Recently, a displacement-based nonlocal bar model has been developed. The model is based on the assumption that nonlocal forces can be modeled as viscoelastic (VE) long-range interactions mutually exerted by nonadjacent bar segments due to their relative motion; the classical local stress resultants are also present in the model. A finite element (FE) formulation with closed-form expressions of the elastic and viscoelastic matrices has also been obtained. Specifically, Caputo's fractional derivative has been used in order to model viscoelastic long-range interaction. The static and quasi-static response has been already investigated. This work investigates the stochastic response of the nonlocal fractional viscoelastic bar introduced in previous papers, discretized with the finite element method (FEM), forced by a Gaussian white noise. Since the bar is forced by a Gaussian white noise, dynamical effects cannot be neglected. The system of coupled fractional differential equations ruling the bar motion can be decoupled only by means of the fractional order state variable expansion. It is shown that following this approach Monte Carlo simulation can be performed very efficiently. For simplicity, here the work is limited to the axial response, but can be easily extended to transverse motion.
Issue Section:
Special Section Papers
References
1.
Lakes
, R. S.
, 1991
, “Experimental Micro Mechanics Methods for Conventional and Negative Poisson's Ratio Cellular Solids as Cosserat Continua
,” ASME J. Eng. Mater. Technol.
, 113
(1
), pp. 148
–155
.2.
Aifantis
, E. C.
, 1994
, “Gradient Effects at Macro, Micro, and Nano Scales
,” J. Mech. Behav. Mater.
, 5
(3
), pp. 355
–375
.3.
Qian
, D.
, Wagner
, G. J.
, Liu
, W. K.
, Yu
, M.-F.
, and Ruoff
, R. S.
, 2002
, “Mechanics of Carbon Nanotubes
,” ASME Appl. Mech. Rev.
, 55
(6
), pp. 495
–533
.4.
Arash
, B.
, and Wang
, Q.
, 2012
, “A Review on the Application of Nonlocal Elastic Models in Modeling of Carbon Nanotubes and Graphenes
,” Comput. Mater. Sci.
, 51
(1
), pp. 303
–313
.5.
Wang
, L. F.
, and Hu
, H. Y.
, 2005
, “Flexural Wave Propagation in Single-Walled Carbon Nanotube
,” Phys. Rev. B
, 71
(19
), pp. 195412
–195418
.6.
Tang
, P. Y.
, 1983
, “Interpretation of Bend Strength Increase of Graphite by the Couple Stress Theory
,” Comput. Struct.
, 16
(1–4
), pp. 45
–49
.7.
Poole
, W. J.
, Ashby
, M. F.
, and Fleck
, N. A.
, 1996
, “Micro-Hardness of Annealed and Work-Hardened Copper Polycrystals
,” Scr. Mater.
, 34
(4
), pp. 559
–564
.8.
Lam
, D. C. C.
, Yang
, F.
, Chong
, A. C. M.
, Wang
, J.
, and Tong
, P.
, 2003
, “Experiments and Theory in Strain Gradient Elasticity
,” J. Mech. Phys. Solids
, 51
(8
), pp. 1477
–1508
.9.
McFarland
, A. W.
, and Colton
, J. S.
, 2005
, “Role of Material Microstructure in Plate Stiffness With Relevance to Microcantilever Sensors
,” J. Micromech. Microeng.
, 15
(5
), pp. 1060
–1067
.10.
Russell
, D. L.
, 1992
, “On Mathematical Models for the Elastic Beam With Frequency-Proportional Damping
,” Control and Estimation in Distributed Parameter Systems
, H. T.
Banks
, ed., SIAM
, Philadelphia, PA
, pp. 125
–169
.11.
Banks
, H. T.
, and Inman
, D. J.
, 1991
, “On Damping Mechanism in Beams
,” ASME J. Appl. Mech.
, 58
(3
), pp. 716
–723
.12.
Lei
, Y.
, Friswell
, M. I.
, and Adhikari
, S.
, 2006
, “A Galerkin Method for Distributed Systems With Non-Local Damping
,” Int. J. Solids Struct.
, 43
(11–12
), pp. 3381
–3400
.13.
Friswell
, M. I.
, Adhikari
, S.
, and Lei
, Y.
, 2007
, “Non-Local Finite Element Analysis of Damped Beams
,” Int. J. Solids Struct.
, 44
(22–23
), pp. 7564
–7576
.14.
Eringen
, A. C.
, 1972
, “Linear Theory of Non-Local Elasticity and Dispersion of Plane Waves
,” Int. J. Eng. Sci.
, 10
(5
), pp. 425
–435
.15.
Eringen
, A. C.
, 1983
, “On Differential Equations of Non-Local Elasticity and Solutions of Screw Dislocation and Surface Waves
,” J. Appl. Phys.
, 54
(9
), pp. 4703
–4710
.16.
Lu
, P.
, Lee
, H. P.
, Lu
, C.
, and Zhang
, P. Q.
, 2007
, “Application of Nonlocal Beam Models for Carbon Nanotubes
,” Int. J. Solids Struct.
, 44
(16
), pp. 5289
–5300
.17.
Pradhan
, S. C.
, and Phadikar
, J. K.
, 2009
, “Nonlocal Elasticity Theory for Vibration of Nanoplates
,” J. Sound Vib.
, 325
(1–2
), pp. 206
–223
.18.
Park
, S. K.
, and Gao
, X. L.
, 2006
, “Bernoulli-Euler Beam Model Based on a Modified Couple Stress Theory
,” J. Micromech. Microeng.
, 16
(11
), pp. 2355
–2359
.19.
Kong
, S.
, Zhou
, S.
, Nie
, Z.
, and Wang
, K.
, 2008
, “The Size-Dependent Natural Frequency of Bernoulli-Euler Micro-Beams
,” Int. J. Eng. Sci.
, 46
(5
), pp. 427
–437
.20.
Ma
, H. M.
, Gao
, X.-L.
, and Reddy
, J. N.
, 2008
, “A Microstructure Dependent Timoshenko Beam Model Based on a Modified Couple Stress Theory
,” J. Mech. Phys. Solids
, 56
(12
), pp. 3379
–3391
.21.
Challamel
, N.
, and Wang
, C. M.
, 2008
, “The Small Length Scale Effect for a Non-Local Canti-Lever Beam: A Paradox Solved
,” Nanotechnology
, 19
(34
), p. 345703
.22.
Zhang
, Y. Y.
, Wang
, C. M.
, and Challamel
, N.
, 2010
, “Bending, Buckling, and Vibration of Micro/Nanobeams by Hybrid Nonlocal Beam Model
,” ASCE J. Eng. Mech.
, 136
(5), pp. 562
–574
.23.
Payton
, D.
, Picco
, L.
, Miles
, M. J.
, Homer
, M. E.
, and Champneys
, A. R.
, 2012
, “Modelling Oscillatory Flexure Modes of an Atomic Force Microscope Cantilever in Contact Mode Whilst Imaging at High Speed
,” Nanotechnology
, 23
(26
), p. 265702
.24.
Murmu
, T.
, and Adhikari
, S.
, 2012
, “Nonlocal Frequency Analysis of Nanoscale Biosensors
,” Sens. Actuators, A
, 173
(1
), pp. 41
–48
.25.
Chen
, C.
, Ma
, M.
, Liu
, J.
, Zheng
, Q.
, and Xu
, Z.
, 2011
, “Viscous Damping of Nanobeam Resonators: Humidity, Thermal Noise, and a Paddling Effect
,” J. Appl. Phys.
, 110
(3
), p. 034320
.26.
Lee
, J.
, and Lin
, C.
, 2010
, “The Magnetic Viscous Damping Effect on the Natural Frequency of a Beam Plate Subject to an In-Plane Magnetic Field
,” ASME J. Appl. Mech.
, 77
(1
), p. 011014
.27.
Di Paola
, M.
, Failla
, G.
, and Zingales
, M.
, 2013
, “Non-Local Stiffness and Damping Models for Shear-Deformable Beams
,” Eur. J. Mech., A: Solids
, 40
, pp. 69
–83
.28.
Di Paola
, M.
, Failla
, G.
, and Zingales
, M.
, 2014
, “Mechanically Based Nonlocal Euler-Bernoulli Beam Model
,” J. Micromech. Microeng.
, 4
(1), p. A4013002.29.
Failla
, G.
, Sofi
, A.
, and Zingales
, M.
, 2015
, “A New Displacement-Based Framework for Non-Local Timoshenko Beams
,” Meccanica
, 50
(8
), pp. 2103
–2122
.30.
Alotta
, G.
, Failla
, G.
, and Zingales
, M.
, 2014
, “Finite Element Method for a Nonlocal Timoshenko Beam Model
,” Finite Elem. Anal. Des.
, 89
, pp. 77
–92
.31.
Fuchs
, M. B.
, 1991
, “Unimodal Beam Elements
,” Int. J. Solids Struct.
, 27
(5
), pp. 533
–545
.32.
Fuchs
, M. B.
, 1997
, “Unimodal Formulation of the Analysis and Design Problems for Framed Structures
,” Comput. Struct.
, 63
(4
), pp. 739
–747
.33.
Phadikar
, J. K.
, and Pradhan
, S. C.
, 2010
, “Variational Formulation and Finite Element Analysis for Nonlocal Elastic Nanobeams and Nanoplate
,” Comput. Mater. Sci.
, 49
(3
), pp. 492
–499
.34.
Pradhan
, S. C.
, 2012
, “Nonlocal Finite Element Analysis and Small Scale Effects of CNTs With Timoshenko Beam Theory
,” Finite Elem. Anal. Des.
, 50
, pp. 8
–20
.35.
Alotta
, G.
, Failla
, G.
, and Zingales
, M.
, 2015
, “Finite-Element Formulation of a Nonlocal Hereditary Fractional-Order Timoshenko Beam
,” ASCE J. Eng. Mech.
, 143
(5
), p. D4015001.36.
Podlubny
, I.
, 1999
, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications
, Academic Press
, New York
.37.
Mainardi
, F.
, 2010
, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models
, World Scientific
, Singapore
.38.
Alotta
, G.
, Di Paola
, M.
, and Pirrotta
, A.
, 2014
, “Fractional Tajimi–Kanai Model for Simulating Earthquake Ground Motion
,” Bull. Earthquake Eng.
, 12
(6
), pp. 2495
–2506
.39.
Di Paola
, M.
, Fiore
, V.
, Pinnola
, F. P.
, and Valenza
, A.
, 2014
, “On the Influence of the Initial Ramp for a Correct Definition of the Parameters of Fractional Viscoelastic Materials
,” Mech. Mater.
, 69
(1
), pp. 63
–70
.40.
Bagley
, R. L.
, and Torvik
, P. J.
, 1983
, “A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity
,” J. Rheol.
, 27
(3
), pp. 201
–210
.41.
Bagley
, R. L.
, and Torvik
, P. J.
, 1986
, “On the Fractional Calculus Model of Viscoelastic Behavior
,” J. Rheol.
, 30
(1
), pp. 133
–155
.42.
Failla
, G.
, and Pirrotta
, A.
, 2012
, “On the Stochastic Response of a Fractionally-Damped Duffing Oscillator
,” Commun. Nonlinear Sci. Numer. Simul.
, 17
(12
), pp. 5131
–5142
.43.
Bagley
, R. L.
, and Torvik
, P. J.
, 1985
, “Fractional Calculus in the Transient Analysis of Viscoelastically Damped Structures
,” AIAA J.
, 23
(6
), pp. 918
–925
.44.
Di Paola
, M.
, Pinnola
, F. P.
, and Spanos
, P. D.
, 2014
, “Analysis of Multi-Degree-of-Freedom Systems With Fractional Derivative Elements of Rational Order
,” International Conference on Fractional Differentiation and Its Applications (ICFDA
), Catania, Italy, June 23–25, pp. 1–6.45.
Di Paola
, M.
, Failla
, G.
, and Zingales
, M.
, 2010
, “The Mechanically Based Approach to 3D Non-Local Linear Elasticity Theory: Long-Range Central Interactions
,” Int. J. Solids Struct.
, 47
(18–19
), pp. 2347
–2358
.46.
Altan
, B. S.
, 1989
, “Uniqueness of the Initial-Value Problems in Non-Local Elastic Solids
,” Int. J. Solids Struct.
, 25
(11
), pp. 1271
–1278
.47.
Polizzotto
, C.
, 2001
, “Non Local Elasticity and Related Variational Principles
,” Int. J. Solids Struct.
, 38
(42–43
), pp. 7359
–7380
.48.
Samko
, S. G.
, Kilbas
, A. A.
, and Marichev
, O. I.
, 1993
, Fractional Integral and Derivatives: Theory and Applications
, Gordon and Breach Science Publishers
, Philadelphia, PA.49.
Di Paola
, M.
, Failla
, G.
, and Zingales
, M.
, 2009
, “Physically-Based Approach to the Mechanics of Strong Non-Local Linear Elasticity Theory
,” J. Elasticity
, 97
(2
), pp. 103
–130
.50.
Reddy
, J. N.
, 1999
, “On the Dynamic Behaviour of the Timoshenko Beam Finite Elements
,” Sādhanā
, 24
(3
), pp. 175
–198
.51.
Pinnola
, F. P.
, 2016
, “Statistical Correlation of Fractional Oscillator Response by Complex Spectral Moments and State Variable Expansion
,” Commun. Nonlinear Sci. Numer. Simul.
, 39
, pp. 343
–359
.52.
Shinozuka
, M.
, and Deodatis
, G.
, 1988
, “Stochastic Process Models for Earthquake Ground Motion
,” Probab. Eng. Mech.
, 3
(3
), pp. 114
–123
.Copyright © 2017 by ASME
You do not currently have access to this content.
