金属微结构和残余应力测量的超声波理论
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摘要
构件中的残余应力会改变弹性波传播速度，这种现象称之为声弹性效应。近的八十年来，利用声弹性效应进行构件中残余应力的无损检测是研究的热点。在声弹性的无损检测中，一般情况下无法将金属构件作为各向同性材料。实际上，金属构件是由大量晶粒集合而成的多晶体，这些多晶体材料构件在它们的加工成型(压轧、拉伸、铸造)过程中会使得其中的晶粒具有某种偏爱取向，产生织构。晶粒各向异性和多晶体的织构使得金属构件的力学性能具有各向异性。
大量的实验和计算结果表明：多晶体的织构对波速的影响比初始应力的影响大许多。为了确定多晶体材料中的初始应力，需要找出同时考虑多晶体的织构和初始应力效应的多晶体材料弹性本构关系。多年来，各种超声波技术已经应用于多晶体的织构和初始应力测量。例如，电磁声换能器(EMAT)和压电换能器(PZT)可以用来发射与接收多晶体材料中的偏振波，超声波的波速通过测量波的传播距离与时间间隔来确定。本硕士论文所讨论的问题为：
1)在经典线弹性领域，应力和应变呈线性本构关系。若考虑到残余应力的影响，经典的线弹性Hooke定理将变得不再适用，必须考虑本构关系中的非线性项。Barsch(1968)阐述了Vogit模型下各向同性材料的非线性本构关系。为了建立考虑织构和残余应力效应的多晶体声弹性本构方程，本论文将推导出比Barsch的立方晶粒各向同性集合多晶体材料非线性本构关系结果更一般的、立方晶粒各向异性集合多晶体材料非线性本构关系。
2) Man(1996，1999，2000)给出了立方晶粒正交集合的、考虑微结构和残余应力效应的声弹性本构关系张量，该本构关系中包含十二个待定的材料常数和七个织构系数(用来描述晶粒的偏爱取向)。本论文将利用立方晶粒各向异性集合的非线性弹性本构关系(在前面部分得到的)，推导立方晶粒各向异性集合的声弹性本构关系。
3) Tanuma(1994)利用Stroh理论得到各向同性和横观各向同性材料的静力表面阻抗张量和格林函数。对于弱织构问题，Huang(2003，2004)给出了立方晶粒正交集合和各向异性集合下格林函数的显表达式。本论文利用Barnett和Lothe的积分方法给出各向异性集合多晶体材料的静力表面阻抗张量和格林函数。
4) Tanuma和Man(2002)给出了Stroh理论下Rayleigh波速表达式，该表达式包含织构和残余应力线性项的效应，其结果为Rayleigh波测量构件的残余应力提供了理论依据，但他们的表达式适用于立方晶粒正交集合的金属板材的织构和残余应力测量。本论文得到立方晶粒各向异性集合的声弹性本构方程和Rayleigh波速表达式。
The stress in a body changes the speeds of elastic waves propagating in the body.This phenomenon is called the acoustoelastic effect. The possibility of using theacoustoelastic effect for nondestructive evaluation of stress has been the impetus formuch of the research in acoustoelasticity during the last four decades. In thedevelopment of acoustoelasticity as a tool for the nondestructive evaluation of stress,it has become increasingly apparent that the characterization of the structuralmaterials being examined as isotropic is overly simplistic. Most of these materials areactually polycrystalline aggregates whose manufacturing procedures (rolling, drawing,and forging, etc.) make grains have certain preferred orientation which causes theaggregates to be anisotropic.
Empirical experiments and computational results suggest that the influence ofthe certain preferred orientation on ultrasonic wave speeds is stronger than that ofinitial stresses. To determine the initial stresses in the polycrystalline, we have to findthe constitutive relation of the polycrystalline with effects of certain preferredorientation and initial stresses. Over the years, various ultrasonic techniques havebeen developed to measure the wave speed changes due to the microstructure andinitial stress. Electromagnetic transducers (EMAT) and piezoelectric transducers(PZT) have been used to generate and detect polarized waves in a polycrystalline.Knowing the path length and the time interval required for the waves to pass, one candetermine wave speeds. Our work mainly includes:
1) In classic linear elasticity, one often takes a constitutive relation as linearfunction of strains. When initial stresses are very high, the classicconstitutive relation is challenged and the nonlinear effect of theconstitutive relation has to be considered. Barsch (1968) gave the nonlinearconstitutive relation for isotropic materials based on Vogit's model. Herein,we will derive a general constitutive relation on nonlinearly elasticpolycrystals.
2) Man(1998) gave the constitutive relation of a weak-textured orthorhombicaggregate of cubic crystallites with effects of the microstructure and initialstresses. Man's constitutive relation contains twelve material constants andseven texture coefficients. In this paper, we obtain a general constitutive relation of a weakly-textured anisotropic polycrystals based on theory ofnonlinear elasticity.
3) The surface-impedance tensors and Green's function of isotropic andtransversely isotropic materials are obtained by Tanuma (1994) by themethod of Stroh's eigenvalues. Here, we study the surface-impedancetensors and Green's function for weakly anisotropic materials via theintegration expressions introduced by Barnett and Lothe.
4) Tanuma and Man (2002) derived an angular dependence of Rayleigh-Wavevelocity in prestressed polycrystalline medial with monoclinic texture byStroh's formalism. By the acoustoelastic tensor we obtained in section 4, wederive Rayleigh-Wave velocity with the effect of material anisotropic andresidual stress.
大量的实验和计算结果表明：多晶体的织构对波速的影响比初始应力的影响大许多。为了确定多晶体材料中的初始应力，需要找出同时考虑多晶体的织构和初始应力效应的多晶体材料弹性本构关系。多年来，各种超声波技术已经应用于多晶体的织构和初始应力测量。例如，电磁声换能器(EMAT)和压电换能器(PZT)可以用来发射与接收多晶体材料中的偏振波，超声波的波速通过测量波的传播距离与时间间隔来确定。本硕士论文所讨论的问题为：
1)在经典线弹性领域，应力和应变呈线性本构关系。若考虑到残余应力的影响，经典的线弹性Hooke定理将变得不再适用，必须考虑本构关系中的非线性项。Barsch(1968)阐述了Vogit模型下各向同性材料的非线性本构关系。为了建立考虑织构和残余应力效应的多晶体声弹性本构方程，本论文将推导出比Barsch的立方晶粒各向同性集合多晶体材料非线性本构关系结果更一般的、立方晶粒各向异性集合多晶体材料非线性本构关系。
2) Man(1996，1999，2000)给出了立方晶粒正交集合的、考虑微结构和残余应力效应的声弹性本构关系张量，该本构关系中包含十二个待定的材料常数和七个织构系数(用来描述晶粒的偏爱取向)。本论文将利用立方晶粒各向异性集合的非线性弹性本构关系(在前面部分得到的)，推导立方晶粒各向异性集合的声弹性本构关系。
3) Tanuma(1994)利用Stroh理论得到各向同性和横观各向同性材料的静力表面阻抗张量和格林函数。对于弱织构问题，Huang(2003，2004)给出了立方晶粒正交集合和各向异性集合下格林函数的显表达式。本论文利用Barnett和Lothe的积分方法给出各向异性集合多晶体材料的静力表面阻抗张量和格林函数。
4) Tanuma和Man(2002)给出了Stroh理论下Rayleigh波速表达式，该表达式包含织构和残余应力线性项的效应，其结果为Rayleigh波测量构件的残余应力提供了理论依据，但他们的表达式适用于立方晶粒正交集合的金属板材的织构和残余应力测量。本论文得到立方晶粒各向异性集合的声弹性本构方程和Rayleigh波速表达式。
The stress in a body changes the speeds of elastic waves propagating in the body.This phenomenon is called the acoustoelastic effect. The possibility of using theacoustoelastic effect for nondestructive evaluation of stress has been the impetus formuch of the research in acoustoelasticity during the last four decades. In thedevelopment of acoustoelasticity as a tool for the nondestructive evaluation of stress,it has become increasingly apparent that the characterization of the structuralmaterials being examined as isotropic is overly simplistic. Most of these materials areactually polycrystalline aggregates whose manufacturing procedures (rolling, drawing,and forging, etc.) make grains have certain preferred orientation which causes theaggregates to be anisotropic.
Empirical experiments and computational results suggest that the influence ofthe certain preferred orientation on ultrasonic wave speeds is stronger than that ofinitial stresses. To determine the initial stresses in the polycrystalline, we have to findthe constitutive relation of the polycrystalline with effects of certain preferredorientation and initial stresses. Over the years, various ultrasonic techniques havebeen developed to measure the wave speed changes due to the microstructure andinitial stress. Electromagnetic transducers (EMAT) and piezoelectric transducers(PZT) have been used to generate and detect polarized waves in a polycrystalline.Knowing the path length and the time interval required for the waves to pass, one candetermine wave speeds. Our work mainly includes:
1) In classic linear elasticity, one often takes a constitutive relation as linearfunction of strains. When initial stresses are very high, the classicconstitutive relation is challenged and the nonlinear effect of theconstitutive relation has to be considered. Barsch (1968) gave the nonlinearconstitutive relation for isotropic materials based on Vogit's model. Herein,we will derive a general constitutive relation on nonlinearly elasticpolycrystals.
2) Man(1998) gave the constitutive relation of a weak-textured orthorhombicaggregate of cubic crystallites with effects of the microstructure and initialstresses. Man's constitutive relation contains twelve material constants andseven texture coefficients. In this paper, we obtain a general constitutive relation of a weakly-textured anisotropic polycrystals based on theory ofnonlinear elasticity.
3) The surface-impedance tensors and Green's function of isotropic andtransversely isotropic materials are obtained by Tanuma (1994) by themethod of Stroh's eigenvalues. Here, we study the surface-impedancetensors and Green's function for weakly anisotropic materials via theintegration expressions introduced by Barnett and Lothe.
4) Tanuma and Man (2002) derived an angular dependence of Rayleigh-Wavevelocity in prestressed polycrystalline medial with monoclinic texture byStroh's formalism. By the acoustoelastic tensor we obtained in section 4, wederive Rayleigh-Wave velocity with the effect of material anisotropic andresidual stress.
引文
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[7]. Mojia Huang, Tenglong Zhen, Weifu Huang, Constitutive Relation of Anisotropic Cubic Polycrystal on Ultrasonic Measurement of Texture Coefficients, International Symposium on Mechanical Waves in Solids, MAY 15-18, 2006 HANGZHOU CHINA.
[8]．黄模佳，Man C-S，立方晶粒各向异性集合的弹性本构关系[C]．力学即其在土木工程中的应用．，清华大学出版，2002，107-116．
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[16]．詹华，黄模佳，林秀巧，唐海(2006)，立方晶粒各向异性集合金属的非线性弹性本构关系，南昌大学学报(工科版)，第28卷第4期，379-385．
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[2]．董大勤，2002化工设备机械基础[M]，化工工业出版社，
[3]．J．杰拉德[英]，超声检测新技术[S]．北京：科学技术出版社，1991(1)
[4]. H J Bunge, Texture Analysis in Material: Mathematical Methord [M]. London: Butterworths, 1982
[5]. R J Roe, Inversion of pole figures for materials having cubic crystal symmetry [J]. J Appl Phys, 1966, 37: 2069-2072.
[6]. Sayers,C.M.(1982)Ultrasonic velocities in anisotropic polycrystalline aggregates. J.Phys.D, 15:2157-2167. 15:2157-2167.
[7]. Mojia Huang, Tenglong Zhen, Weifu Huang, Constitutive Relation of Anisotropic Cubic Polycrystal on Ultrasonic Measurement of Texture Coefficients, International Symposium on Mechanical Waves in Solids, MAY 15-18, 2006 HANGZHOU CHINA.
[8]．黄模佳，Man C-S，立方晶粒各向异性集合的弹性本构关系[C]．力学即其在土木工程中的应用．，清华大学出版，2002，107-116．
[9]. Voigt, W., 1889. UQ ber die Beziehungzwischen den beiden ElastiziaQtskonstanten isotroper KoQrper. Wied.Ann. 38, 573-587.
[10]. Reuss, A., 1929. Berchungder Fiessgrenze von Mischkristallen auf Grund der PlastiziQatsbedingung fQur Einkristalle. Z. Angew. Math. Mech. 9, 49-58.
[11]. Man C.-S., (1998), Hartig's law and linear elasticity with initial stress. Inverse Problems 14: 313—319.
[12]. Barsch,G.R.,(1968)Relation between third-order elastic constants of single crystals and polycrystals.Journal of Applied Physics,39:3780-3793.
[13]. V A Lubarda, New estimates of the third-order elastic constants for isotropic aggregates of cubic crystals [J]. Journal of the Mechanics and Physics of Solids, 1997,45: 471-490.
[14]. Man C.-S. and Paroni R.: On the separation of stress-induced and texture-induced birefringence in acoustoelasticity. J. Elasticity 45:91—116 (1996)
[15]. Paroni R. and Man C.-S.: Two Micromechanical Models in Acoustoelasticity: a Comparative Study. Journal of Elasticity 59:145—173 (2000)
[16]．詹华，黄模佳，林秀巧，唐海(2006)，立方晶粒各向异性集合金属的非线性弹性本构关系，南昌大学学报(工科版)，第28卷第4期，379-385．
[17]. C.-S. Man, On the constitutive equations of some weakly-textured materials. Arch. Rational Mech. Anal. 143 (1998) 77-103.
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