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A thermoelastoplastic theory for special Cosserat rods. (English) Zbl 07225902
The authors develop a unified dynamic model for the coupled thermoelastoplastic motions of special Cosserat rods using a direct approach. The kinematics, dynamics and the first and second laws of thermodynamics are discussed. The one-dimensional form of the energy balance and the entropy balance are presented for the rods and their constitutive equations are given. The evolution equation of the temperature-like one-dimensional field variable is obtained. The yield function, flow rule and hardening law are introduced. The total strain measures of the rod are assumed to decompose additively into elastic and plastic parts. The stored thermoelastic energy of the rod is assumed to depend only on the elastic part of the total strain. Then, the most general quadratic form of the thermoelastic stored energy for hemitropic rods is obtained. The total material strains are postulated to be additively decomposed into elastic and plastic parts. The thermoelastoplastic constitutive relations are recorded by using a special form of the Helmholtz energy density. The yield function, the associative flow rule and the hardening law are obtained. The model is developed without any connection to the three dimensional basic equations of thermoelastoplasticity. The engineering applications remain to the determination of material constants. This paper may be of interest to mathematically minded engineers.
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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[1] [1] Manning, RS, Maddocks, JH, Kahn, JD. A continuum rod model of sequence-dependent DNA structure. J Chem Phys 1996; 105(13): 5626-5646.
[2] [2] Swigon, D, Coleman, BD, Tobias, I. The elastic rod model for DNA and its application to the tertiary structure of DNA minicircles in mononucleosomes. Biophys J 1998; 74(5): 2515-2530.
[3] [3] Bozec, L, van der Heijden, G, Horton, M. Collagen fibrils: Nanoscale ropes. Biophys J 2007; 92: 70-75.
[4] [4] Marko, JF, Neukirch, S. Competition between curls and plectonemes near the buckling transition of stretched supercoiled DNA. Phys Rev E 2012; 85(1): 011908.
[5] [5] Kumar, A, Mukherjee, S, Paci, JT. A rod model for three dimensional deformations of single-walled carbon nanotubes. Int J Sol Struct 2011; 48: 2849-2858.
[6] [6] Kumar, A, Kumar, S, Gupta, P. A helical Cauchy-Born rule for special Cosserat rod modeling of nano and continuum rods. J Elast 2016; 124(1): 81-106. · Zbl 1338.74012
[7] [7] Gupta, P, Kumar, A. Effect of material nonlinearity on spatial buckling of nanorods and nanotubes. J Elast 2017; 126(2): 155-171. · Zbl 1354.74023
[8] [8] Audoly, B, Pomeau, Y. Elasticity and geometry: From hair curls to the non-linear response of shells. Oxford: Oxford University Press, 2010. · Zbl 1223.74001
[9] [9] McMillen, T, Goriely, A. Tendril perversion in intrinsically curved rods. J Nonlinear Sci 2002; 12(3): 241-281. · Zbl 1100.74575
[10] [10] Costello, GA. Theory of wire rope. New York: Springer-Verlag, 1997.
[11] [11] Goyal, S, Perkins, C, Lee, CL. Nonlinear dynamics and loop formation in Kirchoff rods with implications to the mechanics of DNA and cables. J Comp Phys 2005; 209: 371-389. · Zbl 1329.74154
[12] [12] Chang, J, Pan, J, Zhang, J. Modelling rod-like flexible biological tissues for medical training. In: Magnenat-Thalmann, N (ed.) Modelling the physiological human 3DPH 2009 (Lecture Notes in Computer Science, vol. 5903). Berlin: Springer, 2009, 51-61.
[13] [13] Love, AEH. A treatise on the mathematical theory of elasticity. Mineola: Dover Books, 2000.
[14] [14] Cosserat, E, Cosserat, F. Theory of deformable bodies. Paris: Scientific Library A. Hermann and Sons, 1909. · JFM 38.0693.02
[15] [15] Green, AE, Naghdi, PM, Wenner, ML. On the theory of rods. I. Derivations from the three-dimensional equations. Proc R Soc London, Ser A 1974; 337(1611): 451-483. · Zbl 0325.73053
[16] [16] Rubin, MB. Cosserat theories: Shells, rods and points. Dordrecht: Springer Science & Business Media, 2000. · Zbl 0984.74003
[17] [17] Kumar, A, Mukherjee, S. A geometrically exact rod model including in-plane cross-sectional deformation. J App Mech 2011; 78: 011010.
[18] [18] Antman, SS. Nonlinear problems of elasticity. New York: Springer-Verlag, 1995. · Zbl 0820.73002
[19] [19] Ericksen, JL. Hypo-elastic potentials. Q J Mech Appl Math 1958; 11(1): 67-72. · Zbl 0083.18801
[20] [20] Green, AE. Micro-materials and multipolar continuum mechanics. Int J Eng Sci 1965; 3(5): 533-537.
[21] [21] Hodges, DH. Nonlinear composite beam theory. Prog Astronaut Aeronaut 2006; 213: 304.
[22] [22] Berdichevsky, V. Variational principles of continuum mechanics: I. Fundamentals. Heidelberg: Springer, 2009. · Zbl 1183.49002
[23] [23] DeSilva, CN, Whitman, AB. Thermodynamical theory of directed curves. J Math Phys 1971; 12(8): 1603-1609. · Zbl 0242.73004
[24] [24] Kafadar, CB. On the nonlinear theory of rods. Int J Eng Sci 1972; 10(4): 369-391. · Zbl 0243.73023
[25] [25] Green, AE, Naghdi, PM. On thermal effects in the theory of rods. Int J Solids Struct 1979; 15(11): 829-853. · Zbl 0419.73004
[26] [26] Simmonds, JG. A simple nonlinear thermodynamic theory of arbitrary elastic beams. J Elast 2005; 81(1): 51-62. · Zbl 1090.74034
[27] [27] Beyrouthy, J, Neff, P. A viscoelastic thin rod model for large deformations: Numerical examples. Math Mech Solids 2011; 16(8): 887-896. · Zbl 1269.74134
[28] [28] Linn, J, Lang, H, Tuganov, A. Geometrically exact Cosserat rods with Kelvin-Voigt type viscous damping. Mech Sci 2013; 4(1): 79-96.
[29] [29] Altenbach, H, Bîrsan, M, Eremeyev, VA. On a thermodynamic theory of rods with two temperature fields. Acta Mech 2012; 223(8): 1583-1596. · Zbl 1401.74163
[30] [30] Cao, DQ, Song, MT, Tucker, RW. Dynamic equations of thermoelastic Cosserat rods. Commun Nonlinear Sci Numer Simul 2013; 18(7): 1880-1887. · Zbl 1311.80003
[31] [31] Simo, JC, Kennedy, JG. On a stress resultant geometrically exact shell model. Part V. Nonlinear plasticity: Formulation and integration algorithms. Comp Methods App Mech Eng 1992; 96(2): 133-171. · Zbl 0754.73042
[32] [32] Saje, M, Planinc, I, Turk, G. A kinematically exact finite element formulation of planar elastic-plastic frames. Comp Methods App Mech Eng 1997; 144 (1-2): 125-151. · Zbl 0892.73070
[33] [33] Štok, B, Halilovi, M. Analytical solutions in elasto-plastic bending of beams with rectangular cross section. Appl Math Modell 2009; 33(3): 1749-1760.
[34] [34] Simo, JC, Hjelmstad, KD, Taylor, RL. Numerical formulations of elasto-viscoplastic response of beams accounting for the effect of shear. Comp Methods App Mech Eng 1984; 42(3): 301-330. · Zbl 0517.73074
[35] [35] Park, MS, Lee, BC. Geometrically non-linear and elastoplastic three-dimensional shear flexible beam element of von-Mises-type hardening material. Int J Numer Methods Eng 1996; 39(3): 383-408. · Zbl 0846.73067
[36] [36] Healey, TJ. Material symmetry and chirality in nonlinearly elastic rods. Math Mech Solids 2002; 7: 405-420. · Zbl 1090.74610
[37] [37] Drucker, DC. The effect of shear on the plastic bending of beams. J Appl Mech 1956; 28: 509. · Zbl 0074.19205
[38] [38] Neal, BG. The effect of shear and normal forces on the fully plastic moment of a beam of rectangular cross section. J Appl Mech 1961; 28(2): 269-274. · Zbl 0095.39304
[39] [39] Duan, L, Chen, W. A yield surface for doubly symmetrical sections. Eng Struct 1990; 12(2): 114-119.
[40] [40] Gendy, AS, Saleeb, AF.
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