Peridynamic Higher-Order Beam Formulation
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Peridynamic Higher-Order Beam Formulation Zhenghao Yang 1 & Erkan Oterkus 1
& Selda Oterkus
1
Received: 26 April 2020 / Accepted: 16 September 2020/ # The Author(s) 2020
Abstract
In this study, a novel higher-order peridynamic beam formulation is presented. The formulation is obtained by using Euler-Lagrange equations and Taylor’s expansion. To demonstrate the capability of the presented approach, several different beam configurations are considered including simply supported beam subjected to distributed loading, simply supported beam with concentrated load, clampedclamped beam subjected to distributed loading, cantilever beam subjected to a point load at its free end and cantilever beam subjected to a moment at its free end. Transverse displacement results along the beam obtained from peridynamics and finite element method are compared with each other and very good agreement is obtained between the two approaches. Keywords Peridynamics . Higher-order . Beam . Euler-Lagrange equation
1 Introduction Peridynamic (PD) theory was introduced by Silling [1] to overcome the limitations of widely used classical continuum mechanics (CCM). PD equations do not contain spatial derivatives as opposed to their usage in CCM which allows its equations to be valid even if the displacement field is not continuous due to the existence of cracks. Moreover, it has a length scale parameter called horizon which does not exist in CCM, so that it can capture nonlocal effects. PD has been applied to many different material systems including metals [2], composite materials [3] and concrete [4]. Moreover, PD is not limited to structural analysis, but can also be used to analyse heat transfer [5], diffusion [6], porous flow [7], fluid flow [8], etc. An extensive review of PD research is given in Javili et al. [9]. Original PD formulation is suitable to analyse 1-Dimensional, 2Dimensional and 3-Dimensional models by assigning translational degrees of freedom
* Erkan Oterkus [email protected]
1
PeriDynamics Research Centre, Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, 100 Montrose Street, Glasgow G4 0LZ UK, Scotland
Journal of Peridynamics and Nonlocal Modeling
to each material point. However, for certain shapes including beams, plates and shells, such formulations can be computational expensive. Alternatively, simplified beam, plate and shell formulations with additional rotational degrees of freedom per material point can be utilised. There are several PD formulations available in the literature suitable for beam, plate and shell structures. Amongst these, Diyaroglu et al. [10] proposed a state-based Euler-beam formulation for slender beams. This formulation was further extended by Yang et al. [11] to analyse Kirchhoff plates. O’Grady and Foster [12] developed a non-ordinary state-based model to represent the bending behaviour of Euler-Bernoulli beam. They also introduced a non-ordinary state-based peridynamic model for Kirchhoff-Love plate [13]. In order to take into account
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