A state-based peridynamic formulation for functionally graded Kirchhoff plates

Yang, Zhenghao; Öterkuş, Erkan; Oterkus, Selda

doi:10.1177/1081286520963383

Cited by 12 publications

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“…PD theories for components including rods and beams [9–13], plates [14–19] and shells [20–22] are developed. Within the CCM, these theories provide an efficient and robust approach for the analysis of structures.…”

Section: Introductionmentioning

confidence: 99%

“…Many numerical studies of complex fracture processes illustrate the ability of PD to capture crack initiation [2-4], crack branching [5], crack kinking [6] and crack interaction with initial heterogeneities, such as holes and pores [7,8]. PD theories for components including rods and beams [9-13], plates [14][15][16][17][18][19] and shells [20][21][22] are developed. Within the CCM, these theories provide an efficient and robust approach for the analysis of structures.…”

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confidence: 99%

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Some analytical solutions to peridynamic beam equations

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Peridynamics (PD) has been introduced to account for long range internal force/moment interactions and to extend the classical continuum mechanics (CCM). PD equations of motion are derived in the form of integro-differential equations and only few analytical solutions to these equations are presented in the literature. The aim of this paper is to present analytical solutions to PD beam equations for both static and dynamic loading conditions. Applying trigonometric series, general solutions for the deflection function are derived. For several examples in the static case including simply supported beam and cantilever beam, the coefficients in the series are presented in a closed analytical form. For the dynamic case, the solution is derived for a simply supported beam applying the variable separation with respect to the time and the axial coordinate. Several numerical cases are presented to illustrate the derived solutions. Furthermore, PD results are compared against results obtained from the classical beam theory (CBT). A very good agreement between these two different approaches is observed for the case of the small horizon sizes (HSs), which shows the capability of the current approach. INTRODUCTIONPeridynamics (PD) is a non-local theory, which operates with long-range force/moment interactions [1]. Unlike the classical continuum mechanics (CCM), the deformation gradient, its higher gradients or gradients of internal state variables are not introduced. PD equations of motion are integro-differential equations, in contrast to CCM where partial differential equations are introduced. This makes PD attractive in analysis of discontinuities such as cracks. Many numerical studies of complex fracture processes illustrate the ability of PD to capture crack initiation [2-4], crack branching [5], crack kinking [6] and crack interaction with initial heterogeneities, such as holes and pores [7,8]. PD theories for components including rods and beams [9-13], plates [14][15][16][17][18][19] and shells [20][21][22] are developed. Within the CCM, these theories provide an efficient and robust approach for the analysis of structures. Indeed many analytical solutions for beams, plates and shells are available providing a direct insight into the deformation and stress states. On the other hand, they are useful to validate the numerical methods and computer codes, such as the finite element method. However, unlike the CCM, only few analytical solutions to PD equations are available.

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Section: Introductionmentioning

confidence: 99%

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Some analytical solutions to peridynamic beam equations

et al. 2022

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“…Peridynamic formulations for elementary structures including rods and beams [3,9,23], plates [10,13,[24][25][26] and shells [4,12,17] are available. Based on the CCM, the theories of beams, plates and shells provide an efficient and robust approach for the structural analysis.…”

Section: Introductionmentioning

confidence: 99%

Beam buckling analysis in peridynamic framework

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Peridynamics is a non-local continuum theory which accounts for long-range internal force/moment interactions. Peridynamic equations of motion are integro-differential equations, and only few analytical solutions to these equations are available. The aim of this paper is to formulate governing equations for buckling of beams and to derive analytical solutions for critical buckling loads based on the nonlinear peridynamic beam theory. For three types of boundary conditions, explicit expressions for the buckling loads are presented. The results are compared with the classical results for buckling loads. A very good agreement between the non-local and the classical theories is observed for the case of the small horizon sizes which shows the capability of the current approach. The results show that with an increase of the horizon size the critical buckling load slightly decreases for the fixed overall stiffness of the beam.

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“…Yang vd. [27][28][29][30] BB ve SDB PD teorilerini kullanarak FK kirişlerin ve plakaların mekanik yükler altındaki davranışlarını incelemişlerdir. PD teorisinden yararlanarak yapılan malzemelerin hasar analiz çalışmalarına son zamanlarda yayımlanan kitaplardan erişilebilmektedir [10,18,31].…”

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Investigation of fracture behaviour of one-dimensional functionally graded plates by using peridynamic theory

KAYA>

OLMUŞ

Dördüncü

2022

Gazi Üniversitesi Mühendislik Mimarlık Fakültesi Dergisi

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Composite materials are widely used in aerospace, military, and nuclear engineering fields due to their desirable properties such as lightness and high strength. The mismatch of the stiffness at the interfaces between distinct materials results in a stress concentration; thus, crack nucleation and inter-layer separations can be observed. The concept of functionally graded materials (FGMs) aims to achieve a structure whose material properties continuously vary in one or more coordinate directions. This continuous variation is obtained for the material properties of the FG structure.This situation provides a way to minimize the stress concentrations which may occur at the interfaces between the two different materials. FGMs are one of the most important structures in defense and aerospace industries owing to their superior properties. In order to design FG structures safely, it is very crucial to understand and investigate possible damages under different loads to increase the reliability of these structures. Since structural testing and analysis techniques may be costly, there is a necessity to use improved and accurate computational tools to predict the deformation and stress fields of the FG structures. Numerical modeling of the fracture and damage in FGMs remains a formidable challenge in computational mechanics due to the non-symmetric material variations in the FGMs. In PeriDynamic (PD) theory, the equations of classical continuum mechanics (CCM) are reformulated by replacing the derivatives with volumetric integral expressions. Hence, the equilibrium equations of PD theory are still valid even if the material includes a discontinuity, unlike CCM. In this study, the influence of the material variations in the FG plates on the crack nucleation and propagation is investigated by means of PD theory. As a result of the analysis, it is observed that the material distributions have an evident effect on the fracture behaviour of the plate, and the plate strength can be increased by tailoring the material properties in the FG plates.

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A state-based peridynamic formulation for functionally graded Kirchhoff plates

Cited by 12 publications

References 23 publications

Some analytical solutions to peridynamic beam equations

Some analytical solutions to peridynamic beam equations

Beam buckling analysis in peridynamic framework

Investigation of fracture behaviour of one-dimensional functionally graded plates by using peridynamic theory

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