Non-destructive testing of multilayer medium by the method of velocity of elastic waves hodograph

The method of velocity of elastic waves hodograph, aimed at non-destructive testing of structurally heterogeneous composite materials and products based on them, as well as multilayer products and constructions, is considered. The theoretical basis for determining the propagation velocity of elastic waves in a multilayer medium by the hodograph method is given. Based on the studies, recommendations are given for determining the propagation velocity of elastic waves in each individual layer of a multilayer medium, which allows non-destructive testing of the physicomechanical characteristics of each layer of a multilayer medium. It is shown that in addition to simple multiple reflections in a homogeneous medium, in a multilayer medium with parallel interfaces consisting of two or more layers, complex types of multiple reflected waves and mixed waves (reflected-refracted and refracted-reflected) can arise. The main task of applying the low-frequency ultrasonic method is to determine the acoustic parameters of the propagation of elastic waves (velocities, amplitudes, spectra). The main methods for determining the elastic wave velocities are considered, based on the hodograph equation of the indicated reflected waves in a multilayer medium.

The first mention of this method for nondestructive testing of thick-walled large-sized constructions and structuresmade of composite materials is given in [16]. There are no other publications on this method in relation to non-destructive testing of these objects.
It should be noted that the propagation velocities of longitudinal and transverse elastic waves are the most important parameters for non-destructive testing of physicomechanical characteristics (elastic and strength) of medium materials [15].
Methodology. For a two-layer medium with a parallel interface, in which the propagation velocity of elastic waves in the first upper layer is much lower than the wave velocity in the second layer ( Fig. 1), the hodograph equation takes the following form: where t 1 -the propagation time of the elastic wave in the first layer of the medium vertically.
In this medium, in addition to reflected waves, refracted waves can also be observed, while the maximum refraction effect is observed only if the layer thicknesses are equal. Hodograph equation for refracted waves w 121 will be as follows: where  0 -ultimate angle of refraction, When constructing the hodograph of refracted waves at a distance X 1 from the origin (Fig.2), a kink of the straight line will be observed: W v is, the farther from the origin is the break point of fracture of the hodograph of refracted waves. In this case, the first part of the line before the hodograph fracture corresponds to the propagation of a direct longitudinal wave in the first layer, the second -in the second layer.
It can be seen from Fig.2 and formulas (1) and (2) that the hodographs of reflected and refracted waves differ significantly from each other. Neglecting the interface, it can be shown that the shape of the hodograph of the reflected waves will not qualitatively change (see Fig.1). So, the hodograph equation for a two-layer medium has the following form:  Fig.2. Scheme for constructing a hodograph of refracted waves in a two-layer medium Rec1, Rec2 -position of the refracted waves receivers; vw 1 , vw 2 -the speed of refracted waves in the first and second layer, respectively where δ = δ 1 = δ 2 ; The approximation error is determined by the formula where n = v 1 /v 2 ; t 0 -time at the emitter point. Usually, the value of the approximation error Δt is taken to be equal to the error of measuring the time of the instrument. Associating the approximation error with the ratio of the distance between the emitter and the receiver to the thickness of the product, we obtain Thus, formula (5) expresses the homogenization region of a two-layer medium into a homogeneous one, at which the hodograph of the reflected waves will not qualitatively differ.
For a multilayer isotropic medium with a parallel interface, the equations of the longitudinal linear hodograph of reflected waves are as follows: where  1 -the angle between the direction of wave propagation (beam) and the normal in the first upper layer. An analysis of expressions (6) and (7) shows that in order to obtain the equations of the linear hodograph of reflected waves in explicit form, it is necessary to expand expression (7) in a row, after which we obtain: It should be noted that with X/(n) < 0,5 the value of the third term of the equation can be neglected and limited to two terms of the expansion.
Of considerable interest is also the equation of the hodograph of refracted waves for a multilayer medium: In the process of propagation of elastic waves, especially longitudinal waves in a single-layer and multi-layer media, except for the once reflected, multiply reflected, refracted and mixed waves can also be observed. Consider the most characteristic features of the propagation of multiple waves, the main types of which are shown in Fig.3.
In the general case of a homogeneous medium with parallel media interfaces, the equation of the hodograph of multiple reflected waves (R 101 , R 10101 , ...., R (10)n-1 ) has the following form (Fig.3, а): where n -number of reflections.
In order to correctly distinguish between normal (once) reflected waves from multiple reflected waves, which also have a hodograph of a hyperbolic shape (8), it is necessary to know some relationships between these waves. So, the abscissas of the minimum of the normal hodograph Х 01 and multiple hodographs Х 0n are interconnected by the ratio Х 0n = nх 01, and the ratio of the propagation time at the point of the emitter for normal t 01 and multiple t 0n waves has the form t 0n = nt 0n . It should be noted that the main conditions for distinguishing normal waves from multiple reflected waves are that the hodograph of multiple waves located at the same times as the hodograph of normal reflections from the upper interfaces are more abrupt. An effective means of recognizing normal waves is also an analysis of the amplitudes of normal and multiple waves at known coefficients of reflections from the interfaces.
The ratio of the amplitudes of multiple A n and normal A 1 reflections for a medium with one intermediate interface can be written as follows: where k 1 и k 2 -reflection coefficients, respectively, from the upper reflecting media interface and from the intermediate interface; n -multiplicity indicator (number of reflections). The dependence (9) most clearly shows the ratio of amplitudes of multiple and normal waves if we compare the graphs for different reflection coefficients (Fig.4). So, with k x = 0.2 and k 2 = 1 triple reflection should have an amplitude 25 times lower than a single reflection, and at k x = 0.2 and k 2 = 0.5 already 100 times smaller.
Thus, to highlight normal waves, you can use the following methods: • tracking reflections by moving the receiver relative to the emitter; • assessment of the ratio of the times of arrival of waves and the shapes of the in-phase axes;  • selection of optimal X/ values when tracking reflected waves in a layered and homogeneous medium; • analysis of the ratio of amplitudes of normal and multiple reflections. In addition to simple multiple reflections in a homogeneous medium, in the case of a multilayer medium with parallel interfaces consisting of two or more layers, complex types of multiple reflected waves and mixed waves (reflected-refracted and refracted-reflected) can arise.
For a two-layer medium with wave R 102 (see Fig.3, b), using the previously considered principle, i.e., replacing the complex wave path with a simpler one and averaging the velocity values for each of the interfaces v av1 and v av2 , we obtain the following expressions for determining the distance between the emitter Em and the receiver Rec and the wave propagation time: where  -angle of wave reflection at the interface. Excluding , we obtain where av1 2 . Expression (10) is a hyperbole with a minimum at the origin. In this case, the time at the point of radiation for normal and multiple reflections from each medium interface is related by the dependence where (1) и (2) -medium interfaces. For a two-layer medium with wave R 212 , the equation of the hodograph of the reflected waves has the following form: time at the point of radiation for multiple and normal waves is determined by the dependence Based on the principle of reciprocity, the hodograph of the wave R m will coincide with the hodograph of the wave R 102 .
The hodograph equation for a wave has the following form: the relationship between the times at the point of emission In addition to the considered types of multiple waves, in a controlled medium can be observed waves of a mixed type -reflected-refracted (R 1212 > R 3231323 and others). For this type of wave, we consider a medium in which there is an interface that is simultaneously reflective and refractive (see Fig.3, c). For such a medium, the equation of the hodograph of the reflected-refracted wave will have the following form: where  =  1 +  2 +  3 ; N -number of media interfaces. Analysis of the hodograph equation of the reflected-refracted wave shows that on the time axis (at the point of radiation of Em) the hodograph of the refracted and reflected wave will cut off the following value: ; v 1 , v 2 -velocity in the upper and underlying layer.
Thus, the smaller the difference between v 1 и v 2 the greater the difference between t refr and t refl is. Under real conditions, there may be a case where the refracting and reflecting boundaries do not coincide. This may cause wave interference. For this case, the condition for the intersection of the hodographs of the reflected and refracted waves can be written in the following form: where  refr и  refl -thickness of the layer of the medium in which the wave is refracted or reflected.
Having made some conversions we obtain where av.refl av.refr refl av av.refl . Solving equation (11) with respect to а, we obtain: . sin cos cos sin sin So, for the intersection of hodographs, reflected and refracted waves, to take place, the condition must be met: When analyzing the properties of direct and reflected waves, it was noted that for a homogeneous medium, the hodograph of a direct wave is parallel to the asymptotes of the hodograph of the reflected wave. However, in real conditions, the intersection of the hodograph of the direct and reflected waves can be observed. In this case, the equation for determining the distance X x from the emitter, at which the intersection will occur, will have the form: The main task of applying the low-frequency ultrasonic method is to determine the acoustic parameters of the propagation of elastic waves (velocities, amplitudes, spectra). The solution to the problem will clarify the determination of the coordinates of acoustic emission sources (AE) or the correct determination of the resource based on the registration of AE signals [1][2][3][4]