A Refined Nonlocal Strain Gradient Theory for Assessing Scaling-Dependent Vibration Behavior of Microbeams
A size-dependent Euler–Bernoulli beam model, which
accounts for nonlocal stress field, strain gradient field and higher
order inertia force field, is derived based on the nonlocal strain
gradient theory considering velocity gradient effect. The governing
equations and boundary conditions are derived both in dimensional
and dimensionless form by employed the Hamilton principle. The
analytical solutions based on different continuum theories are
compared. The effect of higher order inertia terms is extremely
significant in high frequency range. It is found that there exists
an asymptotic frequency for the proposed beam model, while for
the nonlocal strain gradient theory the solutions diverge. The effect
of strain gradient field in thickness direction is significant in low
frequencies domain and it cannot be neglected when the material
strain length scale parameter is considerable with beam thickness.
The influence of each of three size effect parameters on the natural
frequencies are investigated. The natural frequencies increase with
the increasing material strain gradient length scale parameter or
decreasing velocity gradient length scale parameter and nonlocal
parameter.
[1] L. Li, Y. Hu, and L. Ling, “Flexural wave propagation in small-scaled
functionally graded beams via a nonlocal strain gradient theory,”
Composite Structures, vol. 133, pp. 1079–1092, 2015.
[2] L. Li, Y. Hu, and L. Ling, “Wave propagation in viscoelastic
single-walled carbon nanotubes with surface effect under magnetic field
based on nonlocal strain gradient theory,” Physica E: Low-dimensional
Systems and Nanostructures, vol. 75, pp. 118–124, 2016.
[3] L. Li and Y. Hu, “Wave propagation in fluid-conveying viscoelastic
carbon nanotubes based on nonlocal strain gradient theory,”
Computational Materials Science, vol. 112, pp. 282–288, 2016.
[4] P. Khodabakhshi and J. Reddy, “A unified beam theory with strain
gradient effect and the von k´arm´an nonlinearity,” ZAMM-Journal
of Applied Mathematics and Mechanics/Zeitschrift f¨ur Angewandte
Mathematik und Mechanik, 2016.
[5] L. Li and Y. Hu, “Nonlinear bending and free vibration analyses of
nonlocal strain gradient beams made of functionally graded material,”
International Journal of Engineering Science, vol. 107, pp. 77–97, 2016.
[6] F. Ebrahimi and M. R. Barati, “Magneto-electro-elastic buckling analysis
of nonlocal curved nanobeams,” The European Physical Journal Plus,
vol. 131, no. 9, p. 346, 2016.
[7] L. Li and Y. Hu, “Buckling analysis of size-dependent nonlinear beams
based on a nonlocal strain gradient theory,” International Journal of
Engineering Science, vol. 97, pp. 84–94, 2015.
[8] F. Ebrahimi and M. R. Barati, “On nonlocal characteristics of curved
inhomogeneous euler–bernoulli nanobeams under different temperature
distributions,” Applied Physics A, vol. 122, no. 10, p. 880, 2016.
[9] F. Ebrahimi and M. R. Barati, “A nonlocal strain gradient refined beam
model for buckling analysis of size-dependent shear-deformable curved
fg nanobeams,” Composite Structures, vol. 159, pp. 174–182, 2017.
[10] S. Guo, Y. He, D. Liu, J. Lei, L. Shen, and Z. Li, “Torsional vibration
of carbon nanotube with axial velocity and velocity gradient effect,”
International Journal of Mechanical Sciences, 2016.
[11] D. Zhang, Y. Lei, and Z. Shen, “Vibration analysis of horn-shaped
single-walled carbon nanotubes embedded in viscoelastic medium under
a longitudinal magnetic field,” International Journal of Mechanical
Sciences, vol. 118, pp. 219–230, 2016.
[12] L. Li, Y. Hu, and X. Li, “Longitudinal vibration of size-dependent rods
via nonlocal strain gradient theory,” International Journal of Mechanical
Sciences, vol. 115, pp. 135–144, 2016.
[13] L. Li, X. Li, and Y. Hu, “Free vibration analysis of nonlocal strain
gradient beams made of functionally graded material,” International
Journal of Engineering Science, vol. 102, pp. 77–92, 2016.
[14] A. C. Eringen, “On differential equations of nonlocal elasticity and
solutions of screw dislocation and surface waves,” Journal of Applied
Physics, vol. 54, no. 9, pp. 4703–4710, 1983.
[15] E. C. Aifantis, “On the role of gradients in the localization of
deformation and fracture,” International Journal of Engineering Science,
vol. 30, no. 10, pp. 1279–1299, 1992.
[16] R. D. Mindlin, “Micro-structure in linear elasticity,” Archive for Rational
Mechanics and Analysis, vol. 16, no. 1, pp. 51–78, 1964.
[17] C. W. Lim, G. Zhang, and J. N. Reddy, “A higher-order nonlocal
elasticity and strain gradient theory and its applications in wave
propagation,” Journal of the Mechanics and Physics of Solids, vol. 78,
pp. 298–313, 2015.
[18] Y. Zhang, C. Wang, and N. Challamel, “Bending, buckling, and vibration
of micro/nanobeams by hybrid nonlocal beam model,” Journal of
Engineering Mechanics, vol. 136, no. 5, pp. 562–574, 2009.
[19] J. Zang, B. Fang, Y.-W. Zhang, T.-Z. Yang, and D.-H. Li, “Longitudinal
wave propagation in a piezoelectric nanoplate considering surface effects
and nonlocal elasticity theory,” Physica E, vol. 63, pp. 147–150, 2014.
[20] B. Wang, Z. Deng, H. Ouyang, and J. Zhou, “Wave propagation analysis
in nonlinear curved single-walled carbon nanotubes based on nonlocal
elasticity theory,” Physica E, vol. 66, pp. 283–292, 2015.
[21] M. Z. Nejad, A. Hadi, and A. Rastgoo, “Buckling analysis of
arbitrary two-directional functionally graded euler–bernoulli nano-beams
based on nonlocal elasticity theory,” International Journal of
Engineering Science, vol. 103, pp. 1–10, 2016. (Online). Available:
http://www.sciencedirect.com/science/article/pii/S0020722516300180
[22] A. Daneshmehr, A. Rajabpoor, and A. Hadi, “Size dependent
free vibration analysis of nanoplates made of functionally graded
materials based on nonlocal elasticity theory with high order theories,”
International Journal of Engineering Science, vol. 95, pp. 23–35, 2015.
[23] F. Ebrahimi and M. R. Barati, “A nonlocal higher-order shear
deformation beam theory for vibration analysis of size-dependent
functionally graded nanobeams,” Arabian Journal Forence &
Engineering, vol. 41, no. 5, pp. 1679–1690, 2015.
[24] N. Challamel and C. Wang, “The small length scale effect for a non-local
cantilever beam: a paradox solved,” Nanotechnology, vol. 19, no. 34, p.
345703, 2008.
[25] J. Fern´andez-S´aez, R. Zaera, J. Loya, and J. Reddy, “Bending of
euler–bernoulli beams using eringens integral formulation: a paradox
resolved,” International Journal of Engineering Science, vol. 99, pp.
107–116, 2016.
[26] N. Challamel, Z. Zhang, C. Wang, J. Reddy, Q. Wang, T. Michelitsch,
and B. Collet, “On nonconservativeness of eringens nonlocal elasticity
in beam mechanics: correction from a discrete-based approach,” Archive
of Applied Mechanics, vol. 84, no. 9-11, pp. 1275–1292, 2014.
[27] E. Benvenuti and A. Simone, “One-dimensional nonlocal and gradient
elasticity: closed-form solution and size effect,” Mechanics Research
Communications, vol. 48, pp. 46–51, 2013.
[28] F. Yang, A. Chong, D. Lam, and P. Tong, “Couple stress based strain
gradient theory for elasticity,” International Journal of Solids and
Structures, vol. 39, no. 10, pp. 2731–2743, 2002. [29] N. Fleck, G. Muller, M. Ashby, and J. Hutchinson, “Strain gradient
plasticity: theory and experiment,” Acta Metallurgica et Materialia,
vol. 42, no. 2, pp. 475–487, 1994.
[30] N. Fleck and J. Hutchinson, “A phenomenological theory for strain
gradient effects in plasticity,” Journal of the Mechanics and Physics of
Solids, vol. 41, no. 12, pp. 1825 – 1857, 1993. (Online). Available:
http://www.sciencedirect.com/science/article/pii/002250969390072N
[31] C. Polizzotto, “A gradient elasticity theory for second-grade materials
and higher order inertia,” International Journal of Solids and Structures,
vol. 49, no. 15, pp. 2121–2137, 2012.
[32] C. Polizzotto, “A second strain gradient elasticity theory with second
velocity gradient inertia–part i: Constitutive equations and quasi-static
behavior,” International Journal of Solids and Structures, vol. 50, no. 24,
pp. 3749–3765, 2013.
[33] C. Polizzotto, “A second strain gradient elasticity theory with second
velocity gradient inertia–part ii: Dynamic behavior,” International
Journal of Solids and Structures, vol. 50, no. 24, pp. 3766–3777, 2013.
[34] H. Askes and E. C. Aifantis, “Gradient elasticity in statics and dynamics:
an overview of formulations, length scale identification procedures, finite
element implementations and new results,” International Journal of
Solids and Structures, vol. 48, no. 13, pp. 1962–1990, 2011.
[35] H. Askes and E. C. Aifantis, “Gradient elasticity and flexural wave
dispersion in carbon nanotubes,” Physical Review B, vol. 80, no. 19,
p. 195412, 2009.
[36] S. Papargyri-Beskou, D. Polyzos, and D. Beskos, “Wave dispersion in
gradient elastic solids and structures: a unified treatment,” International
Journal of Solids and Structures, vol. 46, no. 21, pp. 3751–3759, 2009.
[37] D. Polyzos and D. Fotiadis, “Derivation of mindlins first and second
strain gradient elastic theory via simple lattice and continuum models,”
International Journal of Solids and Structures, vol. 49, no. 3, pp.
470–480, 2012.
[38] S. N. Iliopoulos, D. G. Aggelis, and D. Polyzos, “Wave dispersion in
fresh and hardened concrete through the prism of gradient elasticity,”
International Journal of Solids and Structures, vol. 78, pp. 149–159,
2016.
[1] L. Li, Y. Hu, and L. Ling, “Flexural wave propagation in small-scaled
functionally graded beams via a nonlocal strain gradient theory,”
Composite Structures, vol. 133, pp. 1079–1092, 2015.
[2] L. Li, Y. Hu, and L. Ling, “Wave propagation in viscoelastic
single-walled carbon nanotubes with surface effect under magnetic field
based on nonlocal strain gradient theory,” Physica E: Low-dimensional
Systems and Nanostructures, vol. 75, pp. 118–124, 2016.
[3] L. Li and Y. Hu, “Wave propagation in fluid-conveying viscoelastic
carbon nanotubes based on nonlocal strain gradient theory,”
Computational Materials Science, vol. 112, pp. 282–288, 2016.
[4] P. Khodabakhshi and J. Reddy, “A unified beam theory with strain
gradient effect and the von k´arm´an nonlinearity,” ZAMM-Journal
of Applied Mathematics and Mechanics/Zeitschrift f¨ur Angewandte
Mathematik und Mechanik, 2016.
[5] L. Li and Y. Hu, “Nonlinear bending and free vibration analyses of
nonlocal strain gradient beams made of functionally graded material,”
International Journal of Engineering Science, vol. 107, pp. 77–97, 2016.
[6] F. Ebrahimi and M. R. Barati, “Magneto-electro-elastic buckling analysis
of nonlocal curved nanobeams,” The European Physical Journal Plus,
vol. 131, no. 9, p. 346, 2016.
[7] L. Li and Y. Hu, “Buckling analysis of size-dependent nonlinear beams
based on a nonlocal strain gradient theory,” International Journal of
Engineering Science, vol. 97, pp. 84–94, 2015.
[8] F. Ebrahimi and M. R. Barati, “On nonlocal characteristics of curved
inhomogeneous euler–bernoulli nanobeams under different temperature
distributions,” Applied Physics A, vol. 122, no. 10, p. 880, 2016.
[9] F. Ebrahimi and M. R. Barati, “A nonlocal strain gradient refined beam
model for buckling analysis of size-dependent shear-deformable curved
fg nanobeams,” Composite Structures, vol. 159, pp. 174–182, 2017.
[10] S. Guo, Y. He, D. Liu, J. Lei, L. Shen, and Z. Li, “Torsional vibration
of carbon nanotube with axial velocity and velocity gradient effect,”
International Journal of Mechanical Sciences, 2016.
[11] D. Zhang, Y. Lei, and Z. Shen, “Vibration analysis of horn-shaped
single-walled carbon nanotubes embedded in viscoelastic medium under
a longitudinal magnetic field,” International Journal of Mechanical
Sciences, vol. 118, pp. 219–230, 2016.
[12] L. Li, Y. Hu, and X. Li, “Longitudinal vibration of size-dependent rods
via nonlocal strain gradient theory,” International Journal of Mechanical
Sciences, vol. 115, pp. 135–144, 2016.
[13] L. Li, X. Li, and Y. Hu, “Free vibration analysis of nonlocal strain
gradient beams made of functionally graded material,” International
Journal of Engineering Science, vol. 102, pp. 77–92, 2016.
[14] A. C. Eringen, “On differential equations of nonlocal elasticity and
solutions of screw dislocation and surface waves,” Journal of Applied
Physics, vol. 54, no. 9, pp. 4703–4710, 1983.
[15] E. C. Aifantis, “On the role of gradients in the localization of
deformation and fracture,” International Journal of Engineering Science,
vol. 30, no. 10, pp. 1279–1299, 1992.
[16] R. D. Mindlin, “Micro-structure in linear elasticity,” Archive for Rational
Mechanics and Analysis, vol. 16, no. 1, pp. 51–78, 1964.
[17] C. W. Lim, G. Zhang, and J. N. Reddy, “A higher-order nonlocal
elasticity and strain gradient theory and its applications in wave
propagation,” Journal of the Mechanics and Physics of Solids, vol. 78,
pp. 298–313, 2015.
[18] Y. Zhang, C. Wang, and N. Challamel, “Bending, buckling, and vibration
of micro/nanobeams by hybrid nonlocal beam model,” Journal of
Engineering Mechanics, vol. 136, no. 5, pp. 562–574, 2009.
[19] J. Zang, B. Fang, Y.-W. Zhang, T.-Z. Yang, and D.-H. Li, “Longitudinal
wave propagation in a piezoelectric nanoplate considering surface effects
and nonlocal elasticity theory,” Physica E, vol. 63, pp. 147–150, 2014.
[20] B. Wang, Z. Deng, H. Ouyang, and J. Zhou, “Wave propagation analysis
in nonlinear curved single-walled carbon nanotubes based on nonlocal
elasticity theory,” Physica E, vol. 66, pp. 283–292, 2015.
[21] M. Z. Nejad, A. Hadi, and A. Rastgoo, “Buckling analysis of
arbitrary two-directional functionally graded euler–bernoulli nano-beams
based on nonlocal elasticity theory,” International Journal of
Engineering Science, vol. 103, pp. 1–10, 2016. (Online). Available:
http://www.sciencedirect.com/science/article/pii/S0020722516300180
[22] A. Daneshmehr, A. Rajabpoor, and A. Hadi, “Size dependent
free vibration analysis of nanoplates made of functionally graded
materials based on nonlocal elasticity theory with high order theories,”
International Journal of Engineering Science, vol. 95, pp. 23–35, 2015.
[23] F. Ebrahimi and M. R. Barati, “A nonlocal higher-order shear
deformation beam theory for vibration analysis of size-dependent
functionally graded nanobeams,” Arabian Journal Forence &
Engineering, vol. 41, no. 5, pp. 1679–1690, 2015.
[24] N. Challamel and C. Wang, “The small length scale effect for a non-local
cantilever beam: a paradox solved,” Nanotechnology, vol. 19, no. 34, p.
345703, 2008.
[25] J. Fern´andez-S´aez, R. Zaera, J. Loya, and J. Reddy, “Bending of
euler–bernoulli beams using eringens integral formulation: a paradox
resolved,” International Journal of Engineering Science, vol. 99, pp.
107–116, 2016.
[26] N. Challamel, Z. Zhang, C. Wang, J. Reddy, Q. Wang, T. Michelitsch,
and B. Collet, “On nonconservativeness of eringens nonlocal elasticity
in beam mechanics: correction from a discrete-based approach,” Archive
of Applied Mechanics, vol. 84, no. 9-11, pp. 1275–1292, 2014.
[27] E. Benvenuti and A. Simone, “One-dimensional nonlocal and gradient
elasticity: closed-form solution and size effect,” Mechanics Research
Communications, vol. 48, pp. 46–51, 2013.
[28] F. Yang, A. Chong, D. Lam, and P. Tong, “Couple stress based strain
gradient theory for elasticity,” International Journal of Solids and
Structures, vol. 39, no. 10, pp. 2731–2743, 2002. [29] N. Fleck, G. Muller, M. Ashby, and J. Hutchinson, “Strain gradient
plasticity: theory and experiment,” Acta Metallurgica et Materialia,
vol. 42, no. 2, pp. 475–487, 1994.
[30] N. Fleck and J. Hutchinson, “A phenomenological theory for strain
gradient effects in plasticity,” Journal of the Mechanics and Physics of
Solids, vol. 41, no. 12, pp. 1825 – 1857, 1993. (Online). Available:
http://www.sciencedirect.com/science/article/pii/002250969390072N
[31] C. Polizzotto, “A gradient elasticity theory for second-grade materials
and higher order inertia,” International Journal of Solids and Structures,
vol. 49, no. 15, pp. 2121–2137, 2012.
[32] C. Polizzotto, “A second strain gradient elasticity theory with second
velocity gradient inertia–part i: Constitutive equations and quasi-static
behavior,” International Journal of Solids and Structures, vol. 50, no. 24,
pp. 3749–3765, 2013.
[33] C. Polizzotto, “A second strain gradient elasticity theory with second
velocity gradient inertia–part ii: Dynamic behavior,” International
Journal of Solids and Structures, vol. 50, no. 24, pp. 3766–3777, 2013.
[34] H. Askes and E. C. Aifantis, “Gradient elasticity in statics and dynamics:
an overview of formulations, length scale identification procedures, finite
element implementations and new results,” International Journal of
Solids and Structures, vol. 48, no. 13, pp. 1962–1990, 2011.
[35] H. Askes and E. C. Aifantis, “Gradient elasticity and flexural wave
dispersion in carbon nanotubes,” Physical Review B, vol. 80, no. 19,
p. 195412, 2009.
[36] S. Papargyri-Beskou, D. Polyzos, and D. Beskos, “Wave dispersion in
gradient elastic solids and structures: a unified treatment,” International
Journal of Solids and Structures, vol. 46, no. 21, pp. 3751–3759, 2009.
[37] D. Polyzos and D. Fotiadis, “Derivation of mindlins first and second
strain gradient elastic theory via simple lattice and continuum models,”
International Journal of Solids and Structures, vol. 49, no. 3, pp.
470–480, 2012.
[38] S. N. Iliopoulos, D. G. Aggelis, and D. Polyzos, “Wave dispersion in
fresh and hardened concrete through the prism of gradient elasticity,”
International Journal of Solids and Structures, vol. 78, pp. 149–159,
2016.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:75051", author = "Xiaobai Li and Li Li and Yujin Hu and Weiming Deng and Zhe Ding", title = "A Refined Nonlocal Strain Gradient Theory for Assessing Scaling-Dependent Vibration Behavior of Microbeams", abstract = "A size-dependent Euler–Bernoulli beam model, which
accounts for nonlocal stress field, strain gradient field and higher
order inertia force field, is derived based on the nonlocal strain
gradient theory considering velocity gradient effect. The governing
equations and boundary conditions are derived both in dimensional
and dimensionless form by employed the Hamilton principle. The
analytical solutions based on different continuum theories are
compared. The effect of higher order inertia terms is extremely
significant in high frequency range. It is found that there exists
an asymptotic frequency for the proposed beam model, while for
the nonlocal strain gradient theory the solutions diverge. The effect
of strain gradient field in thickness direction is significant in low
frequencies domain and it cannot be neglected when the material
strain length scale parameter is considerable with beam thickness.
The influence of each of three size effect parameters on the natural
frequencies are investigated. The natural frequencies increase with
the increasing material strain gradient length scale parameter or
decreasing velocity gradient length scale parameter and nonlocal
parameter.", keywords = "Euler-Bernoulli Beams, free vibration, higher order
inertia, nonlocal strain gradient theory, velocity gradient.", volume = "11", number = "3", pages = "551-11", }
