1. The probe sound fields of this model are derived from design concepts for focussing and planar probes existing at the BAM since many years [5,6,7,8 ]. Different probe characteristics can easely be investigated, even with curvature adaptation of the probe wedge.
At the modeling software the probe soundfield is described by an analytical approximation for the farfield and a special treatment of the nearfield area (see Fig. 3 with the simplified soundfield pattern of an angle beam probe). This approximation contains the wave length dependent geometrical directivity pattern of the coupling surface excited by the probe and a point directivity pattern. The nearfield is taken into account by a constant beam diameter in its area. Included is also the wave length dependent distance law on the beam axis with some assumptions concerning the shape and orientation of the wave front in each point of the space reached by this soundfield.
For both wave modes ( longitudinal and shear waves ) we are using the asymptotic approximations for the point directivity pattern according to e.g. Miller and Pursey [3]. Creeping longitudinal waves can be included with a special point directivity pattern for the longitudinal waves. Rayleigh waves are not taken into account.
Focussing and the influence of curved coupling surfaces are introduced as described in several articles of the authors and others [4,5,6,7,8].
The receiving characteristic of a probe is derived from the transmitted soundfield assuming a complete reciprocity between reception and transmission.
| Figure 2 shows the basic ingredients of our model:
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The medium of wave propagation is isotrop and homogenious. Special cases of anisotropy can be taken into account with the introduction of a special ray tracing approach based on the summation of the group time of flight contributions of all the differently oriented grains between the probe and a defect. For the case of an isotropic material the sound velocities for longitudinal and shear waves are considered to be constant and independent from the wavelength and the orientation. The attenuation is considered to be frequency dependent.
- The spectrum and the shape of the transmitter pulse is derived with the help of a quadrupol model of the probe using the KLM approach [8,9] and taking into account the special data of the probe construction (e.g. thickness of the transducer element, its size, material, delay path material, attenuation material, different layers of glueing and lambda/4 -adaptations etc.) - see also Fig. 4.
- The geometry is described with three dimensions. For the numerical integration, the reflector is divided into different surface elements of which the shape and size can be adapted to the defect geometry. Side-drilled holes, flat bottom-holes, elliptical or rectangular shaped cracklike reflectors with arbitrary orientation and position can be considered as well as corner effect reflectors. The three dimensional geometry of our model is limiting its application to blocks with parallel or inclined surfaces and to curved blocks with concentric surfaces.
- The reflection of an ultrasonic beam on an opposite or other surface (may it be planar or curved) is considered as a reflection at an exact mirror for an incoming beam, that means that an integration across surface elements is replaced by the assumption of a reflected ray. In case of curved surfaces or mode conversion, ray tracing with Fermat´s principle is applied. Angle dependent reflection and mode conversion factors for unlimited planar waves are taken into account for the reflection at a bottom surface.
- Special attention is payed to the interaction between the ultrasonic waves and the defect:
For each surface element of the defect this interaction is superimposed by partial waves, each one propagating along special sound pathes or rays and being affected by reflection or mode conversion depending on the specific sound path considered. Each wave part is weighted with a geometric directivity pattern of the surface element depending on the wavelength and on the size and shape of the element and in addition with one out of four different point source directivity patterns. Those patterns - differentiated according to the specific case of incoming and observed wave modes and angles : trans -trans, long - long, trans - long and long - trans - are formulated according to the approximation of the physical elastodynamic [10, 11, 13] and are depending on the considered wave mode and the impinging and observation angle.
The figures 5 and 6 introduce an experimental verification of the different point source directivity patterns[16]. At the test block of figure 5, an angle beam probe for long. or shear waves excites a quasi real crack. The electrodynamic pick up receives at an observation angle the waves reflected, mode converted or diffracted at the brittle fracture crack. The results (e.g. Figure 7) showed an excellent agreement with the theory of the physical elastodynamic [16]. Each sound path between the transmitting and the receiving function of the ultrasonic probes (corresponding to the rays of a ray tracing approach) is generating a partial Echo E(½), which contributes to the total echo amplitude. The different factors F(½) of those partial Echos are depending on the probe soundfield, the geometry of the object and the position of the defect. Probe soundfields and the reflection factors at the different surfaces of the object are described with approximations. They are in general phase dependent , that means they are complex values and are depending on the wavelength respective frequency. One calculates the sum of the partial echos for a sufficiently large number of frequencies, generating thus a discret system function of the ultrasonic inspection arrangement under consideration. This system function is multiplied with the complex spectrum of the transmitter pulse generated by the quadrupol model, which gives a frequency weighted system function The A-scan is then obtained by an inverse Fourier transformation of this weighted function The number of frequencies to be calculated is reduced with the help of an apriori information about the probable sound path, at which the maximum echo pulse will occur. That means that the exact position of the echo within the time domain is based on the time of flight of the ray corresponding to the maximum echo pulse. Based on the A-scans different kinds of gray scale or colour coded B- and TD (Time Displacement) -scans can be produced. All the results are presented in terms of A-scans, B-scans or TD-scans and are directly comparable to results gained with mechanized scanners.
We have preferred to calculate the A-s cans based on frequency domain samples and an inverse Fourier transformation, because this allows a simple introduction of individual pulse spectra from different probe designs.
The theoretical results can very easily be interpreted, because the different partial echos contributing to the indication of interest can be switched on or of.
Fig 4a:
Quadrupol-Chain of an ultrasonic probe
Fig 4b: Spectrum and time function of the probe pulse
Fig 5:
Test block for the Point Source Directivity
Fig 6:
The Point Source Directivity at the reflector
Fig 7:
Observation Angle |
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