吕炎, 林晓磊, 高杰, 何存富. 基于级数法的热弹各向异性层合板兰姆波频散特性分析. 力学学报, 2023, 55(9): 1939-1949 doi: 10.6052/0459-1879-23-234 引用本文: 吕炎, 林晓磊, 高杰, 何存富. 基于级数法的热弹各向异性层合板兰姆波频散特性分析. 力学学报, 2023, 55(9): 1939-1949 doi: 10.6052/0459-1879-23-234 Lyu Yan, Lin Xiaolei, Gao Jie, He Cunfu. Analysis of Lamb wave dispersion characteristics of thermoelastic anisotropic laminates based on the polynomial method. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(9): 1939-1949 doi: 10.6052/0459-1879-23-234 Citation: Lyu Yan, Lin Xiaolei, Gao Jie, He Cunfu. Analysis of Lamb wave dispersion characteristics of thermoelastic anisotropic laminates based on the polynomial method. Chinese Journal of Theoretical and Applied Mechanics , 2023, 55(9): 1939-1949 doi: 10.6052/0459-1879-23-234 吕炎, 林晓磊, 高杰, 何存富. 基于级数法的热弹各向异性层合板兰姆波频散特性分析. 力学学报, 2023, 55(9): 1939-1949 doi: 10.6052/0459-1879-23-234 引用本文: 吕炎, 林晓磊, 高杰, 何存富. 基于级数法的热弹各向异性层合板兰姆波频散特性分析. 力学学报, 2023, 55(9): 1939-1949 doi: 10.6052/0459-1879-23-234 Lyu Yan, Lin Xiaolei, Gao Jie, He Cunfu. Analysis of Lamb wave dispersion characteristics of thermoelastic anisotropic laminates based on the polynomial method. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(9): 1939-1949 doi: 10.6052/0459-1879-23-234 Citation: Lyu Yan, Lin Xiaolei, Gao Jie, He Cunfu. Analysis of Lamb wave dispersion characteristics of thermoelastic anisotropic laminates based on the polynomial method. Chinese Journal of Theoretical and Applied Mechanics , 2023, 55(9): 1939-1949 doi: 10.6052/0459-1879-23-234 基于勒让德级数法, 联立Green-Nagdhi热弹性理论, 建立了含温度场各向异性层合板的超声导波频散特性理论模型, 揭示温度场环境下多层复合材料中超声导波的传播过程. 同时, 构建了温度场环境下多层各向同性与各向异性层合板的声学频域仿真模型, 以提取特定温度下层合板超声导波的频散曲线. 通过对比仿真数据与理论计算的结果, 验证了所提理论方法的有效性. 随后, 以不同铺层方向单向纤维材料组成的层合板为例, 分析了相同温度条件下中间层纤维角度对各向异性层合板超声导波频散曲线的影响规律, 并细节分析了特定频率处A0模态的位移及应力波结构的分布特征. 此外, 着重考虑温度场变化对碳纤维复合材料层合板中超声导波频散特性的影响机理, 指出导波基础模态的偏移规律, 并详细列举了不同频率与温度下的基础模态相速度值. 最后, 利用不同温度工况下的相速度差值, 提取多层各向异性层合板相速度温度敏感度变化曲线, 探究不同频率下对称和反对称模态的相速度温度敏感度, 为多层复合材料力学性能的超声无损检测与评估提供了理论基础.

勒让德级数法 /  单向纤维层合板 /  温度场 /  超声导波 / Abstract: A theoretical model of the dispersion characteristics of ultrasonic guided waves in anisotropic laminates with temperature field is developed based on Legendre's polynomial method and Green-Nagdhi thermoelasticity theory to reveal the propagation process of ultrasonic guided waves in multilayered composites under temperature field environment. At the same time, the acoustic frequency domain simulation model of multilayer isotropic and anisotropic laminates in a temperature field environment is constructed to extract the dispersion curves of ultrasonic guided waves of the laminates at specific temperatures. The validity of the proposed theoretical method is verified by comparing the simulation data with the theoretical calculations. After that, the dispersion curves of the ultrasonic guided waves of anisotropic laminates are analyzed by taking laminates composed of unidirectional fiber materials with different layup directions as an example, and the distribution characteristics of the displacement and stress wave structure of the A0 modes at a specific frequency are analyzed in detail concerning the fiber angle of the intermediate ply at the same temperature condition. In addition, the mechanism of the influence of temperature field changes on the dispersion characteristics of ultrasonic guided waves in carbon fiber composite laminates is focused on, the shift laws of the ultrasonic guided wave fundamental modes are pointed out, and the values of the fundamental mode phase velocities at different frequencies and temperatures are listed in detail. In the end, the phase velocity temperature sensitivity change curves of multilayer anisotropic laminates are extracted by utilizing the phase velocity difference values at different temperature conditions, and the phase velocity temperature sensitivity of symmetric and antisymmetric modes at different frequencies is explored, which provides a theoretical basis for ultrasonic nondestructive testing and evaluation of the mechanical properties of multilayer composites.

Key words: Legendre polynomial method /  unidirectional fiber laminate /  temperature field /  ultrasonic guided wave /  dispersion curve  Materials ρ /
(g·cm ⁻³ ) C ₁₁ /
GPa C ₁₂ /
GPa C ₄₄ /
GPa C _e /
(J·kg·
(°)·m ⁻¹ ) β _i /
(10 ⁶ N·
(°) ⁻¹ ·m ⁻² ) K _i /
(W/
m·K) steel7.9281.72131.1284.2987104.051.34copper8.0153.9179.2937.3139050.115.0 Materials ρ /(g·cm ⁻³ ) C ₁₁ /GPa C ₁₂ /GPa C ₁₃ /GPa C ₂₂ /GPa C ₂₃ /GPa C ₃₃ /GPa C ₄₄ /GPa C ₅₅ /GPa C ₆₆ /GPa C _e /
(J·kg·
(°)·m ⁻¹ ) β _i /
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(W/
m·K) T300/9141.56143.86.26.213.36.513.33.65.75.767014.524.1 (a) 0°/0°/0°单向纤维层合板 (单位: km/s)
(a) 0°/0°/0° unidirectional fiber laminates (unit: km/s)TemperatureFrequency100 kHz200 kHz300 kHz400 kHz 443 K1.546531.709531.774941.80711413 K1.547411.715311.776281.80752383 K1.548301.716831.776551.80772353 K1.549201.716861.776691.80807323 K1.550081.717401.777021.80874293 K1.550981.717961.778331.80888(b) 0°/90°/0°单向纤维层合板 (单位: km/s)
(b) 0°/90°/0° unidirectional fiber laminates (unit: km/s)TemperatureFrequency100 kHz200 kHz300 kHz400 kHz443 K1.431831.582651.645761.67983413 K1.432671.583681.646141.68010383 K1.433521.584191.646511.68041353 K1.434371.584711.646881.68070323 K1.435211.585231.647261.68099293 K1.436061.586931.647631.68129 (a) 0°/0°/0°单向纤维层合板 (单位: km/s)
(a) 0°/0°/0° unidirectional fiber laminates (unit: km/s)TemperatureFrequency100 kHz200 kHz300 kHz400 kHz 443 K9.500269.487089.453119.32452413 K9.500289.487239.453259.32486383 K9.500299.487259.453289.32519353 K9.500319.487279.453529.32575323 K9.500329.487319.453539.32581293 K9.500349.487509.453669.32666(b) 0°/90°/0°单向纤维层合板 (单位: km/s)
(b) 0°/90°/0° unidirectional fiber laminates (unit: km/s)TemperatureFrequency100 kHz200 kHz300 kHz400 kHz443 K7.854527.703597.347146.010361413 K7.854537.704157.347386.013454383 K7.854567.704287.347636.016093353 K7.854577.704357.347856.018984323 K7.854597.704427.348066.021760293 K7.854617.704487.348306.024584 何存富, 孙雅欣, 吴斌等. 高频纵向导波在钢杆中传播特性的研究. 力学学报, 2016, 37(8): 1713-1735 (He Cunfu, Sun Yaxin, Wu Bin, et al. Propagation characteristics of high frequency longitudinal guided waves in steel rod. Chinese Journal of Theoretical and Applied Mechanics , 2016, 37(8): 1713-1735 (in Chinese)

He Cunfu, Sun Yaxin, Wu Bin, et al. Propagation characteristics of high frequency longitudinal guided waves in steel rod[J]. Chinese Journal of Theoretical and Applied Mechanics , 2016, 37(8): 1713-1735. (in Chinese)) Knopoff L. A matrix method for elastic wave problems. Bulletin of the Seismological Society of America , 1964, 54(1): 431-438 doi: 10.1785/BSSA0540010431 Kausel E, Roësset JM. Stiffness matrices for layered soils. Bulletin of the seismological Society of America , 1981, 71(6): 1743-1761 doi: 10.1785/BSSA0710061743 Kirilenko AA, Senkevich SL, Steshenko SO. Application of the generalized scattering matrix technique for the dispersion analysis of 3 D slow-wave structures. Telecommunications and Radio Engineering , 2015, 74(17): 1497-1511 Zheng MF, et al. State-vector formalism and the Legendre polynomial solution for modelling guided waves in anisotropic plates. Journal of Sound and Vibration , 2018, 412: 372-388 Gao J, Lyu Y, Zheng MF, et al. Modeling guided wave propagation in multi-layered anisotropic composite laminates by state-vector formalism and the Legendre polynomials. Composite Structures , 2019, 228: 111319 doi: 10.1016/j.compstruct.2019.111319 Lefebvre JE, Zhang V, Gazalet J, et al. Legendre polynomial approach for modeling free-ultrasonic waves in multilayered plates. Journal of Applied Physics , 1999, 85(7): 3419-3427 doi: 10.1063/1.369699 Wang XH, Li FL, Zhang XM, et al. Thermoelastic guided wave in fractional order functionally graded plates: An analytical integration Legendre polynomial approach. Composite Structures , 2021, 256: 112997 doi: 10.1016/j.compstruct.2020.112997 Chen T, Zhang XM, Zhou HM, et al. Characteristics of complete circumferential guided wave in a piezoelectric semiconductor cylindrical shell. Journal of Intelligent Material Systems and Structures , 2023, 34(6): 733-748 doi: 10.1177/1045389X221121910 王现辉, 李方琳, 刘宇建等. 板中热弹波传播: 一种改进的勒让德多项式方法. 力学学报, 2020, 52(5): 1277-1285 (Wang Xianhui, Li Fanglin, Liu Yujian, et al. Thermoelastic wave propagation in plates: an improved legendre polynomial approach. Chinese Journal of Theoretical and Applied Mechanics , 2020, 52(5): 1277-1285 (in Chinese)

Wang Xianhui, Li Fanglin, Liu Yujian, et al. Thermoelastic wave propagation in plates: an improved legendre polynomial approach[J]. Chinese Journal of Theoretical and Applied Mechanics , 2020, 52(5): 1277-1285. (in Chinese)) 李妍, 何天虎, 田晓耕. 超短激光脉冲加热薄板的广义热弹扩散问题. 力学学报, 2020, 52(5): 1255-1266 (Li Yan, He Tianhu, Tian Xiaogeng. A generalized thermoelastic diffusion problem of thin plate heated by the ultrashort laser pulses. Chinese Journal of Theoretical and Applied Mechanics , 2020, 52(5): 1255-1266 (in Chinese)

Li Yan, He Tianhu, Tian Xiaogeng. A generalized thermoelastic diffusion problem of thin plate heated by the ultrashort laser pulses[J]. Chinese Journal of Theoretical and Applied Mechanics , 2020, 52(5): 1255-1266. (in Chinese)) Al-Toki MHZ, Ali HAK, Ahmed RA, et al. A numerical study on vibration behavior of fiber-reinforced composite panels in thermal environments. Structural Engineering and Mechanics , 2022, 82(6): 691 吴楠, 郝旭峰, 史耀辉等. 高精度碳纤维增强树脂复合材料夹层天线面板热变形影响参数仿真与实验. 复合材料学报, 2020, 7(7): 1619-1628 (Wu Nan, Hao Xufeng, Shi Yaohui. Simulation and experiment on thermal deformation influence parameters of high accuracy carbon fiber reinforced plastic sandwiched antenna panels. Acta Materiae Compositae Sinica , 2020, 7(7): 1619-1628 (in Chinese)

Wu Nan, Hao Xufeng, Shi Yaohui. Simulation and experiment on thermal deformation influence parameters of high accuracy carbon fiber reinforced plastic sandwiched antenna panels[J]. Acta Materiae Compositae Sinica , 2020, 7(7): 1619-1628. (in Chinese)) Green AE, Naghdi PM. Thermoelasticity without energy dissipation. Journal of Elasticity , 1993, 31(3): 189-208 doi: 10.1007/BF00044969 Green AE, Naghdi PM. On undamped heat waves in an elastic solid. Journal of Thermal Stresses , 1992, 15(2): 253-264 doi: 10.1080/01495739208946136 Green AE, Naghdi PM. A re-examination of the basic postulates of thermomechanics. Proceedings of the Royal Society of London. Series A : Mathematical and Physical Sciences , 1991, 432(1885): 171-194 doi: 10.1098/rspa.1991.0012 Al-Qahtani H, Datta SK. Thermoelastic waves in an anisotropic infinite plate. Journal of Applied Physics , 2004, 96(7): 3645-3658 doi: 10.1063/1.1776323 Verma KL, Hasebe N. Dispersion of thermoelastic waves in a plate with and without energy dissipation. International Journal of Thermophysics , 2001, 22: 957-978 doi: 10.1023/A:1010743519828 Li CL, Han Q. Thermoelastic wave characteristics in a hollow cylinder using the modified wave finite element method. Acta Mechanica , 2016, 227(6): 1711-1725 doi: 10.1007/s00707-016-1578-5 Yu JG, Zhang XM, Xue TL. Generalized thermoelastic waves in functionally graded plates without energy dissipation. Composite Structures , 2010, 93(1): 32-39 doi: 10.1016/j.compstruct.2010.06.020 Yu JG, Xue TL. Generalized thermoelastic waves in spherical curved plates without energy dissipation. Acta Mechanica , 2010, 212(1-2): 39-50 doi: 10.1007/s00707-009-0238-4 Dodson JC, Inman DJ. Thermal sensitivity of Lamb waves for structural health monitoring applications. Ultrasonics , 2013, 53(3): 677-685 Gandhi N, Michaels JE, Lee SJ. Acoustoelastic Lamb wave propagation in biaxially stressed plates. The Journal of the Acoustical Society of America , 2012, 132(3): 1284-1293 Yang ZY, Liu KH, Zhou K, et al. Investigation of thermo-acoustoelastic guided waves by semi-analytical finite element method. Ultrasonics , 2020, 106: 106141 doi: 10.1016/j.ultras.2020.106141 Sharma JN, Pathania V. Generalized thermoelastic waves in anisotropic plates sandwiched between liquid layers. Journal of Sound and Vibration , 2004, 278(1-2): 383-411 doi: 10.1016/j.jsv.2003.10.010 Gomez Garcia P, Fernández-Álvarez JP. Floquet-bloch theory and its application to the dispersion curves of nonperiodic layered systems. Mathematical Problems in Engineering , 2015, 2015: 475364 Huang YH, Wu Z, Li XC. Development of simple thermal expansion coefficient measurement apparatus and its application to several materials. CIESC Journal , 2016, 67(S2): 38-45 王海楼, 曹淼, 孙宝忠等. 三维编织碳纤维/环氧树脂复合材料横向压缩性质的温度效应. 复合材料学报, 2018, 35(3): 607-615 (Wang Hailou, Cao Miao, Sun Baozhong, et al. Temperature effect on transverse compressive behaviors of 3D braided carbonnfiber/epox ycomposites. Acta Materiae Compositae Sinica , 2018, 35(3): 607-615 (in Chinese)

Wang Hailou, Cao Miao, Sun Baozhong, et al. Temperature effect on transverse compressive behaviors of 3 D braided carbonnfiber/epox ycomposites[J]. Acta Materiae Compositae Sinica , 2018, 35(03): 607-615. (in Chinese) Yang ZY, Wu ZJ, Zhang JQ, et al. Acoustoelastic guided wave propagation in axial stressed arbitrary cross-section. Smart Materials and Structures , 2019, 28(4): 045013 doi: 10.1088/1361-665X/aadb6e