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

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Abstract

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.

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Acknowledgements

This work is supported by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

Funding

Project supported by the National Natural Science Foundation of China (No. 12172169), the China Scholarship Council (CSC) (No. 202006830038), and the Natural Sciences and Engineering Research Council of Canada (No. RGPIN-2017-03716115112)

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Authors and Affiliations

  1. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Pei Zhang & Hai Qing

  2. Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, T6G 1H9, Canada

    Pei Zhang & P. Schiavone

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  1. Pei Zhang
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  2. P. Schiavone
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  3. Hai Qing
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Correspondence to Hai Qing.

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Citation: ZHANG, P., SCHIAVONE, P., and QING, H. Unified two-phase nonlocal formulation for vibration of functionally graded beams resting on nonlocal viscoelastic Winkler-Pasternak foundation. Applied Mathematics and Mechanics (English Edition), 44(1), 89–108 (2023) https://doi.org/10.1007/s10483-023-2948-9

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Zhang, P., Schiavone, P. & Qing, H. Unified two-phase nonlocal formulation for vibration of functionally graded beams resting on nonlocal viscoelastic Winkler-Pasternak foundation. Appl. Math. Mech.-Engl. Ed. 44, 89–108 (2023). https://doi.org/10.1007/s10483-023-2948-9

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  • DOI: https://doi.org/10.1007/s10483-023-2948-9

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