Study and numerical simulation of ultrasonic signals for uniaxial stressed specimen.
M. Mekideche*, A. Bouhadjera, T. benkdidah, M. Grimess, S. Haddad NDT_Lab, Jijel University, Ouled Aissa, Jijel 18000, Algeria.
* mekmohammed@yahoo.fr Abstract
In some fields of engineering construction, the aluminum alloys become attractive materials due to their mechanical properties. Taking this fact into account, a prism-shaped specimen of aluminum AA2017A is investigated under uniaxial stress and the measurements are performed considering two methods. For ultrasonic wave propagating normal to the loading direction, the ultrasonic mode conversion is considered.
But for the waves propagating parallel to loading direction, the normal incidence direct contact transducer is adapted. Then, we present a numerical simulation of ultrasonic signal of received echo, and highlight the acoustoelasticity effect that involves a linear dependence between time delay separating the received ultrasonic signals and the undergoing forces.
1. Introduction
In nondestructive testing and evaluation, the elastic waves are widely used for characterizing materials. The acoustoelasticity theory exploits the nonlinear behavior of a medium to establish the relation between stress and ultrasonic velocity [1, 2]. The propagation characteristics of ultrasonic waves depend on many physical properties of the propagating medium such as: the different elastic constants, material density, residual or applied strains, and wave polarization.
The analysis of patterns of detected echoes can reveal more details about the physical structure along the propagation path, and give valuable information about the interaction of elastic waves with the inside of the propagating medium. The simulation approach is a tool that may be an alternative to experimentation, in the case where the results can be validated practically. For example, in [3, 4] the simulating tool is applied to show ultrasonic signals using a valid model. Note that the accuracy of results can be achieved, only when the suitable tools concerning signal processing and experimentation are adopted.
Nowadays, Metallic alloys are extensively used in some fields of construction engineering, due to their mechanical properties. As an example, some authors [5, 6] interested in investigation of aluminum alloys, because they are promising new materials for advanced applications. These materials are very serious candidates of modern construction industry of aeronautics, transport, and civil engineering. This fact is due to some good properties such as: high stiffness, low weight, high temperature load stability, and resistance to corrosion by oxidation.
These metallic parts are often subjected to varying stresses, and heavy loads. For safety reasons, it can be of paramount importance to be able to constantly monitor the level of these stresses. Taking this fact into account, a prism-shaped specimen of aluminum AA2017A is investigated under uniaxial stress and the measurements are performed using the prism technique [7]. As a first step, it is crucial to determine by experimentation, the effect of stress upon the ultrasonic waves propagating parallel and perpendicular to the applying forces. In a second step, we derive a model for simulating the received waveforms using the pulse- echo technique.
2. Experimental setup
In the experimentation, the prism-technique [7] is used, and a specimen of aluminum alloy (AA2017A) is investigated. A system of three perpendicular axes numbered 1, 2, and 3 is chosen for indexing directions of propagation and polarization. The direction of the compressive uniaxial stress is that of axis 1. The main aim of this experience is to study and simulate the behavior of ultrasonic waves propagating inside a stressed medium. This study may be more significant when two different directions of wave propagation are considered. Therefore, in order to realize our objective, the measurements are achieved using both mode conversion and direct transmission methods. The mode conversion method is used for ultrasonic waves propagating normal to stress. Ultrasonic waves are generated by an immersion transducer: 2.5 MHz, which also acts as a transmitter/receiver (Fig. 1). The transducer is mounted on a support allowing it revolving around the origin of the system of axes, in the plane formed by direction 2 and 3. The mode conversion is accomplished as soon as the critical angle is reached.
The longitudinal wave propagating in the liquid is mode converted through the interface liquid-specimen.
Inside the specimen, the resulting longitudinal and shear waves are propagating along different paths. The measurements are accomplished aligning in each time, the propagating path of the desired longitudinal or shear wave on direction 2. In all the following, the subscript is used to indicate the direction of wave propagation, while subscript is used to indicate the direction of wave polarization. In addition, the state in which no stress is applied to the specimen is referred to by initial state. Then, for both longitudinal and shear waves, the time of flight in the specimen in stressed state is
(1) Where, is the time of flight in water, is the time of flight in the specimen in initial state, is the time shift between ultrasonic waves in stressed state and ultrasonic waves in initial state.
The direct transmission method is used for studying the effect of applying forces upon the longitudinal waves propagating parallel to stress in direction 1. In this case, the longitudinal ultrasonic waves are generated by a direct contact transducer: 1MHz that acts as a transmitter/receiver (Fig. 2).The transducer is
Stress (σ)
3 2
Spring Transducer
Figure 2 : Direct transmission scheme 1
Specimen Transducer
Stress (σ)
1
3
2 Water
Specimen
Figure 1: Mode conversionby immersion scheme
placed on the bottom face of the specimen using a spring to guarantee a homogenous contact. In this case, the time of flight in the specimen in stressed state is given by:
(2) 3. Measurements
The transducers are driven with a voltage generator of PANAMETRICS, type 5077PR, Mark SOFRANEL (voltage pulser: 200v, PRF: 500 kHz), using for data acquisition a digital storage oscilloscope Mark Tektronix TDS 1002 linked to a PC.
Starting with the propagation perpendicular to stress, the propagating path of the desired wave (longitudinal or shear) obtained by mode conversion is aligned along direction 2 by simple rotation of the immersion transducer. Concerning longitudinal waves, the direction 2 is also that of polarization. Hence, the time of flight is given on setting 2 in equation (1). The waveforms of the received echoes corresponding to longitudinal waves are plotted in (Fig. 3) where the solid line plot corresponds to the reference waveform, the dashed line plot corresponds to a load of 10 kN, and the dotted line plot corresponds to a load of 40 kN.
Then, for shear waves, the time of flight is also given by equation (1) but on setting 2 and 3, the corresponding waveforms of received echoes are plotted in (Fig. 4) where the solid line plot corresponds to the reference waveform, the dashed line plot corresponds to a load of 5 kN, and the dotted line plot corresponds to a load of 45 kN.
In the case of propagation parallel to stress, the time of flight of longitudinal waves is given on setting 1 in equation (2), the corresponding waveforms of received echoes are plotted in (Fig.5) where the solid line plot corresponds to the reference waveform, the dashed line plot corresponds to a load of 25 kN, and the dotted line plot corresponds to a load of 50 kN.
Given the compressive stress 0 and the time of flight, we can determine the elastic constants exploiting acoustoelasticity theory. For a uniaxially stressed medium in direction 1, the velocity of longitudinal and shear waves propagating normal to stress can be written in terms of stress and elastic constants [8] as follows:
2 2 2! (3) " # $ 2! (4)
Figure 3: Experimental waveforms obtained for longitudinal waves propagating perpendicular to stress.
Figure 4: Experimental waveforms obtained for shear waves propagating perpendicular to stress.
Figure 5: Experimental waveforms obtained for longitudinal waves propagating parallel to stress.
For longitudinal waves propagating parallel to stress the equation is:
%% 2 2 # 4 4 10! (5) Where, is the medium density, , are Lamé constants, , , $ are Murnaghan constants, and ( is the bulk modulus. Each of equations (3), (4), (5) can be simplified in a linear form such as:
1 ( (6) Where, is the velocity in stressed state, is the velocity in initial state, and ( is the acoustoelastic coefficient.
4. System model and Simulation
Assuming the specimen subjected to elastic waves excitation generated by an ultrasonic transducer as a system. Let ) be the excitation voltage applied to the active element of the transducer. If * designates the received echo in initial state and * designates the received echo in stressed state.
Introducing the impulse responses, we may have the following forms of convolution:
*= +* ) (7)
*= +* ) (8)
Where both + and + are impulse responses of the system in initial and stressed state respectively. In frequency domain it follows that
,- H-/- (9) ,- H0-/- (10) Where H- and H0- are the fourier transform of + and + respectively. Eliminating /- by combination of equations (9) and (10) yields
,- H0-/H- S- 3-S- (11) Therefore, the system can be represented as the following:
In virtue of [9], the transfer functions H- and H0- are written of the form
H- )45678. )4678 (12)
H0- )45:78. )4:78 (13)
* + *
Figure 6: representative scheme of the system
Substituting equations (12) and (13) in equation (11) and rearranging we obtain
3- )4 5:745678. )4:74678 (14)
Where ; is the propagating path, <- and <- are the attenuation coefficients in initial and in stressed state, (- is the wave number in initial state related to the phase velocity by (-= 2=-/ , (- is the wave number in stressed state related to the phase velocity by (-= 2=-/. Introducing the wave velocity in equation (14) yields
3- )4 5:745678. )4>7%/?:4%/?68 (15) Note that the transfer function 3- is a product of two terms:
A real term related to a proportional action that is
3@AB@- 3- )4 5:745678 (16) And, a complex term related to the dispersion in the material that is
3CD@- )4>7%/?:4%/?68 (17)
If we designate by <- <- <- and by 1/ 1/;, then, on substituting in equations (16) and (17) yields
3- )4578 . )4>7E (18)
Then, we can write
,- )4578 . )4>7E S- (19) The term
)
4>7 S0- represents the fourier transform of * . Substituting in equation (19) yields ,-)
<-; . FGH * I (20) And equivalently in time domain* FG4%
)
<J-K;!L
* (21) We can conclude that the signal * results from the signal *shifted in time space by an amount that can be positive or negative and subjected to a proportional action)
457; that can be greater or less than one.Based on [10], the coefficient <- is computed by the following relation:
<-MN/O P QR10|, -|/|,-| (22) Where T is the total length path of propagation, , - and ,- the frequency spectra of * and
* respectively. For simulation, the reference signal * is digitalized and stored in a database.
For propagation perpendicular to stress, the waveforms of the received echoes corresponding to longitudinal waves are represented in (Fig. 6) where the solid line plot corresponds to the reference waveform, the dashed line plot corresponds to a load of 10 kN, and the dotted line plot corresponds to a load of 40 kN.
Then, for shear waves, the corresponding waveforms of received echoes are plotted in (Fig. 7) where the solid line plot corresponds to the reference waveform, the dashed line plot corresponds to a load of 5 kN, and the dotted line plot correspond to a load of 45 kN. Then, For propagation parallel to stress the waveforms of received echoes are plotted in (Fig.8) where the solid line plot corresponds to the reference waveform, the dashed line plot corresponds to a load of 25 kN, and the dotted line plot corresponds to a load of 50 kN.
5. Discussion
• Propagation perpendicular to stress
Note that in the case of propagation perpendicular to stress, whatever the type of ultrasonic waves longitudinal or shear, the waveforms obtained in the stressed state * are time delayed in comparison with that obtained without stress * . We notice an increase in time delay , while a decrease in velocity . All these variations are proportional to the applying forces. This fact can be proved if we rewrite the equation (1) on
Substituting by T/ , and by T/ yields
T / (23) In equation above, the sign of is also the sign of the difference . The fact that waveforms in stressed state are time delayed leads to U 0 in * , which implies U , hence the velocity decrease with the loading forces. The amplitudes of waveforms in stressed state are less attenuated than that of the waveform in initial state, because the proportional factor in this case is greater than one.
• Propagation parallel to stress
In contrast to the preceding case, the waveforms corresponding to longitudinal waves obtained in the stressed state * are time advanced in comparison with that obtained without stress * . We notice an increasing in both absolute value of time delay and velocity proportionally to the applying forces.
By the same manner to the above, this fact can be proved if we rewrite now the equation (2) on Substituting by T/ , and by T/ we obtain the same equation (23).
Figure 6: Simulated waveforms obtained for longitudinal waves propagating perpendicular to stress.
Figure 7: Simulated waveforms obtained for shear waves propagating perpendicular to stress.
Figure 8: Simulated waveforms obtained for longitudinal waves propagating parallel to stress.
The fact that waveforms in stressed state are time advanced leads to 0 in * , which implies , hence the velocity increases with the loading forces. The amplitudes of waveforms in stressed state are more attenuated than that of the waveform in initial state, because the proportional factor in this case is less than one.
5. Conclusion
In this work, a specimen of aluminum alloy is investigated under stress. The measurements are performed in two directions of wave propagation. In the case of propagation perpendicular to stress, whatever longitudinal or shear waves, the waveforms of received echoes in stressed state when compared with that of reference, are les attenuated, and present a time delay leading to a decrease in wave velocity. However, in the case of longitudinal waves propagating parallel to the applying forces, the waveforms of received echoes in stressed state are more attenuated relative the reference, and present a time delay leading to an increase in velocity.
In fact, these variations in amplitude and stress are proportional to the applying forces.
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