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Pure Mathematics 2024
一类可压缩超弹性材料的圆柱空穴现象的定性分析
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Abstract:
本文主要研究了一类可压缩超弹性材料的实心圆柱体的径向有限变形问题，通过理论解和数值模拟，我们得到了一些有趣的结论。存在一个临界拉伸值，当径向拉伸小于临界值时，圆柱体保持为实心圆柱体，当径向拉伸大于临界值时，圆柱体轴线上会形成圆柱空穴。应力集中和突变现象进一步表明，可压缩超弹性圆柱体中的空穴现象符合实际的物理背景。
In this paper, the finite radial deformation of a solid cylinder made of a class of compressible hyperelastic materials is studied. Through theoretical solutions and numerical simulations, we obtain some interesting conclusions. However, there is a critical stretching value. When the radial stretching is less than the critical value, the cylinder remains a solid cylinder. When the radial stretching is greater than the critical value, a cylindrical cavity will form on the axis of the cylinder. Furthermore, the stress concentration and abrupt change phenomenon shows that the cavity phenomenon in the compressible hyperelastic cylinder is in line with the actual physical background.
| [1] | Gent, A.N. and Lindley, P.B. (1958) Internal Rupture of Bonded Rubber Cylinders in Tension. Proceedings of the Royal Society of London. Series A, 249, 195-205. https://doi.org/10.1098/rspa.1959.0016 |
| [2] | Ball, J.M. (1982) Discontinuous Equilibrium Solutions and Cavitations in Nonlinear Elasticity. Philosophical Transactions of the Royal Society of London. Series A, 306, 557-611. https://doi.org/10.1098/rsta.1982.0095 |
| [3] | Horgan, C.O. and Abeyaratne, R. (1986) A Bifurcation Problem for a Compressible Nonlinear Elastic Medium: Growth of a Micro-Void. Journal of Elasticity, 16, 189-200. https://doi.org/10.1007/BF00043585 |
| [4] | Horgan, C.O. and Polignone, D.A. (1995) Cavitation in Nonlinear Elastic Solids: A Review. Applied Mechanics Review, 48, 471-485. https://doi.org/10.1115/1.3005108 |
| [5] | Yuan, X.G., Zhu, Z.Y. and Cheng, C.J. (2005) Study on Cavitated Bifurcation Problems for Spheres Composed of Hyper-Elastic Materials. Journal of Engineering Mathematics, 51, 15-34. https://doi.org/10.1007/s10665-004-1242-2 |
| [6] | Yuan, X.G. and Wei, F.L. (2011) Cavitation at the Axile Line of a Class of Compressible Hyperelastic Cylinder. Acta Mechanica Soida Sinica, 10, 115-119. |
| [7] | Murphy, J.G. and Biwa, S. (1997) Nonmontonic Cavity Growth in Finite, Compressible Elasticity. International Journal of Solids and Structures, 34, 3859-3872. https://doi.org/10.1016/S0020-7683(96)00237-5 |
| [8] | Shang, X.C. and Cheng, C.J. (2001) Exact Solution for Cavitated Bifurcation for Compressible Hyper-Elastic Materials. International Journal of Engineering Science, 39, 1101-1117. https://doi.org/10.1016/S0020-7225(00)00090-2 |
| [9] | Yuan, X.G., Zhu, Z.Y. and Cheng, C.J. (2004) Void Formation and Growth for a Class of Compressible Hyper-Elastic Sphere. Journal of Shanghai University, 8, 13-18. https://doi.org/10.1007/s11741-004-0004-8 |
| [10] | 尚新春, 程昌钧. 超弹性材料中的球形空穴分岔[J]. 力学学报, 1996, 28(4): 751-756. |
| [11] | Carroll, M.M. (1988) Finite Strain Solutions in Compressible Isotropic Elasticity. Journal of Elasticity, 20, 65-92. https://doi.org/10.1007/BF00042141 |
