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Random vibrations of stress-driven nonlocal beams with external damping

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Abstract

Stochastic flexural vibrations of small-scale Bernoulli–Euler beams with external damping are investigated by stress-driven nonlocal mechanics. Damping effects are simulated considering viscous interactions between beam and surrounding environment. Loadings are modeled by accounting for their random nature. Such a dynamic problem is characterized by a stochastic partial differential equation in space and time governing time-evolution of the relevant displacement field. Differential eigenanalyses are performed to evaluate modal time coordinates and mode shapes, providing a complete stochastic description of response solutions. Closed-form expressions of power spectral density, correlation function, stationary and non-stationary variances of displacement fields are analytically detected. Size-dependent dynamic behaviour is assessed in terms of stiffness, variance and power spectral density of displacements. The outcomes can be useful for design and optimization of structural components of modern small-scale devices, such as micro- and nano-electro-mechanical-systems.

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References

  1. Pourasghar A, Chen Z (2019) Effect of hyperbolic heat conduction on the linear and nonlinear vibration of CNT reinforced size-dependent functionally graded microbeams. Int J Eng Sci 137:57–72

    MathSciNet  MATH  Google Scholar 

  2. Xia X, Weng GJ, Hou D, Wen W (2019) Tailoring the frequency-dependent electrical conductivity and dielectric permittivity of CNT-polymer nanocomposites with nanosized particles. Int J Eng Sci 142:1–19

    MathSciNet  MATH  Google Scholar 

  3. Mojahedi M (2017) Size dependent dynamic behaviour of electrostatically actuated microbridges. Int J Eng Sci 111:74–85

    MathSciNet  MATH  Google Scholar 

  4. Moradweysi P, Ansari R, Hosseini K, Sadeghi F (2018) Application of modified Adomian decomposition method to pull-in instability of nano-switches using nonlocal Timoshenko beam theory. Appl Math Model 54:594–604

    MathSciNet  MATH  Google Scholar 

  5. Hosseini SM (2018) Analytical solution for nonlocal coupled thermoelasticity analysis in a heat-affected MEMS/NEMS beam resonator based on Green–Naghdi theory. Appl Math Model 57:21–36

    MathSciNet  MATH  Google Scholar 

  6. De Bellis ML, Bacigalupo A, Zavarise G (2019) Characterization of hybrid piezoelectric nanogenerators through asymptotic homogenization. Comput Methods Appl Mech Eng 355:1148–1186

    MathSciNet  MATH  Google Scholar 

  7. Natsuki T, Urakami K (2019) Analysis of vibration frequency of carbon nanotubes used as nano-force sensors considering clamped boundary condition. Electronics 8(10):1082

    Google Scholar 

  8. Mohammadian M, Abolbashari MH, Hosseini SM (2019) Application of hetero junction CNTs as mass nanosensor using nonlocal strain gradient theory: an analytical solution. Appl Math Model 76:26–49

    MathSciNet  MATH  Google Scholar 

  9. Tran N, Ghayesh MH, Arjomandi M (2018) Ambient vibration energy harvesters: a review on nonlinear techniques for performance enhancement. Int J Eng Sci 127:162–185

    MathSciNet  MATH  Google Scholar 

  10. Basutkar R (2019) Analytical modelling of a nanoscale series-connected bimorph piezoelectric energy harvester incorporating the flexoelectric effect. Int J Eng Sci 139:42–61

    MathSciNet  MATH  Google Scholar 

  11. Ghayesh MH, Farokhi H (2020) Nonlinear broadband performance of energy harvesters. Int J Eng Sci 147:103202

    MathSciNet  MATH  Google Scholar 

  12. Farajpour A, Ghayesh MH, Farokhi H (2018) A review on the mechanics of nanostructures. Int J Eng Sci 133:231–263

    MathSciNet  MATH  Google Scholar 

  13. Bauer S, Pittrof A, Tsuchiya H, Schmuki P (2011) Size-effects in TiO2 nanotubes: diameter dependent anatase/rutile stabilization. Electrochem Commun 13:538–541

    Google Scholar 

  14. Kiang C, Endo M, Ajayan P, Dresselhaus G, Dresselhaus M (1998) Size effects in carbon nanotubes. Phys Rev Lett 81:1869–1872

    Google Scholar 

  15. Xiao S, Hou W (2006) Studies of size effects on carbon nanotubes’ mechanical properties by using different potential functions. Fuller Nanotub Carbon Nanostruct 14:9–16

    Google Scholar 

  16. Zienert A, Schuster J, Streiter R, Gessner T (2010) Transport in carbon nanotubes: contact models and size effects. Physica Status Solidi B Basic Solid State Phys 247:3002–3005

    Google Scholar 

  17. Chowdhury R, Adhikari S, Wang C, Scarpa F (2010) A molecular mechanics approach for the vibration of single-walled carbon nanotubes. Comput Mater Sci 48:730–735

    Google Scholar 

  18. Tang C, Meng L, Sun L, Zhang K, Zhong J (2008) Molecular dynamics study of ripples in graphene nanoribbons on 6H-SiC(0001): temperature and size effects. J Appl Phys 104:113536

    Google Scholar 

  19. Marotti de Sciarra F (2009) A nonlocal model with strain-based damage. Int J Solids Struct 46(22–23):4107–4122

    MathSciNet  MATH  Google Scholar 

  20. Wang LF, Hu HY (2005) Flexural wave propagation in single-walled carbon nanotube. Phys Rev B 71(19):195412–195418

    Google Scholar 

  21. Lu P, Lee HP, Lu C, Zhang PQ (2007) Application of nonlocal beam models for carbon nanotubes. Int J Solids Struct 44(16):5289–5300

    MATH  Google Scholar 

  22. Alotta G, Failla G, Zingales M (2014) Finite element method for a nonlocal Timoshenko beam model. Finite Elem Anal Des 89:77–92

    Google Scholar 

  23. Marotti de Sciarra F (2014) Finite element modelling of nonlocal beams. Physica E 59:144–149

    Google Scholar 

  24. Ansari R, Sahmani S, Arash B (2010) Nonlocal plate model for free vibrations of single-layered graphene sheets. Phys Lett A 375:53–62

    Google Scholar 

  25. Murmu T, Adhikari S (2011) Nonlocal vibration of carbon nanotubes with attached buckyballs at tip. Mech Res Commun 38:62–67

    MATH  Google Scholar 

  26. Ansari R, Sahmani S (2012) Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models. Commun Nonlinear Sci Numer Simul 17:1965–1979

    MathSciNet  Google Scholar 

  27. Lakes RS (1991) Experimental micro mechanics methods for conventional and negative Poissons ratio cellular solids as Cosserat continua. J Eng Mater Technol 113(1):148–155

    Google Scholar 

  28. Arash B, Wang Q (2012) A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comput Mater Sci 51(1):303–313

    Google Scholar 

  29. Rogula D (1965) Influence of spatial acoustic dispersion on dynamical properties of dislocations. Bull Acad Pol Sci Ser Sci Tech 13:337–343

    Google Scholar 

  30. Rogula D (1982) Introduction to nonlocal theory of material media. In: Rogula D (ed) Nonlocal theory of material media. CISM courses and lectures, vol 268. Springer, Wien, pp 125–222

    Google Scholar 

  31. Eringen AC (1972) Linear theory of nonlocal elasticity and dispersion of plane waves. Int J Eng Sci 10:425–435

    MATH  Google Scholar 

  32. Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703

    Google Scholar 

  33. Tricomi FG (1957) Integral equations. Reprinted by Dover Books on Mathematics, Interscience, New-York, USA, p 1985

  34. Polyanin AD, Manzhirov AV (2008) Handbook of integral equations, 2nd edn. Chapman & Hall/CRC, Boca Raton

    MATH  Google Scholar 

  35. Romano G, Barretta R, Diaco M, Marotti de Sciarra F (2017) Constitutive boundary conditions and paradoxes in nonlocal elastic nano-beams. Int J Mech Sci 121:151–156

    Google Scholar 

  36. Challamel N, Wang CM (2008) The small length scale effect for a non-local cantilever beam: a paradox solved. Nanotechnology 19:345703

    Google Scholar 

  37. Fernández-Sáez J, Zaera R, Loya JA, Reddy JN (2016) Bending of Euler–Bernoulli beams using Eringen’s integral formulation: a paradox resolved. Int J Eng Sci 99:107–116

    MathSciNet  MATH  Google Scholar 

  38. Borino G, Failla B, Parrinello F (2003) A symmetric nonlocal damage theory. Int J Solids Struct 40:3621–3645

    MATH  Google Scholar 

  39. Khodabakhshi P, Reddy JN (2015) A unified integro-differential nonlocal model. Int J Eng Sci 95:60–75

    MathSciNet  MATH  Google Scholar 

  40. Lam DCC, Yang F, Chong ACM, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51(8):1477–1508

    MATH  Google Scholar 

  41. Challamel N, Reddy JN, Wang CM (2016) Eringen’s stress gradient model for bending of nonlocal beams. J Eng Mech 142(12):04016095

    Google Scholar 

  42. Numanoǧlu HM, Akgöz B, Civalek Ö (2018) On dynamic analysis of nanorods. Int J Eng Sci 130:33–50

    Google Scholar 

  43. Cornacchia F, Fantuzzi N, Luciano R, Penna R (2019) Solution for cross- and angle-ply laminated Kirchhoff nano plates in bending using strain gradient theory. Compos B Eng 173:107006

    Google Scholar 

  44. Cornacchia F, Fabbrocino F, Fantuzzi N, Luciano R, Penna R (2019) Analytical solution of cross- and angle-ply nano plates with strain gradient theory for linear vibrations and buckling. Mech Adv Mater Struct. https://doi.org/10.1080/15376494.2019.1655613

    Article  Google Scholar 

  45. Lim CW, Zhang G, Reddy JN (2015) A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J Mech Phys Solids 78:298–313

    MathSciNet  MATH  Google Scholar 

  46. Barretta R, Marotti de Sciarra F (2018) Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams. Int J Eng Sci 130:187–198

    MathSciNet  MATH  Google Scholar 

  47. Apuzzo A, Barretta R, Faghidian SA, Luciano R, Marotti de Sciarra F (2018) Free vibrations of elastic beams by modified nonlocal strain gradient theory. Int J Eng Sci 133:99–108

    MathSciNet  MATH  Google Scholar 

  48. Polizzotto C, Fuschi P, Pisano AA (2004) A strain-difference-based nonlocal elasticity model. Int J Solids Struct 41:2383–2401

    MATH  Google Scholar 

  49. Fuschi P, Pisano AA, Polizzotto C (2019) Size effects of small-scale beams in bending addressed with a strain-difference based nonlocal elasticity theory. Int J Mech Sci 151:661–671

    Google Scholar 

  50. Marotti de Sciarra F (2009) On non-local and non-homogeneous elastic continua. Int J Solids Struct 46(3–4):651–676

    MathSciNet  MATH  Google Scholar 

  51. Di Paola M, Zingales M (2008) Long-range cohesive interactions of non-local continuum faced by fractional calculus. Int J Solids Struct 45:5642–5659

    MATH  Google Scholar 

  52. Di Paola M, Pirrotta A, Zingales M (2010) Mechanically-based approach to non-local elasticity: variational principles. Int J Solids Struct 47(5):539–548

    MATH  Google Scholar 

  53. Di Paola M, Failla G, Zingales M (2013) Non-local stiffness and damping models for shear-deformable beams. Eur J Mech A/Solids 40:69–83

    MathSciNet  MATH  Google Scholar 

  54. Failla G, Sofi A, Zingales M (2015) A new displacement-based framework for non-local Timoshenko beams. Meccanica 50(8):2103–2122

    MathSciNet  MATH  Google Scholar 

  55. Romano G, Barretta R (2017) Nonlocal elasticity in nanobeams: the stress-driven integral model. Int J Eng Sci 115:14–27

    MathSciNet  MATH  Google Scholar 

  56. Barretta R, Marotti de Sciarra F, Vaccaro MS (2019) On nonlocal mechanics of curved elastic beams. Int J Eng Sci 144:103–140

    MathSciNet  MATH  Google Scholar 

  57. Romano G, Barretta R (2017) Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams. Compos B Eng 114:184–188

    Google Scholar 

  58. Barretta R, Čanadija M, Feo L, Luciano R, Marotti de Sciarra F, Penna R (2018) Exact solutions of inflected functionally graded nano-beams in integral elasticity. Compos B 142:273–286

    Google Scholar 

  59. Apuzzo A, Barretta R, Luciano R, Marotti de Sciarra F, Penna R (2017) Free vibrations of Bernoulli–Euler nano-beams by the stress-driven nonlocal integral model. Compos B Eng 123:105–111

    Google Scholar 

  60. Lee J, Lin C (2010) The magnetic viscous damping effect on the natural frequency of a beam plate subject to an in-plane magnetic field. J Appl Mech 77:011014

    Google Scholar 

  61. Chen C, Ma M, Liu J, Zheng Q, Xu Z (2011) Viscous damping of nanobeam resonators: humidity, thermal noise, and a paddling effect. J Appl Phys 110:034320

    Google Scholar 

  62. Di Paola M, Fiore V, Pinnola FP, Valenza A (2014) On the influence of the initial ramp for a correct definition of the parameters of the fractional viscoelastic material. Mech Mater 69:63–70

    Google Scholar 

  63. Di Mino G, Airey G, Di Paola M, Pinnola FP, D’Angelo G, Lo Presti D (2016) Linear and nonlinear fractional hereditary constitutive laws of asphalt mixtures. J Civ Eng Manag 22(7):882–889

    Google Scholar 

  64. Calleja M, Kosaka P, San Paulo A, Tamayo J (2012) Challenges for nanomechanical sensors in biological detection. Nanoscale 4:4925–4938

    Google Scholar 

  65. Baidyk T et al (2005) MEMS/NEMS. In: Leondes CT (ed) Handbook techniques and applications. University of California, Los Angeles

    Google Scholar 

  66. Verma VK, Yadava RDS (2016) Stochastic resonance in MEMS capacitive sensors. Sens Actuators B Chem 235:583–602

    Google Scholar 

  67. Roberts JB, Spanos PD (1999) Random vibrations and statistical linearization. Dover Publication Inc, New York

    Google Scholar 

  68. Crandall SH, Mark WD (1963) Random vibration in mechanical systems. Academic Press Inc, New York

    Google Scholar 

  69. Di Paola M, Pirrotta A (1999) Non-linear systems under impulsive parametric input. Int J Non Linear Mech 34(5):843–851

    MathSciNet  MATH  Google Scholar 

  70. Pirrotta A (2005) Non-linear systems under parametric white noise input: digital simulation and response. Int J Non Linear Mech 40(8):1088–1101

    MATH  Google Scholar 

  71. Alotta G, Di Paola M, Pinnola FP (2017) Cross-correlation and cross-power spectral density representation by complex spectral moments. Int J Non Linear Mech 94:20–27

    Google Scholar 

  72. Lei Y, Murmu T, Adhikari S, Friswell MI (2013) Dynamic characteristics of damped viscoelastic nonlocal Euler–Bernoulli beams. Eur J Mech A/Solids 42:125–136

    MathSciNet  MATH  Google Scholar 

  73. Lei Y, Adhikari S, Friswell MI (2013) Vibration of nonlocal Kelvin-Voigt viscoelastic damped Timoshenko beams. Int J Eng Sci 66–67:1–13

    MathSciNet  MATH  Google Scholar 

  74. Alotta G, Failla G, Pinnola FP (2017) Stochastic analysis of a nonlocal fractional viscoelastic bar forced by Gaussian white noise. ASCE-ASME J Risk Uncertain Eng Syst Part B Mech Eng 3(3):030904–030904-7

    Google Scholar 

  75. Alotta G, Di Paola M, Failla G, Pinnola FP (2018) On the dynamics of non-local fractional viscoelastic beams under stochastic agencies. Compos B Eng 137:102–110

    Google Scholar 

  76. Pirrotta A, Cutrona S, Di Lorenzo S, Di Matteo A (2015) Fractional visco-elastic Timoshenko beam deflection via single equation. Int J Numer Meth Eng 104:869–886

    MathSciNet  MATH  Google Scholar 

  77. Alotta G, Failla G, Zingales M (2017) Finite element formulation of a non-local hereditary fractional order Timoshenko beam. J Eng Mech ASCE 143(5):1943–7889.0001035

    Google Scholar 

  78. Meirovitch L (2001) Fundamentals of vibrations. McGraw-Hill International Edition, New York

    Google Scholar 

  79. Di Lorenzo S, Di Paola M, Pinnola FP, Pirrotta A (2014) Stochastic response of fractionally damped beams. Probab Eng Mech 35:37–43

    Google Scholar 

  80. Shinozuka M, Deodatis G (1988) Stochastic process models for earthquake ground motion. Probab Eng Mech 3(3):114–123

    Google Scholar 

  81. Ashby M, Shercliff H, Cebon D (2007) Materials engineering, science, processing and design. Elsevier, Burlington

    Google Scholar 

  82. Ashby M (1999) Materials selection in mechanical design. Butterworth-Heinemann, Woburn

    Google Scholar 

  83. Abazari AM, Safavi SM, Rezazadeh G, Villanueva LG (2015) Modelling the size effects on the mechanical properties of micro/nano structures. Sensors 15:28543–28562

    Google Scholar 

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Acknowledgements

Financial supports from the MIUR in the framework of the Projects PRIN 2015 “COAN 5.50.16.01” (code 2015JW9NJT Advanced mechanical modeling of new materials and structures for the solution of 2020 Horizon challenges) and PRIN 2017 (code 2017J4EAYB Multiscale Innovative Materials and Structures (MIMS); University of Naples Federico II Research Unit) and from the research program ReLUIS 2019 are gratefully acknowledged.

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  1. Department of Structures for Engineering and Architecture, University of Naples Federico II, via Claudio 21, Ed. 6, 80125, Naples, Italy

    Francesco P. Pinnola, Marzia S. Vaccaro, Raffaele Barretta & Francesco Marotti de Sciarra

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  1. Francesco P. Pinnola
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  2. Marzia S. Vaccaro
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  3. Raffaele Barretta
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Correspondence to Raffaele Barretta.

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Pinnola, F.P., Vaccaro, M.S., Barretta, R. et al. Random vibrations of stress-driven nonlocal beams with external damping. Meccanica 56, 1329–1344 (2021). https://doi.org/10.1007/s11012-020-01181-7

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