Symmetric Solitary Waves and Their Existence Conditions in Cubic Nonlinear Microstructured Solids

Naranmandula; Ereduncang

doi:10.3879/j.issn.1000-0887.2014.11.004

Volume 35 Issue 11

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2014 > 35(11): 1210-1217

Naranmandula, Ereduncang. Symmetric Solitary Waves and Their Existence Conditions in Cubic Nonlinear Microstructured Solids[J]. Applied Mathematics and Mechanics, 2014, 35(11): 1210-1217. doi: 10.3879/j.issn.1000-0887.2014.11.004

Citation:

PDF( 3025 KB)

Symmetric Solitary Waves and Their Existence Conditions in Cubic Nonlinear Microstructured Solids

doi: 10.3879/j.issn.1000-0887.2014.11.004

College of Physics and Electronic Information, Inner Mongolia University for the Nationalities, Tongliao, Inner Mongolia 028043, P.R.China

Funds: The National Natural Science Foundation of China（11462019; 10862003）

Received Date: 2014-05-22
Rev Recd Date: 2014-06-22
Publish Date: 2014-11-18

Abstract

Abstract

In view of the macroscale cubic nonlinear effect, the microscale cubic nonlinear effect and the microscale dispersion effect of solid materials, a new model for the longitudinal wave propagation in 1D microstructured solids was established based on the modified Mindlin theory. The qualitative analysis method was applied to the dynamical system, the existence of symmetric bell and anti-bell type solitary waves in the cubic nonlinear microstructured solid was proved under appropriate conditions, and the existence conditions of the 2 solitary waves were given. The microscale cubic nonlinear effect on the bell and anti-bell type solitary waves was analyzed with the numerical method. The results indicate that the widths of the 2 solitary waves decreases (or increases) with the rise (or fall) of the microscale nonlinear effect while the amplitudes of the 2 solitary waves remain unchanged.
- Mindlin theory,
- microstructured solid,
- solitary wave,
- existence condition

FullText(HTML)

References(13)

References

[1]	Mindlin R D. Micro-structure in linear elasticity[J].Archive for Rational Mechanics and Analysis, 1964,16(1): 51-78.
[2]	Engelbrecht J, Khamidullin Y. On the possible amplification of nonlinear seismic waves[J].Physics of the Earth and Planetary Interios,1988,50(1): 39-45.
[3]	Erofeev V I.Wave Processes in Solids With Microstructure [M]. Singapore: World Scientific, 2003: 101-223.
[4]	程昌钧. 理性力学在中国的传播与发展[J]. 力学与实践, 2008,30(1): 10-17.（CHENG Chang-jun. The dissemination and development of rational mechanics in China[J].Mechanics in Engineering,2008,30(1): 10-17.(in Chinese)）
[5]	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 Tian-min. Restudy of theories for elastic solids with microstructure[J].Applied Mathematics and Mechanics,2002,23(8): 771-777.(in Chinese))
[6]	陈少华, 王自强. 应变梯度理论进展[J]. 力学进展, 2003,33(2): 207-216.(CHEN Shao-hua, WANG Zi-qiang. Advances in strain gradient theory[J].Advances in Mechanics, 2003,33(2): 207-216.(in Chinese))
[7]	胡更开, 刘晓宁, 荀飞. 非均匀微极介质的有效性质分析[J]. 力学进展, 2004,34(2): 195- 214.(HU Geng-kai, LIU Xiao-ning, XUN Fei. Micromechanics of heterogeneous micropolar mediums[J].Advances in Mechanics,2004,34(2): 195-214.(in Chinese))
[8]	Engelbrecht J, Pastrone F. Wave in microstructured solids with nonlinearities in microscale[J].Proceedings of the Estonian Academy of Sciences, Physics, Mathematics,2003,52(1): 12-20.
[9]	Porubov A V, Pastrone F. Nonlinear bell-shaped and kink-shaped strain waves in microstructured solids[J].International Journal of Non-Linear Mechanics,2004,39(8): 1289-1299.
[10]	Porubov A V, Aero E L, Maugin G A. Two approaches to study essentially nonlinear and dispersive properties of the internal structure of materials[J].Physical Review E, 2009,79: 046608-046620.
[11]	Janno J, Engelbrecht J. An inverse solitary wave problem related to microstructured materials[J].Inverse Problems,2005,21(6): 2019-2034.
[12]	Janno J, Engelbrecht J. Solitary waves in nonlinear microstructured materials[J].Journal of Physics A: Mathematical and General,2005,38(23): 5159-5172.
[13]	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.