Analysis of Functionally Graded Timoshenko Beams by Using Peridynamics
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Analysis of Functionally Graded Timoshenko Beams by Using Peridynamics Zhenghao Yang 1 & Erkan Oterkus 1 & Selda Oterkus 1 Received: 7 May 2020 / Accepted: 21 September 2020/ # The Author(s) 2020
Abstract
In this study, a new peridynamic formulation is presented for functionally graded Timoshenko beams. The governing equations of the peridynamic formulation are obtained by utilising Euler-Lagrange equation and Taylor’s expansion. The proposed formulation is validated by considering a Timoshenko beam subjected to different boundary conditions including pinned support-roller support, clamped-roller support and clampedfree boundary conditions. Results from peridynamics are compared against finite element analysis results. A very good agreement is obtained for transverse displacements, rotations and axial displacements along the beam. Keywords Peridynamics . Timoshenko beam . Functionally graded . Nonlocal
1 Introduction Peridynamics was introduced by Silling [1] to overcome the limitations of widely used classical continuum mechanics formulation especially for problems including discontinuities in the displacement field due to existence of cracks. Moreover, it has a length scale parameter, horizon, which gives peridynamics a nonlocal characteristic and defines the range of nonlocal interactions between material points. Peridynamics has been utilised to analyse many different types of material systems including metals [2] and composites [3]. Peridynamics can be utilised for not only structural analysis but also for the solution of other physical fields such as heat transfer [4], porous flow [5], diffusion [6], etc. Peridynamics is not limited to elastic behaviour, but peridynamic plasticity [7], viscoelasticity [8] and viscoplasticity [9] formulations are available. Peridynamic formulations are not limited at macro-scale but can be used for
* Selda Oterkus [email protected]
1
Department of Naval Architecture, Ocean and Marine Engineering, PeriDynamics Research Centre, University of Strathclyde, 100 Montrose Street, Glasgow G4 0LZ, UK
Journal of Peridynamics and Nonlocal Modeling
the analysis of polycrystalline materials [10] and nano-structures [11]. An extensive review of peridynamics is given in Javili et al. [12]. The original formulation of peridynamics considers only translational degrees of freedom for material points and is capable of performing 3-dimensional analysis. However, this approach can be computationally expensive for certain geometries such as beam, plate and shell-type structures. To capture the correct deformation behaviour of such structures, additional rotational degrees of freedom may be necessary. Such formulations are currently available in the literature. Amongst these Taylor and Steigmann [13] introduced a peridynamic formulation for thin plates. O’Grady and Foster [14, 15] developed non-ordinary state-based formulations suitable for Euler-Bernoulli beams and Kirchhoff-Love plate theory. Diyaroglu et al. [16] also proposed a state-based formulation suitable for Euler-Bernoul
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