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.
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@article{CRMECA_2018__346_4_320_0,
author = {No\"el Challamel},
title = {Static and dynamic behaviour of nonlocal elastic bar using integral strain-based and peridynamic models},
journal = {Comptes Rendus. M\'ecanique},
pages = {320--335},
publisher = {Elsevier},
volume = {346},
number = {4},
year = {2018},
doi = {10.1016/j.crme.2017.12.014},
language = {en},
}
TY - JOUR AU - Noël Challamel TI - Static and dynamic behaviour of nonlocal elastic bar using integral strain-based and peridynamic models JO - Comptes Rendus. Mécanique PY - 2018 SP - 320 EP - 335 VL - 346 IS - 4 PB - Elsevier DO - 10.1016/j.crme.2017.12.014 LA - en ID - CRMECA_2018__346_4_320_0 ER -
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/
[1] Influence of spatial acoustic dispersion on dynamical properties of dislocations, Bull. Acad. Pol. Sci. Sér. Sci. Tech., Volume 13 (1965), pp. 337-385
[2] Nichtlokal Elastostatik: Ableitung aus der Gittertheorie, Z. Phys., Volume 196 (1966), pp. 203-211
[3] Model of elastic medium with simple structure and space dispersion, Prikl. Mat. Mekh., Volume 30 (1966), pp. 542-550
[4] On nonlocal elasticity, Int. J. Eng. Sci., Volume 10 (1972) no. 3, pp. 233-248
[5] Nonlocal elasticity and related variational principles, Int. J. Solids Struct., Volume 38 (2001), pp. 7359-7380
[6] Uniqueness of initial–boundary value problems in nonlocal elasticity, Int. J. Solids Struct., Volume 25 (1989) no. 11, pp. 1271-1278
[7] Nonlocal Continuum Field Theories, Springer, New York, 2002
[8] Continuum Mechanics Through the Twentieth Century – A Concise Historical Perspective, Springer, 2013
[9] Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, Volume 48 (2000), pp. 175-209
[10] Deformation of a peridynamic bar, J. Elast., Volume 73 (2003), pp. 173-190
[11] Peridynamics via finite element analysis, Finite Elem. Anal. Des., Volume 43 (2007), pp. 1169-1178
[12] Physically-based approach to the mechanics of strong non-local linear elasticity theory, J. Elast., Volume 97 (2009), pp. 103-130
[13] On fractional peridynamic deformations, Arch. Appl. Mech., Volume 86 (2016) no. 12, pp. 1987-1994
[14] Bending of Euler–Bernoulli beams using Eringen's integral formulation: a paradox resolved, Int. J. Eng. Sci., Volume 99 (2016), pp. 107-116
[15] Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams, Int. J. Mech. Sci., Volume 121 (2017), pp. 151-156
[16] Analytical solutions of peristatic and peridynamic problems for a 1D infinite rod, Int. J. Solids Struct., Volume 49 (2012), pp. 2887-2897
[17] Wave dispersion and basic concepts of peridynamics compared to classical nonlocal damage models, J. Appl. Mech., Volume 83 (2016)
[18] Peristatic solutions for finite one- and two-dimensional systems, Math. Mech. Solids, Volume 22 (2017) no. 8, pp. 1639-1653
[19] A symmetric nonlocal damage model, Int. J. Solids Struct., Volume 40 (2003) no. 13, pp. 3621-3645
[20] Relation between non-local elasticity and lattice dynamics, Cryst. Lattice Defects, Volume 7 (1977), pp. 51-57
[21] One-dimensional nonlocal and gradient elasticity: closed-form solution and size effect, Mech. Res. Commun., Volume 48 (2013), pp. 46-51
[22] On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys., Volume 54 (1983), pp. 4703-4710
[23] Finite Difference Equations, Dover, 1992
[24] Nonlocal or gradient elasticity macroscopic models: a question of concentrated or distributed microstructure, Mech. Res. Commun., Volume 71 (2016), pp. 25-31
[25] 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] On the role of the virtual boundary layer in 1D fractional elasticity problems, J. Eng. Mech., Volume 143 (2017), p. 9
[27] Theory of nonlocal elasticity and some applications, Res. Mech., Volume 21 (1987), pp. 313-342
[28] The small length scale effect for a non-local cantilever beam: a paradox solved, Nanotechnology, Volume 19 (2008)
[29] A dispersive wave equation using non-local elasticity, C. R. Mécanique, Volume 337 (2009), pp. 591-595
[30] 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] Gradient-enhanced damage for quasi-brittle materials, Int. J. Numer. Methods Eng., Volume 39 (1996), pp. 3391-3403
[32] Handbook of Integral Equations, CRC Press, Taylor and Francis Group, 2008
[33] Analysis of non-local models through energetic formulations, Int. J. Solids Struct., Volume 40 (2003), pp. 2905-2936
[34] Micromorphic approach for gradient elasticity, viscoplasticity, and damage, J. Eng. Mech., Volume 135 (2009), pp. 117-131
[35] Bending, buckling and vibration of hybrid nonlocal beams, J. Eng. Mech., Volume 136 (2010) no. 5, pp. 562-574
[36] A unified integro-differential nonlocal model, Int. J. Eng. Sci., Volume 95 (2015), pp. 60-75
[37] Exact solutions for the static bending of Euler–Bernoulli beams using Eringen's two phase local/nonlocal model, AIP Adv., Volume 6 (2016)
[38] Closed form solution for a nonlocal elastic bar in tension, Int. J. Solids Struct., Volume 40 (2003), pp. 13-23
[39] 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] A nonhomogeneous nonlocal elasticity model, Eur. J. Mech. A, Solids, Volume 25 (2006), pp. 308-333
[41] A different approach to Eringen's nonlocal stress model with application for beams, Int. J. Solids Struct., Volume 112 (2017), pp. 222-238
[42] Axial vibration of the nanorods with the nonlocal continuum rod model, Physica E, Volume 41 (2009), pp. 861-864
[43] Nonlocal theories for bending, buckling and vibration of beams, Int. J. Eng. Sci., Volume 45 (2007), pp. 288-307
[44] 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] Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams, Compos. Struct., Volume 105 (2013), pp. 351-368
[46] Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory, Int. J. Mech. Sci., Volume 115–116 (2016), pp. 135-144
[47] Longitudinal vibrations of a beam: a gradient elasticity approach, Mech. Res. Commun., Volume 23 (1996), pp. 35-40
[48] Static and dynamic analysis of a gradient elastic bar in tension, Arch. Appl. Mech., Volume 72 (2002), pp. 483-497
[49] On Eringen's stress gradient model for bending of nonlocal beams, J. Eng. Mech. (2016)
[50] 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] , 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] Nonlinear Waves in Elastic Crystals, Oxford University Press, 1999
[53] Discrete systems behave as nonlocal structural elements: bending, buckling and vibration analysis, Eur. J. Mech. A, Solids, Volume 44 (2014), pp. 125-135
[54] 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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