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M. Bentahar C.S.C, BP 64, Route de Dely Brahim, Chéraga. Alger. Tel &Fax: (02)361850
Email: m_bentahar@usa.net A. BadidiBouda, A. Benchaala
C.S.C, BP 64, Route de Dely Brahim, Chéraga. Alger. Tel &Fax: (02)361850
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Abstract:
By their efficiency and non destructive character, ultrasonic methods have taken an important place in the mechanical characterisation field. The knowledge of the elastic constants is of great importance for the ultrasonic stress state determination. The second order elastic constants are determined with the linear elastic theory while the third order constants require the use of the acoustoelastic theory. As this latter describes the influence of the applied stress on ultrasonic waves velocities, the used specimen must be stressed.
For this, an experimental set which consists of a waterfilled tank associated with a tensile machine has been conceived in order to characterise an isotropic material ( A37 Steel ). Two unfocused longitudinalwave transducers were used in a through mode at 5 MHz operating frequency. The receiving transducer diameter, is big enough to inform us clearly about the ultrasonic waves attenuation, and hence the influence of the material texture when the incidence angle is changed. Agreement of the experiment with the acoustoelasticity predictions has been found.
1. Introduction:
In the quality testing, the detection of flaws in materials is very important. Most of flaws which are harmful to component parts, are favoured by the residual stresses within the material. The ultrasonic study of these stresses cannot be done without a previous knowledge of the second and third order elastic constants.
The principle of ultrasonic methods is simple: from the different measurements of ultrasonic waves velocities, we can find the elastic constants with the help of the propagation equations.
The first ultrasonic methods used are the contact methods [1]. However, the multiple reflections in the coupling layer and the necessity of cutting many samples (in different directions), having been subjected to the same heat treatment, complicate the use of the aforementioned method.
These difficulties vanish once the immersion technique is used [4,14,13,15]. Indeed, the first advantage of this method is to eliminate the multiple reflections in the coupling layer. Furthermore, we are not obliged to cut more than one sample: the access to one side of the sample, even if it is thin, is sufficient to find the ultrasonic waves velocities in different planes, since a sweeping is guaranteed by the use of two goniometers. Also, the reproducibility of the amplitude and attenuation measurements in the liquidsolid interface is more convenient than the contact method, even if the transducer sample coupling is done with a great care [15].
For this purpose, an ultrasonic immersion system adapted to a tensile machine, has been conceived in order to characterise an isotropic material ( A33 steel). This device will allow us to show the anisotropy induced by the application of a stress and to evaluate the second and third order elastic constants.
2. Linear elasticity:
For an elastic unstressed material, the propagation velocities of ultrasonic waves within the material, depend only on the second order elastic constants and the material density. In the case of an isotopic material we have:
 (1)

Where r
is the material density, l
and m
are the Lamé constants.
When the material is stressed, the relations (1) cannot describe the change in the ultrasonic wave velocities due to the applied stress. New equations, which take into account these changes, are indispensable for every mechanical behaviour study of materials under stress.
3. Non linear elasticity:
Applied stresses influence on ultrasonic waves velocities in materials, is described by the acoustoelasticity theory. This latter, superimpose an ultrasonic perturbation i.e. variable in time and space, on a homogeneous elastic deformation.
In this case, Hook's law is given by:
 (2)

where C_{ijkl} and C_{ijklmn} are the second and third order elastic constants, e_{ij} is the strain tensor and s_{ij} is the stress tensor.
The propagation equation given in this case, is the same as the one given in the unstressed case, except that the tensor A_{ij }, which is called acoustoelasticity tensor, depends on the second and third order elastic constants, the applied stresses and the ultrasonic wave direction of propagation [13].
 (3)

When the material is isotropic, the A_{ij} tensor elements depend only on the considered direction of propagation, the material's state of deformation and the second and third order
Lamé constants l
, m
, n_{1}, n_{2}and n_{3}.
4. Induced anisotropy:
The application of an uniaxial stress on an initially isotropic material, induces a directionally anisotropy in the material (figure 1).
Fig 1: Planes of isotropy P^
and anisotropy P_{//
} 
Two orthogonal planes, where the first passes by the applied stress, while the second is perpendicular to it, originate and become an inevitable passage for every study of ultrasonic waves propagation under stress.
The relative variations of the ultrasonic waves velocities, in the presence and in the absence of stress, expressed in these two planes are given by the equations(4):
Where (V_{T},V_{TO}) and (V_{L},V_{LO}) are the transverse and longitudinal ultrasonic velocities with and without stress, n_{1}, n_{2}and n_{3}are the third order Lamé constants and _{3} is the direction of propagation projection along the applied stress axis.
5. Description of the experimental apparatus:
The experimental apparatus consists of a water filled tank associated with a tensile machine. We can vary the incidence angle without moving the sample, which is held by the tensile machine and immersed in the coupling liquid. The incidence is the result of a horizontal rotation (q
angle) and a vertical one (j
angle). Both are made with two goniometers mounted on the same vertical axis on which the probesholder is fixed. (Figure 2)
Fig 2: The experimental apparatus 
Equations (1), show that it is useless to apply any stress on the material in order to determine the second order elastic constants, because they are independent of the material mechanical state. However, equations (4) require the application of a stress for the third order Lamé constants determination. But in both cases, the determination of the ultrasonic waves velocities is indispensable.
6. Experimental techniques:
In order to evaluate the different elastic constants, we have chosen an isotropic material (A33steel), having a classical prismatic shape. Contrary to the destructive testing, we are not obliged to use standardised samples, since the characterisation is based on the knowledge of the time of flight, the sample thickness (in its different states), the incidence angles and the water temperature.
6.a. Time of flight measurement:
The time of flight is calculated with the help of the cross correlation method [16]. The use of this latter has been allowed by verifying that the reference signal, and the measurement signal have the same shape, but are temporally shifted. The reference signal is taken in the absence of the sample and stored for the rest of the measurements.
The obtained signals, which stemmed from the same transducers, are the result of the passage of the ultrasonic waves through the material in different incidence angles.( Figure 4)
Fig 4: Time of flight principle measurement 
6.b. Material thickness measurement:
Since the measurement zone is small, we cannot measure the sample thickness mechanically, so we have applied an ultrasonic method which requires measurements in normal incidence. The principle of this method is described in figure 5.[17]
Fig 5: Ultrasonic thickness measurement principle 
With the help of this method, we have made immersion thickness acoustical measurements, for different materials whose thickness have been mechanically measured. The obtained results show the efficiency of this method since the difference between the mechanical and the acoustical results does not exceed 0.05mm (figure 6).
Fig 6: Comparison between mechanical and acoustical thickness measurements 
6.c. Ultrasonic waves velocity in water and incidence angles:
In order to increase the measurements accuracy , we have used an interpolation method [18], which gives ultrasonic waves velocity in a pure water. The velocity is given by the following equation:
 (5)

Where the a_{i} are the coefficients of the interpolation polynomial, and T is the water temperature at the moment of measurement.
The incidence angles accuracy, depends on the goniometers in one hand, and on the normal incidence in the other hand. The normal incidence is obtained with the help of an interpolation polynomial which gives a good accuracy, and from the zero angle the other incidences are obtained with 0.01° accuracy, for each incidence.
The last thing to do is the determination of the ultrasonic waves velocities within the material. The problem is that these velocities depend on the refraction angle. This latter cannot be known without a previous knowledge of the ultrasonic velocities in the material. To overcome this difficulty, a relation linking the ultrasonic wave velocity (V), the incidence angle (i), the ultrasonic wave velocity in water (V_{W}), the time of flight of the ultrasonic wave in the material ( t) and the material thickness (e) has been made [13]. This latter is given by:
 (6)

7. Simulation of the longitudinal velocity variations in the P^ and P_{//} planes:
In this simulation we have varied the incidence angle in each plane and this for different applied stresses. The longitudinal velocity variations versus the refraction angle in the material in the P^
plane, show that this plane keeps its isotropic behaviour, and this for different stresses (figure 7). The same operation has been made in the P_{//} plane. In this plane we can readily notice the induced anisotropy due to the applied stress, which appears in the change of the lines slopes, as shown in figure 8.
In this simulation we have taken into account the texture effect, and this by considering the experimental velocities calculated in the materials free stress states, and this for every incidence angle.
Fig 7: Longitudinal velocity variations in the P^
plane 
Fig 8: Longitudinal velocity variations in the P_{//} plane

8. Experimental results :
For the experimental process two unfocused longitudinalwave transducers have been used in a through mode at 5 MHz operating frequency. In order to get a better results, the diameter of the emitter transducer has been chosen smaller than the receiver one. This will allow a better receiving of the ultrasonic wave refracted by the material.
The first step in the experimental procedure, is to determine the second order Lamé constants. This can be done in the material unstressed state for the above mentioned reason. Since the material is isotropic, we have changed the incidence angle in the two planes (P^
, P_{//}), in order to get a sufficient number of measurements for an accurate Lamé constants. The results have been in a good accordance with those given in the literature [19] :
l=(112 ± 1)GPa
 m=(81.2 ± 0.7)GPa


For the third order Lamé constants, the experiment is still going on and the results are partially obtained.
9. Conclusion:
In this work, we have presented a procedure for the determination of the second and third order elastic constants. We have discussed the different steps for the determination of each parameter. We have also given a simulation which gives an idea of the ultrasonic velocities variations, due to the stress and the texture. As the separation of these two parameters is not possible, we have to make a 3D plot: the velocity versus the refraction angle and the applied stress, for a better understanding of the ultrasonic wave velocity variations.
In the light of the promising results obtained for the second order elastic constants, we intend to carry on the work for the third order Lamé constants determination.
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