×

Equilibrium paths of a three-bar truss in finite elasticity with an application to graphene. (English) Zbl 1446.74155

Summary: This paper presents the formulation of the equilibrium problem of a three-bar truss in the nonlinear context of finite elasticity. The bars are composed of a homogeneous, isotropic, and compressible hyperelastic material. The equilibrium equations in the deformed configuration are derived under the assumption of homogeneous deformations and the stability of the solutions is assessed through the energy criterion. The general formulation is then specialized for a compressible Mooney-Rivlin material. The results for both vertical and horizontal load cases show unexpected post-critical behaviors involving several branches, stable asymmetrical configurations, bifurcation, and snap-through. The three-bar truss studied here is not only a benchmark test for the numerical analysis of nonlinear truss structures, but also a representative system for the unit cell of the graphene hexagonal lattice. Therefore, an application to graphene is performed by simulating the covalent bonds between carbon atoms as the bars of the truss, characterized by the modified Morse potential. The results provide insights on the internal mechanisms that take place when graphene undergoes large in-plane deformations, whose influence should be considered when developing molecular mechanics and continuum models in nonlinear elasticity.

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74K99 Thin bodies, structures
74B20 Nonlinear elasticity
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Mises, R. Über die stabilitätsprobleme der elastizitätstheorie. Z Angew Math Mech 1923; 3(6): 406-422. · JFM 49.0594.02
[2] Mises, R, Ratzersdorfer, J. Die Knicksicherheit von Fachwerken. Z Angew Math Mech 1925; 5(3): 218-235. · JFM 51.0639.05
[3] Bažant, ZP, Cedolin, L. Stability of structures: Elastic, inelastic, fracture and damage theories. Singapore: World Scientific, 1991. · Zbl 0744.73001
[4] Savi, MA, Pacheco, PM, Braga, AM. Chaos in a shape memory two-bar truss. Int J Non Linear Mech 2002; 37(8): 1387-1395. · Zbl 1346.74030
[5] Bellini, PX. The concept of snap-buckling illustrated by a simple model. Int J Non Linear Mech 1972; 7(6): 643-650.
[6] Psotny, M, Ravinger, J. Von Mises truss with imperfection. Slovak J Civ Eng 2003; 11: 1-7.
[7] Bazzucchi, F, Manuello, A, Carpinteri, A. Interaction between snap-through and Eulerian instability in shallow structures. Int J Non Linear Mech 2017; 88: 11-20.
[8] Pecknold, D, Ghaboussi, J, Healey, T. Snap-through and bifurcation in a simple structure. J Eng Mech 1985; 111(7): 909-922.
[9] Kwasniewski, L. Complete equilibrium paths for Mises trusses. Int J Non Linear Mech 2009; 44(1): 19-26.
[10] Ligarò, SS, Valvo, PS. Large displacement analysis of elastic pyramidal trusses. Int J Solids Struct 2006; 43(16): 4867-4887. · Zbl 1120.74582
[11] Pelliciari, M, Tarantino, AM. Equilibrium paths for von Mises trusses in finite elasticity. J Elast. Epub ahead of print 27 March 2019. DOI: 10.1007/s10659-019-09731-1. · Zbl 1434.74027
[12] Rezaiee-Pajand, M, Naghavi, A. Accurate solutions for geometric nonlinear analysis of eight trusses. Mech Based Des Struct Mach 2011; 39(1): 46-82.
[13] Cengiz Toklu, Y, Temür, R, Bekdaş, G. Computation of nonunique solutions for trusses undergoing large deflections. Int J Comput Methods 2015; 12(03): 1550022.
[14] Lee, C, Wei, X, Kysar, JW, et al. Measurement of the elastic properties and intrinsic strength of monolayer graphene. Science 2008; 321(5887): 385-388.
[15] Marianetti, CA, Yevick, HG. Failure mechanisms of graphene under tension. Phys Rev Lett 2010; 105(24): 245502.
[16] Bu, H, Chen, Y, Zou, M, et al. Atomistic simulations of mechanical properties of graphene nanoribbons. Phys Lett A 2009; 373(37): 3359-3362.
[17] Zhang, P, Huang, Y, Geubelle, P, et al. The elastic modulus of single-wall carbon nanotubes: A continuum analysis incorporating interatomic potentials. Int J Solids Struct 2002; 39(13-14): 3893-3906. · Zbl 1049.74753
[18] Odegard, GM, Gates, TS, Nicholson, LM, et al. Equivalent-continuum modeling of nano-structured materials. Compos Sci Technol 2002; 62(14): 1869-1880.
[19] Chang, T, Gao, H. Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model. J Mech Phys Solids 2003; 51(6): 1059-1074. · Zbl 1049.74034
[20] Arroyo, M, Belytschko, T. Finite crystal elasticity of carbon nanotubes based on the exponential Cauchy-Born rule. Phys Rev B 2004; 69(11): 115415.
[21] Guo, X, Wang, J, Zhang, H. Mechanical properties of single-walled carbon nanotubes based on higher order Cauchy-Born rule. Int J Solids Struct 2006; 43(5): 1276-1290. · Zbl 1119.74534
[22] Scarpa, F, Adhikari, S, Phani, AS. Effective elastic mechanical properties of single layer graphene sheets. Nanotechnol 2009; 20(6): 065709.
[23] Georgantzinos, S, Katsareas, D, Anifantis, N. Graphene characterization: A fully non-linear spring-based finite element prediction. Physica E 2011; 43(10): 1833-1839.
[24] Berinskii, I, Borodich, FM. Elastic in-plane properties of 2D linearized models of graphene. Mech Mater 2013; 62: 60-68.
[25] Tserpes, K, Papanikos, P. Finite element modeling of single-walled carbon nanotubes. Composites Part B 2005; 36(5): 468-477.
[26] Meo, M, Rossi, M. Prediction of Young’s modulus of single wall carbon nanotubes by molecular-mechanics based finite element modelling. Compos Sci Technol 2006; 66(11-12): 1597-1605.
[27] Belytschko, T, Xiao, S, Schatz, GC, et al. Atomistic simulations of nanotube fracture. Phys Rev B 2002; 65(23): 235430.
[28] Lanzoni, L, Tarantino, AM. Equilibrium configurations and stability of a damaged body under uniaxial tractions. Z Angew Math Phys 2015; 66(1): 171-190. · Zbl 1317.74036
[29] Tarantino, AM. Homogeneous equilibrium configurations of a hyperelastic compressible cube under equitriaxial dead-load tractions. J Elast 2008; 92(3): 227. · Zbl 1179.74012
[30] Tarantino, AM. Equilibrium paths of a hyperelastic body under progressive damage. J Elast 2014; 114(2): 225-250. · Zbl 1428.74189
[31] Ziegler, H. Principles of structural stability, vol. 35. Basel: Birkhäuser, 2013.
[32] Ciarlet, PG, Geymonat, G. Sur les lois de comportement en élasticité non linéaire compressible. CR Acad Sci Paris Sér II 1982; 295: 423-426. · Zbl 0497.73017
[33] Peng, Z, Yonggang, H, Geubelle, PH, et al. On the continuum modeling of carbon nanotubes. Acta Mech Sin 2002; 18(5): 528-536.
[34] Tersoff, J. New empirical approach for the structure and energy of covalent systems. Phys Rev B 1988; 37(12): 6991.
[35] Brenner, DW. Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. Phys Rev B 1990; 42(15): 9458.
[36] Genoese, A, Genoese, A, Rizzi, NL, et al. On the derivation of the elastic properties of lattice nanostructures: The case of graphene sheets. Composites Part B 2017; 115: 316-329.
[37] Xiao, J, Gama, B, Gillespie, J An analytical molecular structural mechanics model for the mechanical properties of carbon nanotubes. Int J Solids Struct 2005; 42(11-12): 3075-3092. · Zbl 1144.74028
[38] Ni, Z, Bu, H, Zou, M, et al. Anisotropic mechanical properties of graphene sheets from molecular dynamics. Physica B 2010; 405(5): 1301-1306.
[39] Gao, Y, Hao, P. Mechanical properties of monolayer graphene under tensile and compressive loading. Physica E 2009; 41(8): 1561-1566.
[40] Xu, M, Paci, JT, Oswald, J, et al. A constitutive equation for graphene based on density functional theory. Int J Solids Struct 2012; 49(18): 2582-2589.
[41] Lu, Q, Huang, R. Nonlinear mechanics of single-atomic-layer graphene sheets. Int J Appl Mech 2009; 1(3): 443-467.
[42] Singh, S, Patel, B. Nonlinear elastic properties of graphene sheet under finite deformation. Compos Struct 2015; 119: 412-421.
[43] Zhou, J, Huang, R. Internal lattice relaxation of single-layer graphene under in-plane deformation. J Mech Phys Solids 2008; 56(4): 1609-1623. · Zbl 1171.74334
[44] Wang, J, Guo, X, Zhang, H, et al. Energy and mechanical properties of single-walled carbon nanotubes predicted using the higher order Cauchy-Born rule. Phys Rev B 2006; 73(11): 115428.
[45] Lanzoni, L, Tarantino, AM. Damaged hyperelastic membranes. Int J Non Linear Mech 2014; 60: 9-22.
[46] Lanzoni, L, Tarantino, AM. A simple nonlinear model to simulate the localized necking and neck propagation. Int J Non Linear Mech 2016; 84: 94-104.
[47] Lanzoni, L, Tarantino, AM. Finite anticlastic bending of hyperelastic solids and beams. J Elast 2018; 131(2): 137-170. · Zbl 1390.74034
[48] Falope, FO, Lanzoni, L, Tarantino, AM. Bending device and anticlastic surface measurement of solids under large deformations and displacements. Mech Res Commun 2019; 97: 52-56.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.