Static and dynamic behaviour of nonlocal elastic bar using integral strain-based and peridynamic models

Noël Challamel

doi:10.1016/j.crme.2017.12.014

Static and dynamic behaviour of nonlocal elastic bar using integral strain-based and peridynamic models

Noël Challamel ¹

¹ Université de Bretagne Sud, EA 4250, Institut de recherche Dupuy-de-Lôme, Centre de recherche, rue de Saint-Maudé, BP 92116, 56100 Lorient, France

Comptes Rendus. Mécanique, Volume 346 (2018) no. 4, pp. 320-335.

Résumé

The static and dynamic behaviour of a nonlocal bar of finite length is studied in this paper. The nonlocal integral models considered in this paper are strain-based and relative displacement-based nonlocal models; the latter one is also labelled as a peridynamic model. For infinite media, and for sufficiently smooth displacement fields, both integral nonlocal models can be equivalent, assuming some kernel correspondence rules. For infinite media (or finite media with extended reflection rules), it is also shown that Eringen's differential model can be reformulated into a consistent strain-based integral nonlocal model with exponential kernel, or into a relative displacement-based integral nonlocal model with a modified exponential kernel. A finite bar in uniform tension is considered as a paradigmatic static case. The strain-based nonlocal behaviour of this bar in tension is analyzed for different kernels available in the literature. It is shown that the kernel has to fulfil some normalization and end compatibility conditions in order to preserve the uniform strain field associated with this homogeneous stress state. Such a kernel can be built by combining a local and a nonlocal strain measure with compatible boundary conditions, or by extending the domain outside its finite size while preserving some kinematic compatibility conditions. The same results are shown for the nonlocal peridynamic bar where a homogeneous strain field is also analytically obtained in the elastic bar for consistent compatible kinematic boundary conditions at the vicinity of the end conditions. The results are extended to the vibration of a fixed–fixed finite bar where the natural frequencies are calculated for both the strain-based and the peridynamic models.

Reçu le : 2017-03-04
Accepté le : 2017-12-06
Publié le : 2018-01-10

DOI : 10.1016/j.crme.2017.12.014

Mots-clés : Nonlocal mechanics, Continuum mechanics, Peridynamic models, Integro-differential equation, Functional equation, Scale effects

Affiliations des auteurs :

Noël Challamel ¹

¹ Université de Bretagne Sud, EA 4250, Institut de recherche Dupuy-de-Lôme, Centre de recherche, rue de Saint-Maudé, BP 92116, 56100 Lorient, France

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Noël Challamel. Static and dynamic behaviour of nonlocal elastic bar using integral strain-based and peridynamic models. Comptes Rendus. Mécanique, Volume 346 (2018) no. 4, pp. 320-335. doi : 10.1016/j.crme.2017.12.014. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2017.12.014/

Bibliographie
Cité par

[1] D. Rogula Influence of spatial acoustic dispersion on dynamical properties of dislocations, Bull. Acad. Pol. Sci. Sér. Sci. Tech., Volume 13 (1965), pp. 337-385

[2] E. Kröner; B.K. Datta Nichtlokal Elastostatik: Ableitung aus der Gittertheorie, Z. Phys., Volume 196 (1966), pp. 203-211

[3] I.A. Kunin Model of elastic medium with simple structure and space dispersion, Prikl. Mat. Mekh., Volume 30 (1966), pp. 542-550

[4] A.C. Eringen; D.G.B. Edelen On nonlocal elasticity, Int. J. Eng. Sci., Volume 10 (1972) no. 3, pp. 233-248

[5] C. Polizzotto Nonlocal elasticity and related variational principles, Int. J. Solids Struct., Volume 38 (2001), pp. 7359-7380

[6] B.S. Altan Uniqueness of initial–boundary value problems in nonlocal elasticity, Int. J. Solids Struct., Volume 25 (1989) no. 11, pp. 1271-1278

[7] A.C. Eringen Nonlocal Continuum Field Theories, Springer, New York, 2002

[8] G.A. Maugin Continuum Mechanics Through the Twentieth Century – A Concise Historical Perspective, Springer, 2013

[9] S.A. Silling Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, Volume 48 (2000), pp. 175-209

[10] S.A. Silling; M. Zimmermann; R. Abeyaratne Deformation of a peridynamic bar, J. Elast., Volume 73 (2003), pp. 173-190

[11] R.W. Macek; S.A. Silling Peridynamics via finite element analysis, Finite Elem. Anal. Des., Volume 43 (2007), pp. 1169-1178

[12] M. Di Paola; G. Failla; M. Zingales Physically-based approach to the mechanics of strong non-local linear elasticity theory, J. Elast., Volume 97 (2009), pp. 103-130

[13] A.K. Lazopoulos On fractional peridynamic deformations, Arch. Appl. Mech., Volume 86 (2016) no. 12, pp. 1987-1994

[14] J. Fernández-Sáez; R. Zaera; J.A. Loya; J.N. Reddy Bending of Euler–Bernoulli beams using Eringen's integral formulation: a paradox resolved, Int. J. Eng. Sci., Volume 99 (2016), pp. 107-116

[15] G. Romano; R. Barretta; M. Diaco; F. Marotti de Sciarra Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams, Int. J. Mech. Sci., Volume 121 (2017), pp. 151-156

[16] Y. Mikata Analytical solutions of peristatic and peridynamic problems for a 1D infinite rod, Int. J. Solids Struct., Volume 49 (2012), pp. 2887-2897

[17] Z.P. Bažant; W. Luo; V.T. Chau; M.A. Bessa Wave dispersion and basic concepts of peridynamics compared to classical nonlocal damage models, J. Appl. Mech., Volume 83 (2016)

[18] V.V. Nishawala; M. Ostoja-Starzewski Peristatic solutions for finite one- and two-dimensional systems, Math. Mech. Solids, Volume 22 (2017) no. 8, pp. 1639-1653

[19] G. Borino; B. Failla; F. Parrinello A symmetric nonlocal damage model, Int. J. Solids Struct., Volume 40 (2003) no. 13, pp. 3621-3645

[20] A.C. Eringen; B.S. Kim Relation between non-local elasticity and lattice dynamics, Cryst. Lattice Defects, Volume 7 (1977), pp. 51-57

[21] E. Benvenuti; A. Simone One-dimensional nonlocal and gradient elasticity: closed-form solution and size effect, Mech. Res. Commun., Volume 48 (2013), pp. 46-51

[22] A.C. Eringen On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys., Volume 54 (1983), pp. 4703-4710

[23] H. Levy; F. Lessman Finite Difference Equations, Dover, 1992

[24] N. Challamel; C.M. Wang; I. Elishakoff Nonlocal or gradient elasticity macroscopic models: a question of concentrated or distributed microstructure, Mech. Res. Commun., Volume 71 (2016), pp. 25-31

[25] P. Seleson; Q. Du; M.L. Parks On the consistency between nearest-neighbor peridynamic discretizations and discretized classical elasticity models, Comput. Methods Appl. Mech. Eng., Volume 311 (2016), pp. 698-722

[26] W. Sumelka On the role of the virtual boundary layer in 1D fractional elasticity problems, J. Eng. Mech., Volume 143 (2017), p. 9

[27] A.C. Eringen Theory of nonlocal elasticity and some applications, Res. Mech., Volume 21 (1987), pp. 313-342

[28] N. Challamel; C.M. Wang The small length scale effect for a non-local cantilever beam: a paradox solved, Nanotechnology, Volume 19 (2008)

[29] N. Challamel; L. Rakotomanana; L. Le Marrec A dispersive wave equation using non-local elasticity, C. R. Mécanique, Volume 337 (2009), pp. 591-595

[30] C.W. Lim; G. Zhang; J.N. Reddy A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation, J. Mech. Phys. Solids, Volume 78 (2015), pp. 298-313

[31] R.H.J. Peerlings; R. de Borst; W.A.M. Brekelmans; J.H.P. de Vree Gradient-enhanced damage for quasi-brittle materials, Int. J. Numer. Methods Eng., Volume 39 (1996), pp. 3391-3403

[32] A.D. Polyanin; A.V. Manzhirov Handbook of Integral Equations, CRC Press, Taylor and Francis Group, 2008

[33] E. Lorentz; S. Andrieux Analysis of non-local models through energetic formulations, Int. J. Solids Struct., Volume 40 (2003), pp. 2905-2936

[34] S. Forest Micromorphic approach for gradient elasticity, viscoplasticity, and damage, J. Eng. Mech., Volume 135 (2009), pp. 117-131

[35] Y.Y. Zhang; C.M. Wang; N. Challamel Bending, buckling and vibration of hybrid nonlocal beams, J. Eng. Mech., Volume 136 (2010) no. 5, pp. 562-574

[36] P. Khodabakhshi; J.N. Reddy A unified integro-differential nonlocal model, Int. J. Eng. Sci., Volume 95 (2015), pp. 60-75

[37] Y.B. Wang; X.W. Zhu; H.H. Dai Exact solutions for the static bending of Euler–Bernoulli beams using Eringen's two phase local/nonlocal model, AIP Adv., Volume 6 (2016)

[38] A.A. Pisano; P. Fuschi Closed form solution for a nonlocal elastic bar in tension, Int. J. Solids Struct., Volume 40 (2003), pp. 13-23

[39] A.A. Pisano; P. Fuschi Reply to the discussion on the paper “Closed form solution for a nonlocal elastic bar in tension”, Int. J. Solids Struct., Volume 62 (2015), p. 273

[40] C. Polizzotto; P. Fuschi; A.A. Pisano A nonhomogeneous nonlocal elasticity model, Eur. J. Mech. A, Solids, Volume 25 (2006), pp. 308-333

[41] C.C. Koutsoumaris; K.G. Eptaimeros; G.J. Tsamasphyros A different approach to Eringen's nonlocal stress model with application for beams, Int. J. Solids Struct., Volume 112 (2017), pp. 222-238

[42] M. Aydogdu Axial vibration of the nanorods with the nonlocal continuum rod model, Physica E, Volume 41 (2009), pp. 861-864

[43] J.N. Reddy Nonlocal theories for bending, buckling and vibration of beams, Int. J. Eng. Sci., Volume 45 (2007), pp. 288-307

[44] J. Song; J. Shen; X.F. Li Effects of initial axial stress on waves propagating in carbon nanotubes using a generalized nonlocal model, Comput. Mater. Sci., Volume 49 (2010), pp. 518-523

[45] N. Challamel Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams, Compos. Struct., Volume 105 (2013), pp. 351-368

[46] L. Li; Y. Hu; X. Li Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory, Int. J. Mech. Sci., Volume 115–116 (2016), pp. 135-144

[47] B.S. Altan; H. Evensen; E.C. Aifantis Longitudinal vibrations of a beam: a gradient elasticity approach, Mech. Res. Commun., Volume 23 (1996), pp. 35-40

[48] K.G. Tsepoura; S. Papargyri-Beskou; D. Polyzos; D.E. Beskos Static and dynamic analysis of a gradient elastic bar in tension, Arch. Appl. Mech., Volume 72 (2002), pp. 483-497

[49] N. Challamel; J.N. Reddy; C.M. Wang On Eringen's stress gradient model for bending of nonlocal beams, J. Eng. Mech. (2016)

[50] J.L. Lagrange Recherches sur la nature et la propagation du son, Miscellanea Philosophico-Mathematica Societatis Privatae Taurinensis I, 1759 2nd pagination, i-112 (see also Œuvres, Tome 1, pp. 39–148)

[51] J.L. Lagrange, Mallet-Bachelier, Gendre et successeur de Bachelier, Imprimeur-libraire du bureau des longitudes, de l'École polytechnique, de l'École centrale des arts et manufactures, Paris (1853), p. 367 (Paris, 1788)

[52] G.A. Maugin Nonlinear Waves in Elastic Crystals, Oxford University Press, 1999

[53] N. Challamel; C.M. Wang; I. Elishakoff Discrete systems behave as nonlocal structural elements: bending, buckling and vibration analysis, Eur. J. Mech. A, Solids, Volume 44 (2014), pp. 125-135

[54] G.P. Lignola; F.R. Spena; A. Prota; G. Manfredi Exact stiffness-matrix of two nodes Timoshenko beam on elastic medium. An analogy with Eringen model of nonlocal Euler–Bernoulli nanobeams, Comput. Struct., Volume 182 (2017), pp. 556-572

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Armin D. Berecki; Valentin B. Glavardanov; Nenad M. Grahovac; Miodrag M. Zigic Bifurcation analysis of a nanotube through which passes a nanostring, Acta Mechanica, Volume 235 (2024) no. 11, p. 6867 | DOI:10.1007/s00707-024-04076-w
Guobing Wang; Hui Wang; Chunjiang Li; Dehong Zhang; Minjun Zeng Analysis of the effect of nonlocal factors on the vibration of nanobeams, Journal of Mechanics, Volume 40 (2024), p. 665 | DOI:10.1093/jom/ufae033
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Rami Ahmad El-Nabulsi; Waranont Anukool Fractal nonlocal thermoelasticity of thin elastic nanobeam with apparent negative thermal conductivity, Journal of Thermal Stresses, Volume 45 (2022) no. 4, p. 303 | DOI:10.1080/01495739.2022.2041517
Seyed Mojtaba Hozhabrossadati; Noël Challamel; Mohammad Rezaiee-Pajand; Ahmad Aftabi Sani Free vibration of a nanogrid based on Eringen’s stress gradient model, Mechanics Based Design of Structures and Machines, Volume 50 (2022) no. 2, p. 537 | DOI:10.1080/15397734.2020.1720720
Paulina Stempin; Wojciech Sumelka Formulation and experimental validation of space-fractional Timoshenko beam model with functionally graded materials effects, Computational Mechanics, Volume 68 (2021) no. 3, p. 697 | DOI:10.1007/s00466-021-01987-6
Andrea Francesco Russillo; Giuseppe Failla; Gioacchino Alotta; Francesco Marotti de Sciarra; Raffaele Barretta On the dynamics of nano-frames, International Journal of Engineering Science, Volume 160 (2021), p. 103433 | DOI:10.1016/j.ijengsci.2020.103433
Paulina Stempin; Wojciech Sumelka Dynamics of Space-Fractional Euler–Bernoulli and Timoshenko Beams, Materials, Volume 14 (2021) no. 8, p. 1817 | DOI:10.3390/ma14081817
Mohammad Ali Maneshi; Esmaeal Ghavanloo; S. Ahmad Fazelzadeh One-Dimensional Well-Posed Nonlocal Elasticity Models for Finite Domains, Size-Dependent Continuum Mechanics Approaches (2021), p. 149 | DOI:10.1007/978-3-030-63050-8_5
Mohammad A. Maneshi; Esmaeal Ghavanloo; S. Ahmad Fazelzadeh Well-posed nonlocal elasticity model for finite domains and its application to the mechanical behavior of nanorods, Acta Mechanica, Volume 231 (2020) no. 10, p. 4019 | DOI:10.1007/s00707-020-02749-w
V. S. Mutnuri; S. Gopalakrishnan On causality of wave motion in nonlocal theories of elasticity: a Kramers–Kronig relations study, Annals of Solid and Structural Mechanics, Volume 12 (2020) no. 1-2, p. 165 | DOI:10.1007/s12356-020-00056-6
H. Zhang; C.M. Wang; N. Challamel; W.H. Pan Calibration of Eringen's small length scale coefficient for buckling circular and annular plates via Hencky bar-net model, Applied Mathematical Modelling, Volume 78 (2020), p. 399 | DOI:10.1016/j.apm.2019.09.052
V. S. Mutnuri; S. Gopalakrishnan A Re-Examination of Wave Dispersion and on Equivalent Spatial Gradient of the Integral in Bond-Based Peridynamics, Journal of Peridynamics and Nonlocal Modeling, Volume 2 (2020) no. 3, p. 243 | DOI:10.1007/s42102-020-00033-y
Vittorio Gusella; Giuseppina Autuori; Patrizia Pucci; Federico Cluni Dynamics of Nonlocal Rod by Means of Fractional Laplacian, Symmetry, Volume 12 (2020) no. 12, p. 1933 | DOI:10.3390/sym12121933
G. Mikhasev; E. Avdeichik; D. Prikazchikov Free vibrations of nonlocally elastic rods, Mathematics and Mechanics of Solids, Volume 24 (2019) no. 5, p. 1279 | DOI:10.1177/1081286518785942
N. Challamel; H. Zhang; C.M. Wang; J. Kaplunov Scale effect and higher-order boundary conditions for generalized lattices, with direct and indirect interactions, Mechanics Research Communications, Volume 97 (2019), p. 1 | DOI:10.1016/j.mechrescom.2019.04.002
C. Chr. Koutsoumaris; K. G. Eptaimeros A research into bi-Helmholtz type of nonlocal elasticity and a direct approach to Eringen’s nonlocal integral model in a finite body, Acta Mechanica, Volume 229 (2018) no. 9, p. 3629 | DOI:10.1007/s00707-018-2180-9
Seyed Mojtaba Hozhabrossadati; Noël Challamel; Mohammad Rezaiee-Pajand; Ahmad Aftabi Sani Application of Green's function method to bending of stress gradient nanobeams, International Journal of Solids and Structures, Volume 143 (2018), p. 209 | DOI:10.1016/j.ijsolstr.2018.03.009
H. Zhang; C.M. Wang; N. Challamel Modelling vibrating nano-strings by lattice, finite difference and Eringen's nonlocal models, Journal of Sound and Vibration, Volume 425 (2018), p. 41 | DOI:10.1016/j.jsv.2018.04.001

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