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Tensor Stresses Field Reconstruction in Product Volume by the Ultrasonic Computerized Tomography MethodsKoshovyy V., Krivin Ye., Nazarchuk Z., Romanyshyn I.,
Karpenko Physico-Mechanical Institute of the National Academy of Sciences of Ukraine.
In the given paper the results of development of the ultrasonic (US) computerized tomography (CT) integral methods for reconstruction of the stresses tensor field SD have been presented. Two directions of this task solution have been proposed. The first direction is connected with determination of the stresses tensor (ST) components SD by the USCT methods. On its basis the necessary stresses field characteristics may be determined. The second direction is connected with reconstruction of the stresses tensor invariants SD. At that case, the task of invariants SD reconstruction have been reduced to reconstruction of the material scalar physico-mechanical characteristics (PMC). Special attention at the UST methods development was being devoted a possibility of their practical application and technical realization.
The tensor stresses inhomogeneous distribution is being usually observed jointly with the material inhomogeneous distribution. Therefore, possibilities of tomographic technology for division of the material scalar and tensor inhomogeneities, will be considered in the paper also.
The integral method of stresses tensor field (STF) components reconstruction.
Acoustic diagnostics of the material deflected mode (DM) is being based on the acoustic elasticity (AE) effect. Acoustic methods of technical diagnostics of the material DM are being intensively developed, especially at last years. But, determination of the STF SD by AE method at material inhomogeneous DM is a problem, far from resolving, satisfactory for engineering practice[2,3]. The AE matrix theory is the most convenient for engineering calculations. Possible changes of the material stresses cause only very small changes of the material acoustic characteristics (AC). Therefore, at consideration of the US waves spreading process in the NDT tasks the restriction to approach of geometric acoustics and beam rectilinear spreading is admissible.
In the AE matrix theory the relative changes of the different types US waves spread time (slowness) dtik and velocities dnik are being considered for purposes of the acoustical tensometry, where i characterizes a spread direction of the US wave, k - direction of its polarization. Taking into account AE effect smallnes, the influence of stresses on the US waves information parameters dtik and dnik one can describe by linear dependence. Besides, the resulting influence of the complex DM on the elastic waves are being determined by summing of corrections from each component (normal and shear stress). Thus, relative changes of the US wave spread time are proportional to the ST components and are being discribed as:
where s ln - stresses tensor, aikln - AE coefficients (AEC), which are being expressed through the material elasticity characteristics or determined by experimental way at one axial DM.
At inclined (to axises of the coordinate system, in which ST s ln is set) material sounding for determination of dnik on the basis (1) it is necessary previously to determine the ST components in the coordinate system, connected with the US wave spread beam, on the basis of relation:
where R(r) - matrix of coordinates transformation at coordinate system turn up to comparison with the coordinates system, connected with the US wave spread beam; r characterizes a set of turn angles.
In the acoustical tensometry for ST components determination on the basis of relations (1) and (2) it is selected the most convenient sounding directions.
One of the effective CT methods is the sum image (SI) method. Initial data for the SI method are beam projections, which in the two-dimensional case are being presented as:
where - the true SD, - ort in direction of beam spreading.
The sum image are being determined by the relation
and is only the low frequency copy of the true SD. The SI and true SD are being connected by the resultant relation. The resultant kernel are being determinated by the data collection scheme.
Different data collection schemes are possible. Let's consider a situation, when data collection are being fulfilled in planes, perpendicular to axises of the chosen coordinate basis. Relative changes of the spreaad times of the US longitudinal dni and transverse waves with polarization in the data collection plane dnt are being used as projecting data. Connection between projecting data, measured in the plane, perpendicular to axis OZ, with the ST components are being obtained on the basis (1) - (3) and has the following view:
where - radius-vector of a point on the sounding plane. Dependences between projecting data and the ST components at data collection in the planes x=const and y=const have a similar view.
On the first stage of tomographic reconstruction the sum images (4) on the basis of projecting data of the kind (5), (6), measured in the corresponding planes, are being constructed by the way of inverse projecting. For anisotropic medium, when an inhomogeneity have the tensor character, a beam projection depends not only from the tensor components, but also from spread direction and polarization of the sounding US wave . That results in more complex relations between the SI and true SD than in the scalar situation. These relations depend from data collection plane orientation. Placing relations for the beam projections (5), (6) in (4), we shall obtain relation between the SI and true SD.
Let we have the parameter distribution in the plane z=const and take into account only the first component in (5). In this situation the constructed two-dimensional SI d T is being discribed by the following resultant:
In the scalar SD case the resultant kernel has the view . Thus, in the situation (7) the function are being modulated by the angle a dependence a zz cos2a + a xx sin2a .
For the SI set in parallel planes (z=const) the following relation is true:
The two-dimensional image, which is being conformed to projecting data (5), one can note as:
The similar relations may be obtained for the sum images of the slownesses relative changes in the other planes for the longitutinal and transverse US waves. As result we shall obtain 6 equations of type (9) concerning of 6 ST components. On the basis of this equations system we shall obtain in the spectral domain:
three dimensional Fourier transformation, A - matrix (6x6), which elements are functions of the spatial frequencies, determining only by the data collection scheme. In the spatial frequencies domain the relation (10) reflects the ST components SD in the following view:
Thus, the true ST components SD are being presented in view of resultants sum of the plane parallel set of the sum images. Functions, resultanting with these SI, depend only from the data collection procedure.
Division of the scalar and tensor inhomogeneities.
In practical tasks both the scalar inhomogeneities and the tensor inhomogeneities (stresses), influence on the US signals informative parameters. It being known, that this influence of the scalar inhomogeneity may be significantly more than influence of stresses. Mistaken interpretation of the experimental results may result in significant errors at stresses estimation. Influence of the material scalar inhomogeneities on determanation of the stresses tensor field one can significantly decrease by the tomographic methods. Division of scalar and tensor inhomogeneities influence in the process of tomographic reconstruction are being based on independence of the scalar inhomogeneity tomographic image from the data collection plane, as against of the tensor one.
Now we shall present the algorithm of stresses tensor field reconstruction on the scalar inhomogeneity background. Let us assume that data collection are being fulfilled in the planes, perpendicular to axises of chosen coordinate system, and relative changes of the longitudinal and two transverse US waves slownesses are being measured.
Relative change of the slowness of the US wave along beam may be approximately presented in view of two components. The first component is connected with material stresses, and the second one is caused by influence of scalar inhomogeneities. For example, in plane, perpendicular to Z axis, relative change of longitudinal US wave slowness one can write as:
where is being determined by expression (5), - relative changes of longitudinal US wave slowness, caused by scalar inhomogeneity influence. Results of measuring in other planes one can present by the similar way.
Determination of the ST components SD on the basis of projections (11) is being carried out by the following way. At first, the SI set are being constructed by inverse projecting:
Each sum image may be presented as:
Tomographic reconstruction of the first stresses invariant.
The material deflected mode characteristics don't depend from coordinate system choice. They may be determinated through invariants of stresses tensor (IST). In some situations the task of IST reconstruction one can lead to the task of reconstruction of the scalar parameters SD. That task requires of lesser volume of calcullations comparatively with reconstruction of the ST components SD.
The first IST SD reconstruction method are being based on formation of the invariant beam projection on the basis of measuring results of slownesses relative changes of the longitudinal and transverse US waves with mutually perpendicular polarization .
Invariant is a scalar parameter and the spatial distribution of this parameter may be reconstructed by the reconstruction method for the material scalar characteristics. A beam projection of the first IST may be obtained on the basis of the measured data by the following way:
The developed methods of determination of the stresses tensor field components are not complex and their practical realisation have been foreseen in the developed ultrasonic computerised tomograph for material characterisation UST-2000.
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