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Elastic waves in a circular cylinder and cylindrical annulus for a subclass of implicit constitutive equations. (English) Zbl 1446.74149

Summary: The propagation of elastic waves in a circular cylinder and cylindrical annulus for two types of power-law constitutive equations is investigated. These power-law constitutive equations can describe elastic responses where the linearised strain and stress are nonlinearly related. These constitutive equations are a subclass of the more general class of implicit constitutive equations and are characterised by expressing the strain as a non-invertible function of the stress. Pseudo-solitary stress wave solutions for both types of constitutive equations in the circular cylinder and cylindrical annulus are derived. We find that for the power-law constitutive equation of Type I, a shock front will develop at the back of the wave while for the power-law constitutive equation of Type II, a shock front will develop at the front of the wave. Estimates of the times at which the shock front will develop are given. Standing wave solutions for both types of constitutive equations in the circular cylinder and cylindrical annulus are also obtained and the periods of oscillation are compared.

MSC:

74J30 Nonlinear waves in solid mechanics
74J35 Solitary waves in solid mechanics
74B20 Nonlinear elasticity
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[1] Saito, T, Furuta, T, Hwang, JH, et al. Multifunctional alloys obtained via a dislocation-free plastic deformation mechanism. Science 2003; 300(5618): 464-467.
[2] Talling, R, Dashwood, R, Jackson, M, et al. Determination of \(C_{11}-C_{12}\) in Ti-36Nb-2Ta-3Zr-0.3O (wt
[3] Withey, E, Jin, M, Minor, A, et al. The deformation of “gum metal” in nanoindentation. Mater Sci Eng A—Struct Mater 2008; 493: 26-32.
[4] Zhang, S, Li, S, Jia, M, et al. Fatigue properties of a multifunctional titanium alloy exhibiting nonlinear elastic deformation behavior. Scr Mater 2009; 60: 733-736.
[5] Rajagopal, KR . On the nonlinear elastic response of bodies in the small strain range. Acta Mechanica 2014; 225(6): 1545-1553. · Zbl 1401.74045
[6] Devendiran, V, Sandeep, R, Kannan, K, et al. A thermodynamically consistent constitutive equation for describing the response exhibited by several alloys and the study of a meaningful physical problem:. Int J Solids Struct 2016; 108.
[7] Kulvait, V, Málek, J, Rajagopal, K. Modeling gum metal and other newly developed titanium alloys within a new class of constitutive relations for elastic bodies. Arch Mech 2017; 69: 223-241.
[8] Rajagopal, KR . On implicit constitutive theories. Applic Math 2003; 48(4): 279-319. · Zbl 1099.74009
[9] Rajagopal, KR . The elasticity of elasticity. Z angew Math Phys 2007; 58(2): 309-317. · Zbl 1113.74006
[10] Rajagopal, K, Srinivasa, A. On the response of non-dissipative solids. Proc R Soc A: Math Phys Eng Sci 2007; 463: 357-367. · Zbl 1129.74010
[11] Rajagopal, K . Non-linear elastic bodies exhibiting limiting small strain. Math Mech Solids 2011; 16(1): 122-139. · Zbl 1269.74026
[12] Rajagopal, K, Walton, J. Modeling fracture in the context of a strain-limiting theory of elasticity: A single anti-plane shear crack. Int J Fracture 2011; 169(1): 39-48. · Zbl 1283.74074
[13] Gou, K, Mallikarjunaiah, M, Rajagopal, K, et al. Modeling fracture in the context of a strain-limiting theory of elasticity: A single plane-strain crack. Int J Eng Sci 2014; 88.
[14] Kulvait, V, Málek, J, Rajagopal, KR. Anti-plane stress state of a plate with a v-notch for a new class of elastic solids. Int J Fracture 2013; 179(1): 59-73.
[15] Ortiz, A, Bustamante, R, Rajagopal, KR. A numerical study of a plate with a hole for a new class of elastic bodies. Acta Mechanica 2012; 223(9): 1971-1981. · Zbl 1356.74122
[16] Bulíćek, M, Málek, J, Rajagopal, K, et al. On elastic solids with limiting small strain: Modelling and analysis. EMS Surv Math Sci 2014; 1: 293-342. · Zbl 1314.35184
[17] Itou, H, Kovtunenko, VA, Rajagopal, KR. Nonlinear elasticity with limiting small strain for cracks subject to non-penetration. Math Mech Solids 2017; 22(6): 1334-1346. · Zbl 1371.74245
[18] Bustamante, R, Rajagopal, K. Solutions of some simple boundary value problems within the context of a new class of elastic materials. Int J Non-Lin Mech 2011; 46: 376-386.
[19] Bustamante, R, Rajagopal, K. Solutions of some boundary value problems for a new class of elastic bodies undergoing small strains. Comparison with the predictions of the classical theory of linearized elasticity: Part I. Problems with cylindrical symmetry. Acta Mechanica 2014; 226: 1815-1838. · Zbl 1317.74034
[20] Rajagopal, K, Umakanthan, S. Spherical inflation of a class of compressible elastic bodies. Int J Non-Lin Mech 2011; 46: 1167.
[21] Sandeep, R, Kannan, K, Rajagopal, K. Numerical and approximate analytical solutions for cylindrical and spherical annuli for a new class of elastic materials. Arch Appl Mech 2016; 86(11): 1815-1826.
[22] Sakaguch, N, Niinomi, M, Akahori, T. Tensile deformation behavior of Ti-Nb-Ta-Zr biomedical alloys. Mater Trans 2004; 45(4): 1113-1119.
[23] Kannan, K, Rajagopal, K, Saccomandi, G. Unsteady motions of a new class of elastic solids. Wave Motion 2014; 51(5): 833-843. · Zbl 1456.74017
[24] Kambapalli, M, Kannan, K, Rajagopal, K. Circumferential stress waves in a nonlinear cylindrical annulus in a new class of elastic materials. Quart J Mech Appl Math 2014; 67: 193-203. · Zbl 1347.74045
[25] Magan, A, Mason, D, Harley, C. Two-dimensional nonlinear stress and displacement waves for a new class of constitutive equations. Wave Motion 2017; 77(12): 003.
[26] Rajagopal, K, Saccomandi, G. Circularly polarized wave propagation in a class of bodies defined by a new class of implicit constitutive relations. Z Angew Math Phys 2013; 65: 1003-1010. · Zbl 1308.74078
[27] Huang, SJ, Rajagopal, K, Dai, HH. Wave patterns in a nonclassic nonlinearly-elastic bar under Riemann data. Int J Non-Lin Mech 2017; 91: 76-85.
[28] Abramowitz, M, Stegun, IA. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1964. · Zbl 0171.38503
[29] McMahon, J . On the roots of the Bessel and certain related functions. Ann Math 1895; 9: 23-30. · JFM 25.0842.02
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