The present state-of-the-art article is devoted to the analysis of new trends and recent results carried out during the last in the field of fractional calculus application to dynamic problems of solid mechanics. This review involves the papers dealing with study of dynamic behavior of linear and nonlinear 1DOF systems, systems with two and more DOFs, as well as linear and nonlinear systems with an infinite number of degrees of freedom: vibrations of rods, beams, plates, shells, suspension combined systems, and multilayered systems. Impact response of viscoelastic rods and plates is considered as well. The results obtained in the field are critically estimated in the light of the present view of the place and role of the fractional calculus in engineering problems and practice. This articles reviews 337 papers and involves 27 figures.
1.
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, and Shitikova
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, and Rogers
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, and Shitikova
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.7.
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, M.
, and Tan
, W.
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.8.
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.9.
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, I.
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.10.
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, and Trujillo
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.11.
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, R.
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,” Fractals and Fractional Calculus in Continuum Mechanics
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.12.
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, and Schanz
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, and Gaul
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.14.
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, Y.
, Vinagre
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, and Podlubny
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.15.
Diethelm
, K.
, Ford
, N. J.
, Freed
, A. D.
, and Luchko
, Yu.
, 2005, “Algorithms for the Fractional Calculus: A Selection of Numerical Methods
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, and Korotkin
, I. A.
, 2006, “Numerical Methods for Some One-Dimensional Equations With Fractional Derivatives
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, O. P.
, and Kumar
, P.
, 2007, “Comparison of Five Numerical Schemes for Fractional Differential Equations
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, J.
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, and Agrawal
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, and Torvik
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, and Torvik
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, and Calico
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.24.
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, and Bagley
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.25.
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, Swinney
, D. V.
, and Griffin
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, 1993, “Fractional Order Calculus Model of the Generalized Theodorsen Function
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.26.
Ciskowski
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, Sonnad
, V.
, and Xie
, K. J.
, 1991, “Parallel Implementation on the ICAP/3090 of a BEM Formulation for Fractional Operator Modeled Viscoelastodynamic Structure
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.27.
Xie
, K. J.
, Ciskowski
, R. D.
, and Royster
, L. H.
, 1988, “Investigation of Wave Propagation in Viscoelastic Media Modeled by Fractional Derivative Using the Boundary Element Method
,” Boundary Element Method in Applied Mechanics
, M.
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, pp. 523
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.28.
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, K. J.
, Royster
, L. H.
, and Ciskowski
, R. D.
, 1989, “A Boundary Element Method Formulation for Fractional Operator Modeled Viscoelastodynamic Structures
,” Advances in Boundary Elements
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, C. A.
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, pp. 55
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.29.
Cortés
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, and Elejabarrieta
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, 2006, “An Approximate Numerical Method for the Complex Eigenproblem in Systems Characterised by a Structural Damping Matrix
,” J. Sound Vib.
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, pp. 166
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.30.
Cortés
, F.
, and Elejabarrieta
, M. J.
, 2007, “Finite Element Formulations for Transient Dynamic Analysis in Structural Systems With Viscoelastic Treatments Containing Fractional Derivative Models
,” Int. J. Numer. Methods Eng.
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.31.
Cortés
, F.
, and Elejabarrieta
, M. J.
, 2007, “Homogenised Finite Element for Transient Dynamic Analysis of Unconstrained Layer Damping Beams Involving Fractional Derivative Models
,” Comput. Mech.
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, pp. 313
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.32.
Cortés
, F.
, and Elejabarrieta
, M. J.
, 2008, “Structural Vibration of Flexural Beams With Thick Unconstrained Layer Damping
,” Int. J. Solids Struct.
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, pp. 5805
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.33.
Eldred
, L. B.
, Baker
, W. P.
, and Palazotto
, A. N.
, 1995, “Kelvin-Voigt vs Fractional Derivative Model as Constitutive Relations for Viscoelastic Materials
,” AIAA J.
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), pp. 547
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.34.
Palazotto
, A. N.
, and Birman
, V.
, 1995, “Environmental and Viscoelastic Effects on Stresses in Adhesive Joints
,” J. Aerosp. Eng.
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), pp. 107
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.35.
Eldred
, L. B.
, Baker
, W. P.
, and Palazotto
, A. N.
, 1996, “Numerical Application of Fractional Derivative Model Constitutive Relations for Viscoelastic Materials
,” Comput. Struct.
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), pp. 875
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.36.
Enelund
, M.
, and Josefson
, B. L.
, 1997, “Time-Domain Finite Element Analysis of Viscoelastic Structures With Fractional Derivatives Constitutive Relations
,” AIAA J.
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), pp. 1630
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.37.
Enelund
, M.
, Fenander
, A.
, and Olsson
, P.
, 1997, “Fractional Integral Formulation of Constitutive Equations of Viscoelasticity
,” AIAA J.
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), pp. 1356
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, M.
, and Lesieutre
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, 1999, “Time Domain Modeling of Damping Using Anelastic Displacement Fields and Fractional Calculus
,” Int. J. Solids Struct.
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), pp. 4447
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.39.
Enelund
, M.
, Mähler
, L.
, Runesson
, K.
, and Josefson
, B. L.
, 1999, “Formulation and Integration of the Standard Linear Viscoelastic Solid With Fractional Order Rate Laws
,” Int. J. Solids Struct.
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), pp. 2417
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.40.
Enelund
, M.
, and Olsson
, P.
, 1999, “Damping Described by Fading Memory—Analysis and Application to Fractional Derivative Models
,” Int. J. Solids Struct.
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Adolfsson
, K.
, Enelund
, M.
, and Larsson
, S.
, 2003, “Adaptive Discretization of an Integro-Differential Equation With a Weakly Singular Convolution Kernel
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Adolfsson
, K.
, Enelund
, M.
, and Larsson
, S.
, 2004, “Adaptive Discretization of Fractional Order Viscoelasticity Using Sparse Time History
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, pp. 4567
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.43.
Adolfsson
, K.
, Enelund
, M.
, and Larsson
, S.
, 2008, “Space-Time Discretization of an Integro-Differential Equation Modeling Quasi-Static Fractional-Order Viscoelasticity
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), pp. 1631
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, J.
, and Ricles
, J. M.
, 1998, “The Fractional Order Elastic-Viscoelastic Equations of Motion: Formulation and Solution Methods
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, Dëu
, J.-F.
, and Ohayon
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, 2004, “Finite Element Formulation of Viscoelastic Sandwich Beams Using Fractional Derivative Operators
,” Comput. Mech.
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, pp. 282
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, Dëu
, J.-F.
, and Ohayon
, R.
, 2005, “A Fractional Derivative Viscoelastic Model for Hybrid Active-Passive Damping Treatments in Time Domain—Application to Sandwich Beams
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), pp. 33
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.47.
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, A. C.
, Dëu
, J.-F.
, Mengue
, S.
, and Dubois
, F.
, 2006, “An Adaptation of the Gear Scheme for Fractional Derivatives
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, A. C.
, Dëu
, J.-F.
, and Ohayon
, R.
, 2007, “Hybrid Active-Passive Damping Treatment of Sandwich Beams in Nonlinear Dynamics
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), pp. 851
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.49.
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, J.-F.
, Galucio
, A. C.
, and Ohayon
, R.
, 2008, “Dynamic Responses of Flexible-Link Mechanisms With Passive/Active Damping Treatment
,” Comput. Struct.
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), pp. 258
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.50.
Galucio
, A. C.
, Dëu
, J.-F.
, and Dubois
, F.
, 2008, “The Gα-scheme for Approximation of Fractional Derivatives: Application to the Dynamics of Dissipative Systems
,” J. Vib. Control
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), pp. 1597
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.51.
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, P.
, and Kempfle
, S.
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, and Schanz
, M.
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, M.
, 1999, “A Boundary Element Formulation in Time Domain for Viscoelastic Solids
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, and Gaul
, L.
, 2003, “Implementation von Stoffgesetzen mit fractionalen Ableitungen in die Finite Elemente Methode
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, and Gaul
, L.
, 2006, “On a Critique of a Numerical Scheme for the Calculation of Fractionally Damped Systems
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, and Gaul
, L.
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, and Berti
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, 1992, “A Fractional Derivative Model for Single-Link Mechanism Vibration
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, and Schmidt
, L. C.
, 1995, “Dynamic Response of a Damped Large Space Structure: a New Fractional-Spectral Approach
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), pp. 113
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, and Schmidt
, L. C.
, 1996, “A Fractional-Spectral Method for Vibration of Damped Space Structures
,” Eng. Struct.
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, and Schmidt
, L. C.
, 1996, “Frequency Domain Dynamic Analysis of Large Space Structures With Added Elastomeric Dampers
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, and Schmidt
, L. C.
, 1997, “Complex Fractional-Spectral Method for Space Curved Struts: Theory and Application
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, and Schmidt
, L. C.
, 1999, “Dynamic Analysis of Large Shell Structures: Complex Damped Spectral Element Method
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, and Suzdalnitsky
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, and Suzdalnitsky
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, and Cheng
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, G. G.
, 2002, “Dynamic Behaviors of Timoshenko Beam With Fractional Derivative Constitutive Relation
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, Li
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, and Cheng
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, 2002, “Quasi-Static and Dynamical Analysis for Viscoelastic Timoshenko Beam With Fractional Derivative Constitutive Relation
,” Appl. Math. Mech.
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, Zhu
, Z. Y.
, and Cheng
, C. J.
, 2003, “Application of Galerkin Method to Dynamical Behavior of Viscoelastic Timoshenko Beam With Finite Deformation
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, Li
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, and Cheng
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, 2003, “A Numerical Method for Fractional Integral With Applications
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, and Padovan
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, and Sawicki
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, and Sawicki
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, 1999, “Frequency Driven Phasic Shifting and Elastic-Hysteretic Partitioning Properties of Fractional Mechanical System Representation Schemes
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, and Nemes
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, 1998, “Comparison of Viscoelastic Models in Simulating the Transient Response of a Slewing Polymer Arm
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, Nemes
, J. A.
, and Piedboeuf
, J. C.
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, Pagé
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, and Potvin
, M. J.
, 1999, “Efficient Simulation of a Multilayer Viscoelastic Beam Using an Equivalent Homogeneous Beam
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, C.
, and Ruge
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