent system of equilibrium equations governing the deformations of the layers are derived and subjected to the variationally consistent edge boundary conditions and the constitutive boundary conditions associated with the stress-driven integral convolution. The formulation is applied to multilayered and sandwich nanobeams and the effects of the interfacial imperfections on the displacement fields and the interfacial displacement jumps are studied. It is found that the interfacial imperfections have greater impact on the field variables of multilayered nanobeams than that of the multilayered beams with the large-scale dimensions.

Słowa kluczowe:
layered nanobeam, discrete layer approach, size-effect, imperfect interface, nonlocal elasticity

Streszczenie:
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.

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

Streszczenie:
In this work, variational formulations are proposed for solving numerically the problem of bending and buckling of Timoshenko nano-beams. The present work belongs to research branch in which the non-local theory of elasticity has been used for analysis of beam-like elements in smart materials, micro-electro-mechanical (MEMS) or nano-electro-mechanical systems (NEMS). In fact, the local beam theory is not adequate to describe the behavior of beam-like elements of smart materials at the nano-scale, so that different non-local models have been proposed in last decades for nano-beams. The nano-beam model considered in this work is a convex combination (mixture) of local and non-local phases. In the non-local phase, the kinematic entities in a point of the nano-beam are expressed as integral convolutions between internal forces and an exponential kernel. The aim is to construct a functional whose stationary condition provides the solution of the problem. Two different functionals are defined: one for the pure non-local model, where the local fraction of the mixture is absent, and the other for the mixture with both local and non-local phases. The Euler equations of the two functionals are derived; then, attention focuses on the mixture model. The functional of the mixture depends on unknown Lagrange multipliers and the Euler equations of the functional provide not only the governing equations of the problem but also the relationships between these Lagrange multipliers and the other variables on which the functional depends. In fact, approximations of the variables of the functional can not be chosen arbitrarily in numerical analyzes but have to satisfy suitable conditions. The Euler equations involving the Lagrange multipliers are essential in the numerical analyzes and suggest the correct approximations that have to be adopted for Lagrange multipliers and the other unknown variables of the functional. The proposed method is verified by comparing numerical solutions with exact solutions in bending problem. Finally, the method is used to determine the buckling load of Timoshenko nano-beams with mixture of phases.

Słowa kluczowe:
non-local elasticity, variational methods, Timoshenko beam, buckling load, smart materials