Bending analysis of functionally graded nanobeams based on stress-driven nonlocal model incorporating surface energy effects

Penna, Rosa

doi:10.1016/j.ijengsci.2023.103887

Cited by 24 publications

(7 citation statements)

References 50 publications

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“…The use of Hamilton’s principle allows us to obtain the governing equation of the free vibrations problem [ 44 ]

where

the z-component of the unit normal vector

, which is the outward normal to the cross-section lateral surface [ 43 ].…”

Section: Problem Formulationmentioning

confidence: 99%

“…In this section, we provide a brief review of the surface stress-driven nonlocal model (SSDM) as outlined in [ 43 ]. Assuming a purely elastic constitutive behavior, the formulation of the surface stress-driven nonlocal model involves defining the bending curvature, χ, through the integral convolution, as detailed in the same reference [ 43 ]

where

and

are the positions of points of the domain of the Euclidean space occupied by the FG nanobeam at time t ;

is an averaging kernel depending on the characteristic length of material,

;

and M * are, respectively, the equivalent (bulk and surface) bending stiffness and the applied bending moment, defined as

being

and

…”

Section: Surface Stress-driven Model For Free Vibrations Analysismentioning

confidence: 99%

“…As widely recognized, a specific function kernel, denoted as

, is chosen to be

for smooth source fields

in the domain

; the elastic curvature χ, as expressed in Equation (17), is equivalent to the following second-order differential equations, as outlined in [ 43 ]

…”

Section: Surface Stress-driven Model For Free Vibrations Analysismentioning

confidence: 99%

“…n z is the z-component of the unit normal vector n, which is the outward normal to the cross-section lateral surface [43].…”

Section: Governing Equationsmentioning

confidence: 99%

“…Furthermore, Penna [ 43 ] recently extended the SDM model by coupling it with the SET to create the Surface Stress-Driven Model (SSDM). This model, well-posed mathematically, not only captures long-range interactions but also addresses surface effects.…”

Section: Introductionmentioning

confidence: 99%

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Application of Surface Stress-Driven Model for Higher Vibration Modes of Functionally Graded Nanobeams

Lovisi,

Feo,

Lambiase

et al. 2024

Nanomaterials

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This paper employs a surface stress-driven nonlocal theory to investigate the synergistic impact of long-range interaction and surface energy on higher vibration modes of Bernoulli–Euler nanobeams made of functionally graded material. It takes into account surface effects such as the surface modulus of elasticity, residual surface stresses, surface density, and rotary inertia. The governing equation is derived through the application of Hamilton’s principle. The novelty of this work lies in its pioneering approach to studying higher-order vibrations, carefully considering the combination of long-range interactions and surface energy in nanobeams of functionally graded materials through a well-posed mathematical model of nonlocal elasticity. This study conducts a parametric investigation, examining the effects of the nonlocal parameter and the material gradient index for four static schemes: Cantilever, Simply-Supported, Clamped-Pinned and Clamped-Clamped nanobeams. The outcomes are presented and discussed, highlighting the normalized nonlocal natural frequencies for the second through fifth modes of vibration in each case under study. In particular, this study illustrates the central role of surface effects in the dynamic response of nanobeams, emphasizing the importance of considering them. Furthermore, the parametric analysis reveals that the dynamic response is influenced by the combined effects of the nonlocal parameter, the material gradient index, the shapes of the cross-sections considered, as well as the static scheme analyzed.

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“…The use of Hamilton’s principle allows us to obtain the governing equation of the free vibrations problem [ 44 ]

where

the z-component of the unit normal vector

, which is the outward normal to the cross-section lateral surface [ 43 ].…”

Section: Problem Formulationmentioning

confidence: 99%

where

and

are the positions of points of the domain of the Euclidean space occupied by the FG nanobeam at time t ;

is an averaging kernel depending on the characteristic length of material,

;

and M * are, respectively, the equivalent (bulk and surface) bending stiffness and the applied bending moment, defined as

being

and

…”

Section: Surface Stress-driven Model For Free Vibrations Analysismentioning

confidence: 99%

“…As widely recognized, a specific function kernel, denoted as

, is chosen to be

for smooth source fields

in the domain

; the elastic curvature χ, as expressed in Equation (17), is equivalent to the following second-order differential equations, as outlined in [ 43 ]

…”

Section: Surface Stress-driven Model For Free Vibrations Analysismentioning

confidence: 99%

“…n z is the z-component of the unit normal vector n, which is the outward normal to the cross-section lateral surface [43].…”

Section: Governing Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Application of Surface Stress-Driven Model for Higher Vibration Modes of Functionally Graded Nanobeams

Lovisi,

Feo,

Lambiase

et al. 2024

Nanomaterials

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Size‐dependent buckling loads of non‐uniform Bernoulli–Euler beams with elastic boundary conditions and thermal effects based on two‐phase local/nonlocal elastic model

Xu,

Chen,

Zhang

2024

Z Angew Math Mech

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We present predictive models of the size‐dependent buckling loads of non‐uniform Bernoulli–Euler beams under thermal effects based on the two‐phase local/nonlocal elastic model. The beam ends are assumed to be constrained by elastic springs with translational and rotational stiffness to simulate general boundary conditions. In contrast to most literature in this field, both the bending and thermal deformations of the beams are simultaneously considered to be two‐phase local/nonlocal of two phases, that is, the thermal effect is taken as equivalent to a size‐dependent thermal load. By using the fully equivalent differential form of the local/nonlocal equation with a set of constitutive boundary conditions, the problem is solved numerically with the aid of the generalized differential quadrature method (GDQM). Through conducting validation study, several parametric studies are given for examining the effects of the slope of beams’ thickness variation, nonlocal parameter, and elastically supported conditions on the buckling loads of non‐uniform beams. The results show that constrained stiffness has a drastic influence on the critical bucking loads of the beams. Furthermore, the consideration of the two‐phase thermal load will further reduce the actual buckling load of the beams.

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Wave solutions in nonlocal integral beams

Barretta,

Iuorio,

Luciano

et al. 2024

Continuum Mech. Thermodyn.

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Wave propagation in slender beams is addressed in the framework of nonlocal continuum mechanics. The elastodynamic problem is formulated exploiting consistent methodologies of pure integral, mixture and nonlocal strain gradient elasticity. Relevant wave solutions are analytically provided, with peculiar attention to reflection and near field phenomena occurring in presence of boundaries. Notably, the solution field is got as superimposition of incident, reflected, primary near field and secondary near field waves. The latter contribution represents a further effect due to the size dependent mechanical behaviour. Limit responses for vanishing nonlocal parameter are analytically evaluated, consistently showing a zero amplitude of the secondary near field wave. Parametric analyses are carried out to show how length scale parameter, amplitude of incident wave and geometric and elastic properties of the beam affect the amplitudes of reflected, primary near field and secondary near field waves. The results obtained exploiting different nonlocal integral elasticity approaches are compared and discussed.

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Bending analysis of functionally graded nanobeams based on stress-driven nonlocal model incorporating surface energy effects

Cited by 24 publications

References 50 publications

Application of Surface Stress-Driven Model for Higher Vibration Modes of Functionally Graded Nanobeams

Application of Surface Stress-Driven Model for Higher Vibration Modes of Functionally Graded Nanobeams

Size‐dependent buckling loads of non‐uniform Bernoulli–Euler beams with elastic boundary conditions and thermal effects based on two‐phase local/nonlocal elastic model

Wave solutions in nonlocal integral beams

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