The ultrasonic methods for detection of internal (residual) stresses are based on acoustoelastic effect, namely on stress-induced velocity variations of acoustic waves propagating in a predeformed solid. The stress dependence of wave velocities is caused by a finite deformation of a body and, in addition, by a nonlinear stress-strain relation of the material. By introducing a second-order effect of strain ( third-order elastic constants - TOEC) and separating the small motion superimposed on a large deformed body, the theory of acoustoelasticity has been developed . The TOEC's are the nonlinear macroscopic result of anharmonic microscopic phenomenon. For this reason, the development of acoustoelasticity have been stimulated by interest in the TOEC's evaluation in single crystals. Because the acoustoelastic effect is very small, other factors such as temperature, micro-inhomogeneity, texture, related weak anisotropy and plasticity of a material cannot be neglected in ultrasonic measurements of residual stress in engineering materials . On the other hand, modern techniques for wave velocity measurements based on digital signal processing can improve acoustoelastic experiments.
2 Theory of acoustoelasticity
Stress and strain free state in a body is denoted as its natural state. The body undergoes a
static deformation (prestress) which leads to the initial state and after that additional dynamic disturbances (acoustic waves) cause the final state, Fig.1. Mathematical form of expressions for wave velocity dependence on the prestress may take various formats according to the kinematic description adopted. Following results in the natural frame of reference (Wo) are derived for the longitudinal wave propagation in the x ş x1 direction, using the infinitesimal description, the updated and the total Lagrangian formulations, and Logarithmic strain tensor, respectively.
The infinitesimal strain tensor gives
Fig 1: Natural,initial and final configuration of predeformed body with superimposed acoustic wave.|
the Green-Lagrange tensor in the updated Lagrangian formulation (ULF)
the Green-Lagrange tensor in the total Lagrangian formulation (TLF)
and the Logarithmic strain tensor yields
in which the stretches - and the deformation gradients . The free energy Y for a given deformation measure Eij may be generally expressed as where cijkl and cijklmn are the second (SOEC) and third (TOEC) order elastic constants, respectively. In the isotropic case, only two (l,m) SOEC and three (l, m, n) TOEC are independent.
The formulation using the logarithmic strain tensor usually gives simpler form of the energy function, than other descriptions do. Its use is popular e.g. in plasticity .
Linearisation of the resulting equations for uniaxial stress state s leads to
for a small value , where index (i) represents a propagation axis and second (j) wave polarization. The quantities S are known as the acoustoelastic coefficients (AEC). A similar relation may be derived for wave velocities in configuration WI.
The velocity difference of two transversal waves polarized in the principal stress (s2,s3) directions, which propagate in the perpendicular direction x1 (s1ş 0), represents acoustoelastic birefringence equation
where cT is the shear wave velocity in a stress-free material.
Acousto-elastic coupling properties (S) must be known to quantify either thermal or residual stresses from wave velocity changes.
The AEC measurements were realized during compression tests of the prismatic specimens
(30x30x60mm3 and 30x30x30mm3) using the testing machine Tiratest 2300, Fig.2. The studied materials were :
The materials i) and iii) could be considered as isotropic and the material, see Tab.1, ii) displays anisotropy of elastic properties, see Tab.2. The elastic properties of stress-free materials were evaluated from pulse-echo measurements and the mass density was determined by weighing in air and water. The polycrystalline and the amorphous materials variations and influence of the texture orientation is analyzed.
- special prepared (multidirectional rolling, temperature treatment) of AlMg3 alloy,
- commercial product from AlCu4BiPb alloy (42 4254.61), where the extrude axis is equal to x3,
- optical glass.
|Longitudinal wave velocity cL
|Transverse wave velocity cT
|Mass density r
|Young modulus E
|Poisson ratio n
|Table 1: Mechanical properties of isotropic materials in the stress-free state and
evaluated TOEC's : l; m;n. |
|Table 2: Mechanical properties of the material AlCu4BiPb. The hexagonal class
of anisotropy is assumed.|
Longitudinal (L) or transversal (T) wave were induced during the loading and the unloading steps. For L-waves, Aerotech Gamma Hf 10MHz-0.25'', normal incidence L-transducers were used with a glycerine couplant, whereas for T-waves, Ultran SWC25-5 (5MHz) normal incidence shear transducers were used with a processed molasses couplant. The time shift t of the transmitted or reflected wave from free-stress state was evaluated by following techniques. The instantaneous wave flight distance is L = Lo(1 - ns/E) and L = Lo(1 + s/E) for the wave propagating in the perpendicular and parallel directions with respect to the loading axis, resp. Thus, the resulting wave is given by V = L/(t + Lo/c).
Fig 2: The scheme of compression tests,wave propagation and US transducers configuarations.|
The experiments has not been realized under temperature control. Nevertheless, the temperature
of the specimen was monitored and temperature changes were smaller than 1oC during each test and temperature differences among the individual tests were not higher than 5oC.
3.1 Impulse technique
Ultrasonic card SFT4003B was used for the pulse-echo technique. The rf-signals were recorded by
digital oscilloscope LeCroy 9304AM (sampling rate 10GS/s, RIS mode, and resolution 10.5bits) during the specimen loading. The signal records were further analyzed in frequency domain, Fig.3.
Fig 3: Pulse - echo technique, frequency analysis of rf-signals|
Unwrapped phase characteristics q(f) in an optimally selected frequency band is linearized by the least square method. The wave-path time shift t may be determined by relations qlin = 2p(const.-t.f) and t = t(s) - t(0). The deviation of the phase q(f) from the straight line qlin is plotted in lower part of fig.3. Small fluctuations around lincarized phase characteristics are visible in the case of the transmitted T-wave.
3.2 Continuous wave technique
The evaluation procedure for the continuous wave (c.w.) technique uses a principle of Phase
Lock Loop (PLL) for the phase and magnitude measurements. The reference harmonic signal
uR = UR sin(wRt + qR) with frequency wR is mixed with the input signal uI in a mixer M1. Input signal component UI sin(wRt + qI) with the frequency wR appears on the mixer output in a form uM1 = 1/2 URUI[cos(qR - qI) - cos(2wR + qR + qI)]. The first component of the mixer output represents the low-frequency part whereas the second component oscillates with high frequency (2wR). Thus a low pass filter applied following the mixer has the response uM1+LPF = 1/2 URUIcos(qR - qI), which is proportional to both the phase difference qR - qI and the input signal amplitude UI. There is only one equation with two unknowns. Adding the other mixer (M2) in which the signal is mixed with the phase shifted (by p/2) reference signal and after another low-pass filtering, we can obtain the result uM2+LPF = 1/2 URUIcos(qR - qI - p/2)= 1/2 URUIsin(qR - qI). Using both the outputs uM1+LPF and uM2+LPF, the amplitude (RMS value) R and phase q of the input signal may be determined by the relations
The (PLL-based) lock-in amplifier SR844 is capable to process the signals up to 200MHz. The real-time calculation of R and q is realized using Digital Signal Processor (DSP), .
The reference signal swept in a fixed frequency band enters into the US emitter. The signal from the US receiver is fed to the lock-in amplifier input. The above described procedure is applied for each frequency step f, and after that the values of R(f), q(f) are evaluated. The actual frequency dependencies of the amplitude R and unwrapped phase q are shown in Fig.2 b.
Time shift t between the reference and input signal may be determined as a slope of the straight line qlin(f) fitted to unwrapped phase q(f), thus .
The saw-tooth amplitude characteristics R(f) is caused by the influence of multiple wave reflection in a specimen. Let function x(t) represents the primary waveform and -x(t - tR)
is the reflected wave from the opposite (free) surface, delayed by tR. Thus, resulting signal from the receiver is xB(t) = x(t) - x(t - tR). Its fourier transform may be expressed as with magnitude and phase
From the inverse fourier transform of R(f), we can obtain delay tR as
The last procedure is evident in Fig.4.
During the loading and unloading process, 21 ultrasonic measurements were performed by both
impulse and c.w. methods. Relative velocity changes are shown in Fig.5. The difference between the results obtained by the impulse and c.w. method is about 5% and 10% for L-wave and T-wave (y = 0o) respectively. The c.w. results "oscillate" along the straight line.
The fluctuations may be caused by temperature drift in the lock-in amplifier.
Fig 4: |
a)Magnitude R(f) of the swept harmonic L- wave generated in a specimen.
b)Evaluation of the first L-wave reflection time delay tR.
c)Comparision of phase oscillations q - q lin with its theoretical prediction (8) of f(f) assuming tR = L0/c
Fig 5: Relative velocity changes induced by applied stress
3.3 Results of acoustoelastic measurements
The acoustoelastic coefficients S11, S12 and S13 has been evaluated from the measurements (Fig.1 a) and the related wave velocity dependencies on the stress s are plotted in Fig.6. For isotropic materials i) and iii), the TOEC's l, m, n has been calculated using the equations derived from TLF and under the assumption of nonlinear elastic material. The TOEC's l, m for the glass are positive value, whereas the TOEC's for AlMg3 are negative. The AEC S33 in AlMg3 is calculated from l, m, n and the result is compared with the value obtained from the direct measurement (Fig.1 b) of longitudinal wave propagation in the loading direction. The differences between the calculated and measured values are better than 10%. The stress-induced velocities of glass have opposite trend than in the case of the duraluminium. The sensitivity of wave propagation in the glass along the loading axis is obviously dominant.
Fig 6: Measued stress-dependence of (V2 - c2)/c2 during eximpression tests|
3.4 Measurement of the acoustical birefringence induced by uniaxial loading
Let us assume that the plane stress is parallel with the plane (x2, x3) and the principal stress components are (0, 0, s). The shear (T) wave propagates in the x1 direction and its polarization vector is oriented under the angle y with respect to the x2 axis, Fig.1. The displacement vector of the T-wave may be divided into two components in the directions x2 and x3 propagating with speeds V2 and V3, respectively. The speeds difference is caused by the loading s. The phase shift DF between both components, arising during the wave propagation along the length L, is expressed as DF= wL(1/V13 - 1/V12). The phase shift evokes modulation of the spectral amplitude of the echo signal which may be derived as M(jw) = [1 - sin2(2y)sin2(DF/2)]1/2, . Spectral modulation M(jw) has minima for DF= (2n - 1)p, n = 1,2,3..., which corresponds to frequencies fmin = (2n - 1)V12V13/[2L(V12 - V13)]. Using eqn(6), then the coefficients S'T may be expressed in dependence on the first minima frequency fmin as
for a isotropic and a weak anisotropic material, resp., where Dc = cT2 - cT3 is a difference of stress-free wave velocities and c T = (cT2 + cT3)/2 is a mean stress-free velocity in the anisotropic material.
In experiments, Fig.1, the incident shear waves are polarized at the angle y = p/4. The stress-induced reversible changes of the rf-signal waveform caused by loading and unloading are plotted in Fig.7, where the isotropic and anisotropic Al materials are compared. Using Fourier transform of the above mentioned signals, the minima position fmin drift caused by applied stress is shown. It is clear from comparing the both studied materials, that we cannot neglect the initial velocity difference Dc for anisotropic material AlCu4BiPb. The dependencies -S'Ts are evaluated from eqn. (10). The stress sensitivity of acoustic birefringence determined by the spectral technique may be compared with the factor (V12 - V13)/cT, where velocities V12(s) and V13(s) are measured separately. The birefringence coefficient S'T determined by both methods differs about 10% and 1% in case of AlMg3 and AlCu4BiPb, resp. In the case of "isotropic" material AlMg3, the weak anisotropy about Dc/cT ť 0.3% is evident.
Fig 7: S°33 calculated upon the formulations (a)-(c) and using the adopted material data|
The assumption of isotropic properties of metals seems to be very strong.
Magnitude of frequency spectrum and fmin position changes due to applied stress
Fig 7b:Comparision spectral and velocity difference techniques.
Fig 7: Comparision of acoustoelastic birefringence measurements for isotropic and anisotropic material.|
4 Remarks on using various deformation measures
An influence of various deformation measures, mentioned in the introduction, is demonstrated on
the own experimental data (Tab.1) and on the elastic properties of isotropic materials adopted from Table 1 in reference , p.531-532. The expressions for longitudinal wave propagation in the solid are derived for hydrostatic (SoK) and uniaxial stress (along x3), where the wave propagation axis is perpendicular (So11) or is parallel (So33) to the loading axis.
The values of the AEC's measured in AlMg3 and in the glass are given in Tab.3. Calculated
results of So33 from the adopted material properties are shown in Fig.7. The acoustoelastic behaviour of metals and glass is substantially different. Inspection of Fig.7 suggests that although a nonlinear material law has to be used in general, in order to describe the AEC's accurately, the logarithmic formulation provides the best basis for the construction of the free energy function. Even with linear stress-logarithmic strain relation, the results exhibit the desired tendency.
|Table 3: Acoustoelastic constants evaluated for hydrostatic stress and uniaxial stress perpendicular or parallel to the direction of longitudinal wave propagation in the frame of Wo,10-4[1/MPa], where is denoted (a)TLF,Linear elastic material (b)TLF, nonlinear elastic material (including constants l,m,n); this formulation was used for evaluation of l,m,n from experimental data- refernce formulation. (c) Logarithimic formulation, linear elastic material.|
Various measurement techniques of ultrasonic velocity changes induced by prestress are briefly described in this paper. The ultrasonic measurements during compression tests of prismatic specimen are used for evaluation of acoustoelastic coefficients. The problem with inherent material anisotropy of the tested commercial Al-alloy is discussed. Two approaches (impulse and continuous wave method) are also critically compared. Resulting differences are less than 10%. Continuous wave (c.w.) method, based on PLL is more time consuming and less accurate. Multiple wave reflections in a specimen may affect the c.w. results.
Acoustoelasticity is a weak and nonlinear effect. The impulse technique requires high resolution in time base, i.e. high sampling rates. The pulse-reflection method uses double wave-path and so it can improve the measurement accuracy.
Acoustical birefringence may be also measured directly using by shear-wave transducer under
polarization angle 45o with respect to the principal stress directions. The difference of the presented spectral technique from separate velocity measurements is less than 0.6%.
The total Lagrangian and logarithmic formulations describing finite deformation, were compared
using the measured and reference data. Introducing the third elastic constants, both formulations are equivalent. Nevertheless, the logarithmic formulation is promising for extension of acoustoelastic theory into plasticity.
This work was supported by GA CR under the postdoctorand grant No.106/97/P135, the project 101/99/0834 and the project COST P4 (No.P4.3).
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