Reflection of Elastic Waves at Imperfect Interface in a Couple-Stress Half-Space Under the External Magnetic Field

Wang Changda; Zhou Yang; Wang Ziyi; Li Yueqiu

doi:10.21656/1000-0887.460152

Volume 47 Issue 8

Aug. 2026

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2026 > 47(8): 1009-1018

Wang Changda, Zhou Yang, Wang Ziyi, Li Yueqiu. Reflection of Elastic Waves at Imperfect Interface in a Couple-Stress Half-Space Under the External Magnetic Field[J]. Applied Mathematics and Mechanics, 2026, 47(8): 1009-1018. doi: 10.21656/1000-0887.460152

Citation:

PDF( 2810 KB)

Reflection of Elastic Waves at Imperfect Interface in a Couple-Stress Half-Space Under the External Magnetic Field

doi: 10.21656/1000-0887.460152

Wang Changda^{1, 2
,
,},
Zhou Yang^{1, 2
,},
Wang Ziyi¹,
Li Yueqiu^3
,

1.
School of Earthquake Engineering and Building Safety, Institute of Disaster Prevention, Sanhe, Hebei 065201, P.R. China
2.
Hebei Technology Innovation Center for Multi Hazard Resilience and Emergency Handling of Engineering Structures, Sanhe, Hebei 065201, P.R. China
3.
College of Science, North China Institute of Science and Technology, Sanhe, Hebei 065201, P.R. China

Received Date: 2025-08-28
Rev Recd Date: 2025-11-11
Publish Date: 2026-08-01

Abstract

Abstract

The reflection problem of elastic waves at the elastic interface in couple-stress solids was studied under the influence of an external magnetic field. Firstly, based on the couple stress theory and Maxwell's electromagnetic theory, the governing equations for the propagation of elastic waves and 3 types of waves (the P wave, the SV wave and the SS wave) were derived. It was also found that the external magnetic field affects the propagation of elastic waves through the Lorentz force, but no new wave patterns are generated. Secondly, based on the elastic interface conditions, the corresponding linear algebraic equations were obtained, and the amplitude ratios of various reflected waves relative to the incident wave in the case of incident P waves and SV waves were derived. Then, the reflection coefficient was defined with the energy flow ratio, and the effects of the external magnetic field, 3 elastic constants, couple stress parameters, and the angular frequency of the incident wave on the elastic wave reflection coefficient, were discussed for the incident P wave. Finally, the accuracy of the numerical results was verified through the conservation of normal energy for each wave.
- external magnetic field,
- couple stress,
- elastic interface,
- reflection,
- energy conservation

FullText(HTML)

References(30)

References

[1]	Knopoff L. The interaction between elastic wave motions and a magnetic field in electrical conductors[J]. Journal of Geophysical Research, 1955, 60(4): 441-456. doi: 10.1029/JZ060i004p00441
[2]	Dunkin J W, Eringen A C. On the propagation of waves on electromagnetic elastic solids[J]. International Journal of Engineering Science, 1963, 1(4): 461-495. doi: 10.1016/0020-7225(63)90004-1
[3]	Yu C P, Tang S. Magneto-elastic waves in initially stressed conductors[J]. Journal of Applied Mathematics and Physics, 1966, 17(6): 766-775.
[4]	Lilley F E M, Smylie D E. Elastic wave motion and a nonuniform magnetic field in electrical conductors[J]. Journal of Geophysical Research, 1968, 73(20): 6527-6533. doi: 10.1029/JB073i020p06527
[5]	Suchtelen J V. Product properties: a new application of composite materials[J]. Philips Research Reports, 1972, 27(784): 28-37.
[6]	Chattopadhyay A, Choudhury S. Propagation, reflection and transmission of magnetoelastic shear waves in a self-reinforced medium[J]. International Journal of Engineering Science, 1990, 28(6): 485-495. doi: 10.1016/0020-7225(90)90051-J
[7]	岳田田, 李月秋, 王长达. 外磁场调控下弹性波的反射和透射[J]. 力学季刊, 2023, 44(4): 1012-1023. Yue Tiantian, Li Yueqiu, Wang Changda. Reflection and transmission of elastic waves regulated by external magnetic field[J]. Chinese Quarterly of Mechanics, 2023, 44(4): 1012-1023. (in Chinese)
[8]	Othman M I A, Song Y Q. The effect of rotation on the reflection of magneto-thermoelastic waves under thermoelasticity without energy dissipation[J]. Acta Mechanica, 2006, 184(1-4): 189-204. doi: 10.1007/s00707-006-0337-4
[9]	Othman M I A, Song Y Q. Reflection of magneto-thermoelastic waves with two relaxation times and temperature dependent elastic moduli[J]. Applied Mathematical Modelling, 2007, 32(4): 483-500.
[10]	Othman M I A. Generalized electro-magneto-thermoelasticity in case of thermal shock plane waves for a finite conducting half-space with two relaxation times[J]. Mechanics and Mechanical Engineering, 2010, 14(1): 5-30.
[11]	Kumar S, Pal P C, Majhi S. Reflection and transmission of plane SH-waves through an anisotropic magnetoelastic layer sandwiched between two semi-infinite inhomogeneous viscoelastic half-spaces[J]. Pure and Applied Geophysics, 2015, 172(10): 2621-2634. doi: 10.1007/s00024-015-1048-3
[12]	Kumar S, Majhi S, Pal P C. Reflection and transmission of plane SH-waves in two semi-infinite anisotropic magnetoelastic media[J]. Meccanica, 2015, 50(9): 1-10.
[13]	Wang C D, Wei P J, Zhang P, et al. Influences of a visco-elastically supported boundary on reflected waves in a couple-stress elastic half-space[J]. Archives of Mechanics, 2017, 69(2): 131-156.
[14]	Singh P, Chattopadhyay A, Singh A.K. Propagation of Love-type wave in functionally graded pre-stressed magneto-visco-elastic fiber-reinforced composite structure[J]. Waves in Random and Complex Media, 2019, 31(5): 1-30.
[15]	Gunghas A, Sheorad D, Kumar S, et al. Waves in a magneto-thermoelastic diffusive half-space with microconcentrations[J]. Waves in Random and Complex Media, 2020, 32(2): 708-727.
[16]	Li Y Q, Bian X Y, Wang C D, et al. The influences of external magnetic field on the reflection and transmission waves at the interface of two dipolar gradient elastic solids[J]. Applied Mathematical Modelling, 2023, 121: 524-541. doi: 10.1016/j.apm.2023.05.020
[17]	Li Y Q, Yue T T, Bian X Y, et al. Reflection and transmission of elastic waves through a sandwich structure with the microstructural effects and Lorentz force considered[J]. Acta Mechanica, 2024, 235(6): 3831-3848. doi: 10.1007/s00707-024-03907-0
[18]	Li Y, Li Y Q, Han Y, et al. Propagation of coupled waves across a magneto-electro-thermo-elastic interface with consideration of fractional order thermoelasticity and microstructural effect[J]. Acta Mechanica, 2024, 235(9): 5469-5488. doi: 10.1007/s00707-024-04000-2
[19]	Li Y, Li Y Q, Wang H, et al. Reflection and transmission of thermoelastic waves under an external magnetic field based on GN-Ⅱ heat conduction and dipolar gradient elasticity[J]. Acta Mechanica, 2025, 236(4): 2583-2598. doi: 10.1007/s00707-025-04292-y
[20]	Li Y, Li Y Q, Yue T T, et al. Effects of external magnetic field on the reflection and transmission of thermoelastic coupled waves with consideration of fractional order thermoelasticity[J]. Mechanics of Advanced Materials and Structures, 2025, 32(10): 2381-2392. doi: 10.1080/15376494.2024.2379504
[21]	Li X H, Shen J, Li C L, et al. The impact of periodicity in functionally graded materials on the attenuation of elastic shear waves[J]. Applied Mathematical Modelling, 2025, 144: 116106. doi: 10.1016/j.apm.2025.116106
[22]	Toupin R A. Elastic materials with couple-stresses[J]. Archive for Rational Mechanics and Analysis, 1962, 11(1): 385-414. doi: 10.1007/BF00253945
[23]	Mindlin R D, Tiersten H F. Effects of couple-stress in linear elasticity[J]. Archive for Rational Mechanics and Analysis, 1962, 11(1): 415-448. doi: 10.1007/BF00253946
[24]	Mindlin R D. Micro-structure in linear elasticity[J]. Archive for Rational Mechanics and Analysis, 1964, 16(1): 51-78. doi: 10.1007/BF00248490
[25]	Mindlin R D. Second gradient of strain and surface-tension in linear elasticity[J]. International Journal of Solids and Structures, 1965, 1(4): 417-438. doi: 10.1016/0020-7683(65)90006-5
[26]	张立民, 张波, 张旭, 等. 面内压缩荷载作用下双层微板系统的同步/异步屈曲[J]. 应用数学和力学, 2023, 44(2): 160-167. doi: 10.21656/1000-0887.430306 Zhang Limin, Zhang Bo, Zhang Xu, et al. Synchronous/asynchronous buckling of double-layered microplate systems[J]. Applied Mathematics and Mechanics, 2023, 44(2): 160-167. (in Chinese) doi: 10.21656/1000-0887.430306
[27]	赵英治, 唐怀平, 赖泽东, 等. 弹性地基上多孔二维功能梯度材料微梁自由振动研究[J]. 应用数学和力学, 2023, 44(11): 1354-1365. doi: 10.21656/1000-0887.440050 Zhao Yingzhi, Tang Huaiping, Lai Zedong, et al. Free Vibration analysis of porous 2D functionally graded material microbeams on Winkler's foundation[J]. Applied Mathematics and Mechanics, 2023, 44(11): 1354-1365. (in Chinese) doi: 10.21656/1000-0887.440050
[28]	Su C, Guan W, Yin Y G, et al. Elastic waves in fluid-saturated porous materials with a couple-stress solid phase[J]. Journal of Sound and Vibration, 2024, 569(117993): 1-18.
[29]	吕鑫, 柯燎亮, 苏洁. 基于偶应力理论的压电材料轴对称接触问题[J]. 应用数学和力学, 2024, 45(10): 1268-1278. doi: 10.21656/1000-0887.450190 LÜ Xin, Ke Liaoliang, Su Jie. An axisymmetric contact problem of piezoelectric materials based on the couple stress theory[J]. Applied Mathematics and Mechanics, 2024, 45(10): 1268-1278. (in Chinese) doi: 10.21656/1000-0887.450190
[30]	张漫哲, 顾水涛, 冯志强. 基于修正偶应力理论的应力最小化双向渐进结构拓扑优化方法[J]. 应用数学和力学, 2025, 46(1): 12-28. doi: 10.21656/1000-0887.450038 Zhang Manzhe, Gu Shuitao, Feng Zhiqiang. Bidirectional evolutionary topology optimization for stress minimization based on the modified couple stress elasticity[J]. Applied Mathematics and Mechanics, 2025, 46(1): 12-28. (in Chinese) doi: 10.21656/1000-0887.450038