E-Journal of Advanced Maintenance Vol.2 (2010/2011) 160-167
Japan Society of Maintenology

Numerical Simulation of Residual Stress Measurement with Acoustic Wave
Cuixiang PEI* and Kazuyuki DEMACHI
Department of Nuclear Engineering and Management, University of Tokyo, Tokyo 113-8656, Japan
ABSTRACTIn this work, a FEM (finite element method) simulation program was used in the residual stress measurementwith acoustic wave. The possibility of assessing the residual stress, by using an EMAT (electromagnetic
acoustic transducer) receiver for precise measurement of the acoustoelstic effect on LCR wave, was investigatedby simulation method.
KEYWORDS

ARTICLE INFORMATION

Residual stress, Acoustoelastic effect, Ultrasonic NDE, FEM, EMAT

Article history:
Received 19 August 2010
Accepted 1 February 2011

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 theresidual stress would substantially increase the accuracy of structure life estimation and the security ofoperation. Several nondestructive testing methods, such as X-Ray diffraction techniques, ultrasonicwave techniques and magnetic effect analytical methods, have been applied in the residual stressmeasurement. As the high permeability of ultrasonic wave, the ultrasonic wave method can be used toevaluate 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 andindustrial 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 velocity and change in the polarization of Rayleighwave. Presently, the residual stress measurement with ultrasonic wave is mainly studied withexperiment method [2,5]. And simultaneously, the numerical simulation method can provide anotheruseful tool for the research of stress measurement with ultrasonic wave. In this work, the possibility ofnumerical simulation of residual stress measurement based on the acoustoelastic effect wasinvestigated with FEM.
Now in practical application, the LCR (refracted longitudinal) wave is usually used to measure
the subsurface residual stress in components. 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.

2.

Numerical Simulation Method

2.1

Analytical model for acoustoelastic effect in solids

An 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*

Corresponding author, E-mail: peicx@nuclear.jp
Present address : 2-11-16, Yayoi, Bunkyo-ku, Tokyo 113-0032, Japan†

ISSN-1883-9894/10/11 © 2010/2011 – JSM and the authors. All rights reserved.

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C. PEI, et al./ Numerical Simulation of Residual Stress Measurementwith Acoustic Waveor a dynamic deformation. The position vector ξ defines the position of a point in the natural stateof zero stress and zero strain. X defines the position of a point in the initial state when the bodyundergoes a static deformation due to residual stress during the manufacturing processes or due to theapplied stresses. Similarly x defines the position in the final states when a dynamic deformationsuch as the ultrasonic wave through the body. The displacement of a point from one state to anothercan be described mathematically as:u i (ξ ) = X − ξ ; u f (ξ ) = x − ξ ; u (ξ ) = x − X = u f − u iFig. 1. Coordinates for material point in three states

Based on the three-state theory, the controlling equation of the acoustic wave in a solid withinitial stress can be written as(iδ IK σ JL+ Dˆ IJKL)∂ 2u K∂u∂ 2ui− γ I + f = ρ 0 (1 − ε NN) 2I∂X J X L∂t∂t(1)

Where σ i is the Cauchy stress tensor in the initial state, ε i the initial strain tensor induced by
the residual stress and ρ 0 the mass density in the natural state, γ is the acoustic damping coefficient,
f is the external force loading on the body. And for isotropic material the elastic constant can beexpressed asDˆ IJKL = λδ IJ δ KL + μ (δ IK δ JL + δ ILδ JK ) + [(−λ + v1 )δ IJ δ KLiiδ IJ )+ (− μ + v2 )(δ IK δ JL + δ ILδ JK )]ε NN+ 2(λ + v2 )(ε IJi δ KL + ε KL(2)iii+ 2( μ + v3 )(ε IK
δ JL + ε ILi δ JK + ε JK
δ IL + ε JL
δ IK )

Where λ and μ are the well known Lame constants, vi (i = 1, 2,3) are the third-orderelastic constants (TOE constants), and δ IJ is Kronecker delta.
2.2. Numerical Model
In order to demonstrate the feasibility of this simulation method, a 3D model with initialhomogeneous uniaxial stress σ x (the stress σ xx , uniaxially directed along the x axis) was used forthe simulation of acoustoelasticity, shown as in Fig.2. A PZT angle transducer was used to generatesurface wave, and an EMAT probe was used to receive the time behavior of the wave signal fordifferent initial stress σ x . As shown in table 1[6,8], two kinds of material have been used in thispaper. Where, the sensitivity constant kc is defined as a relative change in wave velocity per unit161

C. PEI, et al./ Numerical Simulation of Residual Stress Measurementwith Acoustic Wavechange in stress. When the longitudinal wave velocity through a wedge is cw = 2720m / s , based on
Snell’s law, the incidence angle should be set at about θ w = 26D to generate LCR wave in the model.

Fig. 2. A pitch-catch setup model in simulation
Table 1 Material parameter of the model

Material

ρ / kg ⋅ m −3

σ e / S ⋅ m −1

λ / Paμ / PaAl Alloy D54s

2719.038.2E649.1E926.0E9Rail Steel

7800.01.1E6

115.8E9

79.9E9Materialv1 / Pa

v2 / Pa

v3 / Pakc / ( MPa )Al Alloy D54s

-379.0E9-198.0E9-80.0E9-8.79E-5

Rail Steel

36.0E9-266.0E9

178.5E9-1.214E-5−12.3. Numerical simulation method for acoustic wave
iWhen there is residual stress in the material, the initial strain ε IJ would be induced. Based on therelationship between the stress and strain in isotropic material, the initial strain can be calculated byε IJi =E=1+υυσ IJ − δ IJ σ kkEEμ ( 3λ + 2 μ ),(λ + μ )υ=(3)λ2 (λ + μ )Substituting Eq. (3) into Eq. (2), the elastic constant related to the residual stress can be decided.
According to FEM, eq. (1) can be calculated by solving the discrete wave motion eq. (4)[ M ]{U} + [C ]{U } + [ K ]{U } = {F }[M ] = ∑ [M e ]wheredenotes

themassmatrix,e(4)[C ] = ∑ [C e ] the damping matrix,e[ K ] = ∑ [ K ] the stiffness matrix, {U } the nodal displacement and {F } the force vector.
ee162

C. PEI, et al./ Numerical Simulation of Residual Stress Measurementwith Acoustic WaveT

[ M e ] = ρ ∫ e [ N ] [ N ]dVVT[C e ] = γ ∫ e [ N ] [ N ]dVVT[ K ] = ∫ e [ B ] [ Dˆ IJKL ][ B ]dVeV

[ N ] is the shape function and [ B ] the strain matrix of the element used in the FEM model. In orderto solve the differential function, Eq. (4) is rearranged as into the following equation by using theexplicit integration method in time-domain[7]⎧{U }t +Δt = {U }t −Δt − 2Δt [ M ]−1 [C ]{U }t − 2Δt [ M ]−1 [ K ]{U }t + 2Δt [ M ]−1 {F }t⎪⎨{U }t +Δt + {U }tΔt⎪{U }t +Δt = {U }t +2⎩(5)2.4. Modeling of PZT angle transducer

To simplify the problem, the modeling approach of the PZT angle transducer presented here
focuses on the transmission of mechanical wave between the transducer and the test-piece. As shownin Fig. 3, the transmission of mechanical wave is modeled as a distribution of force fi (t ) , distributed

only on the excited portion of the test-piece’s boundary surface. The function fi (t ) is givenas⎧ A sin ( 2π fti ) ; 0 ≤ ti ≤ T
fi (t ) = ⎨ti < 0 or ti > T⎩ 0;(i − 1) d sin θ w
ti = t +( N − 1) cw(6)where A is the amplitude of the exciting signal, T the pulse width of the exciting signal , f the central
frequency of the exciting signal and N the number of contributing nodes.

Fig. 3. Modeling of the PZT angle transducer

2.5. Numerical method for EMAT receiver

A 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 ultrasonic163

C. PEI, et al./ Numerical Simulation of Residual Stress Measurementwith Acoustic Wave

wave occurred in the test-piece under the EMAT receiver, the vibration by the ultrasonic wave wouldinteract with the static magnetic flux density of the static magnet B 0 to yield transient eddy current
J e in the metal. J e = σ e v × B 0 , v is the velocity of a particle, σ e is the conductivity of themetal. And a micro current would be induced by the electromagnetic induction effect. The magnetic
flux density B0 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 theBiot-Savart's law, and the voltage signal in pick-up coil can be written as,NJe ⎤∂∂ ⎡fAld
⋅
=∑evv
∫∫
∫∫∫
⎢ cond r dv ⎥ ⋅ dl⎦
i =1 ∂t Γii =1 ∂t Γi ⎣NVe = ∑(7)

where Γi is boundary of surface Si and r is the distance from a current source point in theconductor to a point in the pick-up coil.
3.

Numerical Simulation Results

The 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. 4. The time behavior of theEMAT coil voltage is plotted in Fig. 5. As the EMAT receiver was set at 40 mm away from the PZTtransducer, the velocity of the LCR wave is calculated which is about 6000 m/s. This simulation resultis agreed with the theoretical value.

Fig. 4. Ultrasonic wave field excited by PZTFig. 5. LCR wave received by EMAT

164

C. PEI, et al./ Numerical Simulation of Residual Stress Measurementwith Acoustic WaveIn Fig. 6 and Fig. 7, the close-up of the LCR wave signals for two initial stresses in Al Alloy andrail steel is provided. The acoustoelastic effect with a small phase delay induced by the stress can beobserved. It can be seen that the variation in LCR wave velocity due to acoustoelastic effect in rail
steel is much smaller than that in Al alloy.

Fig. 6. Acoustoelastic effect in Al alloy

Fig. 7. Acoustoelastic effect in rail steel

It can be seen that the time delay induced by the stress is very small in Fig.6 and Fig.7. So it ishard to evaluate the relative change in wave velocity directly. To calculate the relative change in wave
velocity indirectly, a precise method based on the cross correlation algorithm was executed asfollowing.
A. Firstly, the cross correlation coefficient of the two wave signals at the same measuring position
with and without residual stress was calculated by delaying the start time of one signal withEq.(8)Nr (i * dt ) =∑ ⎡⎣( x(k ) − mx ) ⋅ ( y (k − i) − my )⎤⎦(8)

k =1

N

N

∑ ( x(k ) − mx ) ∑ ( y (k − i) − my )
k =1

2

k =1

165

2

C. PEI, et al./ Numerical Simulation of Residual Stress Measurementwith Acoustic Wave

where dt is the interval between the time resolution, r the correlation coefficient at delay i*dt, x
and y are the two series of wave signal, mx and my are the means of the two series. Fig. 8 shows
an example of the result of the cross correlation coefficient of two wave signals. And the delay
time Δt can be precisely decided when the correlation coefficient reaches the maximum.

Fig. 8. Example of the calculation result of the cross correlation between two signals

B. The relative change of the wave velocity can be calculated with Eq. (9)c ΔtΔc c1 − c0==− 0c0c0s + c0 Δt(9)where c0 denotes the wave velocity without residual stress, c1 the velocity with residual stress, s
the distance between the two transducers and Δc the velocity change due to the stress. To
overcome the artificial error in the calculation of wave velocity, the c0 was set as the theoreticalvalue of wave velocity without residual stress here.
Based on the above method, the relative changes in LCR wave for different residual stress weresimulated, as shown in Fig.9. It can be seen that the simulation results show a very good agreementwith the theoretical values.

Fig. 9. Relative change of LCR wave’s velocity in Al Alloy and rail steel

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C. PEI, et al./ Numerical Simulation of Residual Stress Measurementwith Acoustic Wave

4. Conclusion
In this paper, a numerical code is developed for numerical simulation of acoustoelatic effect inpre-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 agreementwith the theoretical values.

Acknowledgement
This work was supported by China Scholarship Council (CSC).

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