微结构固体中孤立波的演变及非光滑孤立波

那仁满都拉; 韩元春; 张芳

doi:10.21656/1000-0887.390069

微结构固体中孤立波的演变及非光滑孤立波

doi: 10.21656/1000-0887.390069

¹内蒙古民族大学物理与电子信息学院, 内蒙古通辽 028000;²内蒙古民族大学数学学院, 内蒙古通辽 028000

基金项目: 国家自然科学基金(11462019)

详细信息

作者简介:
那仁满都拉（1963—）,男,教授,博士,硕士生导师（通讯作者. E-mail: nrmdltl@126.com）.

中图分类号: O331;O347
计量
- 文章访问数: 1071
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- 被引次数: 0
出版历程
- 收稿日期: 2018-02-05
- 修回日期: 2018-12-12
- 刊出日期: 2019-04-01

Solitary Wave Evolution and Non-Smooth Solitary Waves in Microstructured Solids

¹College of Physics and Electronic Information, Inner Mongolia University for Nationalities, Tongliao, Inner Mongolia 028000, P.R.China; ²College of Mathematics, Inner Mongolia University for Nationalities, Tongliao, Inner Mongolia 028000, P.R.China

Funds: The National Natural Science Foundation of China(11462019)

摘要

摘要: 给出了包含宏观应变和微形变的全部二次项以及宏观应变三次项的一种新的自由能函数.利用新自由能函数并根据Mindlin微结构理论，建立了描述微结构固体中纵波传播的一种新模型.利用近来发展的奇行波系统的动力系统理论，分析了系统的所有相图分支，并给出了周期波解、孤立波解、准孤立尖波解、孤立尖波解以及紧孤立波解.孤立尖波解和紧孤立波解的得到，有效地证明了在一定条件下，微结构固体中可以形成和存在孤立尖波和紧孤立波等非光滑孤立波.此结果进一步推广了微结构固体中只存在光滑孤立波的已有结论.
- 模型 /
- 微结构固体 /
- 非光滑孤立波 /
- 演变
Abstract: A new free energy function was given with all quadratic terms of macro strain and micro deformation, as well as cubic terms of macro strain. A new model for description of the longitudinal wave propagation in microstructured solids was established by means of the new free energy function and Mindlin’s microstructure theory. Based on the dynamical system theory for singular traveling wave systems developed recently, all bifurcations of phase portraits of the traveling wave systems were analyzed, and the periodic wave solutions, the solitary wave solutions, the quasi peakon solutions, the peakon solutions and the compacton solutions were given. The obtained peakon and compacton solutions effectively prove that non-smooth solitary waves such as the peakon and the compacton can form and exist in microstructured solids under certain conditions. The results further exceed the conclusion that only smooth solitary waves can exist in microstructured solids.
- model /
- microstructured solid /
- non-smooth solitary wave /
- evolution

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参考文献(16)

[1]	COSSERAT E, COSSERAT F. Theoriedes Corps Deformables [M]. Paris: Hermann, 1909: 32-56.
[2]	TOUPIN R A. Elastic materials with couple-stresses[J]. Archive for Rational Mechanics and Analysis,1962,11(1): 385-414.
[3]	MINDLIN R D. Micro-structure in linear elasticity[J]. Archive for Rational Mechanics and Analysis,1963,16(1): 51-78.
[4]	GREEN A E, RIVLIN R S. Multipolar continuum mechanics[J]. Archive for Rational Mechanics and Analysis,1964,17(2): 113-147.
[5]	ERINGEN A C, SUHUBI E S. Nonlinear theory of simple micro-elastic solids: I[J]. International Journal of Engineering Science,1964,2(2): 189-203.
[6]	ERINGEN A C. On the linear theory of micropolar elasticity[J]. Journal of Mathematics and Mechanics,1969,7(12): 1213-1220.
[7]	戴天民. 对带有微结构的弹性固体理论的再研究[J]. 应用数学和力学, 2002,23(8): 771-777.(DAI Tianmin. Restudy of theories for elastic solids with microstructure[J]. Applied Mathematics and Mechanics,2002,23(8): 771-777.(in Chinese))
[8]	JANNO J, ENGELBRECHT J. Solitary waves in nonlinear microstructured materials[J]. Journal of Physics A: Mathematical and General,2005,38: 5159-5172.
[9]	SALUPERE A, TAMM K. On the influence of material properties on the wave propagation in Mindlin-type microstructured solids[J]. Wave Motion,2013,50(7): 1127-1139.
[10]	JANNO J, ENGELBRECHT J. An inverse solitary wave problem related to microstructured materials[J]. Inverse Problems,2005,21(6): 2019-2034.
[11]	那仁满都拉. 复杂固体并式微结构模型及孤立波的存在性[J]. 应用数学和力学, 2018,39(1): 41-49.(NARANMANDULA. A concurrent microstructured model for complex solids and existence of solitary waves[J]. Applied Mathematics and Mechanics,2018,39(1): 41-49.(in Chinese))
[12]	YU Liqin, TIAN Lixin, WANG Xuedi. The bifurcation and peakon for Degasperis-Procesi equation[J]. Chaos, Solitons & Fractals,2006,30(4): 956-966.
[13]	冯大河, 李继彬. Jaulent-Miodek方程的行波解分支[J]. 应用数学和力学, 2007,28(8): 894-900.(FENG Dahe, LI Jibin. Bifurcations of travelling wave solutions for Jaulent-Miodek equations[J]. Applied Mathematics and Mechanics,2007,28(8): 894-900.(in Chinese))
[14]	LI Jibin, CHEN Guangrong. On a class of singular nonlinear traveling wave equations[J]. International Journal of Bifurcation and Chaos,2007,17(11): 4049-4065.
[15]	LI Jibin. Singular Nonlinear Traveling Wave Equations: Bifurcation and Exact Solutions [M]. Beijing: Science Press, 2013: 1-113.
[16]	LI Jibin, ZHU Wenjin, CHEN Guangrong. Understanding peakons, periodic peakons and compactons via a shallow water wave equation[J]. International Journal of Bifurcation and Chaos,2016,26(12): 1650207.