Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams

When adjoint eigenfunctions are adopted as the test functions in Green's second identity for the Euler‐Bernoulli beam equation, we can develop a quite simple noniterative numerical algorithm to recover an unknown space‐dependent external force H(x) exerted on the beam. The spatial parts of the adjoint eigenfunctions are used as the bases to expand the unknown function H(x), where we view the two end values of H(x) as two unknown coefficients for the simply supported and hinged‐clamped beams, and the left end value of H(x) as an unknown coefficient for the cantilevered beam. We can derive closed‐form solutions of the expansion coefficients, and thus closed‐form series solutions of H(x). Consequently, we have a noniterative method to recover the unknown force H(x) supplemented by the noisy final time displacement data. Numerical examples demonstrate the accuracy, efficiency, and robustness of the novel methods in the recovery of unknown forces on the simply supported, cantilevered, and hinged‐clamped beams.

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“…where the components C ij , i, j = 1, 2 were defined in Equation (19). In this regard, the inversion in Equation (21) can be obtained exactly.…”

Section: Closed-form Expansion Coefficients Methodsmentioning

confidence: 99%

“…[3][4][5][6][7][8] The parameter identification issue is a subclass of inverse problems, which was studied as the inverse spectral problems. [9][10][11] Besides, some researchers studied different models of the Euler-Bernoulli nanobeams [12][13][14][15][16][17][18][19][20][21] in a microscopic scale.…”

Section: Introductionmentioning

confidence: 99%

A simple noniterative method for recovering a space‐dependent load on the Euler‐Bernoulli beam equation

Liu

Jhao

Chang

2018

Math Methods in App Sciences

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“…Nanoscale structures including nanoscale rods [1], rings [2], beams [3][4][5][6][7][8], plates [9,10], and shells [11,12] have been utilised as the fundamental structural parts of many nanoelectromechanical systems (NEMS). Nanomechanical resonators [13,14], nanoscale mass sensors [15], electromechanical nanoactuators [16], and nanoenergy harvesters [17] are salient examples of these NEMS-based devices.…”

Section: Introductionmentioning

confidence: 99%

A review on the mechanics of nanostructures

Farajpour

Ghayesh

Farokhi

2018

International Journal of Engineering Science

192

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Understanding the mechanical behaviour of nanostructures is of great importance due to their applications in nanodevices such as in nanomechanical resonators, nanoscale mass sensors, electromechanical nanoactuators and nanogenerators. Due to the difficulties of performing accurate experimental measurements at nanoscales and the high computational costs associated with the molecular dynamics simulations, the continuum modelling of nanostructures has attracted a considerable amount of attention. Since size influences have a crucial role in the mechanics of structures at nanoscale levels, classical continuum-based theories have been modified to incorporate these effects. Among various modified continuum-based theories, the nonlocal elasticity and the nonlocal strain gradient elasticity have been employed to estimate the mechanical behaviour of nanostructures. In this review paper, first these two modified elasticity theories are briefly explained. Then, the nonlocal motion equations for different nanostructures including nanorods, nanorings, nanobeams, nanoplates and nanoshells are derived. Several papers which reported on the size-dependent mechanical behaviour of nanostructures using modified continuum models are reviewed. Furthermore, important results reported on the vibration, bending and buckling of nanostructures as well as the results of size-dependent wave propagation analyses are discussed.

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“…On the contrary, integral formulations of the stress-driven type [7] are able to properly address issues relating to the fulfilment of equilbrium conditions and paradoxes debated in the scientific community [8,9]. Moreover, it is pointed out that, at least for a certain class of kernel function, the stress-driven integral formulation can be made equivalent to differential problems [10][11][12]. To ensure this equivalence, suitable constitutive boundary conditions must be set in the correct manner.…”

Section: Introductionmentioning

confidence: 99%

Nonlocal integral thermoelasticity: A thermodynamic framework for functionally graded beams

Barretta

Čanađija

Sciarra

2019

Composite Structures

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An effective nonlocal integral formulation for functionally graded Bernoulli-Euler beams in nonisothermal environment is developed. Both thermal and mechanical loadings are accounted for. The proposed model, of stress-driven integral type, is shown to be governed by a thermodynamically consistent differential problem with proper constitutive boundary conditions. The new thermoelastic strategy is illustrated by investigating a set of examples. It is demonstrated that in nonisothermal statically indeterminate problems rather complex structural behaviours can appear and that both the shift of the neutral surface and nonlocality have a dominating influence at small-scales.

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Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams

Cited by 141 publications

References 52 publications

A simple noniterative method for recovering a space‐dependent load on the Euler‐Bernoulli beam equation

A simple noniterative method for recovering a space‐dependent load on the Euler‐Bernoulli beam equation

A review on the mechanics of nanostructures

Nonlocal integral thermoelasticity: A thermodynamic framework for functionally graded beams

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