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在二阶微扰近似条件下, 采用导波模式展开分析方法研究了圆管结构中周向导波的非线性效应. 伴随基频周向导波传播所发生的二次谐波, 可视为由一系列二倍频周向导波模式叠加而成. 从动量定理出发, 结合柱坐标系下非线性应力张量及其散度的数学表达式, 针对圆管中某一基频周向导波模式, 推导出相应的二倍频应力张量及二倍频彻体驱动力的数学表达式, 建立了确定二倍频周向导波模式展开系数的控制方程, 得到了伴随基频周向导波传播所发生的二次谐波声场的形式解. 理论分析和数值计算表明, 当构成二次谐波声场的某一二倍频周向导波模式与基频周向导波的相速度匹配时, 该二倍频周向导波模式的位移振幅表现出随传播周向角积累增长的性质; 当两者的相速度失配时, 二倍频周向导波的振幅随传播周向角表现出“拍”效应.Within the second-order perturbation approximation, the nonlinear effect of primary circumferential guided wave propagation in a circular tube is investigated using modal expansion analysis for waveguide excitation. The nonlinearity of the wave equation governing the wave propagation ensures the second-harmonic generation accompanying primary circumferential guided wave propagation. This nonlinearity may be treated as a second-order perturbation of the linear elastic response. The fields of the second harmonics of primary circumferential guided wave propagation are considered as superpositions of the fields of a series of double frequency circumferential guided wave (DFCGW) modes. Based on the momentum theorem and mathematical formulae of nonlinear stress tensor and its divergence under the cylindrical coordinate system, the mathematical expressions of the corresponding double frequency traction stress tensors and bulk driving forces are deduced for a certain primary circumferential guided wave mode. Subsequently, the equation governing the DFCGW mode expansion coefficient is established. Finally, the mathematical expression of second-harmonic field of the primary circumferential guided wave mode in a tube is derived. The results of the theoretical analyses and numerical calculations indicate that the degree of cumulative growth of the DFCGW mode with circumferential angle is obviously influenced by that of phase velocity matching between the primary and double frequency wave modes. It is found that the amplitude of the DFCGW mode can grow with circumferential angle when its phase velocity matches with that of the primary circumferential guided wave, and that the amplitude of the DFCGW mode will show a beat effect with circumferential angle when its phase velocity is not equal to that of the primary wave mode. The DFCGW mode, whose phase velocity matches with that of the primary wave mode, plays a dominant role in the field of second harmonic generated by the primary wave mode propagation, and the contribution of the other DFCGW modes to the said second-harmonic field is negligible after the primary wave mode has propagated some circumferential angle.
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Keywords:
- circumferential guided wave /
- nonlinear effect /
- second-harmonic generation /
- modal expansion analysis
[1] Gazis D C 1959 J. Acoust. Soc. Am. 31 568
[2] Sun H J, Lin Z, Zhao D Y 2009 Nondestruct. Test. 31 68(in Chinese) [孙海蛟, 林哲, 赵德有 2009 无损检测 31 68]
[3] Xu H, Wang B, Jiang X J 2009 J. Power Eng. 29 1018(in Chinese) [徐鸿, 王冰, 姜秀娟 2009 动力工程 29 1018]
[4] Ta D A, Wang W Q, Wang Y Y 2009 Ultrasound Med. Biol. 35 641
[5] Liu Y, Lissenden C J, Rose J L 2013 Proc. SPIE 8695 86950S-1
[6] Qu J M, Berthelot Y, Li Z 1996 Rev. Prog. Quant. Nondestr. Eval. 15A 169
[7] Liu G, Qu J M 1998 ASME J. Appl. Mech. 65 424
[8] Valle C, Qu J M, Jacobs L J 1997 Int. J. Eng. Sci. 37 1369
[9] Zhang H L, Yin X C 2007 Acta Mech. Solida Sin. 20 110
[10] Liu Y, Li Z, Gong K 2012 Mech. Syst. Signal Pr. 20 157
[11] Deng M X 1998 J. Appl. Phys. 84 3500
[12] Deng M X 1999 J. Appl. Phys. 85 3051
[13] Deng M X, Liu Z Q 2002 Appl. Phys. Lett. 81 1916
[14] Deng M X 2003 J. Appl. Phys. 94 4152
[15] Lima W J, Hamilton M F 2003 J. Sound Vib. 265 819
[16] Deng M X, Xiang Y X 2010 Chin. Phys. B 19 114302
[17] Xiang Y X, Deng M X 2008 Chin. Phys. B 17 4232
[18] Deng M X, Xiang Y X, Liu L B 2011 Chin. Phys. Lett. 28 074301
[19] Srivastava A, Bartoli I, Salamone S, di Lanza S F 2010 J. Acoust. Soc. Am. 127 2790
[20] Deng M X 2006 Nnlinear Lamb Waves in Solid Plates (Beijing: Science Press) pp12-43 (in Chinese) [邓明晰 2006 固体板中的非线性兰姆波 (北京: 科学出版社) 第12–43页]
[21] Chillara V K, Lissenden C L 2013 Ultrasonics 53 862
[22] Deng M X 2005 Acta Acust. 30 132(in Chinese) [邓明晰 2005 声学学报 30 132]
[23] Rose J L 1999 Ultrasonic Waves in Solid Media (Cambridge: Cambridge Univercity Press) pp35-41
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[1] Gazis D C 1959 J. Acoust. Soc. Am. 31 568
[2] Sun H J, Lin Z, Zhao D Y 2009 Nondestruct. Test. 31 68(in Chinese) [孙海蛟, 林哲, 赵德有 2009 无损检测 31 68]
[3] Xu H, Wang B, Jiang X J 2009 J. Power Eng. 29 1018(in Chinese) [徐鸿, 王冰, 姜秀娟 2009 动力工程 29 1018]
[4] Ta D A, Wang W Q, Wang Y Y 2009 Ultrasound Med. Biol. 35 641
[5] Liu Y, Lissenden C J, Rose J L 2013 Proc. SPIE 8695 86950S-1
[6] Qu J M, Berthelot Y, Li Z 1996 Rev. Prog. Quant. Nondestr. Eval. 15A 169
[7] Liu G, Qu J M 1998 ASME J. Appl. Mech. 65 424
[8] Valle C, Qu J M, Jacobs L J 1997 Int. J. Eng. Sci. 37 1369
[9] Zhang H L, Yin X C 2007 Acta Mech. Solida Sin. 20 110
[10] Liu Y, Li Z, Gong K 2012 Mech. Syst. Signal Pr. 20 157
[11] Deng M X 1998 J. Appl. Phys. 84 3500
[12] Deng M X 1999 J. Appl. Phys. 85 3051
[13] Deng M X, Liu Z Q 2002 Appl. Phys. Lett. 81 1916
[14] Deng M X 2003 J. Appl. Phys. 94 4152
[15] Lima W J, Hamilton M F 2003 J. Sound Vib. 265 819
[16] Deng M X, Xiang Y X 2010 Chin. Phys. B 19 114302
[17] Xiang Y X, Deng M X 2008 Chin. Phys. B 17 4232
[18] Deng M X, Xiang Y X, Liu L B 2011 Chin. Phys. Lett. 28 074301
[19] Srivastava A, Bartoli I, Salamone S, di Lanza S F 2010 J. Acoust. Soc. Am. 127 2790
[20] Deng M X 2006 Nnlinear Lamb Waves in Solid Plates (Beijing: Science Press) pp12-43 (in Chinese) [邓明晰 2006 固体板中的非线性兰姆波 (北京: 科学出版社) 第12–43页]
[21] Chillara V K, Lissenden C L 2013 Ultrasonics 53 862
[22] Deng M X 2005 Acta Acust. 30 132(in Chinese) [邓明晰 2005 声学学报 30 132]
[23] Rose J L 1999 Ultrasonic Waves in Solid Media (Cambridge: Cambridge Univercity Press) pp35-41
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