×

Global structure stability for the wave catching-up phenomenon in a prestressed two-material bar. (English) Zbl 1316.35175

Summary: Shock waves in a structure can result in the detachment of an interface and induce microcracks. In a recent study [S.-J. Huang et al., “Mathematical theory and analytical solutions for the wave catching-up phenomena in a nonlinearly elastic composite bar”, Proc. A, R. Soc. Lond. 468, No. 2148, 3882–3901 (2012; doi:10.1098/rspa.2012.0292)], it was shown that for certain nonlinearly elastic materials it is possible to generate a phenomenon in which a tensile wave can catch the first transmitted compressive wave (so the former can be undermined) in an initially stress-free two-material bar. In this study, we consider the wave catching-up phenomenon in a nonlinearly elastic prestressed two-material bar. We use the same method as that used by Huang et al. in the previously mentioned paper to construct solutions. Our main focus is on proving the global structure stability of the solutions in a prestressed (or initially stress-free) two-material bar. We first reduce the corresponding initial boundary value problem into several typical free boundary problems based on the formulation of Riemann invariants. Then, using a constructive method and carefully treating the complexity arising from multiple reflections of waves at the interface in the two-material bar, we successfully prove the global structure stability of the wave catching-up phenomenon.

MSC:

35L50 Initial-boundary value problems for first-order hyperbolic systems
35L65 Hyperbolic conservation laws
74J30 Nonlinear waves in solid mechanics
74M20 Impact in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
35B35 Stability in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J.D. Achenbach, {\it Wave Propagation in Elastic Solids}. North-Holland, Amsterdam, The Netherlands, 1984. · Zbl 0657.73019
[2] I.V. Andrianov, V.V. Danishevs’kyy, H. Topol, and D. Weichert, {\it Homogenization of a 1D nonlinear dynamical problem for periodic composites,} ZAMM Z. Angew. Math. Mech., 91 (2011), pp. 623-654. · Zbl 1316.74043
[3] E.M. Arruda and M.C. Boyce, {\it A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials}, J. Mech. Phys. Solids, 41 (1993), pp. 389-412. · Zbl 1355.74020
[4] A. Berezovski, M. Berezovski, and J. Engelbrecht, {\it Numerical simulation of nonlinear elastic wave propagation in piecewise homogeneous media}, Materials Sci. Engrg. A, 418 (2006), pp. 364-369.
[5] B.E. Clements, J.N. Johnson, and R.S. Hixson, {\it Stress waves in composite materials,} Phys. Rev. E, 54 (1996), pp. 6876-6888.
[6] H.-H. Dai and D.-X. Kong, {\it Global structure stability of impact-induced tensile waves in a rubberlike material,} IMA J. Appl. Math., 71 (2006), pp. 14-33. · Zbl 1114.74024
[7] M. Destrade and N.H. Scott, {\it Surface waves in a deformed isotropic hyperelastic material subject to an isotropic internal constraint}, Wave Motion, 40 (2004), pp. 347-357. · Zbl 1163.74339
[8] Y. Fu and B. Devenish, {\it Effects of pre-stresses on the propagation of nonlinear surface waves in an incompressible elastic half-space}, Q. J. Mech. Appl. Math., 49 (1995), pp. 65-80. · Zbl 0857.73020
[9] A.N. Gent, {\it A new constitutive relation for rubber}, Rubber Chem. Tech., 69 (1996), pp. 59-61.
[10] S.-J. Huang, H.-H. Dai, Z. Chen, and D.-X. Kong, {\it Mathematical theory and analytical solutions for the wave catching-up phenomena in a nonlinearly elastic composite bar,} R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 468 (2012), pp. 3882-3901. · Zbl 1371.74043
[11] J.-F. Jiang, Y. Xu, and H.-H. Dai, {\it A dissipation-rate reserving DG method for wave catching-up phenomena in a nonlinearly elastic composite bar,} J. Comput. Phys., 258 (2014), pp. 405-430. · Zbl 1349.74325
[12] D.-X. Kong, {\it Global structure stability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: Shocks and contact discontinuities,} J. Differential Equations, 188 (2003), pp. 242-271. · Zbl 1015.35069
[13] D.-X. Kong, {\it Global structure instability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: Rarefaction waves,} J. Differential Equations, 219 (2005), pp. 421-450. · Zbl 1093.35049
[14] J.K. Knowles, {\it Impact-induced tensile waves in a rubberlike material,} SIAM J. Appl. Math., 62 (2002), pp. 1153-1175. · Zbl 1041.74010
[15] P.D. Lax, {\it Hyperbolic systems of conservation laws} II, Comm. Pure Appl. Math., 10 (1957), pp. 537-566. · Zbl 0081.08803
[16] T.-T. Li, {\it Global Classical Solutions for Quasilinear Hyperbolic Systems,} Rech. Math. Appl. 32, Wiley-Masson, Paris, 1994. · Zbl 0841.35064
[17] S. Nemat-Nasser and A.V. Amirkhizi, {\it Finite-amplitude shear waves in pre-stressed thin elastomers,} Wave Motion, 43 (2005), pp. 20-28. · Zbl 1231.74198
[18] A.N. Norris and W.J. Parnell, {\it Hyperelastic cloaking theory: Transformation elasticity with prestressed solids,} R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 468 (2012), pp. 2881-2903. · Zbl 1371.74039
[19] A.N. Norris and A.L. Shuvalov, {\it Elastic cloaking theory,} Wave Motion, 48 (2011), pp. 525-538. · Zbl 1283.74024
[20] W.J. Parnell, {\it Effective wave propagation in a pre-stressed nonlinear elastic composite bar,} IMA J. Appl. Math., 72 (2007), pp. 223-244. · Zbl 1118.74025
[21] W.J. Parnell, {\it Nonlinear pre-stress for cloaking from antiplane elastic waves,} R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 468 (2012), pp. 563-580. · Zbl 1364.74049
[22] W.J. Parnell, A.N. Norris, and T. Shearer, {\it Employing pre-stress to generate finite cloaks for antiplane elastic waves,} Appl. Phys. Lett., 100 (2012), 171907.
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.