Numerical Simulation of Residual Stress Measurement with Acoustic Wave
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Cuixiang Pei and Kaziuki Demachi Department of Nuclear Engineering and Management, University of Tokyo,Tokyo 113-8656, Japan
1. Introduction
The residual stress in some mechanical structures increases the likelihood of fatigue cracks, stress corrosion cracks and environmental induced material degradation. The ability to evaluate the residual stress would substantially increase the accuracy of structure life estimation and the security of operation. Several nondestructive testing methods, such as X-Ray diffraction techniques, ultrasonic wave techniques and magnetic effect analytical methods, have been applied in the residual stress measurement. As the high permeability of ultrasonic wave, the ultrasonic wave method can be used to evaluate the residual stress in the interior of the structure as well as at the surface. Besides, being portable and cheap to undertake, this method is well suited to routine inspection procedures and industrial studies of large components[1].The ultrasonic methods for residual stress measurement are mainly based on the acoustoelastic effect related to the variation of ultrasonic wave speed and change in the polarization of Rayleigh wave. Presently, the residual stress measurement with ultrasonic wave is mainly studied with experiment method(2,5]. And simultaneously, the numerical simulation method can provide another useful tool for the research of stress measurement with ultrasonic wave. In this work, the possibility of numerical simulation of residual stress measurement based on the acoustoelastic effect was investigated with FEM.Now in practical application, the LCR (refracted longitudinal) wave is usually used to measure the subsurface residual stress in components[9]. In this paper, the possibility of assessing the stress status, by using an EMAT (electromagnetic acoustic transducer) receiver for precise measurement of LCR (refracted longitudinal) wave, was investigated by simulation.Corresponding authors: Tel: +81 (0)29 287 8431 Fax: +81 (0)29 287 8488. Email: peicx@nuclear.jp, - demachi@nuclear.jp2. Numerical Simulation Method 2.1 Analytical model for acoustoelastic effect in solidsAn ultrasonic wave through a stressed body would give rise to further stresses, and the traditional theory of acoustoelasticity applied in the unstressed medium cannot be used here. Instead, based on the deformation process, Duquennoy et al. defined a theory of three state of a body, shown in Fig. 1 [3,4]. A solid body undergoes a series of deformations from a stress free state to a static deformation or a dynamic deformation. The position vector & defines the position of a point in the natural state of zero stress and zero strain. X defines the position of a point in the initial state when the body undergoes a static deformation due to residual stress during the manufacturing processes or due to the applied stresses. Similarly X defines the position in the final states when a dynamic deformation such as the ultrasonic wave through the body. The displacement of a point from one state to another can be described mathematically as: u'(E)= X - E; u' (5)= x a?“ 5; (E)= x - X = u' a?“ u'Acoustic waveInitial StatePre-stress2Final StateNatural StateFig. 1 Coordinates for material point in three states Based on the three-state theory, the controlling equation of the acoustic wave in a solid with initial stress can be written as(OxOE≫n + A?uxe) ONE -y @, + f = po(1a?“Ex) *u-1at305Where o'is the Cauchy stress tensor in the initial state, & the initial strain tensor and p'the mass density in the natural state, y is the acoustic damping coefficient,f is the external force loading on the body. And for isotropic material the elastic constant can be expressed as A?JKL = 18,0ke + u(Eixon +8,8xx)+[(-2+v;)8a??A?KL+(-4+ v2)OKOL+OLOjK)]a?￢'w +2(a + v2)a?￢OKL + a?￢ KLO) +2(u+v) EROL +a?￢18,K + EKOL + E'LOIK)(2) where 2 and u are the well known Lame constants and v;(i = 1,2,3) are the third-order elastic constants (TOE constants).2.2 Numerical ModelIn order to demonstrate the feasibility of this simulation method, a 3D model with initial homogeneous stress Ox, uniaxially directed along the x axis, was used for the simulation of acoustoelastic effect on LCR wave, shown as in Fig.2. A PZT angle transducer was used to generate surface wave, and an EMAT probe was used to receive the time behavior of the wave signal for different stress. As shown in table 1[6,8], two kinds of material have been used in this paper. Where, the sensitivity constant k is defined as a relative change in wave speed per unit change in stress.EMATPZTMagnet30mm -Coil-60mm25 pmFig. 2 A pitch-catch setup model in simulationTable 1 Material parameter of the modelMaterialp/kg m0/Sm' 2 / Paul Pa2719382000004910000000026000000000Al AlloyD54s Rail Steel7800110000011580000000079900000000MaterialV/Pa| VzI Pa| Vz/ Pak / MPaAl Alloy D54s-379000000000-198000000000-80000000000-0.0000879Rail Steel36000000000-266000000000178.5E91.214E-52.3 Numerical simulation method for acoustic waveAccording to FEM, eq. (1) can be calculated by solving the discrete wave motion eq. (3)[M]{A?}+[C]{U}+[K]{U}={F} (3) where [M] denotes the mass matrix, [C] the damping matrix, [K] the stiffness matrix, {U} the nodaldisplacement and {F} the force vector. Eq. (3) is rearranged as into the following equation by using the explicit integration method in time-domain[7] |{U}rtar = {U}.-os a?“241[M]'[C]{U}, -241 [M]““[K]{U}, +201[M]' {F},{U}iter = {U}, + {U}rts, +{U},2.4 Numerical method for EMAT receiverA typical configuration for an EMAT receiver of acoustic waves in a test-piece is shown in Fig. 2. It is usually composed of a static magnet and a set of coils (named pick-up coil). When the ultrasonic wave occurred in the test-piece under the EMAT receiver, the vibration by the ultrasonic wave would interact with the static magnetic flux density of the static magnet B, to yield transient eddy current J. in the metal.J. = 0,vxB., v is the velocity of a particle, o, is the conductivity of the metal. And a micro current would be induced by the electromagnetic induction effect. The magnetic flux density B. of a permanent magnet can be calculated by equivalent magnetic charge approach [7]. The most direct way to calculate the pick-up signals of the EMAT receiver is to use the Biot-Savart's law, and the voltage signal in pick-up coil can be written as,v=X& SA-d=A?a [SIL - dawhere T, is boundary of surface S; and r is the distance from a current source point in the conductor to a point in the pick-up coil. 3. Numerical Simulation ResultsThe ultrasonic wave was excited by one damped cycle of a 1 MHz sinusoidal. To give a preliminary judgment of the validity of this method, the simulation result of the ultrasonic wave field in a cross section at a given time in the Al alloy model is shown in Fig. 3. The time behavior of the EMAT coil voltage is plotted in Fig. 4. As the EMAT receiver was set at 40 mm away from the PZT transducer, the velocity of the LCR wave is calculated which is about 6000 m/s. This simulation result is agreed with the theoretical value.Rayleigh WaveLCR Waveslasveise Wavekorgitudinal WaveFig. 3 Ultrasonic wave field excited by PZT306EMAT Receiver SignalLCR WaveVoltagel/V0:00:002.0.10a?￢ 4.0x10* 60x1080x10A° 10x10' 1210Time / s1.4x10Fig. 4 LCR wave received by EMATIn Fig. 5 and Fig. 6, the close-up of the LCR wave signals for two initial stresses in Al Alloy and rail steel is provided. The acoustoelastic effect with a small phase delay induced by the stress can be observed. It can be seen that the variation in LCR wave speed due to acoustoelastic effect in rail steel is much smaller than that in Al alloy. The relative change in LCR wave speed for different residual stress is simulated, shown in Fig. 7. It can be seen that the simulation results show a very good agreement to the theoretical values.2.0x10O MPa 200 MPa1.0x10EMAT signal / V0 -1.0x10A° --2.0x102-3.0x10 +5.0x106.0x10%7.0x10$ 8.0x10*Time / s9.0x10%1.0x10%Fig. 5 Acoustoelastic effect in Al alloy3.0x10**--O MPa200 MPa2.0x10* +1.0x104EMAT signal / V0.0 F-1.0x10*-2.0x10-3.0x106.0x 10a?￢7.0x10*9.0x1068.0x 10 Time / sFig. 6 Acoustoelastic effect in rail steel-----O MPa200 MPa7.0x109.0x108.0x106 Time isSimulation value of Al AlloySimulation value of Rail Steel ---- Reference value of Al AlloyReference value of Rail SteelRelative change-0.04.-600400400600-200200 Residual stress / MPaFig. 7 Relative change of LCR wave's speed in Al Alloy andrail steel4. ConclusionIn this paper, a numerical code is developed for numerical simulation of acoustoelatic effect in pre-stressed media. The possibility of assessing the stress status, by using the EMAT for precisely measuring the LCR wave, was investigated. The numerical results showed a very good agreement with the theoretical values.References-1] P. J. Whithers and H. K. H. Bhadeshia, Overview Residual stressPartl. Measurement techniques, Material Science andTechnology, Vol. 17 355-365, (2001). [2] Development of non-contact stress measurement system duringtensile testing using the electromagnetic acoustic transducer,NDT&E 39, 299-303 (2006). [3] Michael Junge, Jianmin Qu, Relationship between Rayleigh wavepolarization and state of stress, Ultrasonics 44, 233-237(2006). [4] Dequennoy M. Ouaftouh M., Ultrsonic evaluation of stresses inorthotropic materials using Rayleigh waves, NDT&E 32,189-199 (1999). [5] Shailesh Gokhale, Determination of applied stress in rails using theacoustoelastic effect of ultrasonic waves, Master's thesis, TexasA&M University ( 2007). 16] Michael Junge, Measurement of applied stress using polarization ofRayleigh waves. Master's thesis, School of Civil and Environmental Engineering, Georgia Institute of Technology,(2003). [7] Cuixiang Pei, zhenmao Chen, Development of simulation methodfor EMAT signals and application to TBC inspection, Int. J. Appl.Electromagn. Mech., Vol.33 (2010). 18] R.T. Smith, R. Stern, and R.W.B Stephens. Third-order elasticmodule of polycrystalline metals from ultrasonic velocity measurements. Journal of Acoustical Society of American,40(5):1002-1008 (1966). [9] Don E. Bray, Wei Tang, Subsurface stress evaluation in steel platesand bars using the LCR ultrasonic wave, Nuclear Engineering and Design, 207:231-240 (2001).507“ “Numerical Simulation of Residual Stress Measurementwith Acoustic Wave“ “Cuixiang Pei,Kaziuki Demachi
1. Introduction
The residual stress in some mechanical structures increases the likelihood of fatigue cracks, stress corrosion cracks and environmental induced material degradation. The ability to evaluate the residual stress would substantially increase the accuracy of structure life estimation and the security of operation. Several nondestructive testing methods, such as X-Ray diffraction techniques, ultrasonic wave techniques and magnetic effect analytical methods, have been applied in the residual stress measurement. As the high permeability of ultrasonic wave, the ultrasonic wave method can be used to evaluate the residual stress in the interior of the structure as well as at the surface. Besides, being portable and cheap to undertake, this method is well suited to routine inspection procedures and industrial studies of large components[1].The ultrasonic methods for residual stress measurement are mainly based on the acoustoelastic effect related to the variation of ultrasonic wave speed and change in the polarization of Rayleigh wave. Presently, the residual stress measurement with ultrasonic wave is mainly studied with experiment method(2,5]. And simultaneously, the numerical simulation method can provide another useful tool for the research of stress measurement with ultrasonic wave. In this work, the possibility of numerical simulation of residual stress measurement based on the acoustoelastic effect was investigated with FEM.Now in practical application, the LCR (refracted longitudinal) wave is usually used to measure the subsurface residual stress in components[9]. In this paper, the possibility of assessing the stress status, by using an EMAT (electromagnetic acoustic transducer) receiver for precise measurement of LCR (refracted longitudinal) wave, was investigated by simulation.Corresponding authors: Tel: +81 (0)29 287 8431 Fax: +81 (0)29 287 8488. Email: peicx@nuclear.jp, - demachi@nuclear.jp2. Numerical Simulation Method 2.1 Analytical model for acoustoelastic effect in solidsAn ultrasonic wave through a stressed body would give rise to further stresses, and the traditional theory of acoustoelasticity applied in the unstressed medium cannot be used here. Instead, based on the deformation process, Duquennoy et al. defined a theory of three state of a body, shown in Fig. 1 [3,4]. A solid body undergoes a series of deformations from a stress free state to a static deformation or a dynamic deformation. The position vector & defines the position of a point in the natural state of zero stress and zero strain. X defines the position of a point in the initial state when the body undergoes a static deformation due to residual stress during the manufacturing processes or due to the applied stresses. Similarly X defines the position in the final states when a dynamic deformation such as the ultrasonic wave through the body. The displacement of a point from one state to another can be described mathematically as: u'(E)= X - E; u' (5)= x a?“ 5; (E)= x - X = u' a?“ u'Acoustic waveInitial StatePre-stress2Final StateNatural StateFig. 1 Coordinates for material point in three states Based on the three-state theory, the controlling equation of the acoustic wave in a solid with initial stress can be written as(OxOE≫n + A?uxe) ONE -y @, + f = po(1a?“Ex) *u-1at305Where o'is the Cauchy stress tensor in the initial state, & the initial strain tensor and p'the mass density in the natural state, y is the acoustic damping coefficient,f is the external force loading on the body. And for isotropic material the elastic constant can be expressed as A?JKL = 18,0ke + u(Eixon +8,8xx)+[(-2+v;)8a??A?KL+(-4+ v2)OKOL+OLOjK)]a?￢'w +2(a + v2)a?￢OKL + a?￢ KLO) +2(u+v) EROL +a?￢18,K + EKOL + E'LOIK)(2) where 2 and u are the well known Lame constants and v;(i = 1,2,3) are the third-order elastic constants (TOE constants).2.2 Numerical ModelIn order to demonstrate the feasibility of this simulation method, a 3D model with initial homogeneous stress Ox, uniaxially directed along the x axis, was used for the simulation of acoustoelastic effect on LCR wave, shown as in Fig.2. A PZT angle transducer was used to generate surface wave, and an EMAT probe was used to receive the time behavior of the wave signal for different stress. As shown in table 1[6,8], two kinds of material have been used in this paper. Where, the sensitivity constant k is defined as a relative change in wave speed per unit change in stress.EMATPZTMagnet30mm -Coil-60mm25 pmFig. 2 A pitch-catch setup model in simulationTable 1 Material parameter of the modelMaterialp/kg m0/Sm' 2 / Paul Pa2719382000004910000000026000000000Al AlloyD54s Rail Steel7800110000011580000000079900000000MaterialV/Pa| VzI Pa| Vz/ Pak / MPaAl Alloy D54s-379000000000-198000000000-80000000000-0.0000879Rail Steel36000000000-266000000000178.5E91.214E-52.3 Numerical simulation method for acoustic waveAccording to FEM, eq. (1) can be calculated by solving the discrete wave motion eq. (3)[M]{A?}+[C]{U}+[K]{U}={F} (3) where [M] denotes the mass matrix, [C] the damping matrix, [K] the stiffness matrix, {U} the nodaldisplacement and {F} the force vector. Eq. (3) is rearranged as into the following equation by using the explicit integration method in time-domain[7] |{U}rtar = {U}.-os a?“241[M]'[C]{U}, -241 [M]““[K]{U}, +201[M]' {F},{U}iter = {U}, + {U}rts, +{U},2.4 Numerical method for EMAT receiverA typical configuration for an EMAT receiver of acoustic waves in a test-piece is shown in Fig. 2. It is usually composed of a static magnet and a set of coils (named pick-up coil). When the ultrasonic wave occurred in the test-piece under the EMAT receiver, the vibration by the ultrasonic wave would interact with the static magnetic flux density of the static magnet B, to yield transient eddy current J. in the metal.J. = 0,vxB., v is the velocity of a particle, o, is the conductivity of the metal. And a micro current would be induced by the electromagnetic induction effect. The magnetic flux density B. of a permanent magnet can be calculated by equivalent magnetic charge approach [7]. The most direct way to calculate the pick-up signals of the EMAT receiver is to use the Biot-Savart's law, and the voltage signal in pick-up coil can be written as,v=X& SA-d=A?a [SIL - dawhere T, is boundary of surface S; and r is the distance from a current source point in the conductor to a point in the pick-up coil. 3. Numerical Simulation ResultsThe ultrasonic wave was excited by one damped cycle of a 1 MHz sinusoidal. To give a preliminary judgment of the validity of this method, the simulation result of the ultrasonic wave field in a cross section at a given time in the Al alloy model is shown in Fig. 3. The time behavior of the EMAT coil voltage is plotted in Fig. 4. As the EMAT receiver was set at 40 mm away from the PZT transducer, the velocity of the LCR wave is calculated which is about 6000 m/s. This simulation result is agreed with the theoretical value.Rayleigh WaveLCR Waveslasveise Wavekorgitudinal WaveFig. 3 Ultrasonic wave field excited by PZT306EMAT Receiver SignalLCR WaveVoltagel/V0:00:002.0.10a?￢ 4.0x10* 60x1080x10A° 10x10' 1210Time / s1.4x10Fig. 4 LCR wave received by EMATIn Fig. 5 and Fig. 6, the close-up of the LCR wave signals for two initial stresses in Al Alloy and rail steel is provided. The acoustoelastic effect with a small phase delay induced by the stress can be observed. It can be seen that the variation in LCR wave speed due to acoustoelastic effect in rail steel is much smaller than that in Al alloy. The relative change in LCR wave speed for different residual stress is simulated, shown in Fig. 7. It can be seen that the simulation results show a very good agreement to the theoretical values.2.0x10O MPa 200 MPa1.0x10EMAT signal / V0 -1.0x10A° --2.0x102-3.0x10 +5.0x106.0x10%7.0x10$ 8.0x10*Time / s9.0x10%1.0x10%Fig. 5 Acoustoelastic effect in Al alloy3.0x10**--O MPa200 MPa2.0x10* +1.0x104EMAT signal / V0.0 F-1.0x10*-2.0x10-3.0x106.0x 10a?￢7.0x10*9.0x1068.0x 10 Time / sFig. 6 Acoustoelastic effect in rail steel-----O MPa200 MPa7.0x109.0x108.0x106 Time isSimulation value of Al AlloySimulation value of Rail Steel ---- Reference value of Al AlloyReference value of Rail SteelRelative change-0.04.-600400400600-200200 Residual stress / MPaFig. 7 Relative change of LCR wave's speed in Al Alloy andrail steel4. ConclusionIn this paper, a numerical code is developed for numerical simulation of acoustoelatic effect in pre-stressed media. The possibility of assessing the stress status, by using the EMAT for precisely measuring the LCR wave, was investigated. The numerical results showed a very good agreement with the theoretical values.References-1] P. J. Whithers and H. K. H. Bhadeshia, Overview Residual stressPartl. Measurement techniques, Material Science andTechnology, Vol. 17 355-365, (2001). [2] Development of non-contact stress measurement system duringtensile testing using the electromagnetic acoustic transducer,NDT&E 39, 299-303 (2006). [3] Michael Junge, Jianmin Qu, Relationship between Rayleigh wavepolarization and state of stress, Ultrasonics 44, 233-237(2006). [4] Dequennoy M. Ouaftouh M., Ultrsonic evaluation of stresses inorthotropic materials using Rayleigh waves, NDT&E 32,189-199 (1999). [5] Shailesh Gokhale, Determination of applied stress in rails using theacoustoelastic effect of ultrasonic waves, Master's thesis, TexasA&M University ( 2007). 16] Michael Junge, Measurement of applied stress using polarization ofRayleigh waves. Master's thesis, School of Civil and Environmental Engineering, Georgia Institute of Technology,(2003). [7] Cuixiang Pei, zhenmao Chen, Development of simulation methodfor EMAT signals and application to TBC inspection, Int. J. Appl.Electromagn. Mech., Vol.33 (2010). 18] R.T. Smith, R. Stern, and R.W.B Stephens. Third-order elasticmodule of polycrystalline metals from ultrasonic velocity measurements. Journal of Acoustical Society of American,40(5):1002-1008 (1966). [9] Don E. Bray, Wei Tang, Subsurface stress evaluation in steel platesand bars using the LCR ultrasonic wave, Nuclear Engineering and Design, 207:231-240 (2001).507“ “Numerical Simulation of Residual Stress Measurementwith Acoustic Wave“ “Cuixiang Pei,Kaziuki Demachi