Unified two-phase nonlocal formulation for vibration of functionally graded beams resting on nonlocal viscoelastic Winkler-Pasternak foundation

• Published in: Applied mathematics and mechanics Vol. 44; no. 1; pp. 89 - 108
• Main Authors: Zhang, Pei, Schiavone, P., Qing, Hai
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2023; Springer Nature B.V; Department of Mechanical Engineering,University of Alberta,Edmonton,Alberta T6G 1H9,Canada%Department of Mechanical Engineering,University of Alberta,Edmonton,Alberta T6G 1H9,Canada%State Key Laboratory of Mechanics and Control of Mechanical Structures,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China; State Key Laboratory of Mechanics and Control of Mechanical Structures,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China
• Edition: English ed.
• Subjects: Applications of Mathematics; Beams (structural); Classical Mechanics; Constraint modelling; Eigenvalues; Elastic deformation; Fluid- and Aerodynamics; Foundations; Functionally gradient materials; Generalized differential quadrature method; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Partial Differential Equations; Quadratures; Stiffness; Vibration damping; Vibration response; Viscoelasticity; 74K10; damping vibration; two-phase nonlocal elasticity; O342; functionally graded (FG) beam; nonlocal viscoelastic Winkler-Pasternak foundation; generalized differential quadrature method (GDQM); generalized differential quadrature method(GDQM); functionally graded(FG)beam
• ISSN: 0253-4827; 1573-2754
• DOI: 10.1007/s10483-023-2948-9

A nonlocal study of the vibration responses of functionally graded (FG) beams supported by a viscoelastic Winkler-Pasternak foundation is presented. The damping responses of both the Winkler and Pasternak layers of the foundation are considered in the formulation, which were not considered in most literature on this subject, and the bending deformation of the beams and the elastic and damping responses of the foundation as nonlocal by uniting the equivalently differential formulation of well-posed strain-driven ( ε -D) and stress-driven ( σ -D) two-phase local/nonlocal integral models with constitutive constraints are comprehensively considered, which can address both the stiffness softening and toughing effects due to scale reduction. The generalized differential quadrature method (GDQM) is used to solve the complex eigenvalue problem. After verifying the solution procedure, a series of benchmark results for the vibration frequency of different bounded FG beams supported by the foundation are obtained. Subsequently, the effects of the nonlocality of the foundation on the undamped/damping vibration frequency of the beams are examined.