Acoustic emission techniques for the analysis of welded structures

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5 Acoustic emission techniques for the analysis of welded structures 1 1

The essence of acoustic emission General

The quality of a structural material is determined by its failure-free operation. From this point of view failure means either fracture or limited damage, loss of tightness or accumulation of excess strain. The most important characteristics of a material quality are those of deformation and fracture resistance. They include, primarily, such characteristics as material yield point, ultimate strength, true rupture resistance and elongation after fracture. On the other hand, the dependence of material behaviour under the service conditions on the above mechanical characteristics can be established by calculation only in the first and rather rough approximation. This is attributable to the following reasons. The mechanical characteristics are of a conditional nature and are determined under the simplified conditions, not corresponding to those of service in terms of geometry of the tested objects, laws of loading, kinds of stress–strain state, the number and laws of variation of various external factors and time factor. Natural variations of the composition and structure as well as their change in service lead to a situation when extrapolation of the results of mechanical characteristics measurement at selective destructive testing of specimens to the material of the working parts, assemblies and units, cannot be performed with a high accuracy. And, finally, the random nature of external impacts makes it impossible to accurately describe the behaviour of a material or structure in service even with the precisely known mechanical characteristics of a material. Possible ways to overcome the above difficulties are: ♦ search for material characteristics more closely related to those physico-chemical processes that predetermine the structural material failure in terms of its resistance to deformation and fracture; ♦ development of non-destructive methods and means of measure­m ent and analysis of these characteristics; ♦ development of the methods and means of continuous monitoring of the change in the structure material condition in service and appearance of indications of a critical situation.

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Solving these problems will permit improvement of the validity of a priori estimates of the material (structure) behaviour in service; establishment of scientifically-based norms of acceptance of industrial products; conducting, if required, a total 100% inspection of the material mechanical properties directly on the working structures; significant reduction or complete elimination of the cases of unforeseen failures, establishment of optimal periods of preventive maintenance. The above problems can be solved, using elastic acoustic waves, arising in material deformation and caused by marked changes in its structure. The process of acoustic wave generation is called acoustic emission (AE), and a variable stochastic physical quantity, carrying information about acoustic emission is called the AE signal. The parameters of AE signals, associated with local changes in the material structure, correlate with parameters of the kinetics of defect propagation and material fracture. One of the sources of deformation signals are the processes of dislocation motion, their AE signals correlating with the discrete mechanism of plastic deformation and jumps on the strengthening diagram. Deformation signals allow quite a reliable detection of phase transformations in materials. Recording deformation signals is one of the methods to study the processes of crack initiation and propagation. The results of physical studies of the acoustic waves, associated with material deformation, demonstrate that this phenomenon allows the creation of efficient methods of nondestructive diagnostic testing of the condition of materials for evaluation of the criticality of the arisen situation and closeness of the moment of failure (fracture). Such prerequisites have already become specifically implemented in the equipment and methods for diagnostics of the materials of structures. Considerable efforts are made currently to apply acoustic emission for nondestructive diagnostics of structural materials for mechanical engineering. The developed methods of diagnostic analysis and equipment, required for this purpose, open up broad capabilities for nondestructive diagnostics of pressure vessels, welded and adhesion bonded joints, detection of fatigue cracks, and changes in the material structure. Obtained results indicate that the acoustic emission method allows measurement of the level of stresses (strains) in the structure material, detection of propagating defects and determination of their coordinates, evaluation of the degree of defect criticality, as well as solving other problems in evaluation of the condition of structures and constructions. These methods, however, have not so far found wide application in practice. This is attributable to their relative novelty, presence of a number of yet unresolved theoretical and practical issues, as well as absence of systematised information, accessible for a broad range of experts on the essence and potential of the methods, their advantages and disadvantages, and rational applications.

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371

Physical essence of acoustic emission

Various physical impacts (mechanical, thermal, etc.) may lead to local changes in the material structure, being exactly the source of acoustic emission. Let us consider some possible mechanisms of formation of this phenomenon. One of the forms of crystal deformation is twinning, which consists in rotation of lattice nodes of one part of the crystal into a position symmetrical to its other part. Lattice transition into a new condition proceeds jumplike with a high (subsonic) velocity, inducing elastic deformation waves. The cause for elastic waves development is also jumplike displacement of dislocations along the slip lines. Dislocation motion in such cases proceeds in the direction of the grain boundaries. A rather large number of dislocations may concentrate at the grain boundaries, forming microcracks. Experiments and calculations show that displacement and coalescence of 300 dislocations are enough for initiation of a crack, capable of propagation. Increase of forces, deforming the crystal, and availability of stress concentrations in the mouth of such a microcrack lead to its discrete growth, accompanied by wave generation. Mechanical or thermal impacts on a material in the solid state may induce phase transformations in it. Transformations of martensitic type are shear collective displacements of atoms and, as a rule, they are accompanied by a change of shape. During this process an impulse action of the transformed structure on the environment is observed, causing mechanical waves in a material. Plastic deformation of material also proceeds in a jumplike manner, this being usually observed in the diagrams, recorded by instrumentation at specimen failure. Investigation of this phenomenon indicates that for some materials, the avalanche of jumplike changes leads to a short-time increment of absolute defor­mation in a short period of time. In terms of dislocation concepts, the mechanism of this phenomenon has the following explanation. At dislocation motion during plastic deformation of the material local dwelling of dislocations at various obstacles occurs. Spontaneous or induced overcoming of such obstacles by dislocations leads to fast local sliding, the sum of which usually yields the observed deformation jump. Thus, a dynamic redistribution of strains and stresses takes place, inducing mechanical waves in the material. The above mechanisms, apparently, do not cover all the causes for development of stress waves in solids at deformation. They, however, have some common features, allowing the notion of acoustic emission (AE) to be defined as follows: acoustic emission is the process of emission of disturbance waves, propagating through the material, caused by local transformation of the structure of materials under the impact of internal stresses, resulting in a change in the crystalline lattice or micro- and macrodefect motion. Having reached the surface of the solid, these

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waves induce displacement of surface points, which may be recorded by appropriate instrumentation. Maximal amplitudes and spectra of the frequencies of the above displacements should depend on pulses causing the emission. So, the amplitudes of displacements, induced by microcrack propagation, should greatly exceed those for the case of individual dislocation motion. Therefore, despite the complex form of each individual implementation of the considered random function in time, individual high-amplitude pulses should be present in case of a wide enough range of material deformations against the background of a relatively weak change of the function. Eventually, the acoustic emission signals can be divided into two main types: ♦ continuous emission, namely oscillations of relatively small amplitudes and with a wide frequency spectrum, with its upper limit reaching 30 MHz, characterise displacements of large groups of dislocations and point to formation of zones with accumulating microdefects. ♦ discrete emission, namely a sequence of short pulses of a complex shape with a steep front and much higher amplitudes. Most of the energy is evolved in the low-frequency part of the spectrum. This type of emission is associated with microdefects developing into micro- and macrocracks, their coalescence and appearance of the main crack. Both types of emission exist either with a time shift or simul­taneously. Currently available equipment receives and processes both types of signals separately, this allowing isolating from the material deformation process the moments, related to the formation of zones, hazardous in the future, and moments, related to fracture processes development in these zones. Photo 1 gives the general oscillogram of AE, arising in stretching of a specimen of 09G2S steel and corresponding to a certain moment of the material structure failure. The oscillogram is typical for the materials experiencing deformation. One can see from the diagram that both AE amplitude and frequency are variable at each moment of time and reflect the complex processes of material deformation and fracture.

Photo 5.1

Typical AE signals.

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373

Physical and mathematical models

The above leads to the main conclusions that practically all the processes, proceeding in materials at fracture, are of a discrete nature, both at microlevel (dislocation displacement) and at macrolevel, when sources of new dislocations, determining the plastic flow in the mouth of the crack and subsequent fracture, form in the mouth of a propagating crack. In this case at all stages of fracture the mathematical model of the process may be based on impulse point sources of the media excitation, and the distribution, intensity and sum of these source may describe the acoustic emission. Investigation of AE at deformation of many materials confirms the validity of the presented model. Figure 5.1 is a schematic of a microcrack formation at coalescence of dislocations on the grain boundary. Figure 5.1a shows dislocations that start moving towards the grain boundary at application of certain forces. The con­d ition of dislocation motion may be written as: σth – τin = nτd.

(5.1)

where σth is the theoretical ultimate strength of the material; τin are the initial stresses in dislocations, caused by structural defects; τ d are the stresses, required for one dislocation displacement along the slip line; n is the number of dislocations in the crystal. The presented depen­d ence is the initial one in quantum fracture mechanics. It determines the conditions for the start of dislocations motion along a slip line and conditions of micro- and macrocrack development,

5.1 Schematic of a microcrack formation at coalescence of dislocations.

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correlating the parameters, characterising the processes of material fracture, with the stress intensity factor. This comparatively simple expression (5.1) is rather complicated for use in calculations, because of indeterminacy of its values for actual materials. Experts on quantum fracture mechanics are currently working to solve this problem, although, similar to any design procedure for a complex object, it will be a quite rough approximation to reality. In case of problems of diagnostics, based on acoustic emission, this drawback is absent, as we are interested not in the quantitative values of the quantities, included into expression (5.1), but in the actual conditions of initiation and development of material defects and their reaching critical values. The latter, due to a discrete nature of their development, are manifested as deformation waves, irrespective of the actual magnitudes, reached by the above quantities. Thus, the solution of the general problem of quantum fracture mechanics with the above values unknown has a meaning for us at least in its second part, namely studying the concurrent with fracture phenomena of generation of elastic waves by the defect sources, thus allowing information to be obtained on the criticality of the fracture processes, proceeding in the material. Considering these emissions at the dislocation level, allows, as was said above, const­r uction of a theoretical description of the processes, involved in material fracture, based on the theory of isolated point sources of forced emission. The fracture processes, accompanied by a summary non-simultaneous action of local sources, are described, using the principle of super­p osition in time and space. The complexity of this approach in the general case is determined only by the complexity of solving the dynamic problem of the theory of elasticity in three dimensions. This disadvantage, however, can be eliminated, using some original techniques in calculations. Such an approach, especially for thin plates, may be based on not solving them ‘point-blank’ by the traditional methods, when direct integral trans­formations of wave equations of mechanics allow in a rather simple manner deriving algebraic equations to determine the decision functions f and ψ in their transformed form. On the other hand, as we know inverse transformations for thin plates involve such huge difficulties in solution that the number of problems, solved by this method so far, is counted in units in the simplest cases, notwithstanding the fact that the authors have created quite a large number of various techniques to simplify the solution of these problems. In our approaches, the solution of dynamic problems of the theory of elasticity was simplified by the fact that the boundary conditions of the problems were satisfied not at the moment of inverse going out of the transformations, but prior to that, at the stage of setting up the equations, this allowing reduction of the number of algebraic equations, determined by the boundary conditions of the problem, from four to three. The latter, fourth boundary condition was determined on the basis of a spectral equation, correlating the relationship of wave number α and circular frequency ω,

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which was derived for the case of one of the stress state conditions at the body boundary being equal to zero, and which determines the spectrum of frequencies and wave lengths which a plate of certain dimensions can ‘transmit’. Another technique, allowing an essential simplification of the problem, was the supposition that the relationship of α and ω has a clearly defined physical meaning and is described by a spatial equation in an extremely narrow range, defined mathematically by delta functions. The latter permitted avoiding the cumbersome processes of triple integration in defining the displacement functions and reducing the process of going out of transformations to a single integration. In view of the above, let us consider the application of the proce­d ure to some solutions, basic for the acoustic emission theory.

2

Theoretical issues of acoustic emission

1 General Experimental studies of deformation of welded joints and struc­tures indicate that the defects present in them propagate in a discrete manner. Energy evolves in pulses, both at defect initiation and during their propagation. It may be assumed that the spectra of the frequencies of signals, generated in this case, their amplitudes, increase and decrease carry certain information on the nature and criticality of the processes of fracture, proceeding in materials. Propagating waves, for instance, in a plate, are accompanied by a periodic change of its thickness, bending or the one and the other simultaneously. Displacements w of the plate surface may be recorded by acoustic transducers (Fig. 5.2). Thus, it appears to be possible, by measuring the parameters characterising the acoustic wave emission, to determine the coordinates of the site of this emission and correlate the emission parameters with these phenomena, procee­d ing at plastic deformation. On the other hand, the transducers, receiving the acoustic signals, are located at different distances from the sites of defect initiation, and the acoustic emission signal, arising at defect initiation, propagates through the plate, which transmits only quite definite frequencies. Thus, from the

5.2

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entire spectrum of frequencies, arising at signal formation, only those will be transmitted to the transducer which correspond to the plate in shape and size. Eventually, the oscillograph screen and the computer memory will not reflect the actual phenomena occurring at defect initiation. In this connection, it is highly important to study the characteristics of acoustic signals, propagating from a defect, which has initiated and emitted the energy, to the point of its measurement, namely the receiver of acoustic emission signals. Significant advantages of evaluation of the material stress–strain state by the method of acoustic wave analysis should be noted, compared to the traditional methods, based on the use of regular strain gauges, measuring the deformation on the material surface only under the gauge. A tremendous number of strain gauges (up to 300 000 pcs. in some cases) would have to be mounted on some large-sized welded structures to obtain the information, sufficient for the analysis to be performed. In practice, the number of gauges is reduced, resulting in the loss of a significant part of the information. Quite different prospects are opened up by the use of acoustic transducers where four transducers allow controlling up to 10 m 2 of the surface. This markedly reduces the total number of the transducers on the object and provides quite reliable information on the strain state of an object (in the considered case the number of transducers is reduced to 80). Such a number of transducers allows effective use of the measuring instruments together with the computing systems and operation in real time. This advantage of the acoustic emission method provides a motive for the development of special equipment and its application as one of the information channels in the system of diagnostics of welded structure reliability. In addition to major advantages the acoustic method of control also has certain above-mentioned drawbacks, requiring a thorough study of the acoustic emission process proper to develop reliable equipment, operating on its basis. This, primarily, is the dependence of the shape and size of the acoustic emission signal on the type of structure, its stress–strain state, type of transducer, method of their fastening and some other factors, related to the capabilities of the measurement–computational system. When developing the equipment and preparing the procedure for operating it, each of the above issues is to be addressed separately, and measures have to be envisaged, eliminating or neutralising the disadvantages of the method. Let us consider the influence of one of the above factors, namely the stress-strain state of the welded structure on propagation of an acoustic flexural wave in a sheet element. Calculation and measurement of residual stresses due to welding show that they can be significant. So, in welding of tanks of an aluminium–magnesium alloy these stresses reach the material proof stress and are extremely non-uniformly distributed. When the acoustic system is set up, shock-type simulators of the acoustic wave are usually used on an object of testing. In addition to Lamb and Rayleigh waves, flexural waves arealso generated in the object in this case. Recording

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such waves is quite possible, their parameters being highly dependent on the stresses in the sheet. Let us consider these phenomena in greater detail.

2

Flexural waves at point impact on the plate surface

1. Derivation of the basic equation and design formula. Let a point source of acoustic signals arise at any point of a welded sheet structure. The acoustic wave, which propagates from this point, changes through dispersion. Let us cut out of the structure a large-diameter ring with a centre at the point where the source of acoustic waves emerges. We will perform analysis of the influence of the stress–strain state on the elastic wave propagation in the case of such a large-diameter circular plate with an impact applied to its centre. In order to simplify analysis and derive expressions for the plate sagging in the analytical form, let us represent the stresses in the plate in the form of uniformly distributed compression or tension. Such a simplification will not distort the actual evaluation of stress influence on plate bending, and will yield quantitative results for the considered specific case. Let us apply the equation (3.89) to determine the sagging in bending of a circular plate under distributed load, assuming that there is no initial sagging of the plate, and initial deformations are uniformly distributed across the plate thickness. With these assumptions, the equation (3.89) can be written as follows: r 1 ∂ 3 w 1 ∂ 2 w 1 ∂w σ r δ ∂w q∗ rdr , + − 2 − = 3 2 ∫ r r D r Dr ∂ ∂ r ∂r ∂r 0

(5.2)



where w is the plate sagging in the direction of the axis z, m; r is the radius along the plate surface, m; t is the current time, s; δ is the plate thickness, m; σ are the total stresses, acting in a plate along radius r, Pa; D = Eδ 3/ 12(1 – ν2) is the cylindrical rigidity of the plate, N ⋅ m; v is the Poisson’s ratio; q* is the distributed load, Pa. Let us now assume that distributed load q* consists of two parts, i.e. is of the following form: q* = q – ρσ(∂ 2w / ∂t 2),

(5.3)

where ρ is the plate material density, kg/m 3. The second term in the righthand part of (5.3) is a dynamic component of load, caused by the forces of inertia of the plate particles, which quickly change their position in the direction of the axis z. Substituting (5.3) into (5.2), we have: r r ∂ 3 w 1 ∂ 2 w 1 ∂w σ r δ ∂w ρδ ∂2 w 1 + − 2 − + r 2 dr = rqdr. 3 2 ∫ ∫ r ∂r D ∂r Dr 0 ∂t Dr 0 ∂r r ∂r



(5.4)

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Integration of such an equation is difficult, because of the presence of an integral of ∂ 2w/∂t 2 function. In order to eliminate the integral in the left-hand part of (5.4), let us rewrite the equation as follows: r σ δ ∂ w ρδ r ∂ 2 w ∂ 2 1 ∇ w− r + r 2 dr = ∫ rq dr , ∫ ∂r D ∂ r Dr 0 ∂ t Dr 0

where 1 ∂  ∂  ∇2 =  r  . r ∂r  ∂r  Then, let us multiply this expression by r and differentiate it with respect to r on the assumption that σ r = const = σ: ∂  ∂ 2  σδ ∂  ∂w  ρδ ∂ 2 w qr  r ∇ w − r  + r 2 =. ∂r  ∂r D  D ∂r  ∂r  D ∂t Dividing this expression by r, we finally obtain: ∇ 2∇ 2 w −

σδ 2 ρδ ∂ 2 w q ∇ w+ =. D D ∂t 2 D

Substituting load q, applied in the centre of the plate in the form of a product of δ +-functions by radius and time, we obtain the basic calculation equation ∇ 2∇ 2 w −

2 Q δ+ (r ) σδ 2 4 ∂ w ∇ w + β= δ + (t ), 2 D 2 πD r ∂t

where Q is the impulse of force at impact on the plate surface, N·s; β 4 = ρδ / D, with the following initial and boundary conditions w = 0 at w = 0 at ∂w / ∂r = 0 at

r = 0; r = ∞; r = ∞ and

r = 0;

Performing direct Hankel transformation with respect to r and Laplace transformation with respect to t, we get: (α 4 + α 2 σδ / D + β4 p 2= ) w Q / 2πD. Then

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Q 1 . 2πD α 4 + α 2 σδ + β4 p 2 D Having performed inverse Laplace transformation, we have: w=

w=

w=

Q 2πDβ2

Q 2πDβ2

 αt sin  2 β α

σδ   D  for σδ 2 α + D α2 +

 αt sh  2  β α

.

α2 +

σδ   D 

σδ α + D

a2 + sd / D > 0;

for a2 + sd / D < 0.

2

Performing inverse Hankel transformation, we have:

w

Q 2πDβ2

 α 2t sin  2 ∞ β

∫ 0

σδ   α2 D  J 0 (αr ) d α σδ α 1+ 2 α D 1+

(5.5a)

for α2 + σδ / D > 0;

w

Q 2πDβ2

 α 2t sh  2 ∞  β

∫ 0

α

1+

σδ α2 D

σδ 1+ 2 α D

  

J 0 (αr ) d α

(5.5b)

for α2 + σδ / D < 0 . In the case, when σ = 0 (no stresses in the plate), the value w may be derived in a closed species: w

∞ J 0 (αr )  α 2  Q sin  2 t  d α. 2πβ2 D ∫0 α β 

(5.6)

The tables do not include such an integral. Let us take it as follows. First, we differentiate expression (5.6) with respect to t. Then, after differentiation and integration

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∞  α2   r 2β2  ∂w Q Q J r t d = α ( α ) cos = α sin    . 0 2 ∂t 2πDβ2 ∫0 4πDβ2 t β   4t 

Integrating over t, we come back to the value w: w =

 r 2β2  dt  r 2β2  Q Q = sin  Si   , 2 ∫ 2 4πDβ 4πDβ  4t  t  4t 

(5.7)

where Si (r2 β2 / 4t) is the integral sine, tending to π/2 at t → 0. Figure 5.3 shows the graphs of the dependence of value w on time t. 2. Analysis of the calculation results. Let us determine the velocity of flexural wave propagation in the direction of coordinate r. With this purpose let us consider the maximum of function (5.7). Calculating the derivative of (5.7)  r 2β2  Q ∂w = − sin   ∂r 2πDβ2 r  4t  and taking ∂w/∂t = 0, we get: sin whence

rm2β2 = 0, 4t

rm2β2 = πn ( n = 1, k ), 4t Q

Displacement w ⋅ 10-2, сm

0.5

100 000 µs

0 10 000 µs

-0.5

1 000 µs

-1.0

100 µs

-1.5

t =10 µs

-2.0 -100

-80

-60

-40

-20

0

20

40

60

80

Distance from the emission source r , сm

5.3

Flexural wave propagation in a steel plate 1 cm thick.

100

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where rm is the value of the coordinate of the point, where the flexural wave amplitude attains an extreme value. Whence 2 πnt . β

= rm

In the case of an isolated impact, replacing β by its value yields: π2 t 2 Dn 2 4 = rm 2= 2 ρδ

4

π2 t 2 E δ 2 n 2 , 12 (1 − ν 2 )ρ

or, if we represent C4 =

E , 12(1 − ν 2 )ρ

then rm = 2 πtC4 δn . Therefore, average velocity υrm of the flexural wave crest propagation will be υrm=

∂rm = ∂t

πC4 δn . t

As we can see, the velocity of the flexural wave crest propagation is not constant and depends on the physical constant C 4, plate thickness δ and time t. In expression (5.5a) coefficient at t can be singled out under the sine sign: ω = (α 2 / β2 ) 1 + σδ / α 2 D , having the dimension of 1/s. Quantity ω is nothing but circular frequency. Going over to phase frequency f, Hz, we obtain: = f

α2 2πβ2

1+

σδ . α2 D

(5.8)

Let us designate by f 0 the expression for the frequency of signals, propagating through the plate in the absence of initial stresses σ: f0 = α2 / 2πβ2. (5.9) Then the influence of the stress state on the nature and propagation of the elastic flexural waves in a plate may be expressed by relative value e (coefficient of influence):

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= e

f − f0 ⋅100 % f

(5.10) The graph in Fig. 5.4 shows the curves of the σ / E dependence on αδ and e . Using this graph for any thickness of plate δ pl and initial stresses σ, it is possible to determine limit wave number α pl in the acoustic signal spectrum, above which all the signals with error e can be considered as independent of σ. For instance, α pl = 0.56 сm –1 for σ / E = 0.001 and e = 0.01 at δ pl = 0.4 cm. Formulas (5.8)–(5.10) allow determination of the frequencies of the signal, propagating through a plate, at which the additions to w, dependent on initial stresses, are significant and should be taken into account. Then having performed integration in formulas (5.5 a) and (5.5 b) not from 0 to ∞, but from α pl to ∞, we obtain a result for w, which is independent of the presence of initial stresses in a plate. This theoretical conclusion has great practical importance, for instance, it allows selection of the lower limit of the working range of frequencies of acoustic transducers so that the initial stress state of the plate does not influence values w and ∂w/∂t. In other words, mounting an acoustic transducer with frequency band from f and higher we will not feel the influence of the initial (residual) stresses and stresses, induced by the external load, on the acoustic signal parameters, and will avoid errors in determination of the coordinates of the sites of intensive plastic deformations, as well as the characteristics of the signal proper. Let us come back to consideration of formulas (5.5a) and (5.5b), representing them approximately as a sum  α i2 t σδ  sin  2 1+ 2  ∞ β α Q i D   w≈ J 0 (α i r )∆α ∑ 2 2πβ D αi = 0 σδ αi 1 + 2 αi D

(5.11a)

5.4. Graph for determination of limit wave number α pl depending on coefficient of influence e and stressed state σ.

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2 for 1 + σδ / α i D > 0;

 α 2t σδ sh  i2 1+ 2 ∞ αi D  β Q w≈ ∑ 2 2πβ D αi = 0 σδ 1+ 2 αi αi D

   J 0 (α i r )∆α

(5.11b)

for 1 + σδ / α i2 D < 0. Formulas (5.11a) and (5.11b) show that the values w at each moment of time for different distances r from the centre of application of force Q are made up by an infinite number of elementary ∆w, each having its own value α i and, therefore, also frequency f i (5.8). Then

∆w ≈

Q 2πβ2 D

 α 2t sin  i2 β αi

σδ   α i2 D  J 0 (α i r )∆α σδ 1+ 2 αi D 1+

for 1 + σδ / α i2 D > 0;

∆w ≈

Q 2πβ2 D

 α 2t sh  i2  β αi

σδ   α i2 D   J (α r )∆α 0 i σδ 1+ 2 αi D 1+

for 1 + σδ / α i2 D < 0. If we plot the graph of dependence of ∆w on distance r to the source of the acoustic signal at different moments of time at different frequencies f, we can see that with the increase of frequency the signal, moving through the plate, intensively attenuates. So, for instance, displacements w, caused by vibrations of frequency f = 15 kH, practically completely attenuate at distance r = 40 cm for a 4 mm thick plate of AMg6 alloy. This situation changes somewhat with time, but does not alter the essence of the phenomenon itself. Thus, selecting the working range of the acoustic transducer frequencies, it is necessary to use the above calculations and not select this range from the high-frequency region if acoustic signals are to be detected at a considerable distance from the transducer. At acoustic transducer operation in a narrow frequency band from flexural wave ∆f, defined by upper and lower limits α1 and α2, displacement w can be found from the following formulas:

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384

 α 2t

w

α2 sin  β  2

Q 2πβ2 D α∫1

α

σδ   α2 D  J 0 (αr ) d α σδ 1+ 2 α D 1+

for 1 + σδ / α2D > 0;

w

 α 2t sh  α2  β2

Q 2πβ2 D α∫1

α

1+ 1+

σδ α2 D

σδ α2 D

   J 0 (αr ) d α

for 1 + σδ / α2D < 0.

3

Propagation of acoustic emission waves in the half-space under the impact of a symmetrical local emission source

1. General equations. A local defect, initiating in the plate in the first approximation, can be described in the cylindrical coordinates by initial ∗ ∗ ∗ deformations ε∗r , εθ , ε z and γ r z , which are the function of coordinates and time. Using equations of equilibrium for this case ∂σ r σ r − σθ ∂τrz ∂ 2u + + = ρ 2; ∂r ∂z r ∂t ∂σ z τr z ∂τrz ∂2 w + + = ρ 2 , ∂z r ∂r ∂t as well as the dependences σ r = λ(e − e∗ ) + 2G (ε r − ε∗r ); σθ = λ(e − e∗ ) + 2G (ε θ − ε∗θ ); σ z = λ(e − e∗ ) + 2G (ε z − ε∗z ); τrz= G ( γ rz − γ ∗rz ), where e = ε r + ε θ + ε z is the sum of relative deformations, we obtain differential equations to determine displacements u and w:

(λ + G )

 ∂ 2u 1 ∂ u u ∂ 2u ∂e +G 2 + − + r ∂ r r2 ∂ z2 ∂r ∂r

 ∂ 2u −ρ 2 = ∂t 

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385

∗

∂ γr z  ∂ ε∗ ε∗ − ε∗θ  ∂ e∗ = 2G  r + r +λ ; +G ∂z ∂r r   ∂r  ∂2 w 1 ∂ w ∂2 w  ∂e ∂2 w + 2 −ρ 2 = (λ + G ) + G  2 + ∂z r ∂r ∂z  ∂t  ∂r

(5.12)

 ∂ γ ∗r z γ ∗r z  ∂ε ∂ e∗ = 2G +G + +λ .   ∂r r  ∂z ∂z  ∗ z

Here u and w are the displacements along radius r and across thickness z, m; λ = νE / [(1 – ν)(1 – 2ν)] is the Lamé constant, Pa; G = E / [2(1 + ν)] is the modulus of elasticity in shear, Pa; E is the modulus of elasticity in tension, Pa. Let us consider the solution of equations (5.12) for the case when ∗ ε r =ε∗θ =ε∗z =ε∗ and γ ∗rz = 0 , i.e. for the case when deformation in the material developed as a discontinuity with micro-displacements, symmetrical relative to its centre, located at point (0, z 0) in Fig. 5.5. The mechanism of initiation and development of deformations in time can be represented by a graph, shown in Fig. 5.6. Deformation ε* develops monotonically from the initial condition, until at the moment of time t0 a discontinuity or instant displacement develops at the weakest point of the material (at point (0, z 0)), characterised by quick relieving of deformation at this point to zero (point a, curve 1), or to a certain value, minimal for a given period of time (point b, curve 2). In the graph in Fig. 5.6 just one moment of time t0 is of interest, when deformation changes its magnitude in a jumplike manner. The process of deformation up to moment t 0 is monotonic and does not make any changes in the problem solution. Dynamic change arises only at moment t 0, and can be described by pulsed function δ +(t – t 0). Let us shift the origin of coor­d inates in time to point t 0; then the distribution of volume, m 3, which has abruptly changed at depth z 0 from the plate surface, can be represent along the axes of coor­d inates and time as a product of δ +-functions:

5.5

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386

5.6

ε∗ = −ε∗0

δ+ (r ) δ + ( z − z0 ) δ + (t ) δ+ [ ω0 − f (αC ) ] . 2πr

(5.13) Here function δ + [ω 0 – f(αC)] gives a correlation between circular frequency ω 0, 1/s, and wave number α, 1/m; function f(αC) will be found further on from the spectral equation, the kind of which is determined by boundary conditions. 2. Some mathematical information. Solution of system (5.12) presents great difficulties. Let us simplify the solution using the Helmholtz expansion theorem, allowing the vector field to be represented as two components, namely irrotational F 1(R) and solenoidal F 2(R) parts: F(R) = F1(R) + F2(R), (5.14) where ∇ × F1(R) = 0 ; ∇ ⋅ F2(R) = 0.

(5.15)

Condition (5.15) is satisfied when and only when function F1(R) may by represented as a gradient of certain scalar function ϕ(R) (in this connection function ϕ(R) is called scalar potential of an irrotational vector field), i.e. F1(R) = ∇ ⋅ ϕ(R), (5.16) and function F2(R) – in the form of rotation of some vector function ψ(R): F2(R) = ∇ × ψ(R).

(5.17)

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Then total displacements F(R) according to (5.14), (5.16) and (5.17), may be written as: F(R) = ∇ ⋅ ϕ(R) + ∇ × ψ(R).

(5.18)

In cylindrical coordinates in the general case ∇= φ

∂φ 1 ∂φ ∂φ i+ j+ k; ∂r ∂z r ∂θ

 1 ∂ψ z ∂ψ θ   ∂ψ r ∂ψ z   1 ∂ (r ψ θ ) 1 ∂ψ r ∇= ×ψ  +  i +  ∂z − ∂r  j +  r ∂θ − r ∂θ ∂θ ∂ r z     

(5.19а)  k . 

(5.19b) Vector ψ may by represented as its derivates in directions r and θ:  1 ∂ψ ∂ψ  ψ=  ,− , χ . ∂r  r ∂θ  Then (5.19b) becomes:  1 ∂χ ∂ 2 ψ   1 ∂ 2 ψ ∂χ  + −  j+ ∇ ×= ψ  i +   r ∂θ ∂r ∂z   r ∂θ∂z ∂r   ∂  ∂ψ  1 ∂ 2 ψ  1 +   −r k. − 2   ∂r  ∂r  r ∂θ  r

(5.19c)

Substituting (5.19а) and (5.19b) into (5.18) and collecting similar terms at i, j and k, we have the following expressions for displace­m ents : ∂φ ∂ 2 ψ 1 ∂χ u= + ; + ∂ r ∂ r ∂z r ∂θ 1 ∂φ 1 ∂ 2 ψ ∂χ ; + − r ∂θ r ∂z ∂θ ∂ r

υ=

∂φ 1 ∂ψ ∂ 2 ψ 1 ∂ 2 ψ − − w= − . ∂z r ∂ r ∂ r 2 r 2 ∂θ2 Allowing for symmetry about axis z (∂ψ / ∂θ = 0 and χ = 0), we get:

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388

∂φ ∂ 2 ψ + ; ∂r ∂r ∂z ∂φ 1 ∂ψ ∂ 2 ψ w= − − . ∂z r ∂r ∂r 2

= u

(5.20)

3. Rayleigh waves in thick plates. Substituting (5.20) into (5.12) and applying the transformations, allowing for the selected model (i.e. assuming ε∗r = ε∗θ = ε∗z = ε∗ and γ ∗r z = 0), we have: ∂ 2 φ 1 ∂φ ∂ 2 φ 1 ∂ 2 φ 1 + ν ∗ + + − = ε ; ∂r 2 r ∂r ∂z 2 C12 ∂t 2 1 − ν

(5.21)

∂ 2 ψ 1 ∂ψ ∂ 2 ψ 1 ∂ 2 ψ + + − = 0, ∂r 2 r ∂r ∂z 2 C22 ∂t 2

where ε* is ‘explosive’ deformation defined by formula (5.13); C 1 and C 2 are the velocities of propagation of the longitudinal and transverse waves: C1 =

E 1− ν ; C2 = ρ (1 + ν)(1 − 2ν)

E 1 . ρ 2(1 + ν)

Differential equations (5.21) at boundary and initial conditions σ z = τ rz = 0 at z = 0 and t = 0 can be solved easily by the method of integral transformations. Performing Hankel transformation for coordinate r and Fourier cosine transformation in time t and denoting 1 + ν ε∗0 ρ∗0 = , 1 − ν 2π we have −α 2 φ +

∂ 2 φ ω02 2 ∗ + 2 φ= − ρ0 δ + ( z − z0 )δ + [ ω0 − f (αC ) ] ; 2 π ∂z C1

∂ 2 ψ ω2 −α ψ + 2 + 02 ψ = 0 . ∂z C2

(5.22)

2



We will solve the first equation of the system (5.22) by applying complex Fourier transformation. Then ∞  2 ω02  e − i γz 2 ∗ dz =  −α − γ + 2  φ = −ρ0 ∫ δ + ( z − z0 ) δ + [ ω0 − f (αC ) ] π C1   −∞

=−

ρ∗0 δ + [ ω0 − f (αC ) ] e − i γz0 . π

(5.23)

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389

Whence φ= and = φ

=

ρ∗0 e − i γz0 δ + [ ω0 − f (αC ) ] 2 π α + γ 2 − ω02 / C12

∞

ρ∗0 π 2π ρ∗0 2π β1

e − i γz0 ei γz dγ = ω02  2 −∞  2 α − 2  + γ C1  

δ + [ ω0 − f (αC ) ] ∫

δ + [ ω0 − f (αC ) ] exp [ − | z0 − z | β1 ] ,

where β1= α 2 − ω02 / C12 > 0. Solution of the second equation of system (5.22) may be written as follows: = ψ A exp [ − | z0 − z | β2 ] + B exp [ | z0 − z | β2 ] , Since at z → ∞ the solution should have a finite value, we assume B = 0. Inverse Hankel transformations for α and Fourier cosine transfor­m ation for ω 0 yield: = φ

ρ∗0 π

∞∞

∫ ∫ αJ

0

(αr ) cos ω0 t ×

0 0

× exp [ − | z0 − z | β1 ] d αd ω0 ; = ψ

2 π

∞∞

∫ ∫ AαJ

0

1 δ + [ ω0 − f (αC ) ] × β1 (5.24)



(αr ) cos ω0 t exp [ − | z0 − z | β2 ] d αd ω0 .

0 0



(5.25)

Constant A is determined from boundary condition τ rz = 0 at z = 0. If τ rz is expressed as displacements u and w:  ∂u ∂w  = τrz G  + ,  ∂z ∂r  and the displacements as f and ψ, using formulas (5.20), we have: = τrz G

∂  ∂φ ∂ 2 ψ ∂ 2 ψ 1 ∂ψ  − −= 2 +  0. ∂ r  ∂z ∂z 2 ∂ r 2 r ∂ r 



(5.26)

390

Fundamentals of evaluation and diagnostics

Substituting (5.24) and (5.24) into (5.26), we have the following expression for coefficient A: A = −ρ∗0

2 exp [ − z0 (β1 − β2 ) ] δ + [ ω0 − f (αC ) ] . π 2α 2 − ω02 / C22



(5.27)

Let us write expression σz as displacements u, w and initial deformations e* and ε*: ∂w σ z = λe + 2G − 2Gε∗z − λe∗ . ∂z In the considered case ε∗r =ε∗θ =ε∗z =ε∗ . Then σ z = λe + 2G

∂w − (3λ + 2G )ε∗ . ∂z

Using formulas (5.20) and equations (5.21) and considering that on boundary z = 0 σ z = 0, after transformations we obtain:  ∂ 2 φ 1 ∂φ 1 ∂ 2 φ ∂  ∂ 2 ψ 1 ∂ψ   σ z =−2G  2 + − 2 2 +  2 +   =0. r ∂r 2C2 ∂t r ∂r   ∂z  ∂r  ∂r

(5.28)

Substituting into (5.28) the values of f and ψ from (5.24) and (5.24) and taking (5.27) into account, we obtain a spectral equation for the half-space: ω02 / 2C22 − α 2 α 2 − ω02 / C12

+ 2 α2

α 2 − ω02 / C22 2α 2 − ω02 / C22

(5.29)

0. =

Equation (5.29) allows correlating ω 0 and α and determining the kind of function δ +[ω 0 – f(αC)] . It has a unique real solution, when ω 0 = αC 3, where C 3 is the velocity of Rayleigh wave propagation. Substituting ω 0 = αC 3 into (5.29), we obtain: (1 + γ 22 ) 2 − 4 γ1 γ 2 =0,

(5.29а)

1 − C32 / C22 . where γ1= 1 − C32 / C12 ; γ 2 = Solution of equation (5.29a) yields C 3 = 0.927C 2 at ν = 0.3. Then, we may write: f(αC) = αC 3. Expression (5.29) indicates that in thick plates only Rayleigh waves propagate in the full spectrum (5.24), (5.25) and, thus, information on the defect is transmitted without distortion. With smaller thickness of the plate the second boundary z = δ starts having an influence, and this influence is the stronger the thinner the plate. Lamb waves are generated, and the plate becomes a waveguide, depending on its

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391

dimensions. In this case, the signal from the defect comes to the sensor greatly distorted 1. On the other hand, analysing expressions (5.24), (5.25), we can see that Rayleigh waves may propagate also in comparatively thin sheets. In these cases, however, the spectrum of the transmitted frequencies no longer corresponds to a single pulse and is determined by exponents in formulas (5.24) and (5.25). On the other hand, these same formulas show that the spectrum is restricted from the lower side at small α and large wavelengths: α = 2π / λ; αС3 = 2πf, where λ is wavelength, m; f is the frequency of propagating waves, 1/s. Thus, only the high-frequency vibrations propagate in thin plates without distortion. Let us determine transverse w and longitudinal u displacements of a plate, caused by a propagating Rayleigh wave. They can be calculated, using formulas (5.20), (5.24), (5.25), (5.27). Presence of restraint δ +[ω 0 – αC 3] allows eliminating the double integral. Then, for z ≤ z 0 = w −

ε∗0 (1 + ν) ∞  αJ 0 (αr ) cos(αC3t ) exp [ −α( z0 − z ) γ1 ] − 2 ∫ 2π (1 − ν) 0 

 2 exp [ −αz0 ( γ1 − γ 2 ) − α( z0 − z ) γ 2 ] d α; 2 1+ γ2 

u= − −

(5.30а)



1 ε∗0 (1 + ν) ∞ αJ1 (αr ) cos(αC3t )  exp [ −α( z0 − z ) γ1 ] − 2 ∫ 2π (1 − ν) 0  γ1

 2 γ 22 exp [ −αz0 ( γ1 − γ 2 ) − α( z0 − z ) γ 2 ] d α. 2 1+ γ2 

(5.30b)



If we assume that z = 0, then for w and u we have: w= −

ε∗0 1 + ν 1 − γ 22 ∞ αJ 0 (αr ) cos(αC3t ) exp[−αz0 γ1 ]d α; 2π2 1 − ν 1 + γ 22 ∫0

u= −

ε∗0 1 + ν  1 2 γ 22  ∞  −  αJ1 (αr ) cos(αC3t ) exp[−αz0 γ1 ]d α. 2 2π 1 − ν  γ1 1 + γ 22  ∫0 (5.31b)



(5.31а)



In the general case two waves propagate over the half-space surface, if their source is not located on the surface (see chapter 3, section 4, item 2). 1

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Formulas (5.31) allow determination of the spectrum of a propagating Rayleigh wave for thin plates and calculation of displace­m ents w and u, associated with it. In practice, we usually have to deal with plates of a complex geometry and variable thickness. One of the plate surfaces, as a rule, remains even, and the minimal thickness of the plate is equal to δ0. In this case Rayleigh waves are of considerable interest, as they propagate in a band of thickness δ 0 with a distortion from the side of lower frequencies, which may be taken into account at signal resto­r ation. Example. Let an acoustic emission source ε* arise in a sheet of a complex geometry of alloy AMg6 with minimal thickness δ 0 = 6 mm at depth z 0 = δ 0/10. Let us find displacement w on the plate surface and determine the spectrum of wave frequencies, transmitted by the sheet to the place of the acoustic receiver location. Suppose the sensitivity of our equipment does not allow receiving signals n times smaller than the nominal signals. Then, with reference to (5.30a) on the assumption that the integrand should be greater than 1/n, at the maximum value of function J 0(αr) cos(αC 3t) , equal to a unity, we have: 1 . α1e −α1 ( δ0 − z0 ) γ1 = n Solving this equation for n = 35.0; γ 1 = 0.868 and z 0 = 0.6 mm, we obtain α 1 ≈ 14 cm –1. Thus, the lower limit of the propagating Rayleigh wave in this case will be the frequency of 0.6 MHz, and in formulas (5.30a) wave number α 1 ≈ 14 cm –1 will be the lower limit of integration. In this case, the shape of the signal, propagating in the upper layers of the plate, changes considerably (Fig. 5.7). As a rule, local deformation ε* does not remain stationary. A defect propagates in a certain direction. In other words, the centre of acoustic emission is moving. Using the superposition principle, it is possible to -6

w ⋅ 10 , сm 8 6 4 2 0

3 1

2

4 r, сm

-2 -4 -6

5.7

δ = 1 0 сm δ = 0 .6 сm

Acoustic emission

393

obtain an expression for displacements, induced by a moving emission source. In this case, for instance, at z – υ zτ < z 0 it may be written for w: t

= w

ε01 1 + ν 0 ∞ αJ 0 α ( x − υ x τ) 2 + ( y − υ y τ) 2  ×   2π2 1 − ν ∫0 ∫0

{

× cos [ αC3 (t − τ) ] exp [ −α(| z0 − z | −υ z τ) γ1 ] − −

2 exp [ −αz0 ( γ1 − γ 2 ) − α( | z0 − z | −υ z τ) γ 2 ] 1 + γ 22

}

d αd τ.

(5.32) The value of w on the plate surface may be derived from (5.32), assuming z = 0: w= −

ε∗01 1 + ν 1 − γ 22 2π2 1 − ν 1 + γ 22

l0 ∞

∫∫α J 0 0

0

α 

( x − υ x τ) 2 + ( y − υ y τ) 2  × 

× cos [ αC3 (t − τ) ] exp [ −α( z0 + υ z τ) γ1 ] d αd τ. Here t0 is the time of action of a moving source of local deformation; υx, υ y, υ z are the projections of the vector of velocity of the source movement on the axis of coordinates. And, finally, application of formulas (5.30а), (5.30b), (5.32) allows calculation of the strength of the impact of the plate surface on the casing of an acoustic emission (AE) transducer: = p ( P / g ) (∂ 2 w / ∂ t 2 ), where P is the weight of the AE transducer, N.

4

Propagation of acoustic emission waves in the half- space under the impact of a non-symmetrical local emission source

1. General equations. A mechanism of propagation of elastic waves at intensive local symmetrical deformation of a material during its loading was considered earlier. It was shown that the generated waves are greatly distorted with the reduction of the sheet thickness. Wave amplitudes decrease with the increase of the distance from the site of their generation. These data are quite essential in the algorithms of processing the acoustic emission information, as they enable, on the one hand, implementing measures to restore the AE signal and, on the other, correctly selecting the working ranges of frequencies for the AE transducer. However, the process of development of intensive local deformations and cracks may not be symmetrical. Therefore, those differences in the travelling elastic wave are of interest, which are caused by asymmetrical nature of deformation ε*.

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Let us revert to the initial differential equations (5.12), which together with the boundary and initial conditions σ z = τ rz = 0 for z = 0 and t = 0 describe the process of elastic wave propagation. Let us assume that they are different from zero and γ rz ≠ 0 . In this general case, functions of displacements f and ψ cannot be introduced just as relatively easily as it was done above. To simplify equations (5.12), let us use the expression of the deformation of rotation = ω

1  ∂u ∂w   − , 2  ∂z ∂r 

(5.33)

and apply the following transformations. Let us add and subtract value ∂2w /∂r∂z in parenthesis of the first equation (5.12). Then (λ + 2G )

∂e ∂ω ∂ 2u + 2G − ρ 2 = 2GS1∗ , ∂r ∂z ∂t

(5.34)

where ∂ε∗ ε∗ − ε∗θ 1 ∂γ ∗rz λ ∂e∗ + + S1∗ = r + r . ∂r r 2 ∂z 2G ∂r In the second equation of system (5.12) let us add and subtract ∂ 2 u 1 ∂u expression in the parenthesis. Then, using equation (5.33), + ∂ ∂ ∂ r z r z we have: (λ + 2G )

∂e 1 ∂ ∂2ω − 2G (r ω) − ρ 2 = 2GS 2∗ , ∂z r ∂r ∂t

(5.35)

where S 2∗ =

∗ ∗ ∂ε∗z 1  ∂γ r z γ r z  λ ∂e∗ +  + . + ∂z 2  ∂r r  2G ∂z

Let us subtract equation (5.35) from (5.34), having first differentiated (5.34) with respect to z and (5.35) with respect to r. We obtain: ∂2ω ∂  1 ∂ ∂ 2 ω ∂S1∗ ∂S 2∗  +  (r ω)  − ρ = − 2 ∂r  r ∂r ∂z ∂r ∂z ∂t 2  or ∇2ω −

ω 1 ∂ 2 ω ∂S1∗ ∂S 2∗ = − , ∂z ∂r r 2 C22 ∂t 2

∂ 2 ω 1 ∂ω ∂ 2 ω 2 ω + + ; C 2 is the velocity of transverse wave propwhere ∇= ∂r 2 r ∂r ∂z 2 agation. On the other hand

Acoustic emission

e = εr + εθ + ε z = 2ω =

∂u ∂w − . ∂z ∂r

1 ∂ ∂w (ru ) + , r ∂r ∂z

395

(5.36) (5.37)

Let us differentiate (5.36) with respect to r, and (5.37) with respect to z. Then, from the derived expressions it follows that: 2 ∂e ∂  1 ∂ ∂ω  ∂u = (ru )  + 2 − 2  ∂r ∂r  r ∂r ∂z ∂ z 

or ∂ 2 u 1 ∂u u ∂ 2 u ∂e ∂ω + − + = +2 , ∂z ∂r 2 r ∂r r 2 ∂z 2 ∂r and ultimately, in view of (5.34), we get: ∂ 2 u 1 ∂u u ∂ 2 u 1 ∂ 2 u 1 ∂ω 1 − 2ν ∗ + − 2+ 2 − 2= + S1 2 2 r ∂r r 1 − ν ∂z 1 − ν ∂r ∂z C1 ∂t or ∇2u −

u 1 ∂ 2u 1 ∂ω 1 − 2ν ∗ − = + S1 , 2 2 2 1 − ν ∂z 1 − ν r C1 ∂t

where С1 is the velocity of longitudinal wave propagation. This technique allowed obtaining two simpler wave equations instead of two complex differential equations (5.12): ∇2ω −

ω 1 ∂ 2 ω ∂S1∗ ∂S 2∗ = − ; ∂z ∂r r 2 C22 ∂t 2

1 ∂ 2u 1 ∂ω 1 − 2ν ∗ u ∇ u − 2 − 2= + S1 . 2 1 − ν ∂z 1 − ν r C1 ∂t

(5.38)

2

2. Solution of equations. Let us solve equations (5.38) for the case when the principal component of initial deformation forms in the direction ∗ ∗ ∗ ∗ 0; thus, e* = 0. With these of axis z, and deformations ε r = ε θ = −ε z / 2, γ r z = assumptions, (5.38) becomes ∇2ω −

3 ∂ 2 ε∗z ω 1 ∂2ω ; − = − 2 ∂r ∂z r 2 C22 ∂t 2

u 1 ∂ 2u 1 ∂ω 1 − 2ν ∂ε∗z . ∇ u − 2 − 2= − r C1 ∂t 2 1 − ν ∂z 2(1 − ν) ∂r 2

(5.39)

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Let us assume that the mechanism of initiation and development of deformation ε*z is the same, as in the case of symmetrically developing deformation ε*. Then ε*z can be also represented by formula (5.13). The first of equations (5.39) after substitution of (5.13) into it can be solved by the method of integral transformations, using Fourier cosine transformation with respect to time t , complex Fourier transformation with respect to coordinate z and Hankel transformation with kernel J 1 with respect to coordinate r:  2 ω02  − i γz 2 ∗ 3α  −α − γ + 2  ω = −ε 0 2 i γe 0 δ + [ ω0 − f (αC ) ] , 2π C2   whence = ω

3αε∗0 2π 2

i γe − i γz0 δ + [ ω0 − f (αC ) ] . ω02 2 2 α − 2 +γ C2

The inverse complex Fourier cosine transformation yields: = ω

∞ ε∗0 i γ e − i γ ( z0 − z ) f ( C ) δ ω − α [ ] 0 + 2 ∫ 2 2 dγ 2π 2 π −∞ β 2 + γ

3α

and after integration = ω

3αε∗0 4π

2 −β2 z0 − z e δ + [ ω0 − f (αC ) ] , π

where, as before, β2=

(5.40)

α 2 − ω02 / C22 .

Knowing the expression for ω , we can find displacements u , solving the second heterogeneous equation of system (5.39). Applying to it Hankel transformation with kernel J1(αr) with respect to radius r and Fourier cosine transformation with respect to time t, we obtain: −α 2 u +

∂ 2 u ω02 1 ∂ω 1 − 2ν αε∗0 u = + − 1 − ν ∂z 1 − ν 4π ∂z 2 C12

2 δ + ( z − z0 )δ + [ω0 − f (αC )]. π

Substituting here instead of its value from (5.40), for z ≤ z0 , we obtain: 3αε∗0 ∂2 u 2 −β2 | z0 − z | 2 u = − β β2 e δ + [ω0 − f (αC )] − 1 4π(1 − ν) π ∂z 2

Acoustic emission

−

1 − 2ν αε∗0 1 − ν 4π

2 δ + ( z − z0 )δ + [ω0 − f (αC )], π

397

(5.41)

where β1= α 2 − ω02 / C12 . We will seek the solution of equation (5.41) in the form of the sum of two solutions – general solution of a homogeneous equation and partial solution of a heterogeneous equation; let us also represent the latter in the form of two summands, corresponding to the two summands in the righthand part of (5.41): = u Ae −β1 | z0 − z | + u r ,1 + u r ,2 . We will seek the first partial solution in the form of: = u r ,1 Be −β2 | z0 − z | δ + [ω0 − f (αC )].

(5.42)

Substituting (5.42) into equation (5.41) (without the second summand in the right-hand part), we find the value of coefficient B : B=

3 αε∗0 β2 4π(1 − ν) β22 − β12

2 . π

We will find the second partial solution, applying in (5.41) (without the first summand in the right-hand part) the complex Fourier transformation with respect to coordinate z: −α 2 u r ,2 − γ 2 u r ,2 +

ω02 1 − 2ν αε∗0 − iγz0 u r ,2 = − e δ + [ω0 − f (αC )]. 2 1 − ν 4π 2 C1

Solving this equation and reverting to variable z by application of inverse complex Fourier transformation, we get: = u r ,2 =

∞ − i γ ( z0 − z ) 1 − 2ν αε∗0 e = d γδ + [ω0 − f (αC )] ∫ 2 1 − ν 4π 2π −∞ β12 + γ 2

1 − 2ν αε∗0 e −β1 | z0 − z | δ + [ω0 − f (αC )]. 1 − ν 4π 2πβ1

Thus, u Ae −β1 | z0 − z | + =

3αε∗0 2 δ + [ω0 − f (αC )] × 4π(1 − ν) π

 β 1 − 2ν −β1 | z0 − z |  ×  2 2 2 e −β2 | z0 − z | + e . 6β1  β2 − β1 

(5.43)

Fundamentals of evaluation and diagnostics

398

We will find constant A from the boundary condition τ rz = 0 at z = 0 .  ∂u ∂w  τrz G  + Then, allowing for (5.33) and the fact that =  , we get:  ∂z ∂ r 

)

(

τ= 2G ∂u / ∂z − = ω 0 at z = 0, rz i.e.

∂u / ∂z − ω = 0 at z = 0. Whence 3αε∗0 2  1 1 β22 1 A=  − 2 2 4π π  β1 1 − ν β1 β2 − β12

 − z0 (β2 −β1 ) − e 

1 − 2ν   δ + [ω0 − f (αC )]. 6(1 − ν)β1 

(5.44)

The second boundary condition has the form of σ z = 0 at z = 0. Expressing σ z as displacements u, w and initial deformations e* and ε*, z we have: σ z = λe + 2G

∂w − 2Gε∗z − λe∗ . ∂z

(5.45)

On the other hand, from the second equation (5.12) at γrz = 0 , it follows that: (λ + G )

∂ε∗ ∂e ∂e∗ ∂2 w +G = ∇ 2 w 2G z + λ +ρ 2 . ∂z ∂z ∂z ∂t

(5.46)

Having differentiated expression (5.45) with respect to z and substituting (5.46) into it, we obtain:  ∂  ∂u ∂w  1  ∂u ∂w  1 ∂ 2 w  ∂σ z = −G   + +  + − 2 2 , ∂z  ∂r  ∂z ∂r  r  ∂z ∂r  C2 ∂t 

(5.47)

or, allowing for (5.33),  ∂  ∂u ∂σ z 1 ∂2 w   1  ∂u  = −2G   − ω  +  − ω  − 2 2  . ∂z  r  ∂z  2C2 ∂t   ∂r  ∂z



(5.48)

For a more convenient application to (5.48) of Hankel trans­f ormation with respect to radius r, let us differentiate it with respect to r:  ∂ 2 φ 1 ∂φ φ ∂2σz 1 ∂3 w  = −2G  2 + − 2 − 2 2 . r ∂r r ∂r ∂z 2C2 ∂t ∂r   ∂r



(5.49)

Acoustic emission

399

Here f = ∂u / ∂z – ω. Let us take in (5.49) Fourier cosine transformation with respect to variable t and Hankel transformation with kernel J 1(αr) with respect to radius r , first substituting instead of ∂w / ∂r its value from (5.33). We get:  ω2 ω2  ∂2 σz = −2G  −α 2 φ + 02 φ − 02 ω . ∂r ∂z 2C2 2C2   Integrating with respect to z , we obtain at z = 0 the boundary condition in the following form:

∫

 2 ω02  α − 2 2C2 

 ω02  φ + ω dz = 0.  2C22  

(5.50)

And finally, after substitution of u and ω into (5.50), we obtain at ω 0 = αC 3 a spectral equation 

2



1

(1 + γ )  γγ 2 2

1 1 1 1  1 − γ 22 + − − =0, 1 − ν γ1 + γ 2 γ 2 γ1  γ2

(5.51)

2

C C32 1 − 32 . and γ 2 = 2 C2 C1 Equation (5.51) has the only real solution, when f(αC) = αC 3 and С 3 = 0.927 С 2, and the δ +-function becomes δ +[ω 0 – αC 3]. Let us calculate displacements w of the plate outer surface in the direction of the axis z . For our case w can be found from (5.33), (5.40), (5.43), (5.44) at z = 0, using inverse Hankel and Fourier cosine transformations: where γ1=

1−

3αε∗0 δ + [ω0 − αC3 ] 2 −β2 z0 ∂ w ∂u e = − 2ω = 4π ∂r ∂z π −

 β22 + − 1  2 2  (1 − ν)(β2 − β1 )

 β22β1 β (1 − 2ν) − z0 (β1 −β2 ) 1 − 2ν − z0 (β1 −β2 ) − 1 + − 2 = e e 2 2 6(1 − ν) (1 − ν)β1 (β2 − β1 ) 6(1 − ν)β1 

= −

3αε∗0 δ + [ω0 − αC3 ] 2 −β2 z0 . e 4π π

Then ∞∞

3αε∗0 δ + [ω0 − αC3 ] 2 −β2 z0 ∂w =− ∫ ∫ e αJ1 (αr ) cos ω0 td αd ω0 , ∂r 4π π 0 0 and, finally, after integration with respect to r,

Fundamentals of evaluation and diagnostics

400

w =

3ε∗0 ∞ αJ 0 (αr ) cos(αC3 t )e −α z0 γ 2 d α. 2π2 ∫0



(5.52)

Formula (5.52) differs from formula (5.31a), derived for the case, when the wave propagates under the impact of spherical source ε*, i.e. when ε∗r =ε∗θ =ε∗z , by the integral coefficient and the fact that it has γ 2 and not γ 1 value as an exponent. This is significant, as displacement w in this case is greater than in (5.31a), and attenuates much slower with increase of distance z 0. Thus, displacement w, just as the rate of its propagation, depends on the kind of defect, and, on the one hand, the amplitude of the signal on the AE transducer can be one of the signs, defining the kind of defect and on the other, it may introduce an indeterminacy into the description of the event proper, if there are no additional parameters, which allow clarifying the ambiguity of the derived AE information. The graphs of the propagating Rayleigh wave in thick and thin plates, depending on the distance to the source of its generation, change significantly with the increase of distance r and depth of location of source z 0, as in the case of development of a symmetrical defect. The shape of the signal proper is also considerably distorted, if the plate is thin. It is difficult to locate such complex signals, because of the possibility of AE transducers reacting to different amplitudes of the same signal. Magnitudes of longitudinal displacements u over the plate surface z = 0 can be calculated by formulas (5.43) and (5.44), taking inverse Hankel and Fourier cosine transformations: 3αε∗0 δ + [ω0 − αC3 ] 2 −β2 z0 e 4π π

= u − =

 1 β22 − −  2 2  β1 (1 − ν)(β2 − β1 )β1

β2 (1 − 2ν)e − z0 (β1 −β2 ) (1 − 2ν)e − z0 (β1 −β2 )  + + = 6(1 − ν)β1 6(1 − ν)β1 (1 − ν)(β22 − β12 )  3αε∗0 4π

 −β2 z0 β22 β2 21 + δ+ [ω0 − αC3 ].  − e π  β1 (1 − ν)(β22 − β12 )β1 (1 − ν)(β22 − β12 ) 

Then  3ε∗0  1 γ 22 γ2 u= − + × 2  2 2 2 2  2π  γ1 (1 − ν)( γ 2 − γ1 ) γ1 (1 − ν)( γ 2 − γ1 )  ∞

×∫ αJ1 (αr ) cos(αC3t )e −αz0 γ 2 d α 0

and u m =

3ε∗0 ∞ αJ1 (αr ) cos(αC3t )e −αz0 γ 2 d α, 2π2 ∫0

Acoustic emission

401

γ2 − γ γ 1 1 + 22 22 1 . γ1 ( γ 1 − γ 2 ) γ1 1 − ν Component of rotation ω is obtained in a similar way:

= where m

= ω

3ε∗0 ∞ 2 α J1 (αr ) cos(αC3t )e −αz0 γ 2 d α. 2π2 ∫0

If the defect does not remain stationary, but develops in a discrete manner with further increase of load, the process of displacement of the centre of acoustic emission may be expressed by formula (5.52), where r is replaced by value 2

r=

n   y 2 +  x − υ ∑ ∆ti  , i =0  

where υ is the rate of defect propagation in the direction of axis r. Then = w

3ε∗0 2π 2

k ∞

∑ ∫ αJ n=0 0

2  n    2  α y + x − υ ∆ t ×   i 0   i =0    

∑

  k −n  × cos αC3  t + ∆ti   e −αz0 γ 2 d α,  i =0  

∑

where k is the number of jumps of a discretely developing defect, formed by the considered moment of time. Thus, sharply concentrated deformations and the associated laws of elastic waves propagation have been considered. It can be anticipated that the finite nature of the dimensions of items, the specific relation between them and plate thickness may lead to corrections in calculations. Let us assume that, as before, the sensitivity of the recording instruments is determined by value n and 1 ; α1e −α1 ( δ0 − z0 ) γ 2 = n 3ε∗0 ∞ w = αJ 0 (αr ) cos(αC3t )e −αz0 γ 2 d α. 2π2 α∫1

(5.53)



(5.54)

Then, if n = 35.0; δ 0 = 6 mm; z 0 = 0.6 mm; γ 2 = 0.375; γ 1 = 0.868; С 2 = 0.293 ⋅ 10 8 сm/s and ε *0= 10 –3 сm 3, we obtain α 1 ≈ 35 сm –1 and f = α 1C 3 / 2π ≈ 1.6 МHz . As can be seen from (5.53), limitation of the spectrum of transmitted frequencies in this case is greater than in the case of a symmetrical source, where f = 0.6 MHz. Calculation of values w by formula (5.54) shows that

Fundamentals of evaluation and diagnostics

402

the limiting thickness of the plate, in which the signal propagates practically without change, is also 50 mm. Formula (5.54) and other similar formulas can be significantly simplified for calculations, if integration is performed between zero and infinity (α 1 = 0). Substituting into formula (5.54) cos(αC 3t) by its representation as a complex variable cos(= αC3t )

1 [exp(iαC3t ) + exp(−iαC3t )] , 2

we obtain for w: = w

3ε∗0 ∞ α {exp[−(αz0 γ 2 − iαC3t )] + exp[−(αz0 γ 2 + iαC3t )]}J 0 (αr )d α. 4π2 ∫0

After integration and going into the area of the real variable 2 we have: 3  3  b cos  φ  + a sin  φ  3ε∗0 2   2  , w= 2 2 2 2 2 2π  (b − a + c ) + 4a 2 b 2  3/ 4   2ba  where a = C3 t ; b = z0γ2 ; c = r; φ =arctg  2 2 2 b −a +c

5

 . 

Influence of the resistance of the medium on acoustic emission wave propagation

The propagation of deformation waves under the actual conditions depends considerably on the resistance of the medium in which they were generated. The influence of such a resistance may be dual. On the one hand, the amplitude of the propagating elastic wave decreases and, on the other hand, in view of the fact that the resistance is proportional to wavelength, dispersion may arise as the wave propagates away from the emission source. In both cases, the magnitude of resistance to wave displacement may have a significant impact on the effectiveness of controlling the fracture processes by the acoustic methods. The second case may be particularly dangerous, when the arising wave dispersion may lead to such a distortion of the waves that not only interpreting the information, associated with the formed and propagating defect, but also the location of the site of its initiation will become impossible. In this connection the issue of studying the influence of the resistance of the medium on the magnitude and shape of the acoustic emission quantum and other integral characteristics of emission becomes particularly important. 2

Representation of the complex variable through its modulus and argument was used: а + ib = (a2 + b2)1/2 exp(iφ); а – ib = (a2 + b2)1/2 exp(–iφ).

Acoustic emission

403

Medium resistance to propagation of a quantum of acoustic emission may be represented as a specific impulse of retarding force b c, N ⋅ s/m 3. Then, the calculation formulas may include the speed of the medium response to such a displacement VT =

E , m/s, bc (1 − ν)(1 + ν)

or, for better understanding, the dynamic coefficient of the medium resistance, introduced in the case of movement of the body particles, S = C / VT , where С is the velocity of propagation of the longitudinal or transverse wave. In view of the above-said, let us write differential equations of a dynamic problem of the theory of elasticity in the following form: ∇2φ −

αS ∂φ 1 ∂ 2 φ 1 + ν ∗ − = ε ; C1 ∂t C12 ∂t 2 1 − ν

αS ∂ψ 1 ∂ 2 ψ ∇ ψ− − =0 , C2 ∂t C22 ∂t 2

(5.55)

2

where, as before, f and ψ are the displacement functions; C1 and C2 are the velocities of propagation of the longitudinal and transverse waves. Value α is the wave number, inverse of wavelength. The position of α in the numerator at the first derivative is indicative of the fact that medium resistance is not the same for all the wavelengths. It is the greater the smaller the wavelength. The value ε* is the relative additional deformation, arising at the origin of the coordinates, which coincides with the centre of an infinite space, which we will consider so as to avoid the wave reflection from the boundaries of an actual body. Let us represent deformation ε* in the already customary manner, using δ +-functions, limiting the change of the material volume by a point with the coordinates r = 0, z = 0: ∗ ε=

Vt∗ δ + (r )δ+ ( z )δ + (t ), 2πr

(5.56)

where V t* is the impulse of an instantaneous change of material volume, cm3 ⋅ s. Let us solve system (5.55) and evaluate the displacements, arising and propagating in this case. First of all, note that we have selected a deformation symmetric about the axes r and z. With such an assumption, no shear deformations will develop in the space, which we will consider.

Fundamentals of evaluation and diagnostics

404

Therefore, the second equation (5.55) will be identically equal to zero. We will solve the first equation with function f, applying Laplace, Fourier and Hankel transformations:  αSp p 2 −  α2 + γ 2 + + C1 C12  whence

 1 + ν Vt ∗ 1 , φ = 1 − ν 2π 2π 

∞ ∞ 1 + ν Vt ∗C1 φ=− α J 0 (α r ) 2 ∫ ∫ 1 − ν 4π 0 −∞

(5.57)

 α SC1t  exp  − 2   × α 2 η2 + γ 2

)

(

sin C1t α 2 η2 + γ 2 exp(i γz )d γd α, where η2 = 1 – S2/4. Assuming z = 0 and considering only this plane, we may essen­tially simplify the latter expression in determination of longitudinal vibrations U (in spherical coordinates) of the material bulk in the direction of the axis r. In this case U =

∞ ∂φ 1 + ν Vt ∗C1  αSC1t  d α. = α 2 J1 (αr )J 0 (αηC1t ) exp  − 2  ∂r 1 − ν 4π ∫0 



(5.58)

The presented integral is quite complicated for computations. The oscillating integrand and presence of an infinite upper limit make the process weakly convergent. Therefore, going over to integral present­a tion of the Bessel function as J 0 (= αηC1t )

2 π

π/ 2

∫ cos(αηC t ⋅ sin θ)d θ, 1

0

after integrating with respect to α between zero and infinity, we obtain 3: U =

1 + ν Vt ∗C1 r 1 − ν 2π 2

where A =

π/ 2

∫ 0

A d θ, B



    2 S η cos θ S 5 + η2 cos 2 θ cos  arctg  2 2 S 4 R2 − η2 cos 2 θ + 2 2   c1 t  4  2

(5.59)

     − arctg  η cos θ   ;     S     

Cosine representation by exponential functions was used, when integrating expression (5.58). 3

Acoustic emission

405

5.8 The diagrams of propagation of AE quantums at а) S = 0.1 and b) S = 2.0. 2  S 2  R2  2 2  B C t  = − η cos θ + 2 2  + S 2 η2 cos 2 θ  c1 t   4  4 4 1

5/ 4

.

This formula may be easily implemented in a personal computer. Calculations by formula (5.59) demonstrated a significant influence of the magnitude of the coefficient of medium resistance on the amplitude of the acoustic elastic vibrations. Figure 5.8 gives the diagrams of propagation of AE quantums at different values of coefficient S. In the graphs two moments of time are considered, at which two positions with U different from zero were recorded in a bulky body. The first moment of time corresponds to value С1t = 3 сm (designated by number 1 in the plot) and the second to value С1t = 6 сm (designated by number 2 in the plot). Calculations are made on the assumption that up to 3 1000 dislocations, concentrated in volume 4 / 3 ⋅ π a1 , where a 1 = 10 –5 сm, take part in the emission process simultaneously. Let us take the averaged time of dislocations displacement in this volume to be equal to 10 –5 s. Then, the impulse of the local change of material volume will be ∗ 3 V = 4 / 3 ⋅ π a= 4 / 3 ⋅ π× 10−15 × 10−5 ≈ 4 × 10−20 , cm3 ⋅ s. 1t t

Assuming ν = 0.3; C 1 = 0.6 × 10 6 сm/s for AMg6 alloy, we obtain U values, presented in Fig. 5.8. It is seen from the Figure that the signal shape is a curve of a sawtooth waveform, having no pronounced discontinuities. This is indicative of the fact that there is practically no dispersion of the signals of acoustic emission waves, related to medium resistance, in signal propagation along radius r. At medium resistance S = 0.1 the wave has a

406

Fundamentals of evaluation and diagnostics

shape typical for deformation waves, propagating in conventional metals (Fig. 5.8a). With the increase of medium resist­a nce S an ever greater volume of material is involved in the vibration process, and the signal shape becomes broader (Fig. 5.8b). It is also seen from the Figures that with the increase of S the positive amplitude of the quantum is significantly decreased and at S = 2 it is practically absent. Medium resistance in this case is so great that an abrupt interruption of the initial impulse impact leads only to a reduction of the AE quantum amplitude. The quantum proper transmits compressive deformation and acquires an ever smoother shape with not very clearly manifested fronts. This is indicative of the fact that use of the AE procedure to control plastic materials with high coefficients of resistance may require a greater number of AE transducers, covering the structure surface with their smaller spacing. It should be noted that the features of fabrication of glass-fibre-reinforced structures with the application of reinforcing winding with strong threads, may significantly change the pattern, especially if AE signals start to be received by these threads and propagate in them. Let us consider the second case of AE signal generation at the initi­ ation of an irreversible defect. In this case in formula (5.58) instead of function δ(t) we will use step function S(t) , expressing the conditions of irreversibility of the initiating defect, assumed by us. Then equation (5.57) will be written in a somewhat different form:  αS p2  1+ ν V * 1 1 −  α2 + γ 2 + p + 2 φ = , C C 1 − ν 2π 2 p 1 1   where V* already is the local change of material volume, cm3, and not a deformation impulse, as it was in expression (5.58). Formula (5.59) becomes π/ 2 r ( S cos φ + 2η sin φ cos θ)  1+ ν V ∗  1 4 U= − d θ ,  2− ∫ 1− ν 4  r π 0 ( S 2 + 4η2 cos θ)r13/ 4 

(5.60)

where  S η cos θ φ =arctg  2 2 S / 4 cos 2 θ + R 2 / C12 t 2 − η 

 ; 

2 = r1 C14 t 4 ( S 2 / 4 − η2 cos 2 θ + R 2 / C12 t 2 ) + η2 S 2 cos 2 θ  .  

Formula (5.60) is characterised by having two parts. The first describes the stationary part of the process (expression 1 / r2 in brackets), concentrated in the vicinity of the defect. This part rapidly decreases with the increase of distance r from the defect.

Acoustic emission

407

The second part of formula (5.60) incorporates a term, dependent on time, and is itself divided into two terms. The first term together with the stationary part describes the kinetics of displacement formation in the vicinity of the defect. With time it tends to zero and, eventually, just the stationary part 1 / r 2 remains, describing the distribution of displace­m ents, when all the non-stationary processes are over. The second term of this expression is a function, describing a wave, propagating from the source centre, with its amplitude decreasing with increase of distance r. The second part of formula (5.60) shows how medium resistance S influences the amplitude of AE quanta. The higher this resistance, the smaller is the quantum amplitude. As in the first case, no signal dispersion occurs. With the coefficient of medium resistance equal to a unity, the amplitude of AE quanta is reduced by practically two orders of magnitude. The derived formulas may be used for analysis of AE events in materials within a broad range of variation of internal resistance to the propagation of elastic deformation waves. So, glass-fibre plastics have a greater resistance to AE wave propagation. These materials are currently used to manufacture a large number of structures from pipelines to components for rocket and aircraft engineering.

6

Acoustic emission wave propagation in plates of arbitrary thickness under the impact of a symmetrical local source of emission

We have considered the propagation of elastic waves from a suddenly initiating defect in thick plates, when the influence of one of its surfaces may be neglected. Based on the results derived in item 3, it can be demonstrated that the thickness of a plate, where such an assumption is justified, is not more than 50 mm. In calculation of thin plates, it is necessary to take into account the influence of the second surface, which leads to the appearance of new components of the wave and greatly distorts the general picture. It is natural that calculation of the field of displacements in this case is more complicated, and the formulas become more cumber­ some and less convenient for calculations. Let us consider a plate of thickness δ, in which an AE signal appeared at the moment t = t 0 at depth z 0. As before, let us assume that the defect initiated in the form of a microexplosion in a local volume at AE wave propagation, symmetric in all the directions (Fig. 5.9) 4. Let us represent the magnitude and nature of AE source distribution in the form of a product of δ-functions. Then the equations, describing the propagation of elastic waves in a plate, may be written as Value of material volume change as a result of defect initiation may be presented as a sphere of radius а1. Value а1 in the first approximation can be taken to be equal to 10–3 cm. In this case V0* = 4/3πа13 = 4.2·10–9 cm3. 4

Fundamentals of evaluation and diagnostics

408

5.9

∇ 2ϕ − ∇ 2ψ −

1 ∂ 2ϕ 1 + ν * δ+ (r ) =− δ+ ( z − z0 )δ+ (t )δ+ ( ω0 − αC ) ; V0 1− ν 2πr C12 ∂t 2

(5.61)

2

1 ∂ψ =0, C22 ∂t 2



where f and ψ are the displacement functions; t is time; ν is Poisson’s ratio; C 1, C 2 are the velocities of propagation of the longitudinal and transverse waves, respectively; Сα, р, α are the parameters of elementary waves in the packet forming the Rayleigh wave (velocity, frequency, wave number); = ∇2

1 ∂  ∂  ∂2 . r + r ∂r  ∂r  ∂z 2

It should be noted that the first equation of system (5.61) incorporates a term in its right-hand part, which correlates the wave frequency, velocity of its propagation and wave number. Let us solve system (5.61) on the assumption that a body is limited from two sides and the following conditions are fulfilled at the boundary: σz = τrz = 0 at

z = 0; δ.

The initial conditions of the problem are determined by function δ(t–t0) , being indicative of an emission source appearing at t = t 0 and disappearing at t = t 0 +0. Applying Laplace transformation with respect to time t (with parameter p) and Hankel transformation with respect to coordinate r (with parameter α ), we obtain:

1 + ν V0∗ ∂2 φ  2 p2  − α + φ = − δ( z − z0 )δ( p ± iαCα ) ,   C12  1 − ν 2π ∂z 2  ∂2 ψ  2 p2  −  α + 2  ψ =0 . C2  ∂z 2  Boundary conditions become:

(5.62)

Acoustic emission

409

1 ∂φ ∂ 2 ψ 2 = τrz 2 + 2 + α= ψ 0, ∂z ∂z G

(5.63)

 p2  1 ∂ψ σz =  α2 + φ + α2 = 0 at z = 0. 2  2G 2C2  ∂z 

In the case, when the velocities of propagation of elementary waves are not higher than C 2, we will seek the solution of boundary problem (5.62), (5.63) in the following form: = φ Ñe zβ1 + = ψ Ae

− zβ 2

P1 −| z − z0 |β1 e , β1 + Be

z β2

,

(5.64)

where β1 =

α 2 + p 2 / C12 ; β2 = = P1

α 2 + p 2 / C22 ;

1 + ν V0∗ δ( p − iαCα ) . 1 − ν 4π

Substituting (5.64) into (5.63), we arrive at a system of algebraic equations to determine the constants A, B, C and the ratio of p, α and С α: − z0 β1 A ( α 2 + β22 ) + B ( α 2 + β22 ) + C ⋅ 2β1 = −2 Pe , 1 − ( δ− z0 ) β1 A ( α 2 + β22 ) e −δβ2 + B ( α 2 + β22 ) eδβ2 + C ⋅ 2β1eδβ1 = 2 Pe , 1

 p2   P  p2 A ( −α 2β2 ) + Bα 2β2 + C  2 + α 2  = − 1  2 + α 2  e − z0β1 , β C C 2 2 1   2  2  2  p  A ( −α 2β2 ) e −δβ2 + Bα 2β2 eδβ2 + C  2 + α 2  eδβ1 = C 2  2  =−

P1  p 2 2  − ( δ− z0 ) β1 .  2 +α e β1  2C2 

(5.65)



The forth equation of system (5.65) determine the correlation of wave numbers α, velocities of elementary waves propagation C α and their frequency characteristics p. This is the so-called spectral equation of a plate or the equation of existence of some specific waves. We solve the system of the first three equations for A, B, C and substitute the found coefficients into the fourth equation. Then, using inverse Laplace

Fundamentals of evaluation and diagnostics

410

transformation (integrating with respect to p between –i ∞ and i ∞ ) we obtain: Aα 3 γ 2 e −αγ 2 δ − Bα 3 γ 2 eαγ 2 δ − C

Pα α2 1 + γ 22 ) eαγ1δ − 1 (1 + γ 22 ) e −αγ1 ( δ− z0 ) =0, ( 2 2 γ1

where A=

2 2P α4 ∆A = − 0 ch[αγ1 (δ − z0 )]  4 γ1 γ 2 − (1 + γ 22 )  ,   ∆ ∆

B=

2 2P α4 ∆B = − 0 ch[αγ1 (δ − z0 )]  4 γ1 γ 2 + (1 + γ 22 )  ,   ∆ ∆

= C

∆ C P0 α 5 = ∆ ∆

1 −αγ z 2 3 −αγ z 2  (1 + γ 2 ) e 1 0 sh(αγ 2 δ) + 4 γ 2 (1 + γ 2 ) e 1 0 ch(αγ 2 δ) + γ  1

}

+ 4 γ 2 (1 + γ 22 ) e −αγ1 ( δ− z0 ) ,

{

}

∆ = α 6 (1 + γ 22 ) 4 γ1 γ 2 eαγ1δ − (1 + γ 22 ) sh(αγ 2 δ) − 4 γ1 γ 2 ch(αγ 2 δ) , γ1=

1−

2

Cα 2 ; γ 2= C12

1−

∗ Cα2 ; P = 1 + ν V0 . 2 0 C2 1 − ν 4π 2

Substituting the derived values we obtain the equation, correlating α and C α / C 2 ratio:

{

}

2 2 2γ 2 ch[αγ1 (δ − z0 )] (1 + γ 22 ) − 4γ1 γ 2  e −αγ 2 δ + (1 + γ 22 ) + 4γ1 γ 2  eαγ 2 δ −     ∆1

5.10 Numerical solution of equation (5.66) for plates of different thickness.

Acoustic emission

−

411

1 1 2 4 αγ ( δ− z ) 2 2 αγ ( δ− z )  (1 + γ 2 ) e 1 0 sh(αγ 2 δ) + 4 γ 2 (1 + γ 2 ) e 1 0 ch(αγ 2 δ) + 2∆1  γ1

}

+ 4 γ 2 (1 + γ 22 ) eαγ1 z0 − 2

1 1 + γ 22 ) e −αγ1 ( δ− z0 ) =0, ( 2 γ1

(5.66)



where ∆1 = − (1 + γ 22 ) sh(αγ 2 δ) − 4 γ1 γ 2 (1 + γ 22 ) ch(αγ 2 δ) − eαγ1δ  . 3

Figure 5.10 presents the results of calculation of the roots of equation (5.66) made by the numerical method using computer means. It follows from the graphs that in physical bodies limited by planes from two sides, unlike the half-space, the formed wave consists of an elementary wave packet propagating at velocities varying from 0.09C 2 up to 0.927C 2. With increase of plate thickness δ the maximum is achieved at a smaller value of wave number αδ (for instance, for δ = 1 cm αδ = 5 cm–1; for δ = 2.6 cm αδ = 2 cm–1). In the range of wave numbers from αδ up to infinity the velocity of wave packet propagation is the same and equal to 0.927C 2, corresponding to the velocity of Rayleigh wave for very thick plates. In other words, the greater the plate thickness, the closer to the ordinate axis the value of velocity of the classical Rayleigh wave, equal to 0.927C 2, is achieved. At δ tending to infinity in the spectral equation (5.66) and after simple transformations, we get a simpler equation for determination of the velocity of wave propagation in thick plates: 4 γ1 γ 2 − (1 + γ 22 ) =0. 2

(5.67)



C22 1 − 2ν into γ 1, at v = 0.3 we obtain = 2 C1 2(1 − ν) 2 2 γ1= 1 − 0, 286 Cα C2 . Solving equation (5.67) with these conditions, we get the velocity of the propagating wave for a plate, the thickness of which tends to infinity. This velocity is designated as С 3, is equal to 0.927С 2, and unlike thin plates it is constant in the entire range of wave numbers from 0 up to infinity. Let us now determine the displacements of plate surface. In terms of f and ψ displacement w is expressed as follows: 2 Substituting C1 from ratio

w( z ) = ∂φ / ∂z + α 2 ψ. Using the first equation (5.64), this formula may be written as follows:

w ( z ) = β1Ce zβ1 − Pe 1

−( z0 − z )β1

+ α 2  Ae − zβ2 + Be zβ2  at z ≤ 0.



(5.68)

Fundamentals of evaluation and diagnostics

412

5.11 Propagation of AE signal in 2 mm thick plate at t = 32 and 96 µs from the moment of its appearance (formula (5.69)).

1 + ν V0∗ δ( p − iαCα ). 1 − ν 4π Applying inverse Laplace and Hankel transformations to expres­s ion (5.68), at z = 0 we obtain:

= P1 where, as before,

w=

∞

∫ α αγ C + P e 1

0

−αγ1 z0

0

+ α 2 ( A + B)  J 0 (αr ) sin(αCα t )d α.



(5.69)

1 + ν V0∗ . 1 − ν 4π 2 We will determine value of displacements w on the plate surface (z = 0) after substitution of values of coefficients A, B and C into formula (5.69) and performing integration within specified limits. Figure 5.11 shows displacements of 2 mm thick plate surface at moments t = 32 and 96 µs from the start of emission. It should be noted that equation (5.66) in the range of wave velocities from 0 up to C 2 in very thick plate (at δ → ∞) allows for the existence of just one wave, propagating with velocity C3 = 0.927С2. Then, in view of the above-said, at large values of αγ 1δ expression (5.69) may be rewritten as: where P0 =

w≈−

1 + ν V0* 1 − γ 22 1 − ν 4π2 1 + γ 22

∞

∫ α exp(−αγ z ) J 1 0

α pl

0

(αr ) sin(αC3t )d α.

(5.70)

As is seen from this formula, integration is limited from the lower side by the magnitude of the limiting value of wave number α pl at which the velocity of wave propagation reaches the calculated value of 0.927C 2. If we assume plate thickness equal to infinity (in practice it is appro­x imately

Acoustic emission

413

5.12 Displacement w of 2.6 cm thick plate surface, caused by Rayleigh wave (С 3/С 2 = 0.927) at the moment when С 3t = 10 cm.

2.6 cm and more), then the lower limit of integration can be taken to be zero. Then integration in the limits from 0 to infinity can be performed in formula (5.70). After integration we have:  3 2 ( γ1 z0 ) + ( C3t ) 1 + ν V 1 − γ  − × w=  3/ 4 2 1 − ν 4π 1 + γ  ( γ1 z0 )2 − ( C3t )2 + r 2  + ( 2 γ1 z0 C3t )2    * 0 2

where

2 2 2 2

{

}

     C3t  3 2 γ1 z0 C3t  × cos arctg    ,  − arctg  2 2  ( γ1 z0 ) − ( C3t ) + r 2    γ1 z 0  2  

γ1=

1−

C32 ; γ 2= C12

1−

C32 ; C3 = C22

(5.71)

0.927C2 .

It should be noted that in formula (5.69) each elementary wave is defined by its velocity of propagation and its limiting wave number α. Thus, in the general case, wave calculations are performed by the same formula (5.69), but at different values of velocities and wave numbers. With increase of plate thickness the strength and shape of the acoustic emission signal change. From a multipeak, for instance in a 0.2 cm plate, with increase of plate thickness it gradually acquires the classical shape of a pulse for a half-space. It is seen from Fig. 5.12 which shows the AE

414

Fundamentals of evaluation and diagnostics

5.13 AE signal from transducer mounted on a plate 1 cm thick at 10 cm distance from an emission source after 250–400 kHz filter.

5.14 AE signal at moment of time 32.25 μs received from a source in a plate 1 cm thick after 200 ± 5 kHz filter.

signal in 2.6 cm thick plate. One can seen that already in this plate the AE signal becomes close to a pulse. Signal in the 2.6 mm plate can be regarded as the limit one transitional in its shape to signals in the half-space. It is also seen in Fig. 5.11 that with longer time since the moment of AE signal initiation, its parameters change significantly as the wave propagates. Decrease of the amplitude becomes particularly noticeable. This should be

Acoustic emission

415

taken into account when assigning the distance between the AE transducers during design of their layout on the objects of control. A feature of the AE signal arriving to the measuring instrument should be also noted. Signal parameters at the input to the measuring instrument strongly depend on the frequency band passed by AE transducers and the measuring instrument proper. Figure 5.13 shows AE signals filtered by transducers with a passband of 250–400 kHz. Comparing the signals in Figures 5.12 and 5.13 one can see that they differ not only by amplitude but also by frequency. These features should be taken into account when designing the AE transducers and measuring instruments. Acoustic emission signals which have passed through narrowband filters have a particularly essential difference. The graph in Fig. 5.14 gives as an example the AE signal in 1 cm plate after filtering in the range of 10 kHz (200±5 kHz). As is seen from the Figure, in this case the signal differs markedly from that shown in Fig. 5.12, and has a simple shape close to that of the Bessel function. Working with such a signal shape at calculation of the AE signal coordinates requires complicated program packages. Analysing equation (5.66), it is readily seen that it has a rigorous solution in the case if αγ 1δ = ∞, i.e. in two limiting cases: 1) at α = ∞; 2) at δ = ∞. In the first case there are no waves in the plate, as integra­tion with respect to α in formula (5.68) is performed between ∞ and ∞ . The second case, as shown above, is represented by the half-space. A propagating wave w does not depend on spectral number α, has the velocity of propagation C 3 and is found from formula (5.66) by proceeding to the limit δ → ∞. The ambiguity of the notion of the infinity in engineering calcu­lations should be noted. As a rule, all the processes, considered in engineering calculations, are finite, and the notion of infinity can be applied in them only with certain assumptions. Analysing expression (5.66), it is to be noted that all the coefficients at the grouped terms are functions, where zero or infinity is achieved by the argument tending to infinity. So, when the plate thickness tends to zero, we have a half-space. Now, what is a half-space for an engineering problem: a very thick plate or a plate of a medium thickness? It is difficult to reply to this question. In view of the above, let us try to explain some processes of wave propagation in plates in physical terms. Formula (5.69) represents the propagating wave as a sum of elementary waves described by the integrand. Elementary waves included into the sum, differ from each other by wavelength, frequency and amplitude. Thus, not all the waves from the packet reach the plate lower boundary. Those waves that do not reach the lower boundary, will satisfy the conditions of pseudoinfinity across the thickness, repre­s enting in sum a wave propagating over the plate upper surface. Considering that such waves will include the shorter waves, the wave packet in expression (5.69) will be truncated from the lower side (by the lower integration limit).

Fundamentals of evaluation and diagnostics

416

5.15 Displacements w and u over the thick plate upper surface (z = 0), induced by longitudinal and transverse waves (formulas (5.71), (5.73)).

Thus, we have simplified formula (5.69) for greater values of αγ 1δ. The value of lower integration limit in (5.70) α pl is the wave number magnitude, at which zero is ensured in spectral equation (5.66) at specified plate thickness Displacement of plate outer surface is made up of two compo­n ents: along axis z – component w induced by transverse waves, and along axis r – component u induced by the action of a longitudinal wave. It can be shown that the second component can be neglected for waves propagating at velocities below C 2 , because of its smallness compared to the first component. Thus, while the longitudinal displace­m ents in a plate under the action of an instant emission source in the transformed form can be written as: = u

∂  ∂ψ  φ+ , ∂r  ∂z   

the formula for determination of displacements u of plate upper surface along axis r can be written as: u=

∞

∫α

α pl

2

  P0 −αγ1 z0 − αγ 2 ( A − B)  sin(αCα t ) J1 (αr )d α, e C + αγ1  

(5.72)

and for plates with great αγ 1 δ values, after substitution of values of coefficients A, B and C, we obtain:

Acoustic emission

= u

1 + ν V0* 1 − γ 22 1 − ν 4 π 2 2 γ1

∞

∫ αJ (αr )sin(αC t )e 1

3

−αγ1 z0

417

d α.

α pl

After integration of this expression between 0 and infinity, we have:  3  2 γ1 z0 C3t  r cos  arctg 2 2 2  ( γ1 z0 ) + r − (C3t )  1+ ν V 1− γ  2 u=  2 1 − ν 4 π 2 γ1  ( γ1 z0 ) 2 + r 2 − (C3t ) 2  + (2 γ1 z0 C3t ) 2   * 0 2

2 2

{

   . 3/ 4  (5.73)   

}

The graphs of displacements u and w given in Fig. 5.15 show that the value of displacements for a thick plate along the axis r induced by the action of the longitudinal wave, is approximately 2 times smaller than the displacements induced by the transverse wave. In the case, when the velocities of propagation of elementary waves are higher than C 2, the solution of the differential equation for ψ in the system (5.62) will take a form different from (5.64). This is attributable to the fact that in the real region the radicand for the value β 2 becomes negative, and the value β 2 proper becomes a complex quantity. The solution for this case may be written as:

= φ Ce zβ1 +

1 + ν V0* e −|z − z0 |β1 δ+ ( p − iαCα ) ; 1 − ν 4π β1

= ψ A sin ( zβ2 ) + B cos( zβ2 ). Substituting the expressions for f and ψ into the equations for the boundary conditions (5.63), we have: − z0 β1 B ( α 2 − β22 ) + C ⋅ 2β1 + 2 Pe =0 ; 1

A ( α 2 − β22 ) sin(δβ2 ) + B ( α 2 − β22 ) cos(δβ2 ) + 2Cβ1eδβ1 − − ( δ− z0 ) β1 0; − 2 Pe = 1

 p2  P  p2  − A α 2β2 − C  2 + α 2  − 1  2 + α 2  e − z0β1 = 0 ;  2C2  β1  2C2 

(5.74)

 p2  − A α 2β2 cos(δβ2 ) + B α 2β2 sin(δβ2 ) − C  2 + α 2  eδβ1 −  2C2  −

P1  p 2 2  − ( δ− z0 ) β1 =0,  2 + α e β1  2C2 



Fundamentals of evaluation and diagnostics

418 where

= P1

1 + ν V0∗ δ( p − iαCα ) ; 1 − ν 4π

α 2 + p 2 / C12 ;

β1=

p 2 / C22 + α 2 .

β2=

Taking the inverse Laplace transformation for the first three equations (5.74) and solving the system, we will find the expression for coefficients A, B, and C: A=

B=

C= where

(

)

4 P0 1 − γ 22 ch αγ1 ( δ − z0 )  ; α 2  4 γ1γ 2 cos ( αδγ 2 ) − eαδγ1 + 1 − γ 22 sin ( αδγ 2 )    16 P0 γ1γ ch αγ1 ( δ − z0 ) 

(

(

) (

)

) ( ) ( ) P (1 − γ ) sin ( αδγ ) − e + , ( cos ( αδγ ) − e ) + (1 − γ ) sin ( αδγ )

α 2 1 − γ 22  4 γ1γ 2 cos ( αδγ 2 ) − eαδγ1 + 1 − γ 22 sin ( αδγ 2 )    2 2

0

4 γ1 γ 2

2

1−

γ1=

Cα2 γ2 ;= C12

2

;

αγ1 z0

2

αδγ1

2 2

2

∗ Cα2 − 1; P = 1 + ν V0 . 2 0 C2 1 − ν 4π 2

Substituting the values of coefficients A, B and C into the fourth equation (5.74), we get a spectral equation of the plate for the case when С α > C 2. After some transformations, we obtain:

(

)

−αγ δ− z −αγ1 z0 2 + 4 γ1γ 2 e 1 ( 0 ) cos ( αδγ 2 ) e −αγ1z0  eαγ1δ 1 − γ 2 sin ( αδγ 2 ) e   1− γ2 − 2 2 γ1

− + −

(

(

)

4 γ 2 1 − γ 22 cos ( αδγ 2 ) ch [ αγ1 (δ − z0 ) ]

(

)

2

4 γ1γ 2 cos ( αδγ 2 ) e −αδγ1  + 1 − γ 22 sin ( αδγ 2 )

+

16 γ1γ 2 sin ( αδγ 2 ) ch [ αγ1 (δ − z0 ) ]

(1 − γ ) 4γ γ ( cos ( αδγ ) e 2 2

1 2

2

−αδγ1

) + (1 − γ ) 2 2

2

)

sin ( αδγ 2 )  

−

(5.75)

1 − γ 22 −αγ1 ( δ− z0 ) = 0. e 2 γ1

Knowing the spectral equation of the plate (5.75), i.e. knowing the correlation between wave number α and the velocity of propagation of elementary waves C, we can determine displacements w, arising in a plate at wave propagation at velocities higher than C 2. Thus, displace­m ents over the surface of the plate at z = 0 with the assumed initial conditions after

Acoustic emission

419

application of inverse Hankel transformation with respect to α and Laplace transformation with respect to p will take the form of:

(

)

2 −αγ δ− z −αγ1 z0 2  + 4 γ1γ 2 e 1 ( 0 ) cos ( αδγ 2 ) e −αγ1z0   1 − γ 2 sin ( αδγ 2 ) e  + w= 2 ∫0 P0 − −αδγ1 2 + 1 − γ 2 sin ( αδγ 2 ) 4 γ1γ 2 cos ( αδγ 2 ) e   16γ1γ 2 ch [ αγ1 (δ − z0 ) ] + + (5.76) 2 2 1 − γ 22  4 γ1γ 2 cos ( αδγ 2 ) e −αδγ1 + 1 − γ 22 sin ( αδγ 2 )    ∞

(

(

)

(

) (

) (

)

)

}

+e −αδ1z0 α J 0 (α r )sin ( αCα t ) d α. Displacements over the plate surface during wave propagation will be equal to the sum of displacements found from formulas (5.69) (Rayleigh wave) and (5.76) (Lamb waves). It should be noted, however, that calculations of displacements by formula (5.72) are made difficult by the need to find the roots of spectral equations to derive the f(α,C α) dependence. This solution requires powerful computing facilities and a lot of machine time for computation, as calculations are made within the limits from 0 to infinity for varying α and C α. The same also pertains to the expressions for w. On the other hand, analysing the graphs of the dependence of roots C α = f(α) for elementary components of Lamb waves (equation (5.75)), shown in Fig. 5.16 (see colour section), it may be seen that function C α = f(α) has a continuous value in terms of wave numbers and is discrete for different orders of roots, i.e. a discreteness of α and C α values by the numbers of the orders of the roots (elementary waves) of the spectral equation is found (5.75). Thus, in the case of wave propagation in thin plates, we deal with piecewise-discontinuous half-spaces of the dependences between α and C α. At the same time, note that all the mode orders given in Fig. 5.16, alongside the individual waves and their short packets, single out one wave in the entire frequency range, propagating at the velocity of 0.4433 cm/μs. This wave exactly represents the main most significant displacement of the plate surface, as is shown in all the drawings for the Lamb wave given below. For calculation of displacements w let us apply the integration in expression (5.76), allowing for the discreteness of C values for continuous values of the wave number α. We will assume that the integrand values will be determined as the sum of its values for each C i, found from expression (5.75). Then

Fundamentals of evaluation and diagnostics

420

(

)

2 −αγ1 z0 2  + 4 γ1γ 2 e −αγ1 ( δ− z0 ) + cos ( αδγ 2 ) e −αγ1z0   1 − γ 2 sin ( αδγ 2 ) e + P0 − ∑ 2 i =1 4 γ1γ 2 cos ( αδγ 2 ) e −αδγ1  + 1 − γ 22 sin(αδγ 2 )  ∞ (5.77) 16 γ1γ 2 ch [ αφ1 (δ − z0 ) ] w=∫ + + 0 2 1 − γ 22  4 γ1γ 2 cos ( αδγ 2 ) e −αδγ1  + 1 − γ 22 sin(αδγ 2 )    k

(

where

+ e −αγ1z0

γ1=

(

) } αJ (α r )sin ( αC t ) d α, 0

1−

Ci2 γ2 ;= C12

(

)

)

α

∗ Ci2 − 1; P = 1 + ν V0 . 2 0 C2 1 − ν 4π 2

Here the values γ 1 and γ 2 also become i-th values. These signs are not incorporated into the above formula to avoid unnecessary ambiguity of the expression. Integration of (5.77) may only be performed by the numerical method and is applied as follows. 1. We find the values of velocities C i from spectral equation (5.75) for each assigned value of wave number α i. 2. For the selected r j we plot the integrand value, summary for k, along axis α i for all k orders of roots C i. 3. We integrate expression (5.77) for the obtained values of the integrand and go over to the next value of r j+1. 4. We repeat operations 1–3 for all values of r j. Now, using the accepted procedure, let us calculate for Lamb waves of the displacements in the plates of alloy 09G2S, when an inner local source of emission appears in them. We find the impulse of disturbance of a local volume of the material at depth z 0 from the plate surface from the following dependence: V0∗=

4 3 4 πR = π ⋅ 0.0013= 4.2 ⋅10−9 , cm3 . 3 3

The graphs in Fig. 5.17 give the results of wave calculation at different moments of time. The graphs in Figures 5.18 and 5.19 show waves at different moments of time, but after filtering by AE transducers with different frequency bands, passing the initial AE signal. Analyzing Fig. 5.17a, we see one wave propagating at velocity C = 0.4382 cm/μs. At the next moment of time equal to 1452 μs, the second wave appears on the plate surface which propagates at a somewhat lower velocity C = 0.427 cm/ μs. The velocities of propagation of the two waves differ only slightly, but the distance between them at the moment when t = 1452 cm/μs, is rather large, and equal to 17 cm. This is quite enough for the AE transducers to operate twice giving inaccurate information about the emission source

Acoustic emission

421

5.17a AE signals on the surface of 1 cm steel plate in the range of frequencies of 0.35–1750 kHz (α = 0.005 – 25 1/cm) at moment of time of 967.7 μs.

5.17b AE signals on the surface of 1 cm steel plate in the range of frequencies of 0.35–1750 kHz (α = 0.005–25 cm –1) at moment of time of 1452 μs.

coordinates. As seen from Fig.5.17b, the distance between the wave peaks becomes greater by the moment when the waves have propagated for a distance of more than 6 m. At the same time, it can be noted that the wave velocities vary about the value 0.44 cm/μs. Here one of the waves propagates slower and slower. It should be noted that the 3 m distance can be regarded as the ‘critical’ distance at which wave splitting can be practically neglected. Fig. 5.18a–d shows the change of the AE signal spectrum depending on the ‘pass’ band of the AE transducer. It is seen that with band displace­ment

422

Fundamentals of evaluation and diagnostics

5.18a AE signals on the surface of 1 cm steel plate in the range of frequencies of 42–70 kHz (α = 0.6–1.0 cm -1) at moment of time of 645 μs.

5.18b AE signals on the surface of 1 cm steel plate in the range of frequencies of 70–140 kHz (α = 1–2 cm –1) at moment of time of 645 μs.

5.18c AE signals on the surface of 1 cm steel plate in the range of frequencies of 210–280 kHz (α = 3–4 cm –1) at moment of time of 645 μs.

Acoustic emission

423

5.18d AE signals on the surface of 1 cm steel plate in the range of frequencies of 420–560 kHz (α = 6–8 cm –1) at moment of time of 645 μs.

5.18e AE signals on the surface of 1 cm steel plate in the range of frequencies of 840–980 kHz (α = 12–14 cm –1) at moment of time of 645 μs.

into a region of higher frequencies, the signal becomes more complicated. It is also interesting to note that in the considered frequency range the displacement amplitude first rises and then drops abruptly with increase of the frequency of the AE transducer ‘pass’ band. In the band of 840–980 kHz the wave amplitude drops to 0.04⋅10 –6 cm, whereas the displacement values for frequencies of 420–560 kHz are equal to 4.2⋅10 –6 cm. For the waves travelling in a fixed frequency range, just as for waves propagating in a wide spectrum, a ‘splitting’ is observed at large distances from the emission source (Fig. 5.19). The depth of location of a defect – emission source essentially affects the amplitude of the AE signal on the plate surface. Figure 5.20 gives the graphs of this dependence for a wave in the range of 0.35–1750 kHz 32 μs after emission. Figure 5.21 gives the dependence of K values on f that was taken into account at selection of the frequency band in calculations of waves given in Figures 5.18 and 5.19.

424

Fundamentals of evaluation and diagnostics

5.19a AE signals on the surface of 1 cm steel plate in the range of frequencies of 244–298 kHz (α = 3–4 cm –1) at moment of time of 1290 μs.

5.19b AE signals on the surface of 1 cm steel plate in the range of frequencies of 244–266 kHz (α = 3–4 cm –1) at moment of time of 2742 μs.

5.20 Elastic wave parameters in the range of 0.35–1750 kHz 32 μs after emission, depending on source location depth

Acoustic emission

425

5.21 The coefficient of electroacoustic transformation K for the AE transducer of the DAE-002Р type.

It should be noted that the presence of the second surface in the plate has a significant role in wave formation. Therefore, with the increase of plate thickness the wave field in the plate does not tend to form one wave, as in the case of the half-space (Rayleigh wave). In thick plates, several types of waves exist simultaneously, propagating at different velocities. Surface displacements for very thick plates are found from a formula derived from (5.73) by proceeding to a limit when plate thickness tends to infinity (δ → ∞): = w

1 − ν V0* 1 + ν 2π 2

∞ k

∫ ∑ αJ 0 i =1

0

 1 (αr ) cos(αCi t ) 1 + 2 2 2 − C i / C2 

  exp(−αz0 γ1 )d α. 

Figure 5.1 a shows a diagram of a wave, propagating from an emission source, which is located at the depth of 0.2 cm from the surface of a plate 1 cm thick in a broad frequency range. Comparing the parameters of this wave with those of a similar wave (Fig.5.18a–e), we can see quite a significant difference. Thus, the limited band of frequencies, in which AE transducers operate, may lead to a significant distortion of the shape of the AE wave and, as a consequence, to improvement or deterioration of the wave recording and the accuracy of determination of AE source coordinates. We have considered the theoretical fundamentals of generation of elastic waves in materials with defects, locally propagating in them at deformation. The motion of an isolated dislocation in our calculations is equated to the motion of a point source. If the motion of a source (dislocation) does not end in an isolated jump, the pattern of a wave field, formed by a moving dislocation or a propagating crack, may be derived by superposition of quanta of each isolated source, with adding up of their impact on each other.

426

Fundamentals of evaluation and diagnostics

7 Waveguides Information is sometimes transferred to the AE transducer through elements, intermediate between the item and the transducer, namely waveguides. Such structural elements of information transfer are required in those cases when the direct contact of the AE transducer with the structure surface is impossible, either for the reason of a high temperature of the material, or because of the impossibility of pene­trating to the location where it is necessary to mount the transducer. Waveguide designs may be different, however, their main parameters, related to signal transfer from the item to the transducer, are identical in the general case. Let us consider one of the typical designs of a waveguide, presented in Fig. 5.22, consisting of the waveguide proper – a thin cylindrical rod and two heads for fastening the waveguide to the item surface and for connecting the acoustic transducer. A thin layer of an optically transparent material should be applied between the AE transducer and the head, to which it is connected, to provide a reliable transmission of signals. Waveguide design is performed to evaluate their transmission properties for subsequent calibration and qualification. 1. General case. If we assume that deformations arise on the end face of a cylindrical waveguide, then the waveguide transfer function can be calculated, using the first equation of system (5.61), which does not incorporate the component of correlation of the wave frequency, its propagation velocity and the wave number: ∇2φ −

1 ∂2φ 1 + ν ∗ δ+ (r ) V0 =− δ( z − z0 )δ + (t ). 2 2 1− ν 2πr C1 ∂t

5.22 AE transducer with a waveguide mounted on the surface of a pressure vessel.

Acoustic emission

427

For this purpose let us multiply both parts of the equation by r and integrate for r going from 0 to r 2 (r 2 is the radius of the waveguide outer surface): r2

r

2  1 ∂  ∂φ  ∂ 2 φ 1 ∂ 2 φ  1 + ν V0∗ ( ) ( ) + − = − δ δ r r dr z t   +   ∫0  r ∂r  ∂r  ∂z 2 C12 ∂t 2  ∫0 δ+ (r )dr. 1 − ν 2π

Assuming that derivatives with respect to z and t in the left-hand part do r r2 ∂φ 2 = 0 and ∫ δ + (r )dr = not depend on r and considering that r 1, we obtain: ∂r 0 0 ∂2φ 1 ∂2φ 1 + ν V0∗ − = − δ( z )δ + (t ). 1 − ν πr22 ∂z 2 C12 ∂t 2

(a)

Taking Laplace transformation with respect to variable t with parameter p and Fourier cosine transformation with respect to variable z with parameter γ:  p2 −  γ2 + 2 C1 

 1 + ν V0∗ φ = −  1 − ν πr22 

2 , π

whence 1+ ν 2 ∗ 1 w0 2 φ= , 1 − ν π γ + p 2 / C12 w0∗ V0∗ / πr22 . where = Inverse Laplace and Fourier transformations will yield: ∞

φ=

dγ 1 + ν 2 w0*C1 1 + ν w0*C1 =− π at 0 ≤ z ≤ C1t; cos( γz )sin( γC1t ) ∫ γ 1− ν π 0 1− ν π at z > C1t.

ϕ=0

This is a rectangular impulse, propagating along axis z of the waveguide (Fig.5.23). The magnitude of displacements in this case is an impulse in the form of a delta-function, propagating through the waveguide along axis z (Fig.5.24), and it will be equal to ∂φ 1+ ν ∗ w= = − w0 C1δ + ( z − C1t ). ∂z 1− ν

(5.78)

If the displacement impulses, transmitted to the waveguide by the object, have a complex shape, the problem may be solved using the superposition

Fundamentals of evaluation and diagnostics

428

5.24

5.23

principle and the equation, establishing a correlation between the plate and the waveguide. So, for instance, if an elementary increment of the shape of the signal, transmitted to the waveguide, is described by expression = dw0∗ wpl∗ exp(−k 2 z02 )dz0 ,



(5.79)

where w*pl is determined by the considerations described below, then differentiating expression (5.78) with respect to w0* and shifting the coordinate z by the value z 0 (the coordinate of synchronising of the processes of displacements in the plate and the waveguide), we obtain first an elementary displacement of the waveguide end face (Fig. 5.24): dw = −

1+ ν ∗ dw0 C1δ( z − z0 − C1t ), 1− ν

and then also displacement w: z

− w=

1+ ν ∗ 1 wï C1 ∫ exp(−k 2 z02 )δ( z − z0 − C1t )dz0 = 1− ν 0

1+ ν * wplC1 exp  −k 2 (C1t − z ) 2  . = − 1− ν It should be noted that relationship (5.79) may be considered as an equation correlating the displacements arising in a plate and trans­ferred by the plate to the waveguide (right-hand part), and displace­m ents, thus arising in the waveguide (left-hand part). The dimensions of w*pl will be cm ⋅ s / cm. Such dimensions indicate that w*pl is an impulse of displacements on the surface of the object of control, distributed along radius r of the object. Equation (5.79) is called the equation of constraint. This equation may be specified with respect to a linear coordinate (in this case z 0) and with respect to time.

Acoustic emission

429

2. Influence of medium resistance. It is of interest to study the influence of the properties of waveguide material in a broader range on the shape and amplitude of a propagating wave. For this purpose let us introduce in differential equation (а) (level 1) dynamic coefficient of the medium resistance S (see §29, item 5). Then, differential equation becomes ∂ 2 φ αS ∂φ 1 ∂ 2 φ 1+ ν − − 2 2 = − w0∗ δ + (t )δ( z ). 2 C t ∂ 1− ν C1 ∂t ∂z 1 Taking for the same boundary and initial conditions Fourier cosine transformation with respect to coordinate z and Laplace transformation with respect to time t, we obtan: φ=

Ñ12 2 ∗ 1+ ν w0 . π 1 − ν α 2 Ñ12 + αSC1 p + p 2

Inverse Laplace transformation gives as:

= φ

2 ∗ 1+ ν w0 π 1− ν

 S2  αSC1t   − 1  at S > 2; exp  − sh  αC1t   2   4  S2  α −1 4

= φ

2 ∗ 1+ ν w0 π 1− ν

S2  αSC1t   α C t − exp  − sin 1  1  2   4  S2 α 1− 4

C1

C1

  at S < 2.  

Having differentiated the expression for f at S < 2 with respect to z and taken the inverse Fourier transformation with respect to α, for diaplacements w, we obtain = w

2 1+ ν π 1− ν

S2 ∂  αC1 St    exp sin 1 − α tC − 1  ∫ 2   4  S 2 0 ∂z 1− 4

w0∗C1

∞

 dα  cos(αz ) .  α 

After differentiation and integration, we finally obtain: w = − w0∗

C13t 2 Sz 2 1+ ν 2 2 π 1− ν  2 2 2     (C St ) 2    ( ) C St S S 1 1  +  C1t 1 − − z  +  C1t 1 − + z     4     4 4 4       

430

Fundamentals of evaluation and diagnostics

5.25 AE wave propagation in a waveguide with different resistance of the material to its propagation.

For values S = 0.1 and S = 1, the calculation results are given in Fig. 5.25. In the graph in Fig. 5.25 the wave, propagating in a waveguide with medium resistance S = 0.1 from an instantly applied displacement impulse, has a sufficiently steep front and unlike the first case with the practically infinite amplitude, it has a limited amplitude decreasing with greater distance from the emission point. After some time, its front becomes more sloping. It is, however, still sufficiently steep in terms of the rise and drop gradient for satisfactory recording by the measuring means. With increase of value S up to 1, the wave front becomes gently sloping and practically unsuitable for its recording by measuring instruments. Derived formulas allow comparing various materials of waveguides mounted on structures with application of complete penetration welding.

8

Simulation of acoustic emission signals

Plate heating by a concentrated laser beam can be one of the methods of simulating the acoustic emission signals (Fig. 5.26). Concentrated heating induces a temperature field with a high gradient in the plate and, consequently, high stresses, leading to the appearance of a zone of plastic deformations. In this case, the acoustic wave may be caused by two processes, namely an impulse process, related to the emergence and disappearance of elastic stresses, and a process related to a sudden development of abruptly induced and remaining plastic deformations. The presence of a particular process is related to the power of the used heat source. Wave processes in a plate, as we know, in the general case may be described by the following system of differential equations:

Acoustic emission

431

5.26 Schematic of a device for simulation of acoustic emission in materials.

∇ 2φ −

1 ∂ 2φ 1 + ν ∗ = εi ; C12 ∂t 2 1 − ν

1 ∂ 2ψ ∇ ψ − 2 2 =0 , C2 ∂t

(5.76)

2



1 ∂  ∂  ∂2 . r + r ∂r  ∂r  ∂z 2 The second equation in a plate of infinite dimensions will be identically equal to zero in connection with the absence of shear deformations. Representing in the plane variant value ε i* as ε 1* for an impulse source and ε*1 for an abruptly arising and remaining source, respectively: = ∇2 where, as before,

δ + (r − r0 ) δ + ( z )δ + (t ) ; 2πr δ (r − r0 ) = ε∗2 V2∗ + δ + ( z ) S + (t ) , 2πr

= ε1∗ V1∗

where V1* is the abruptly arising and disappearing change of material volume (impulse of volumetric expansion of the material) in a ring of radius r 0, cm3 ⋅ s; V2* is the suddenly arising in a ring of radius r0 and remaining change of material volume, cm3; δ+(...) and S+(...) are the delta-function and function of a unit jump, respectively, we successively obtain two solutions. The first differential equation (5.80) may be reduced to a plane problem by integration with respect to the coordinate z between 0 and δ. Assuming that derivative r and t in the left-hand part of (5.80) are independent of z δ ∂φ and considering that = 0, we get: ∂z 0

Fundamentals of evaluation and diagnostics

432

1 ∂ 2 φ 1 + ν ε∗i = . C12 ∂t 2 1 − ν δ

∇2φ −

Substituting the value of function ε∗i into the obtained expression and applying Laplace transformation with respect to time t and Hankel transformation with respect to coordinate r, we get: φ1 =−

1 + ν V1∗C12 J 0 (αr0 ) ; 1 − ν 2πδ α 2 C12 + p 2

(5.81)

1 + ν V2∗C12 J 0 (αr0 ) 1 φ2 =− . 1 − ν 2πδ α 2 C12 + p 2 p

Displacements w acting on AE transducers, mounted on the plate surface, can be found from the following formula: δ

δ

0

0

w = ∫ ε z dz = − ∫

ν νδ (σ r + σθ )dz = − (σ r + σθ ). E E



(5.82)

When it is considered that  ∂2φ 1 ∂2φ 1 ∂2φ  σ r + σθ = −2G  2 + − , r ∂r 2 C22 ∂t 2   ∂r then, substituting into (5.82) the expression for σr + σθ and applying to it Laplace and Hankel transformations, we may write: w =−

νδ  2 p 2   α + 2  φ. 1+ ν  C2 

After substitution of the value of function φ from (5.81) into this expression and taking inverse transformations, we obtain for displacements w 1 and w 2 : = w1

= w2

∞ ν V1*C1  C12  2 1 − 2  ∫ α J 0 (α r ) J 0 (α r0 )sin(α C1t )d α; 1 − ν 2 π  C2  0

(5.82а)

∞   C12   ν V2* α α α J ( r ) J ( r ) cos (αC1t )  d α. (5.82b) 0 0 0 1 −  1 − 2  ∫ 1 − ν 2π 0   C2  

If the heat source is concentrated, the value r 0 can be taken to be equal to zero. In this case, integration in (5.82) is performed quite easily, yielding

Acoustic emission

= w1

ν V1*C1  C12  r 2 + 2C12t 2 ; 1 −  1 − ν 2π  C22  C 2t 2 − r 2 5/2 1

(

)

 ν V2  δ(r )  C12  Ct = + 1 − 2  w2 2 2  1 − ν 2π r  C2  C1 t − r 2 

(

)

  3/2  

433

(5.83)

Inconvenience of analysis of comparatively simple expressions (5.83) consists in infinity being equal to w i at C 12t 2 – r 2 = 0 . In this case it is impossible to trace the decrease of the maximal amplitude of displacements with time and increase of distance r from the heat source. Let us consider the case (Fig. 5.27), when the heat source, acting on the plate, is defocused, and the initiating deformation waves, generated in the plate, may be described by the following expression

ε∗ =αT Θ0 exp(− K 2 r02 ),

(5.84)

where αT is the coefficient of linear expansion of the plate material (1 / °C); Θ0 is the impulse of temperature change in the heating spot centre (°C ⋅ s); K is the coefficient of beam concentration (cm –1). Let us represent an infinitely small impulse of the change of the plate material volume, caused by heating (5.80): dV1∗ = 2πδr0 αT Θ0 ta exp(− K 2 r02 )dr0 , where ta is the time of action of the heat source. Then expression (5.82a) becomes dw1 =

5.27

∞ ν dV1∗C1  C12  2 − 1   α J 0 (αr ) J 0 (αr0 ) sin(αC1t )d α. 1 − ν 2π  C22  ∫0

(5.85)

Fundamentals of evaluation and diagnostics

434

Taking into account formula (5.81), we may write:

w1 =

 C2  ∞ ∞ ν αT Θ0C1δtd 1 − 12  ∫ ∫ α 2 r0 J 0 (αr ) J 0 (αr0 ) × 1− ν  C2  0 0

× sin(αC1t ) exp(− K 2 r0 )dr0 d α. Upon integrating with respect to r 0, we obtain: = w1

∞  α2 ν αT Θ0 C1δta α 2 J 0 (αr ) sin(αC1t ) exp  − 2 2 ∫ 1− ν K  4K 0

  d α. 

(5.86)

The expression, derived for w 1, is quite complicated for integ­r ation. We will proceed as before, substituting instead of Bessel function its integral representation J 0= (αr )

2 π

π/ 2

∫ cos(αr sin θ)d θ. 0

Then, integrating (5.86), we obtain = w1

ν αT Θ0 C1δKta  C12  1 − 2  × 1− ν π  C2 

π/ 2  × ∫   1 − 2 K 2 (r sin θ + C1t ) 2  exp  −(r sin θ + C1t ) 2 K 2  + 0 

 +  1 − 2 K 2 (r sin θ − C1t ) 2  exp  −(r sin θ − C1t ) 2 K 2   d θ. 

(5.87)

Analysing (5.87), it is easy to see that the integrand consists of a sum of two functions, differing in their properties. While the first function describes a certain process, attenuating at greater distance from the source, the second one describes the wave of plate surface displacements, propagating with time. Figure 5.28 shows the distribution of an elastic cylindrical wave, caused by impulse heating (action time t a = 10 –7 s) 5 of 0.5 cm thick plate of an aluminium–magnesium alloy AMg6 by the laser beam at the coefficient of beam concentration K = 20 cm –1 (δ-source). The source acted on an area of a ring of radius r 0 = 0.1 cm. Temperature of plate heating in the spot centre, Θ 0, was 600°C (melting point of alloy AMg6 was equal to 628°C) and was calculated from the following formula: This time is approximately equal to the time of dislocations displacement in the lattice.

5

Acoustic emission

435

5.28 Propagation of cylindrical wave in plate under the impact of impulse heat source.

q Θ0 = , 2πλδ where q is the specific power of the heat source, W (cal / s); λ is the coefficient of heat conductivity of the plate material, W / m ⋅ °С (cal / cm ⋅ s ⋅ °С). For our case it was assumed that λ = 0.33 cal / cm ⋅ s ⋅ °С, q = 624 cal / s . The other calculated values, used in calculation of the waves, are as follows: α Т = 25 ⋅ 10 –6 1 / °С; ν = 0.33; С 1 = 6117.4 m / s ; С 2 = 3058.3 m / s . In the case, when the emission source has the form of suddenly applied heat, expression (5.78 b) may be used to derive the dependence, describing the wave propagation. Then, if dV2∗ = 2πε∗0 δr0 exp(− K 2 r02 )dr0 , upon integrating in (5.78b) we get: w2 =

   C 2  ∞ C tr exp(− K 2 r02 ) ν  dr αT Θ0 δ exp(− K 2 r 2 ) − 1 − 12  ∫ 1 0 0 .   1− ν C2  0 C 2 t 2 − (r − r ) 2  3/ 2  0  1    (5.88)

It is not possible to integrate the second term in the brackets of expression (5.88) with presentation of the result in the form of elementary or special functions. Therefore, as always in such cases, this integral is to be taken numerically, using personal computers. This technique can be easily implemented, considering that coefficient K of the heat source concentration

436

Fundamentals of evaluation and diagnostics

5.29 Propagation of a cylindrical wave in a plate under the impact of suddenly applied heat.

5.30 Propagation of cylindrical wave in plate under the impact of suddenly applied heat (the second case) .

is usually equal to 10 cm –1 and higher. In these cases the upper limit of integration in (5.88) does not exceed 0.4 cm. Figure 5.29 presents the graph of propagation of a cylindrical wave of acoustic emission under the impact of a laser beam on the plate with remaining consequences of this impact. As is seen from the graph, initial deformation from suddenly applied heat, while penetrating through the entire thickness of the plate, remains in the origin of coordinates (first term in the brackets of expression (5.88)). The graph also presents displacements of the plate surface w 2 at the moment of time t = 3 / С 1 ≈ 4.9 ⋅ 10 –6 s . Comparing the graphs in Fig.5.28 and Fig.5.29, it may be seen that they differ significantly. While a propagating wave from an impulse source practically reproduces its action – first compression of the plate material zone and then tension, a suddenly applied source causes a moving uniform compressive impulse.

Acoustic emission

437

If suddenly applied heat is uniformly distributed over a small area, the formula for calculation of displacements in a plate may be derived from (5.88) with replacement of the exponential distribution of heat by a rectangular one. Then, after integration and simple trans­formations we have:  ν  = w2 αT θ0 δ  S (r ) − S (r − a0 ) + 1− ν    C2 + 1 − 12  C2

  C12 t − r 2 − C12 t − (r − a0 ) 2  C1t  

  

  . 

(5.89)

Displacements, calculated for this case, are given in the graph in Fig. 5.30. The initial data, used for calculations, were the same as for the first two cases. There being no re-reflections of the wave from the plate surface (the wave is cylindrical) the change of displacements proceeds quite smoothly, this being convenient for the calibration of AE transducers.

9

Determination of the coordinates of acoustic emission sources



We considered the structure and functional capabilities of typical diagnostic equipment based on acoustic emission. The most important part of the equipment, however, is its software. The software of the diagnostic systems is conditionally divided into three parts: 1. The part, related to obtaining and preliminary processing of initial information, and reducing it to a form, convenient for further processing in a PC. 2. The part, related to determination of time delays and calculation of AE source coordinates. 3. The part, related to evaluation and forecasting of the condition of the objects of control. We have considered the first part of the software in sufficient detail in Section 4. In this paragraph, we consider the principles and main calculation dependences to determine the coordinates of a propa­g ating defect. Determination of the coordinates of AE events is based on measurement of the difference of the times of arrival of AE signals to the transducers, mounted in a certain sequence on the structure surface. Recording of the time of AE transducers responding starts after opera­tion of the first one, which is the closest to AE event. The transducer, which has responded, starts the timer and time count begins, this time being recorded and assigned to each operated transducer. To facilitate and speed up the calculations the transducers are combined into special antennas. Such antennas usually have four transducers, this providing the most optimal operating conditions. Any number of measuring

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438

5.31

antennas may be used, depending on the specific control goals. The most typical configuration of the antennas, are systems with a small number of measurement channels (8–12). Increase of the number of channels makes their maintenance inconvenient, because of a large number of connecting wires and transducers, requiring special storage and servicing. Transducer antennas are mounted on different surfaces of structures and depending on their type, are called linear, flat, cylindrical, spherical, etc. In each of these cases the most effective layout of the transducers is selected. Let us consider some of the most typical cases of determination of the coordinates at different configurations of the measuring antennas.

1

Linear antennas of transducers

Antennas with transducers, located on one line, are mounted on structures, having all the dimensions predominantly along one direction. These are pipelines, columns of truss structures, etc. In this case, the formulas for determination of the coordinates of the arising AE sources may be derived from the following reasoning. Let AE transducers be located at points D 1 and D 2 of the object (Fig. 5.31). Distance b between them is called the measuring base. If an AE event occurs on line D 1 – D 2 at point A act at distance x from the first transducer, and value x < b – x , transducer D 1 will be the first to respond, followed by transducer D 2 after a certain time ∆t 1,2 , called the difference of the time of AE signal arrival to the first and the second transducer. In terms of geometry it is clear that 2х + С∆t1,2 = b,

(5.90)

where C is the velocity of the elastic wave propagation in the structure material. Hence x=

b − C ∆t1,2 2

.



(5.91)

If we move the origin of the coordinates to the centre of line D 1 – D 2 for convenience of calculation, we obtain:

Acoustic emission

439

5.32 To calculation of the coordinates of AE sources by linear antenna with two transducers.

b b − C ∆t1,2 C ∆t1,2 x1 = − =. 2 2 2

(5.92)

Therefore, knowing the velocity of the elastic wave propagation and time ∆t 1,2 , it is possible to quite easily determine the coordinates of AE source. However, if the source is not located on line D 1 – D 2, then calculation of its coordinate by formula (5.91) or (5.92) gives rise to an error, which is sometimes quite significant. Let us consider the issue in greater detail. Let AE source arise in point A act at a certain distance h from base line D 1 – D 2 (Fig. 5.32). A wave, propagating from the source, travelling along beams A actD 1 and A actD 2 will cover distances c and d to the transducers D 1 and D 2, respectively. The difference between these distances is, obviously, equal to the difference of the time of wave arrival to transducers D 1 and D2, multiplied by the velocity C of wave propagation. Let us again combine the origin of the coordinates with transducer D 1. In terms of geometry it is clear that C ∆t1,2 = d − c =

2 h 2 + (b − xact ) 2 − h 2 + xact .



(а)

Formula (a) at fixed b, C and ∆t 1,2 defines a parabola, i.e. the sources, for which the difference of the time of wave arrival from the source to transducers D 1 and D 2 is equal to ∆t 1,2, fall on the respective parabola (see Fig. 5.32). On the other hand, from (5.91) C∆t1,2 = b – 2xappr.

(b)

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440

The line, defined by formula (b) at fixed b, C and ∆t1,2, is a straight line (dashed line in Fig. 5.32). Obviously, it is possible to consider formula (5.91) for a source located on line D 1 – D 2, as a formula for the determination of approximate coordinate x appr of the source, not located on line D 1 – D 2. However, as is easily seen from Fig. 5.25, the error of determination of the coordinate by this formula is the greater the farther from line D 1 – D 2 or (at the same distances from it) the closer to one of the transducers is the emission source. In conclusion it should be again emphasised that formulas (5.91) (or (5.92)) are designed to determine the coordinates of AE source in linear (extended) objects, i.e. in those cases, when the emission source is located on a line, connecting the transducers, or at a very small distance from it.

2

Determination of the coordinates of a flat surface

1

Approximate formulas

If two normal to each other linear antennas (Fig. 5.33), each having two transducers (D 1 – D 2 and D 3 – D 4 ), are to be used, the presence of the second antenna will allow determination of the coordinates of the source, acting at a distance from the line D 1 – D 2. Using the above calcula­tions made for the coordinate x, by analogy we may derive a similar formula for calculation of the coordinate y. Then, if coordinates are counted off from the antenna centre, for a combined action of two antennas, normal to each other in a plane, we obtain: x=

C ∆t1,2 2

; y=

C ∆t3,4 2

.



(5.93)

The quadrant of AE source location is determined by the numbers of the first responding transducer of each antenna. From Fig.5.33 it is seen that the error of calculation of the coordinates of AE sources is largely dependent on the source actual location in a plane (point A act is the actual location of AE source, point A appr is its approximate location). And this error is the greater the farther is this signal from the centre of the antenna and the closer to the diagonal of the quadrant, where the measurements are taken. 2

Exact formulas

Calculation of the coordinates of AE signals in a plane may be performed accurately. To obtain the calculated dependencies, let us make use of the cosine theorem for a skew-angular triangle and of the data in the drawing in Fig.5.34. Then we may write:

Acoustic emission

441

5.33 For calculation of the coordinates of AE source in a plane by two linear antennas.

C 2 t12 = b 2 + r 2 − 2rb cos φ ; C 2 (t1 + ∆t1,2 ) 2= b 2 + r 2 − 2rb cos(90 − φ) ;

(5.94)

C 2 (t1 + ∆t1,3 ) 2= b 2 + r 2 − 2rb cos(180 − φ) ; C 2 (t1 + ∆t1,4 ) 2= b 2 + r 2 − 2rb cos(90 + φ) .



Here t 1 is the time of arrival of elastic AE wave to the first responding transducer of the measuring antenna; ∆t 1,2 , ∆t 1,3 , ∆t 1,4 are the additional times of AE wave propagation from the "wave" front (Fig.5.34) up to each of the transducers in 2, 3, 4 group. The other quantities, included into the equation, are clear from the Figure.

5.34 To derivation of the exact formulas for calculation of the coordinates of AE source in plate using four transducers.

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442

5.35

For calculation of the coordinates of a remote source.

Solving equations (5.94) for r, t 1 and f, a little manipulation yields: t1 = r=

2 2 2 ∆t1,2 + ∆t1,4 − ∆t1,3

2(∆t1,3 − ∆t1,4 − ∆t1,2 )

;

2 C 2 (2t12 + 2t1∆t1,3 + ∆t1,3 ) − 2b

2

 b 2 + r 2 − C 2 t12 φ =arccos  2rb 

 . 

(5.95)

;



The number of equations (5.94) is greater than the number of the defined unknown coordinates and times. The presence of an extra equation allowed, as was mentioned above, a significant simplification of the calculation formulas. 3

Formulas of direction

In some cases it is of interest to calculate the coordinates of emission sources remote from the antenna. The location of AE signals by the above methods gives rise to considerable errors, as was demonstrated here. In these cases, the formulas of the so-called ‘target designation’, directing to the emission source relative to the measuring antenna base, become competitive and may be applied. The source coordinates may then be determined as the point of intersection of target designations of two antennas. Such antennas may fulfil two functions, namely determine the AE coordinates inside the measurement rectangles and together with the second antenna, located at a

Acoustic emission

443

certain distance from the first one, find the source coordinates beyond the measurement rectangle. Let us briefly consider the theory involved. Let an antenna of four transducers, located on bases B , normal to each other (Fig. 5.35), operate by the principle of linear location of each measuring base. Let the emission source be located at a distance from the antenna which is greater compared to the antenna base. Then the propagation of an AE emission wave may be considered in the plane variant. In this case, it is possible to define an angle in relation to the base, for instance D1 – D2, which determines the direction of action of the emission source. In our case from the equality of triangles 4 1D 1D 4D 3 and D 1D 22 1 we may write: tg φ =

∆t3,4 ∆t1,2

; φ =arctg

∆t3,4 ∆t1,2

.

Calculation of the coordinates of AE sources in the actual structures by formulas (5.95) may lead to errors because of the inaccurate determination of the difference of the time of arrival of the AE signal to the transducers. This deficiency is more pronounced in the same locations, namely on the peripheries of the quadrant diagonals, and depends on the class of the equipment, used for control. Conventional AE systems enable obtaining sufficiently reliable results, applying approximate formulas (5.93) with the control area located in the middle part of the measuring antenna. 4

Determination of the coordinates on a spherical surface

Determination of the coordinates of AE events on a spherical surface is significantly simplified due to the features of the sphere, where the shortest distance between any two points is defined uniquely by two geodesic lines. Therefore, whatever the point of the sphere surface, where the AE event initiates, the wave, generated by it, will propagate along the geodesic lines,

5.36 For calculation of coordinates of AE event on a spherical surface.

Fundamentals of evaluation and diagnostics

444

connecting the point of the initiating event with the closest transducers, mounted for receiving acoustic signals. This feature of the undevelopable surface of the sphere allows obtaining the following dependences, when the transducers are mounted on its equator and poles (Fig. 5.36): S1 = R(90 – β);

S2 = R(90 – ε),

where R is the radius of the sphere; S 1 and S 2 are the arcs of a circle on the surface of the sphere along the meridians and parallels, in the direction of which AE waves propagate; ε and β are the angles of sections along the meridians and parallels, at the intersection of which the points of elastic wave emission lie. In this case this is point A. Next, by analogy with (5.90) we will write: 2S1 + C∆t1,3 = πR; 2S2 + C∆t2,4 = πR; then π πR − C ∆t2,4 C ∆t2,4 − = ; 2 2R 2R π πR − C ∆t1,3 C ∆t1,3 β= − = . 2 2R 2R It should be noted that none of the above formulas takes into account the ambiguity of coordinate determination in structures, closed in terms of wave propagation. The coordinates of AE sources for structures, having cylindrical, spherical and other closed surfaces, are determined by applying diverse techniques, such as outlining a limited control zone, registering the transducer which was the first to respond, mounting additional transducersindicators of the zone of the AE event and other measures, providing orientation of calculation algorithms. To avoid duplication of calculation formulas, the method of cyclic rearrangement of indices in the values of the times of signal arrival to AE transducers is applied, where the transducer, which was the first to receive the AE signal, becomes the first transducer. We will not consider these problems in greater detail, as they do not add any new data to the fundamental issues in derivation of the calculation formulas, except for making them more cumbersome, which is related to description of the logical part of the computing algorithms. A large number of welded structures incorporate a non-full sphere as a separate structural element. Railway tank cars, fuel tanks of missiles and aircraft and other similar structures have at their ends spherical bottoms in the form of a truncated sphere. In such cases calculation of the coordinates of the sites of acoustic emission generation becomes more complicated. As we saw for a full sphere, AE signals in all the cases pass through two AE transducers, mounted on the equator. Thus, the arising AE signal always is on one line with the two transducers. This allowed application of the above simple computations. In the case of an incomplete sphere AE transducers are ε=

Acoustic emission

5.37

445

For calculation of AE coordinates on a truncated sphere.

raised above the equator by the value determined by the angle of transducer mounting γ (Fig. 5.37). In view of the above and the fact that the four AE transducers, mounted on parallel γ, make up four areas for determination of acoustic emission coordinates, we may write for the area, shadowed in the Figure (the procedure of co-ordinate calculation is unchanged for other regions of the antenna. Just the indices of time t change, depending on which transducer has received the first AE signal):  Ct1   C (∆t2 + t1 )   2π  = cos  cos   cos   + R  n −1   R     Ct   C (∆t2 + t1 )  + sin  1  sin   cos B ; R  R     C (t + ∆tn −1 )   Ct  π  cos  − γ  cos  1  cos  1 = + R 2   R     Ct   C (t + ∆tn −1 )  + sin  1  sin  1  cos A ; R  R   

(5.96)

 C (t + ∆tn −1 )   C (t + ∆t2 )  π  cos  − γ  = cos  1 cos  1  + R R 2       C (t + ∆t2 )   C (t1 + ∆tn −1 )  + sin  1  sin   cos(2π − A − B) . R R     Here R is the radius of the sphere; C is the velocity of AE wave propagation; n is the number of transducers in the antenna; t 1 is the time of AE signal arrival to the first responding transducer; ∆t 2 , … , ∆t n are the differences of the times of AE signals arrival to the first responding transducer and other transducers in the antenna. Let us find angles A and B from the first two equations:

446

Fundamentals of evaluation and diagnostics   C (t1 + ∆tn −1 )    Ct1  π    cos  2 − γ  − cos  R  cos  R      ; A = arccos     Ct1   C (t1 + ∆tn −1 )  sin   sin     R R         Ct1   C (∆t2 + t1 )    2π    cos  n − 1  − cos  R  cos  R      . B = arccos   Ct1   C (∆t2 + t1 )    sin   sin     R R      

Substituting A and B into the last equation of system (5.92), we obtain an equation to determine time t 1:  C (t + ∆tn −1 )   C (∆t2 + t1 )  π  cos  − γ  − cos  cos  1  + R R 2       C (∆t2 + t1 )   C (t1 + ∆tn −1 )  + sin   sin  × R R        C (t1 + ∆tn −1 )    Ct1  π     cos  2 − γ  − cos  R  cos  R        − × cos 2π − arccos  C t t + ∆ ( ) Ct       n −1  sin  1  sin  1     R R         Ct1   C (∆t2 + t1 )    2π    cos  n − 1  − cos  R  cos  R       − arccos   Ct   C (∆t2 + t1 )    sin  1  sin     R R      

   0. =  

For the determination of the coordinates of the sites of AE events and marking them on the surface of the sphere being tested, we will use the theorem of cosines for a spherical triangle AD1D2 (Fig. 5.37). Then pointing angle α, counted inside the triangle from the tangent to the geodesic line, on which the first responding transducer in the treated antenna is located (transducer D 1 in our case), will be:   C (∆t2 + t1 )   Ct1    2π   − cos  n − 1  cos  R    cos  R      , α =arccos  Ct    2π   1  sin   sin     n R − 1       and the value of the vector of AE coordinate will be l = CT1.

Acoustic emission

5.38

Parameters of propagating wave electric signal.

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448

5.39

Typical design of acoustic emission transducer,

3

Acoustic emission equipment

1

Principles of equipment operation

The above photo 5.1 (see page 372) shows a characteristic acoustic emission signal, appearing at the output of an AE transducer. Analysing the dependence of this signal amplitude on time, it may be seen that the vibration frequency changes chaotically during the entire time interval of signal action. In this case, the most optimal variant of AE signal treatment is processing of its envelope, which often has a quite complicated shape (Fig. 5.38). Highly important is the coincidence of the beginning and end of AE signal action with the start and end of the characteristic process, running in the material at its fracture. Practical experience demonstrated that the most informative parameters of the signal in this case can be: – maximal amplitude of signal-event, – time of increment of the signal up to the maximum value, – signal energy, – frequency of arrival of signals-events from a specific area of the material, – characteristic distribution of signals-events in the area, where the material fracture processes proceed. At the same time, such characteristics of the process as event energy and amplitude of AE signal are parameters, which so far cannot be measured for extended structures, particularly for structures of a complex geometrical shape. Application of these parameters in the evaluation of the load-carrying ability of structures requires special approaches, which allow taking into account the special features of the structures. On the other hand, the absolute majority of acoustic signal transducers are based on sensitive elements of piezoceramics (Fig. 5.39), having certain overall dimensions and, hence, natural resonance frequencies, responding

Acoustic emission

AET1

1

AET8

1

AE1 channel

3

4

5

6

4

5

6

7

8

AE8 channel

3

9

10

13

14

11

12

+5v +12v -12v

LFS1

2

449

15

16

LFS4

2 ∼220v

+12v

To PC

5.40 Block-diagram of ЕМА-3 system.

to different components of AE signals, arising in the material. Experience of studying AE signals, received using transducers, based on piezoceramics and those, using the laser beam as signal converter, demonstrated that AE signals for both cases differ only slightly up to their maximum values and are quite different above them. In this connection measurement of AE signals, received using resonance transducers, becomes point­l ess above their maximum value. After pickup and pre-processing, the AE signal passes through a number of additional filters in the equipment and is finally sent to the computer. Figure 5.40 shows the block-diagram of the EMA-3 system and step-bystep passage of the AE signal through the equipment modules, until the quantitative characteristics of AE event are derived, which represent the fracture processes, proceeding in the material being inspected. AE signals from transducers with built-in preamplifiers 1 are transmitted by cables into the intermediate amplifier block and further on into the analog block of the system to amplifier 3, where their normalisation by amplitude is performed, according to the dynamic range of analog-digital converters. After frequency selection, performed by tunable filters of upper frequencies 4, AE signals arrive to converter 5 of mean-root-square and peak values. Selector 6 forms the adjustable fixed and ‘floating’ thresholds, performs amplitude discrimination of the AE signal, stores the peak value of the AE signal, and records the moment of AE signal arrival. The selector also shapes the strobe of the AE signal, which determines the beginning and end of the AE signal, allowing for its reverberations on the object of control.

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Fundamentals of evaluation and diagnostics

5.41 Shape of discrete AE signal: АМР – peak value of AE impulse amplitude; COOAN – leading edge of AE impulse crossing the discrimina­tion threshold; t – dead time of measurements; MEN – measuring time; ЕСН – time of peak detector switching off; АРК – position in time of peak value of AE impulse amplitude.

Meter 11 of location parameters combines the respective AE channels into location antennas, measures time delays between the fronts of AE signals from the respective channels of the antenna (parameter of the difference of the time of arrival), and module 10 of diagnostic parameter measurement uses an analog-digital converter to measure the maximum amplitude of AE signals. The time between the start of the AE signal and its maximum is measured, and the current time from the start of the experiment to the moment of the first AE signal arrival to the specified location antenna is also recorded. Microprocessor control unit 8 forms a package of data, received by meters 10 and 11, and enters it into a certain area of permanent memory 14. Signals from low-frequency process sensors 2 (pressure, deformation, displacement, force, etc.) come to the scaler and low-frequency filter. The multiplexer by means of cyclic interrogation of eight AE channels (mean root square values of continuous AE) and four low-frequency channels passes the signal of one of the selected channels to low-frequency parameter meter 13, which incorporates the second analogdigital converter. The measured values of the respective signals are entered into a certain area of permanent memory 14. Cyclic interrogation mode and data transfer are conducted by microprocessor control unit 9. In addition

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to the above functions, the control unit carries out data exchange between the digital sub-block and PC directly via interface 16, sets the value of the fixed and ‘floating’ discrimination thresholds in the threshold voltage block 7 and duration of AE signal strobe formation. All these operations are conducted by control unit 9 by the data, entered from PC into a certain area of permanent memory 14, and it performs general synchronisation of the system, using timer 12. Block 8 is a display device and incorporates the controls. Power source 15 enables the system to operate both from an a.c. mains and a storage battery. Figure 5.41 shows the typical technology of AE signal processing. Some features of design of technical diagnostic systems should be noted, which are related to the need to derive certain information to take a decision on the condition of a structure. Such information, as a rule, contains a data block (vector), the total value of which is exactly what determines the ability of the structure material to bear the service load. Usually these are the data on the external impacts on the structure, namely load, temperature, environmental conditions, as well as data, characterising the damage, to which the structure was exposed over the past years of service. The most important is the information on development in some areas of the structure of zones, where the metal properties start changing under the influence of the above factors. Such zones should be monitored so as to be able to take a timely decision on the inadmissibility of further operation of the object in connection with development of negative processes in the metal. In order to solve these tasks, the modern diagnostic systems and equipment should meet a certain number of requirements. This, first of all, is the multi-channelling, yielding a sufficient number of information units, ability to separate the flows of information, characterising the initial stages of fracture and stages, related to its hazardous development. Such a separation yields an isolated independent channel of preliminary information, characterising the possibility of appearance of higher risk zones in the future. Up-to-date diagnostic systems should provide the capability of visual presentation on the monitor of the entire dynamics of the process of initiation and propagation of fracture with indication of more actively developing areas or zones, where these processes may get out of control. Diagnostic equipment should be capable of presenting the entire scope of information, obtained during testing, in a compact manner without any data loss on one sheet of paper of a small format, for instance A4. For this purpose technical diagnostic systems use special software, allowing this procedure to be implemented. One of the features of diagnostic systems is the ability to perform multiple testing of the structure without involving the actual structure, so-called virtual testing. It means the following. Software, used in the equipment, should provide the ability to obtain all the measurement data without screening, however, during testing the monitor only displays the

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information, which has passed the set criterion filters. With poor choice of the filters, this information may yield a not quite accurate idea of the processes, proceeding in a material. Then the change of criterion filters at a repeated virtual testing of the structure will enable a more detailed study of the fracture processes, developing in the structure material. Results of the same test, derived at different settings of the cluster radius, are given below (see Fig. 5.49). As we can see, the difference is considerable. Let us give some characteristics of the EMA-3 diagnostic system, which are significant in the above sense, allowing this equipment to highly successfully perform diagnostic control of objects. The EMA-3 system: – provides the capability of isolating the active zone of the controlled material by its priority; – incorporates special software for processing the data, pre- sented in the graphs, which allows plotting five colour graphs on one sheet of paper of A4 format without changing the information content and visualisation of the proceeding processes, irrespective of their duration; – incorporates vector analysis programs, based on the theory of recognition of patterns of the fracture processes, proceeding in the structure materials, this allowing evaluation of the condition and residual life of the structure materials. Twelve information channels of the basic configuration of the system are combined in terms of design and mathematics into one vector, characterising the material condition; – incorporates high-speed programs of comparison and evalu­a tion of control results; – includes programs of floating clusters of information, tracing the shape of the developing zones of structure material deformation and fracture; – has a more informative real-time screen, allowing the operator during testing to place special marks to note the features, developing during materials deformation; – allows carrying out multiple virtual testing of elements and components of structures after conducting just one physical test. This permits deriving results, which more accurately describe the controlled processes, depending on the conditions of filtering and processing the initial data, which are changed by the operator; – has a high response due to a more perfect software and application of equipment with a high rate of data processing; – has an all-purpose set of location antennas; – incorporates ingenious programs of processing the initial data, allowing continuous monitoring of mobile structural elements.

2

Software of acoustic emission equipment. Diagnostic systems



Systems designed for analysis and acquisition of AE data currently use computers to process it. Only presentation of data in the digital form allows

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coping with the tremendous flow of data, coming from AE transducers during testing. Analog devices do not provide either the required processing methods, or the required high accuracy. Spectral analysis, for instance, does not permit evaluation of either the coordinates of the signal, coming to the transducer, or the degree of criticality of this signal source. On the other hand, a number of characte­ristics of the spectrum, having been appropriately digitised, carry useful information, which may be used, when taking a decision on the condition of the controlled object. Thus, the current level of a particular AE system is largely determined by its software. Revolutionary changes in computer technologies, which have occurred over the recent years, had a direct impact on the requirements to the software for acoustic emission systems. These requirements may be divided into three categories, namely requirements to the algorithmic part, requirements to the interface with the equipment and requirements to user interface. The equipment interface has practically undergone no changes. Various systems use different methods of AE equipment connection to computers and different data communication protocols. The speed of response of switching devices is usually quite high and was just as high already ten years ago. The most important issue in development and use of the interface is application of standard data communication protocols. This is especially important for network systems of continuous monitoring, in particular, those interacting through Internet. The algorithmic part of the software of AE systems is highly diverse and, as a rule, is the subject of know-how of the company, manufacturing the equipment. It usually includes the methods of location of AE sources and of presentation of the derived data in the form of tables and graphs. AE systems of the highest capacity allow displaying diverse information during testing, as well as conducting computer repetition of the recorded test. Software of systems of expert class, in addition to the above-said, should incorporate algorithms for analysis of the current condition of the object of diagnostics and prediction of its residual life. It is obvious that development and existence of such a program become pointless without the appropriate algorithmic part, which is exactly what determines the actual capabilities of the program. User interface is the external presentation of the program. It should be noted that the interface is becoming ever more responsible for the success or failure or a particular program product. Now all the basic standards of the user interface have been developed by Microsoft Company or with its participation. The activities of major research centres in Europe and the United States are focused on investigation of ergonomic, aesthetic and psychological requirements to the interface, through which the computer interacts with man. One of the main goals in designing the user interface is simplification of running the program, which results in minimising the need to perform monotonous operations, all the actions performed being simple

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Fundamentals of evaluation and diagnostics

and intuitively understandable, all the visual program elements containing explanation of their appli­c ation. On the other hand, it is necessary to provide for the user the ability to adapt the interface to his tastes and preferences. Very important is separation of the algorithmic and interface parts in the software. This is achievable by applying multi-level architecture, for instance, based on "client-server" technology. With the use of such architecture, various components of the program are responsible for different tasks, namely data exchange with the user, performing calcu­ lations, providing access to the data base. Thus, only by combining in a program the advanced capabilities in data processing with an up-to-date interface, is it possible to ensure a successful application of AE systems, incorporating this program. The main requirements to the software of systems, processing AE data are outlined below. Requirements to the equipment interface: 1. Compatibility with different types and models of the computer family, selected for data processing. 2. Complete compatibility with the used operating system. 3. Use of standard means for data communication and protocol. 4. Use of standard drivers. Requirements to algorithmic part: 1. Total compatibility with the used operating system. 2. Operation in a multi-task medium. 3. Data exchange with the operating system and other programs being fulfilled, using standard means. 4. Working in heterogeneous computer networks, including Internet. 5. Component-based structure, allowing the program to vary the data processing methods, depending on the changing requirements and to evolve in parts. 6. Use of the "client-server" model, allowing incrementing the computing power though application of network architecture. 7. Interaction with the systems of control of data bases, based on SOL language. Requirements to user interface: 1. Compliance with the accepted standards, which allows starting to use the program without re-training. 2. Convenience for end user, i.e. simplicity and reduction of the number of operations performed. 3. Information content, i.e. displaying a maximum of information, which may be needed. 4. Interactive nature, i.e. change of the program logic, depending on the user actions. 5. Reversibility of operations, i.e. ability to cancel an incorrect action. 6. Availability of a contextual reference system and prompting programs, performing complex operations in the interactive mode instead of the user.

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7. Ability to select various routes for operation performance. 8. Ability to adjust a large number of interface parameters to meet the requirements of a specific user. New generation systems EMA-3 fully meet the above require­ments. They are designed for operation under MS Windows NT 4.0, 2000, Windows XP, Windows 7 operating systems. System architecture provides for their scalability – from a mobile laboratory, based on one set of equipment up to a network monitoring system of 10 to 100 sets of equipment, based on distributed DCOM architecture. The software has a convenient and simple user interface, powerful reference system and training programs. EMA-3 program is much easier to run that the previous versions, developed for MS-DOS operating system. It has a user interface, standard for Microsoft Windows, similar to MS Office programs and is fully integrated with this popular package. The program incorporates a system of recording the data base, improved processes of experimental result processing, has modern appearance and good controllability of the tables and graphs, and supports network operation. An important new feature is the ability to abruptly interrupt testing. Masters, included into the program, allow simplification of testing and processing of the results, easy adjustment of the hardware and making a professionally looking report in a selected format. EMA-3 system may operate in the continuous monitoring mode with keeping of the testing log. The most crucial moment in the design of EMA-3 system is independence of the software and the hardware. Any AE equipment, supporting a standard EMA-3 data communication protocol, is capable of interaction with the programs, which will provide its real-time operation and processing of testing results. Test data, obtained in any equipment, after their conversion into EMA-3 format, may be processed by the system, yielding the entire spectrum of processing results, including a computer repetition of testing. All the results of tests, performed using EMA-2 equipment, are readily processed by the new software. The expert variant of the program allows forecasting the breaking load and residual life of the items. Software of the EMA-3 system, while providing the continuity with respect to EMA-2 both in basic operations, and in methodology of conducting the tests and processing their results, offers the users much wider capabilities, both due to a higher degree of visualisation of the testing process and post-experiment processing, and due to application of totally new methods of information analysis. The new software was developed, taking into account both the world experience of designing such systems and specific wishes of the representatives of local industry. Let us enumerate the main differences of EMA-3 system from EMA-2. The main points of difference of the EMA-3 systems from EMA-2 are as follows:

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 Ability to use any number of AE transducers with their base number, equal to 16.  User interface, similar to Microsoft Office programs, which includes a menu, tool panels, standard dialogue windows and emerging prompts.  Capability of selection of colours and fonts for the basic elements of the program.  Multi-window interface for simultaneous use of diverse infor­mation.  Use of graphic presentation of data and controlling the program, using indicator devices of the type of ‘mouse’ or ‘trackball’.  Accessibility of the greater part of information on system adjustment and results of testing from the main window of the program, without entering additional commands.  Displaying process graphs in real-time during testing.  Plane or three-dimensional, by choice, presentation of the area of the controlled item.  Long file names, using Russian symbols, allowing both a short and descriptive forms to be used, when naming the file.  File extensions are associated with the program. When a file is called, EMA-3 program is immediately loaded in.  The file size is limited to 2000 MB.  Storing system settings, test results, tables and graphs in a format, allowing viewing and editing the data, using standard Windows means.  Standard operations of copying and inserting the data through an exchange buffer, allowing data exchange with other programs.  Integration with the Microsoft Office package, allowing use of its extensive capabilities for data processing.  Reversibility of operations, which may destroy or damage the work results.  Built-in graphics editor, used instead of connecting to a specific graphics package. Saving the graphs in a standard format allows using any graphics package for their updating.  Ability of graphs plotting by default, not requiring any adjust­m ents to be entered.  Use of diverse colour schemes and graded filling, when plotting the graphs.  New types of graphs including step-like and linear difference.  Plotting the testing schematic diagram in the graph.  Plotting generalised graphs for several clusters.  Master programs for setting up the system, conducting the tests and making reports.  Storing most of the adjustment parameters between the operating sessions.  Ability to control the program from another computer through a network, in particular via Internet.

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 Connection to the data base, containing the data on equipment settings and materials tested. Test results may be also entered into it for subsequent storage and processing.  Ability to interrupt recording the test and then resume, without starting from zero.  Operation in the automatic mode, required in continuous monitoring of items with adjustment of data storage parameters.  Recording the sound during the testing process.  The xpert version of the system allows prediction of breaking load and residual life of controlled items.  The expert version of the system issues three warnings of the hazard during testing.  Broader capabilities of computer repetition of testing with adjustment of the repetition speed and ability of stopping the test.  High-quality printing out of test results in the form of text, tables and graphs.  Ability to enter the values of low-frequency parameters, if these have not been stored, and to store them together with the test results.  Mathematical generation of AE events for program testing.  Batch conversion of EMA-2 files into EMA-3 format with their subsequent processing.  The common panel of system configuration.  Internet link.  The interactive question-answering system and manual. The functionality and increased capabilities of the program, described above, would be impossible without applying advanced computer technologies of programming and data processing. The software of EMA-3 systems is completely based on the Microsoft company technologies. These technologies include COM (component object model), ActiveX (modules completely ready for use, controlled through a network or by Internet) and DNA (Distributed Network Architecture). The Visual interface of the EMA-3 program is designed to implement the traditions of EMA-2 systems and requirements to the applied program interface in terms of compatibility with Windows operating system. For most of the time the user works with one of the four windows, which include the mainn window of the program, information widow, testing window and graphics editor window. The main window of the program (Fig. 5.42, see colour section) is constructed by the principle of a multi-window interface, where the sub-windows are the information window, testing window and graphics editor window, which may be open and displayed on the screen simultaneously or alternatively. The upper part of the main window has the menu line, and its main commands are duplicated below in the tool panels. The information window in its appearance is an analog of viewing programs with a mobile separator, where in the left part the user chooses an information section he is interested in, and the information proper is

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reflected in its right part. The tree-like structure of the list shown on the left, gives the user access to hierarchically grouped settings of the system, test results and other information, access to which should be prompt. The capabilities of adjustment of the appearance of the main window and its individual elements are quite broad. The status line, indicating the current mode of the program running, is located in the lower part of the window. The test window of EMA-3 program is visually divided into four parts (Fig. 5.43, see colour section). Similar to the information window, the test widow allows the user to change the parameters of each part. The upper area of the window incorporates the line of the menu and tool panel. The upper left part of the window presents information on the parameters of the object condition during testing and post-experiment computer repetition. Such information includes, in particular, the predicted breaking load and calculated residual life of the object being diagnosed. The lower left and upper right parts show real-time graphs and an indicator of continuous acoustic emission, allowing detailed monitoring of the main parameters of the testing process. The lower right part of the widow shows the diagram of the location antenna, which displays in real time the coordinates of discrete acoustic emission sources, recorded by the system. The antenna diagram may be presented in a plane or three-dimensional form. The user may choose the dimensions or shape of the background for each of the window parts. Before the start of testing, system operation in the continuous monitoring mode may be specified. In this case the system will further operate in a fully automated mode. Plotting the graphs is one of the most important forms of post-experiment data processing. Program configuration allows choosing various types of graphs, and plotting from one to five graphs in one drawing, varying the colours of the lines and background of the graphs, also using a graded change of colours. The program incorporates a multifunctional editor of scan images to provide the capability of changing the plotted graphs, when required (Fig. 5.44, see colour section). The editor allows drawing the main types of lines and geometrical figures, painting, erasing, copying and inserting fragments, colour text, using all the fonts, accessible for the operating system, etc. Graphics editor allows, in addition to the graphs created by the EMA-3 program, to also open and edit the existing scan images, for instance, photos. The EMA-3 program was created using Microsoft Visual Studio package and is based on independent and easily modified modules. Each module is a self-sufficient component of the program code and contains all the data required for its functioning and using it. Accordingly, when system behaviour models were constructed, programmed were not the actions, which were to be sequentially performed (functional approach), but objects to which properties, behaviours and reactions (to the impact of other objects) were attributed, which exactly determine the program flow (object-oriented

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approach). Objects are developed on the basis of description of their classes. Objects of a specific class have the same set of properties, but different magnitudes of one and the same property for each specific object. A common feature for all the objects of one class is also the method of their interaction with each other and with the objects of other classes. Both the interface of the EMA-3 program and its algorithmic part are based on object-oriented approach. The task of processing AE information is extremely complicated. For systems, performing expert assessment of the condition of the controlled items, application of the object-oriented approach simplifies pattern recognition, which is the basis for prediction of the breaking load and residual life of the object of diagnostics. In such problems, connecting the acoustics and mechanics of discontinuous media, the most suitable is the schematic of object modelling, given here. Such a schematic is the basis for plotting the algorithmic part of the EMA-3 program, responsible for prediction of the breaking load and residual life of the object of control. Making reports on the conducted testing should be considered separately. Such reports were traditionally prepared manually, in a free form, with a considerable time interval between the time of conducting the testing and time of preparing the report. This had a negative effect both on the quality of the report, and the time of its submission to the customer. Automation of the process of report preparation is a world-wide tendency. To-day, however, it is applied, mainly, in business and is related to transactions, recording the orders, clients, goods and other commercial issues. Testing industrial products, especially, on a mass scale, requires automation of the report making process just as much. The EMA-3 system includes a program-master of reports, which may be started directly after testing performance. Within several minutes, having asked the user a number of questions, this program generates an electronic report in a standard form. Such a report may be printed out and submitted to the customer, modified or saved as a file, or entered into a data base. It may also be published in the Internet or the local network of the factory. Computer networks, while being a most powerful means of organisation and control of information, distributed between different facilities in one factory or even between different continents, have modified the methods of record keeping, office work and business accounting. Networks are not less useful in organising the work on technical diagnostics. Monitoring of large industrial plants may be performed by a group of work stations, controlling the AE systems, which stations are mounted close to the plants. On the other hand, the system of processing and storage of incoming data, just as the computers with the software, controlling the network and responsible for its flawless functioning and making decisions on the condition of the plant as a whole, will be located in a safe place removed from the plant. In case of a hazard developing, an alarm signal and a command to controlling mechanisms to bring the plant into a safe condition, for instance lower the working load, will be quickly sent from the central control panel.

460

Fundamentals of evaluation and diagnostics

EMA-3 systems are completely ready to operate in networks. Testing process may be controlled from a remote computer. A standard protocol is used, allowing a practically unlimited increase of the number of work stations, control of the acoustic emission and other diagnostic equipment, their connection to and effective use in any networks, including the Internet. Thus, EMA-3 system provides an unprecedented level of scalability, compared to any of its analogs. We should also dwell on the issue of safe storage of the data, obtained in testing and confidentiality of its processing results. Software of EMA-3 system provides several levels of safety, including protection of EMA-3 program proper from unauthorised copying and dissemi­n ation, protection of files, containing the results of testing and post-experiment processing, protection of data bases, when working in networks, different levels of access for different categories of users and data coding. Most of these capabilities become accessible, when the program is installed in computers with Windows NT 4.0 – Windows 2000 operating systems. These powerful operating systems support a high level of safety and information protection, and are reliable and stable in operation. High reliability and failure-safe data storage and functioning of the systems, processing the data, are further achieved by duplication of important information, using back-up tape volumes, development of special servers – data storages, inclusion of stand-by control servers into network architecture. Use of a network protocol with guaranteed delivery, transaction server, and transaction file system, data bases with stored procedures of SQL-Server or Oracle type is also quite important. Thus, software of the EMA-3 system is a fundamentally new product, differing both from the previous versions developed for the EMA-2 systems, and from any local or foreign analog. Offering a higher qualitative level of capabilities, simplicity and comfort during work performance for the users – operators of diagnostic systems, the new approach to construction of program architecture and interface provides, when using the EMA-3 system, 100% control of the objects of diagnostics of any complexity, from a standard specimen for tensile testing to major petrochemical works with a complex process cycle in operation. In this case, the personal hazard for the staff is practically zero, as the system may be controlled from any remote point through a computer network. Software of the EMA-3 system has been developed with the capability of incorporating possible future improvements, it is based on advanced technologies and allows, without changing the existing programs, increasing the computing capabilities hundreds and thousand times, taking a decision on the condition of objects of any degree of complexity practically instantly, and forecasting the breaking load and the residual life of products. Use of the EMA-3 systems allows a significant improvement of the safety of operation of high-cost industrial constructions.

Acoustic emission

461

5.46 The functional plan of an adaptive system of pattern recognition at an assessment of the condition of structures.

3

Identification of the processes proceeding in materials at fracture



One of the most important tasks of modern diagnostic systems is the problem of automatic identification of fracture processes, proceeding in structure materials and taking decisions on this basis on the condition of structures in service, and forecasting the time, during which they will remain operable. Solving this problem became possible with the intro­duction of the acoustic emission method and equipment and of the modern achievements in the field of computer engineering. EMA-3 diagnostic system implemented an algorithm, allowing determination of the breaking load of structure materials already on its first, low level, usually not exceeding 20% of the breaking load. In addition, analysing the material condition and having in the data base the information on the initial level of this condition, the system provides an estimate of the residual life of the structure. This is a significant achievement. Figure 5.45a, b (see colour section) shows images from EMA-3 system monitor screen, obtained during testing. It is interesting to note the presence of signal squares, changing from green to red colour, which are located in the left upper corner of the screen and inform the operator of the stages of fracture and the hazard they create for the structure being controlled. Below the signal boxes the line ‘PREDICTION’ gives the range of the anticipated breaking loads, calculated already at the first stages of structure loading (in this case – at the first warning). Continued loading allows the system to accumulate a sufficient volume of data, so as to determine more precisely the magnitude of the breaking load and calculate the residual life of the material. Figure 5.45b shows the monitor screen at the moment the second warning is given of the risk of loading the structure further. The residual life of the material was calculated at the same moment of time (its magnitude is given under the line of the anticipated breaking load).

Fundamentals of evaluation and diagnostics

462

Thus, the EMA-3 system is capable of replying to the main questions of interest to us, namely: 1. At what load will the structure fail? 2. How long will the structure remain operable with the defects, found at the moment of control? Software functioning in such systems is based on the principles of the theory of pattern recognition (Fig. 5.46), when the features of fracture processes, proceeding in a material, are formed into a vector of material condition. Its further comparison with the training vector and correction of both of them during entering additional data into the decision module allow forming the system decision on the condition of the structure material and forecasting this condition for the period of time, during which the system remains operable. A simple schematic of material condition identification incor­porates two main blocks, namely a transducer and a classifier. A transducer is a device, converting the physical characteristics of the processes, occurring in the object, into sets of criteria, characterising these processes. In our case such processes are local transformations of the material structure in the object, as well as external conditions, inducing such transformations. Measurements yield AE and low-frequency data, which form the vector of material condition (VMC) x = (х1 , х2 , … , хn) ′, where n is the number of the measured characteristics of the process being identified. The classifier is a device which refers each admissible VMC set, coming to its input, to one of a finite number of classes. It is assumed that VMC x belongs to one of M classes of images, ω 1, ω 2, … , ω M, characterising various stages of the process of material fracture. In the general case it may be assumed that apriori probabilities of appearance of the objects of each class are the same, i.e. VMC x may pertain to one or another class with equal probability. In this case, the probability of VMC x actually belonging, for instance to class ω j, is given by the following expression: pj =

M

p( x | ω j )

∑ p( x | ω ) k =1

.

k

The decision function is function d(x), referring VMC x to one of M specified classes. The optimal function is considered to be function d 0(x), which refers VMC x set to the ωi class, in the only case when the following inequality is satisfied: p(x | ωi) > p(x | ωj) , j ≠ i

Acoustic emission

463

or p (x | ωi ) > 1, j ≠ i. p(x | ω j ) Now, let us assume that the measured values of VMC are normally distributed and the covariant matrices, corresponding to them, have the following form: c11 c12 . . . c1n c c . . . c2 n C = 21 22 , .................. cn1 cn 2 . . . cnn where cij is the covariance of the i-th and j-th components of VMC, and cii is the dispersion of i-th component of VMC. Since for normal distribution in n-dimensional space we may write = p (x | ωi )

1 (2π)

2n

C

1/ 2

 1  exp  − (x − m i )′C−1 (x − m i )  ,  2 

where x is the vector of material condition; mi is the vector of mathematical expectation, and the ratio of two distribution densities p(x|ωi) and p(x|ωj) is given by the following expression: p (x | ωi )  1  = exp − (x − m i )′C−1 (x − m i ) − (x − m j )′C−1 (x − m j )   . p(x | ω j ) 2  

5.47 Schematic of identifying the fracture processes during materials deformation.

464

Fundamentals of evaluation and diagnostics

Since the covariant matrix is symmetrical, this expression may be considerably simplified: p (x | ωi ) 1   = exp  x′C−1 (m i − m j ) − (m i + m j )′C−1 (m i − m j )  . p(x | ω j ) 2   Let us introduce the characteristic of plausibility of identifying the processes, proceeding in the materials at fracture r, then: rij = ln

p (x | ωi ) . p (x | ω j )

Then, for the separating function we have the following expression 1 (5.97) rij = (x) x′C−1 (m i − m j ) − (m i + m j )′C−1 (m i − m j ). 2 To find the optimal separating function, it is necessary to calculate the values of function r ij(x) for all i and j (i ≠ j) and select the largest of the obtained values. If it turns out that this maximum is equal to r kj , then VMC x is included into the ω k class. Figure 5.47 gives the schematic of calculation of the criterion parameters. Note that equation 1 x′C−1 (m i − m j ) − (m i + m j )′C−1 (m i − m j ) = 0 2 describes a hyper plane, drawn in the n-dimensional space and separating it into parts, in keeping with VMC belonging to a particular criterion class. Solving equation (5.97), we every time obtain a separation of two classes, namely that of the measured VMC and the reference value, determining the stage of fracture of the structure material.

4

Application of acoustic emission

1 General Successful application of the acoustic emission method in struc­ture testing depends on many factors. This primarily is the correct selection of control equipment and procedures. It should be noted that the acoustic emission method does not have a precise metrology, as it is customary to say about the meter which has a universally recognised standard. Acoustic emission is the propagation of elastic waves in plates, often having a complex geometry. Even a conventional plate, as was seen above, has a great influence on the nature of the wave field, transmitting the information on the structure material condition at the point of the wave origin. Therefore,

Acoustic emission

465

5.48 Dislocation of AE transducers and control zone in specimen tensile testing.

metrological certification of the instruments is highly complicated and requires close co-operation with the developers of equipment and procedures of its application. As a rule, instrument certi­f ication is conducted by joint efforts and depends on the achievements of science and practical work at the moment of certification. Correct selection of the measuring diagnostic system is a pledge of its successful operation. The second important factor, influencing the testing quality, is the level of training and experience of the operator and chief of the testing laboratory. Practical work on diagnostic control showed that such an experience is gained after three to five years of independent work. Thus, availability of a good procedural base, correctly selected equipment and sufficient operating experience allows obtaining reliable results during performance of check testing, result processing and taking the decisions. Let us consider some practical examples of application of acoustic emis­s ion equipment in testing specimens, structural elements and structures proper.

2

Specimen testing

Specimen testing is one of the most widely accepted methods of obtaining information on performance of materials and structures. Therefore, the issue of selection of the shape and dimensions of the specimens should be given special attention. In our case the speci­m ens should provide not only sufficiently complete information on the material, but also the ability to relate this information to the actual structures. Hence, additional requirements are made to the specimens. On the one hand, they should have large dimensions, so as to eliminate the scale factor, and on the other hand, such dimensions, at which the cost of testing and ability to apply the existing testing equipment and proce­d ures of result processing still would allow conducting the testing. In evaluation of the acoustic properties of materials these requirements are met by specimens of the AE-01P type to TU-025E-1986 specification. At tensile testing of such specimens AE transducers are mounted symmetric to its middle part (Fig. 5.48).

Fundamentals of evaluation and diagnostics

466

Table 5.1 Cl. #

X, mm

S(X), mm

N sum.

1 2 3 4 5 6

0215.00 0134.00 0156.00 0137.00 0178.00 0109.00

0000.00 0000.00 0000.00 0002.00 0000.00 0000.00

317.00 127.00 085.00 003.00 146.00 015.00

Four acoustic emission transducers should be mounted on the specimen. When entering the initial data into the measuring instrument, test operator establishes the control zone and the sub-zones – clusters, limiting the surface of the specimen, on which AE data are generalised with one coordinate floating with the cluster. Radii of the clusters of AE signal location are assigned, depending on the requirements, made to the experiment. The larger the radius of the set cluster, the more comprehensive will be the recorded pattern of the material fracture process. And, contrarily, reducing the cluster radius, we start investi­gating local areas of the deforming material. Applied onto a narrow edge of the specimens is a layer of sound-absorbing paste, required to suppress the reflectivity of specimen edges and reducing the associated false AE signals. As was already mentioned, four AE transducers are mounted to receive AE data. This is required to combine two functions of the linear antenna of the transducers, namely determination of the AE signal coor­d inates and elimination of signals, appearing from the side of specimen fastening points. In addition to acoustic emission transducers, sensors to measure displacements, temperature, to observe the change of the deforming material surface and other sensors, required to analyse the fracture processes, are usually mounted in the middle part of the specimens. The magnitude of load, changing at stretching of the specimens, is measured by devices, incorporated into the rupture testing machine. All this data are recorded by the measuring instruments in the form of the vector of the material condition. Before testing AE transducers are set up using an auxiliary transduceremitter, activated by a comparison signal generator, built into the measuring instruments. The setting-up pulse is applied 5 to 6 times, which is followed by more precise determination of the velocity of AE wave propagation, and its value is entered into the catalogue of initial data of testing. As a rule, the transducer-emitter is mounted in the centre of the control zone. Figure 5.49a (see colour section) shows the graphs of deformation in time of two specimens of steel 17GS, obtained using EMA-2 equipment. Specimens are cut from one steel sheet after 27 years of service. The graphs give the sum of amplitudes of acoustic emission events; the number of events per a unit of time; continuous emission signals, characterising

Acoustic emission

467

the processes of extended stages of material fracture, and the load. A significant difference is visible in development of the above charac­teristics of specimens, despite the fact that both the specimens are made of one and the same sheet. This points to the fact that the processes of material fracture proceed in a non-uniform manner. The test results clearly demonstrate the great capabilities presented by acoustic emission in analysis of the mechanisms of material deformation, allowing a more detailed study of the processes of material fracture and providing more detailed information on damage accumu­lation at the early stages of fracture. Let the reader himself analyse the process of fracture of steel 17GS by the presented diagrams and make a conclusion on material performance in this case. Testing, conducted using EMA family equipment, was repeated under other conditions. In this case, as was mentioned above, there was no need to repeat the physical testing. Changed conditions of data reception by EMA equipment, in this case, cluster radii, yielded totally different test results, as localisation of control sites was changed, and instead of one general integral pattern of the controlled area, local zones singled out in this area were studied. Figure 5.49b (see colour section) shows the pattern of AE data dist­ ribution when the location cluster radius was changed from 110 mm, used in data decoding in the first two specimens, to 20 mm in this case. The number of clusters in­c reased up to 6 against one in the first case, which with the number of AE events, equal to 693, was located at distance X = 209 mm from the first transducer (Table 5.1). In the second case 693 acoustic emission events, recorded by EMA-3 equipment, were distributed in six clusters at different distances X from the first transducer. Reduction of the radius of the cluster of AE events grouping allows analysing the process of specimen material fracture in twelve separate local areas of the control zone. This way, naturally, more detailed information on the mechanism of fracture initiation and propagation is provided for analysis.

Photo 5.2

AE transducer on the half-sphere surface.

of the sphere

D4

3 4 1 2

Pump

Receiver

D3 320

4 AE transducer

∅ 915

=

(D2) D4

R

43 8. 3

=2 .6

A

ЕМА-3 diagnostic system

D1

D3

 - AE signals appearing at fourth loading

D1 Welds

View along arrow A

D2 δ

Pressure sensor

1, 2,3, 4 - Discontinuities developing successively as a result of four hydraulic tests

Fundamentals of evaluation and diagnostics

468

5.50

3

Testing of structural elements and components

Testing of half-spheres Somewhat greater difficulties arise in testing the structural elements. In most of the cases these are structures of a complex geometrical shape, this leading to lower accuracy of determination of the coordinates of acoustic emission location. In such cases the structure is divided into sufficiently

Acoustic emission

469

uniform elements, and a measuring antenna of AE transducers is mounted in each of them or in part of them. Such an antenna usually consists of four transducers. Figure 5.50 shows a structural element in the form of a spherical bottom of an AMg6 alloy tank for fuel storage. The bottom wall thickness was 2.7 mm, the sphere radius was 44 cm. For convenience of testing the two half-spheres were connected through a rubber gasket, providing the required tightness when loading the spheres by water. Photo 5.2 shows part of the tested half-sphere, carrying an AE transducer of the measuring antenna. Hydraulic pressure inside the half-spheres, assembled in one block, was applied by a gear-type pump through a special receiver, compensating the pulsations. Pressure was measured by a special membraneslide sensor. Before testing, setting up of the location device of EMA-3 equipment was performed, using a special transducer – AE simulator, on the surface of the controlled upper part of the half-sphere, which had been divided into spherical squares. The accuracy of coordinate determination was evaluated by coincidence of the AE coordinates, deter­m ined by EMA-3 equipment during the simulator operation, with its position on the sphere. This work was performed on a water-filled tank. To simulate defects, welds were deposited on the half-sphere surface, namely two rectilinear welds in the form of a cross with the weld length of 590 mm and one circular weld of 230 mm diameter with the centre at the point of crossing of two rectilinear welds. Half-spheres were tested in stages. Each subsequent test was performed after repair of the tank, which had failed during the previous test. Testing was stopped every time, when a leak developed, and only after that the water was drained, and the fractured section of the tank was repaired. Four tests were conducted to optimise the technology of repair of aluminium alloy structures. Testing results are presented in the right part of Fig. 5.51 (see colour section), which shows AE events in the form of points, located at two welds. The first tests revealed the presence of pores, resulting from low quality of technological welds performance. Already at the pressure of 0.754 MPa testing equipment recorded a burst of continuous and discrete emission, characterising the intensification of the processes of material deformation in welds. Subsequent increase of pressure up to 1.93 MPa led to pore opening and a slight leak developing. This moment was recorded by the equipment in the form of a new more intensive burst of continuous emission. No discrete emission pulses were observed at this moment. Further increase of pressure up to 2.31 MPa led to more intensive outflow of liquid through the widening pores. Testing was carried on after repairing the weld. Testing a pipe knee Application of AE equipment is highly attractive for assessment of the condition and ensuring the safe service of structures of thermal and nuclear

470

Fundamentals of evaluation and diagnostics

5.52 Knee section of piping with AE transducers after hydraulic testing. Bold points indicate sites of acoustic activity of knee material appearing during testing.

power stations, in particular, process piping and pressure vessels. Such structures, operating at high temperatures, and in the case of nuclear plants, also under the conditions of irradiation, require thorough periodical inspection to assess their condition and solve the issue of the possibility of their further service. According to the operating codes of nuclear power stations the reactor units are shut down every four years and their main assemblies are inspected. Unit examination includes nondestructive testing of pressure vessels with their subsequent hydraulic testing at the pressures 1.5 times higher, than the working pressure. As the first example of AE system application in structure testing, let us consider testing of a knee of a feed pipeline of a thermal power station. A knee section of St.20 piping was cut out of an operating high-pressure feed pipeline after 20 years of operation (Fig. 5.52). The figure shows the main dimensions of the knee. Initial wall thickness of the pipe was 0.7 cm; external defect of the type of a corrosion pit 16.3 cm long, 4 cm wide with the maximal depth of 0.4 cm is located on the pipe inner side. Knee testing was conducted at static loading with water up to 25.7 MPa. It was assumed that the fracture will be initiated by a defect. Therefore, to analyse the pipe material condition, acoustic emission transducers were mounted on two linear antennas so that the defect was in the control zone. The first antenna had the base of 70 cm and control zone of 80 cm – 40 cm on each side from the control zone centre. The radius of the initial cluster, forming the region of AE concentration, was set to be equal to 7 cm. The second antenna was mounted normal to the first one on a base, shortened allowing for the pipe curvature, which was equal to 25 cm. Control zone was 50 cm – 25 cm on each side from the base centre. The cluster radius for this antenna was 2.5 cm.

Acoustic emission

471

The pipe was tested in three stages. In the first stage the load was brought up to 5.3 MPa, and an approximate value of breaking pressure was set. It was established that the breaking pressure is in the range of 23.9–28.7 MPa. Repeated testing at up to 13.0 MPa load allowed the breaking load to be more precisely determined as being in the range of 25.8–28.1 MPa. After more precise determination of the breaking load and addi­tional examination of the pipe by nondestructive testing methods, testing was continued up to fracture. At 23.4 MPa pressure AE equipment recorded a leak (t = 1797 s in graph Fig. 5.53 (see colour section), indi­c ating a prefracture condition of the pipe material. A further increase of pressure up to 25.7 MPa, led to defect opening and an abrupt enhancement of water draining through the formed crevice. Fracture occurred. The graphs in Fig. 5.53 give the data on acoustic activity of the pipe material in time for the final third stage of testing. The graph of loading pressure P is also shown there. The graphs show an abrupt burst of continuous AE, indicating development of an initial small leak before fracture. The nature of pipe fracture should be mentioned. As shown by fractography, half of the thickness of the failing wall on the pipe inner side demonstrated brittle fracture, and the second half pointed to tough fracture. This is attributable to embrittlement of the knee material from the working media side to half of the wall thickness during the long time of service. The conducted testing leads to a very important conclusion that a slight leak developed in the pipe before fracture giving a signal of pipe material entering the final stage of fracture. AE equipment has traced with sufficient accuracy the critical points appearing during fracture, this confirming the ability to use the acoustic emission method for a timely detection of the pre-failure condition of a structure and this way preventing its going into this area 6. Structure testing Use of acoustic emission is not limited to control of the condition of materials during their testing or operation. AE is ever wider applied in control of the technological processes, namely welding, melting, proceeding of chemical reactions, packing and dosing, etc. All this gives every ground to believe that acoustic emission will gradually develop into AE technology. AE technology has excellent prospects for future application, as it enables control of the condition of operating plants and of development of various physical processes not by indirect concomi­tant characteristics, which then have to be accepted with varying degrees of errors, but by direct control of the process by its component – the elastic wave, fully characterising For more details see: E.F. Garf, et al., Evaluation of the strength of pipelines, having considerable erosion-corrosion wear, Tekhn. Diagn. Nerazrush. Kontrol', 1999, 31, 58–64. 6

472

Fundamentals of evaluation and diagnostics

5.54 Layout of АE transducers of ЕМА‑2 diagnostic system at estimation of a product pipeline condition (pipe 355 × 8 mm; transducers 4 and 5 are operating in a linear antenna, transducers 3 and 6 – at zone location).

the process proper. So, V.K. Shukho­s tanov demonstrated 7 that AE may significantly influence the techno­logy of manufacturing of enterprise products, eliminating whole blocks of operations, unnecessary in this case. Performed work allowed drastically changing the technology of production of some structures. In fabrication of a spherical tank of 13 m diameter of a steel plate, the manufacturing plant reduced the number of process operations (such as nondestructive testing, preparation to its performance, construction of scaffolding to bring the control equipment to the place of control, etc.) from 74, performed by the traditional sequence, to 22 with transition to AE control. More than 70% reduction was achieved. In the future, in the opinion of V.K.Shukhostanov, application of AE technologies in control of the structure condition during processing operations will allow elimination of control and testing after manufacture. It is natural that it will lead to revolutionary reorganisation of production and will allow freeing large production spaces and funds.

1. Testing of a product line The ability to perform 100% control of pipe material is highly important for practical operation of product pipelines. It is known that such work is currently performed in a very approximate manner in some local sections of the pipeline route, selected by experts proceeding from many years of operation experience. Such a method, however, does not prevent errors, either in selection of the control location, or in taking a decision on the possibility of further operation of the pipeline. In this case, great prospects are opened up by application of diagnostic systems, based on acoustic emission, as they automatically perform 100% control of pipeline condition with successive application along its entire length. Shukhostanov V.K., Intelligent AE technologies of manufacture, operation and repair of modern welded structures, Tekh. Diagnostika i Nerazhrush. Kontr., 1991. No. 3, 42–48. 7

Acoustic emission

473

Table 5.2 Cluster number

Cluster coordinate X, mm

Error S (X), mm

Total number of impulses Nsum

1

62131

0

1

2

66238

102.2497

2

3

58285

125.2498

2

4

52488

95.7991

3

5

24643

0

1

6

27620

0

1

7

60071

53.7517

2

8

39809

0

1

9

51734

70.2496

2

10

15864

10.22

4

11

23344

0

1

12

53351

0

1

13

56340

0

1

14

55111

0

1

15

39104

84.2496

2

16

4545

0

1

17

6944

0

1

18

35882

0

1

19

47261

0

1

20

76338

0

1

21

49076

0

1

22

5209

0

1

23

41406

0

1

Figure 5.54 shows a section of a product pipeline 80 m long, controlled by EMA-2 diagnostic system, using four AE transducers, located on the pipe upper surface in special uncovered holes. Testing of a pipe of product pipeline was performed by a technology, envisaging pressure increase by pumping from a compressor unit, located

Fundamentals of evaluation and diagnostics

474

5.55

Reactor unit of Pack NPS (А, В, С, D, E – AE testing zone).

100 km from the testing site. Such a method of pressure increase ensured absence of acoustic noise from the unit, which increased the pressure. Pipe location under the ground further promoted reduction of the noise background in testing. The gate, the closing of which ensured pressure rise, was located on the opposite side of the pipe at 70 m distance from the testing site. Pressure rose from 4.3 MPa to 5.2 MPa. This turned out to be sufficient to evaluate the pipe load-carrying capacity. During loading EMA-2 equipment recorded 23 clusters of AE activity, distributed along the pipe length, as shown in Table 5.2. As we see, the activity of the pipe material is slight and varies within the range from 1 to 4 events. During pressure rise the system determined the breaking pressure in the pipe, which was equal to 22 MPa (calculated breaking pressure of 21.2 MPa). As the working pressure in the pipe is equal to 4.3 MPa, this being much smaller than the breaking pressure, the pipeline is still operable by this parameter.

2. Testing of nuclear power stations (NPS) A great economic and service effect may be attained when AE technology is applied to assess the condition of NPS. Therefore, let us consider testing of nuclear reactor units, conducted in Hungary in 1987, as the second example of application of AE equipment for assessment of the condition of structures. In 1987 the first reactor unit of WWER-440 type was installed in Pack NPS in Hungary (Fig. 5.55), and its preventive checking and assembly were performed with subsequent application of acoustic emission means during its first testing 8. Pellionis P., Hereb Ya., Procedure and equipment for AE control during proof testing of pressure vessels of a nuclear reactor in Hungary, Tekh. Diagn. i Nerazrush. Kontrol', 1991, No. 3, 14–21. 8

Acoustic emission

475

Time t·10 2, s 5.56 Dependence of AE event number in a welded-in pipeline zone and pressure on time during vessel loading.

The main parameters of the reactor are as follows: 427 cm diameter; 1180 cm height, 14 cm wall thickness, material is high-strength 12SN2MFA steel and number of water circuits is 6. The main goal of AE control consisted in prevention of the possible emergency situation, associated with the potentially hazardous overloads on the reactor body, especially in active radiation zones. AE control of the reactor pressure vessels was conducted in the zone of welded joints, connecting twelve pipelines to the vessel body, as well as the zone of circumferential welded joint at the height of the reactor core region, primary coolant circuits and water purification system. Preliminary check of the condition of the reactor body elements was conducted, using the full range of the conventional control techniques. Acoustic emission signals were recorded during the body heating and its loading by hydraulic pressure, which was 50% higher, than the working pressure, and its subsequent lowering. Control proce­dure envisaged an initial temperature increase up to 120° C and pressure increase up to 19.2 MPa. AE control was performed simultaneously with control of the body element deformations, using strain gauges. Control results are shown in the graphs in Fig. 5.56. As is seen from the graphs, activity of AE emission processes stopped during the period of pressure drop or at its stable magnitude. This is indicative of an absence of AE source activity. In the period between 1988 and 1990, similar work was performed also on the three remaining reactor units, proceeding from the positive experience of AE control application in 1987.

476

Fundamentals of evaluation and diagnostics

5.57 Operating diagram of the expert system of continuous monitoring.

4

Application of AE technology in continuous diagnostic inspection of structures



The complex service conditions of structures and the need for their continuous operation without scheduled shutdowns require development of special monitoring and expert systems of technical diagnostics. Such systems are more complicated and not only fulfil the functions of monitoring the condition of the structure being controlled, but also control the technological process, conducted in this structure in a pre-failure situation right up to taking the structure out of service. This is a comparatively new sphere of activity of diagnostic systems. Conditions for AE application for such equipment have been prepared by the last 25 years of persistent work of scientists and specialists from different countries. Figure 5.57 shows the design of such a system. The theory and technology of data processing in the second and third blocks (numbers 2 and 3 in the diagram) are quite well developed. This is related to the fact that they are not closely based on experiment, but are a purely computational process. The first block (number 1 in the diagram) is the block responsible for preparation of input data for diagnostics system. This is the most critical block, as the volume and accuracy of initial data predetermine the success of the final result. All that will further on happen to the obtained data in the subsequent processing blocks just transforms it, without fundamentally

Acoustic emission

477

5.58 Schematic of a multichannel system of continuous AE monitoring mounted on ammonia storage.

changing it. The effectiveness of this block operation largely depends on the current achievements in the field of methods and means of acquiring diagnostic data directly from the monitored structure. Experiment also has an important role in construction of algorithms for primary processing of the

478

Fundamentals of evaluation and diagnostics

data received from the transducers. So far just one technology has reached a sufficiently high level of satisfying the diagnostic requirements – this is the technology based on acoustic emission appearing in materials during their deformation and subsequent fracture. We will not dwell on interpretation of the technological components of each block. They are quite clearly shown in the diagram. Let us consider only the practical application of this diagram in industrial monitoring. Figures 13, 14, 29-31 (see colour section) present a variant of the expert diagnostic system developed by specialists from the E.O.Paton Electric Welding Institute and Hungary on the basis of the EMA-3 unit. Figure 5.58 gives the connection diagram at system mounting. It envisages a data analysis station removed to any distance from the inspected object. The diagram was implemented in monitoring the condition of four ammonia storages of 34 000 m 2 each. 3500 m 2 of the surface of each storage are continuously monitored by 48 acoustic transducers and using a small amount of additional data characterizing the service properties of the structure (pressure in the vessel, number of cycles of change of stored product level, monitored medium temperature; measurement of storage shell body deformations in the most heavily loaded regions is also envisaged). Sensitivity of measuring instrumentation is so high, that at continuous monitoring of 100% of storage surface, sites were detected, where slight fatigue damage of the material accumulated and continued accumulating with time. Figure 32 shows a scan of the storage side surface, where blue points mark sites of increased acoustic activity in the material induced by vibration of a pipe attached to the body, through which the product is pumped into the storage (Fig. 33). Numbered circles designate AE transducers located in three rows on storage body. The same monitoring was applied on one kilometer section of ammonia line on a bridge across the Dnieper river. Transducers mounted near the bridge on the left and right bank of the Dnieper (Figs. 38, 39) provide data on the condition of the pipe of ammonia line proper, which is covered by an external pipe. As in the case of monitoring the ammonia storages, all the diagnostic data is sent from the monitoring site to a centre located in Kiev (Fig. 34). It should be noted that development of modern means for monitoring the condition of structures, both in the periodic (Fig. 28) and in continuous mode, allows applying a new approach to its organization. High technologies applied in the development of monitoring equipment using acoustic emission, allowed monitoring to be transformed from a routine, inconvenient, inefficient and labour-consuming process into an automatic one performed from one central diagnostic station. All the data on the structure condition flows through communication channels to one station, irrespective of where and at what distance from the monitoring station the diagnosed object is located.

Fig. 13. EMA-3U diagnostic system performing prediction of breaking load, calculation of residual life, continuous monitoring mode. Joint development of the E.O.Paton Electric Welding Institute, and Videoton and Indprom companies.

Fig.14. EMA-3U measuring block.

а) Fig.29. Continuous monitoring of four ammonia storages in OPP (D = 52 m, H = 21 m, Sside = 3431 m2; 48 AE transducers, 6 strain gauges, sensors of pressure, temperature, loading-unloading ) ammonia storages in OPP Fig. 29. Continuous monitoringcycles of four 2 a) EMA-3C instrumentation (D = 52 m, H = 21 m, S side = 3431 m ; 48 AE transducers, 6 strain gauges, sensors of pressure, temperature, loading-unloading cycles) a) EMA-3C instrumentation.

Fig. 30. Equipment for primary processing of AE signal.

Fig.31. Operation control panel Fig. 31. Operation control panel.

Fig. 5.16. Spectral characteristics of AE waves propagating from a source at 0.2 cm depth.

2 nd order

11 th order 20 th order

1 st order

10 th order 19 th order

3 d order

12 th order 21 st order

23 d order

22 d order

5 th order 14 th order

4 th order 13 th order

6 th order 15 th order 24 th order

Wave number α , 1 / cm 7 th order 16 th order 25 th order

8 th order 17 th order 26 th order

9 th order 18 th order

File

Database

View

Testing

Options

Services

Window

Help

Start Options View

Parameters choice

Main parameters information Parameter Antenna # Antenna type Base, mm Control zone 1, mm Control zone 2, mm Cluster radius, mm Formula Filter, kHz Strobe Constant threshold, mV Floating threshold, s Sound velocity, mm/µs Minimum time delay, µs Maximum time delay, µs Transducers # 1,2,3,4

Settings LF 1 LF 3 .Antenna 1 Test results Antenna 1 AE signals Noise LF parameters LF calibrated Marks Antenna 1 - cluster All channels AE signals Noise LF parameters LF calibrated Marks

Network

Value 1 Line1 "--"--"--" 110 60 60 110 1 100 10000 6 3.75 3.9 0 60000

Database

Information mode

Fig. Fig. 5.42. The ofЕМА-3 ЕМА-3program program. 5.42. Themain mainwindow window of EMA – [EMA – 3. Testing] File

Testing

View

Window

Active

ATTENTION, RECORDING!

Parameters of object condition Total number of AE events = 42

Not dangerous

Continuous AE 1-16

Noise: 149 mV

Load: 0 mV .

LF – parameters:

1.20E-05

Load = 0 mV

1.00E-05

XXX = 28 mV

8.00E-06 6.00E-06

Deformation = 90.72 mV

4.00E-06

XXX = 9.3333 mV

2.00E-06 0.00E-06

PREDICTION Objects of prediction - Antenna 1, Load .Event # 42.

Antenna 1 – Line1 "--"--"--" . Load: 0 mV .

1.20E-05 1.00E-05 8.00E-06 6.00E-06 4.00E-06 2.00E-06 0.00E-06

Time, s/10

Testwindow window of ЕМА-3 Fig. Fig.5.43. 5.43. Test ЕМА-3program program.

EMA – [Express-graphics C:\EMA\Results\0-1.ema] File Edit Tools Transformation View Window Help

Pick

Pencil

Line

Rectangle

Ellipse

Fill

Spray

1

1

Antenna # 1

0.9

0.9

Cluster # 1

0.8

0.8

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0.7

0.6

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0

0

127 254

381 509

636

763 890 1017 1144 1271

[Line1 "--"--"--"]

822 AE signals Radius = 110 mm X = 167 mm Y = 0 mm S[X] = 0.027 S[Y] = 0 Tr=870.94 R=25.44 A, mV - Columnar [Max=205] LF3, mV - Linear [Max=8799.998] N – Lin. Sum [Max=822] Avr, mV - Line [Max=36.5]

0

Time, s Text Width

Line

Fill

Font

Eraser L=674[178mm] H=391[103mm]

Lines and Fills

Background

FFig.5.44. i g . 5 . 4 4 .Window W i n d o of w graphics o f g r a peditor h i c s of e dЕМА-3 i t o r o f program ЕМА-3 program for plotting the graphs. for plotting the graphs

EMA – [EMA – 3. Test simulation – D:\Testing\Gas-specimens\Files\011-1.ema] File Testing View Window Slowly

Active Parameters of object condition

Continuous AE 1-16

Total events AE – 316 of 6201

Warning 1: Attention!

PREDICTION

Quickly Noise: 8 mV

LF 3: 993.4066 kg

LF – parameters: LF 1 = 0 σ / p LF 2 = 2.442 mV LF 3 = 989.0111 kg LF 4 = 3.663 mV

Breaking load 5984.4 – 6582.8 kg

Objects of prediction - Antenna 1, LF 3 Event # 0

Antenna 1 - Line "--"--"--"

. LF 3: 993.4066

kg

1.0 0.8 0.6 0.4 0.2 0

Time, s/10 Start

EN

Prediction.doc – Micr…

E M A - [EMA-3. Test si…

EMA – [EMA – 3. Test simulation – D:\Testing\Gas-specimens\Files\011-1.ema] File Testing View Window Quickly

Slowly

Active Parameters of object condition

Total events AE - 1458 of 6201

Warning 2: Greater attention!

PREDICTION

Continuous AE 1-16

Noise: 2 mV

LF – parameters:

.

LF 3: 4641.7568 kg. .

LF 1 = 0.0147 σ / p LF 2 = 2.442 mV LF 3 = 4659.3394 kg LF 4 = 2.442 mV

Breaking load 6131.3 - 6832 kg

Objects of prediction - Antenna 1, LF 3 Event # 3

LF 3: 4641.7568 kg

Antenna 1 - Line "--"--"--"

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0

Time, s/10 Start

EN

Prediction.doc – Micr…

E M A - [EMA-3. Test si…

Fig.5.45, а. 5.45, Images EMA-3 system monitor screen, obtained during Fig. а.from Images from EMA-3 system monitor screen, obtained testing (the first and the second warning) during testing (the first and the second warning).

EMA – [EMA – 3. Test simulation – D:\Testing\Specimens\Files\011-1.ema] File Testing View Window Slowly

Active Parameters of object condition

Continuous AE 1-16

Total events AE – 1946 of 6201

Warning 2: Raise attention!

PREDICTION

Quickly Noise: 1 mV

LF 3: 4997.8003 kg

LF – parameters: LF 1 = 0.8352 σ/p LF 2 = 2.442 mV LF 3 = 5010.9878 kg LF 4 = 2.442 mV

Breaking load 6131.3 - 6832 kg Residual life 24.1074 years

Objects of prediction - Antenna 1, LF 3

. Event # 47 .

LF 3: 4997.8003 kg

Antenna 1 - Line "--"--"--"

.Time, s/10. Start

EN

Prediction.doc – Micr…

E M A - [EMA-3. Test si…

Imageon on EMA-3 EMA-3 system system monitor Fig. Fig.5.45, 5.45, b. b.Image monitorscreen screen obtained during testingduring (in the second warning obtained testing (in the secondrange). warning range)

1

1

0.9

0.9

0.8

0.8

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118

235

353

470

588

705

823

941

1058

1176

Antenna # 1 Cluster # 2

X = 186 mm Y = 186 mm A, mV [Max=106477] N [Max=52] Avr, mV [Max=64.75] LF3, kg [Max=9437.363]

17GS steel, 27 years Cluster radius 11 cm

0

Time, s

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1

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197

393

590

787

984

1180

1377 1574

1770 1967

Antenna # 1 Cluster # 2

X = 209 mm Y = 209 mm A, mV [Max=62792] N [Max=18] Avr, mV [Max=43.75] LF3, kg [Max=9432.968]

17GS steel, 27 years Cluster radius 11 cm

0

Time, s

Fig. 5.49, а. Results of AE testing of 17GS steel specimens after 27 years а. of service. radius is 110 mm. Fig.5.49, Results ofCluster AE testing of 17GS steel specimens

after 27 years of service. Cluster radius is 110 mm

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197

393

590

787

984

1180 1377

1574

Antenna # 1 Cluster # 5

X = 124 mm Y = 124 mm A, mV [Max=2089] N [Max=4] Avr, mV [Max=43.75] LF3, kg [Max=9432.968]

17GS steel, 27 years Cluster radius 2 cm

0

1770 1967

Time, s

Fig.5.49, of AE testing of 17GS steel steel specimens Fig. 5.49,b.b.Results Results of AE testing of 17GS specimens after after 27 of service. 20 mm 27 years of years service. ClusterCluster radiusradius is 20 ismm.

1

1

Antenna # 1

0.9

0.9

Cluster # 1

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28

42

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69

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125

138

[Plane 1 – Square] 15 AE signals Radius = 60 mm X = 50 mm Y = -245 mm S[X] = 0.3891 S[Y] = 0.1952 A, mV – Columnar [Max=233] Avr, mV – Linear [Max=101] LF1, MPa - Linear [Max=1.902649] N – Lin.Sum [Max=15]

0

Time, s

Fig.5.51. Diagram of AE data on the half-sphere surface

Fig. 5.51. Diagram of AE data on the half-sphere surface in the in the area of technological at hydraulic testing area of technological welds welds at hydraulic testing.

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

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0.4

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0.3

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0.2

0.1

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0

0

210

421

631

841

1052

1262

1472

1682

1893 2103

Antenna # 2

[Plane 3 “– “]

Cluster # 1

767 AE signals Radius = 125 mm X = 113 mm Y = 0 mm S[X] = 0.0785 S[Y] = 0 Tr=2241.95 R=39.01 A, mV – Lin.Sum [Max=29918] Avr, mV – Linear [Max=50] N – Columnar [Max=28] Pcr=25.7 MPa

0

Time, s

Fig.5.53. Graph of variation of AE in timeinat time testing Fig. 5.53. Graph of variation of parameters AE parameters at testing of a bend of ahigh feed pipeline. bendpressure of high pressure feed pipeline