ˇTable of Contents ˇMaterials Characterization and testing | Ultrasonic Techniques for Non-destructive Evaluation of Internal StressesM. Landa and J. PleekInstitute of Thermomechanics ASCR, Dolejkova 5, 182 00, Praha 8, Czech Republic Contact |
Fig 1: Natural,initial and final configuration of predeformed body with superimposed acoustic wave. |
(1) |
(2) |
(3) |
(4) |
(5) |
(6) |
AlMg3 | glass | |||
Longitudinal wave velocity c_{L} | 6.405 | 5.637 | mm/ms | |
Transverse wave velocity c_{T} | 3.161 | 3.430 | mm/ms | |
Mass density r | 2.664 | 2.280 | g/cm^{3} | |
SOEC's : | ||||
Young modulus E | 71.29 | 64.70 | GPa | |
Poisson ratio n | 0.339 | 0.206 | ||
TOEC's : | ||||
l | -102.40 | 29.016 | GPa | |
m | -249.69 | 14.724 | GPa | |
n | -288.51 | -26.928 | GPa | |
Table 1: Mechanical properties of isotropic materials in the stress-free state and evaluated TOEC's : l; m;n. |
Direction | c_{L}; c_{qL} | Direction | c_{T} | ||
propagation | polarization | [mm/ms] | propagation | polarization | [mm/ms] |
[100] | [100] | 6.330 | [100] | [001] | 3.089 |
[010] | [010] | 6.330 | [100] | [010] | 3.120 |
[001] | [001] | 6.358 | [010] | [001] | 3.089 |
[101] | [101] | 6.310 | [010] | [100] | 3.119 |
[001] | [100] | 3.086 |
E_{1} | E_{3} | G_{23} | G_{12} | n_{12} | n_{13} | n_{31} | r |
[GPa] | [g/cm^{3}] | ||||||
74.56 | 76.17 | 27.08 | 27.66 | 0.348 | 0.328 | 0.335 | 2.841 |
Table 2: Mechanical properties of the material AlCu4BiPb. The hexagonal class of anisotropy is assumed. |
Fig 2: The scheme of compression tests,wave propagation and US transducers configuarations. |
3.1 Impulse technique
Fig 3: Pulse - echo technique, frequency analysis of rf-signals |
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
u_{R} = U_{R} sin(w_{R}t + q_{R}) with frequency w_{R} is mixed with the input signal u_{I} in a mixer M1. Input signal component U_{I} sin(w_{R}t + q_{I}) with the frequency w_{R} appears on the mixer output in a form u_{M1} = 1/2 U_{R}U_{I}[cos(q_{R} - q_{I}) - cos(2w_{R} + q_{R} + q_{I})]. The first component of the mixer output represents the low-frequency part whereas the second component oscillates with high frequency (2w_{R}). Thus a low pass filter applied following the mixer has the response u_{M1+LPF} = 1/2 U_{R}U_{I}cos(q_{R} - q_{I}), which is proportional to both the phase difference q_{R} - q_{I} and the input signal amplitude U_{I}. 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 u_{M2+LPF} = 1/2 U_{R}U_{I}cos(q_{R} - q_{I} - p/2)= 1/2 U_{R}U_{I}sin(q_{R} - q_{I}). Using both the outputs u_{M1+LPF} and u_{M2+LPF}, the amplitude (RMS value) R and phase q of the input signal may be determined by the relations
(7) |
(8) |
(9) |
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 t_{R}. c)Comparision of phase oscillations q - q _{lin} with its theoretical prediction (8) of f(f) assuming t_{R} = L_{0}/c |
Fig 5: Relative velocity changes induced by applied stress |
3.3 Results of acoustoelastic measurements
The acoustoelastic coefficients S_{11}, S_{12} and S_{13} 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 S_{33} 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 (V^{2} - c^{2})/c^{2} 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 (x_{2}, x_{3}) and the principal stress components are (0, 0, s). The shear (T) wave propagates in the x_{1} direction and its polarization vector is oriented under the angle y with respect to the x_{2} axis, Fig.1. The displacement vector of the T-wave may be divided into two components in the directions x_{2} and x_{3} propagating with speeds V_{2} and V_{3}, 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/V_{13} - 1/V_{12}). The phase shift evokes modulation of the spectral amplitude of the echo signal which may be derived as M(jw) = [1 - sin^{2}(2y)sin^{2}(DF/2)]^{1/2}, [5]. Spectral modulation M(jw) has minima for DF= (2n - 1)p, n = 1,2,3..., which corresponds to frequencies f_{min} = (2n - 1)V_{12}V_{13}/[2L(V_{12} - V_{13})]. Using eqn(6), then the coefficients S'_{T} may be expressed in dependence on the first minima frequency f_{min} as
(10) |
Fig 7: S^{°}_{33} calculated upon the formulations (a)-(c) and using the adopted material data |
Magnitude of frequency spectrum and f_{min} 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. |
AlMg3 | |||
Theory | S_{k}^{o} | S_{11}^{o} | S_{33}^{o} |
(a) | -0.1818 | 0.0951 | -0.3720 |
(b) | 0.4848 | -0.2548 | 0.9944 |
(c) | -0.0452 | -0.1403 | 0.2354 |
Glass | |||
Theory | S_{k}^{o} | S_{11}^{o} | S_{33}^{o} |
(a) | -0.3197 | 0.0637 | -0.4470 |
(b) | -0.6118 | 0.0169 | -0.6455 |
(c) | -0.0908 | -0.1546 | 0.2183 |
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 W^{o},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. |
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