The application of a semielastic lattice model to the description of acoustic emission

Engrn~~rrn~ Printed

Frucrurr

Mechanics

Vol. 23, No. 4, pp. 621-629.

1986

CO13-7944/86 $3.00 + .oO 0 1986 Pergamon Press Ltd.

in Great Britam

THE APPLICATION OF A SEMIELASTIC LATTICE MODEL TO THE DESCRIPTION OF ACOUSTIC EMISSION Faculty

1. GRABEC and J. PETRISIt of Mechanical Engineering, pob. 394, Ljubljana,

Yugoslavia

Abstract--In the article a model of a two-dimensional lattice is formulated which is for the description of acoustic emission in planar samples. The rheological parameters elements are prescribable, so that nonlinear and viscoelastic material properties counted for. The transient phenomena caused by changes in the material properties acterized by the temporal dependence of the displacement field, which is numerically by the Runge-Kutta method. Examples corresponding to acoustic emission due to a stressed strip are presented in the article, and the influence of viscous damping strated. In order to estimate the applicability of the proposed discrete model for the of AE in real systems, numerical results are compared with the signals recorded cantilever beam of PMMA.

convenient of lattice can be acare chardetermined cracking of is demondescription on a bent

INTRODUCTION PHENOMENA

like the deformation and cracking of solid materials, various phase transitions, corrosion, absorption of light pulses, etc. are often accompanied by the generation of elastic waves or acoustic emission (AE)[l, 21. The corresponding changes in the material can be characterized by the detection and analysis of acoustic emission signals, which is suitable for testing purposes. Recorded AE signals usually exhibit a complex structure which is caused not only by the mechanism of wave generation but also by the dispersion due to wave propagation and amplification of the signal in the detection system[2, 31. Because the influence of each particular process in this sequence is experimentally very difficult to specify exhaustively, it is often uncertain which property of the detected signal is a direct consequence of the material changes. For understanding of the experimental results[4] it is therefore instructive to study theoretically the acoustic emission phenomenon. In this task the amplification of a signal in a detection system can be quite simply described by standard methods relating the operation of linear, constant parameter systems[5, 61. More difficult is the description of elastic wave propagation in an object of finite dimensions, where the tensorial nature of elastic waves and mode conversions due to reflections at the surfaces of real samples often represent unsurmountable problems for an analytical treatment. Until now only examples of AE phenomena in samples of very simplified geometry, such as an infinite plate, have been analytically described[4, 7, 81. The most difficult is of course a physical description of the generation mechanism, which in the case of plastic deformation or cracking cannot be represented by a linear time-independent model[3]. The source is therefore often represented by a sudden localized change of material property and described by an appropriate physical quantity like released force[2, 3, 41. In spite of such simplifications the results of an analytical treatment are usually expressed in terms of various transforms, which must ultimately be numerically evaluated[4, 71. The advantage of an analytical treatment is thus lost, and it seems that a direct numerical treatment of elastodynamic equations could simplify the problems related to finite geometry and nonlinear phenomena. The purpose of the present article is therefore to describe a solid material by a discrete semielastic lattice appropriate for the numerical study of acoustic emission phenomena. In order to shorten the computer time needed for the numerical treatment of characteristic examples we apply here only a planar model of a lattice which can without difficulty be generalized to the three-dimensional case. However, nonlinear and viscoelastic properties of lattice elements are retained in order to make feasible the description of such complex phenomena as acoustic emission in plastics. Although we assume that all lattice elements are of the same square form, the rheological properties of each element are prescribable, so that acoustic emission in nonhomogeneous materials like composites can also be studied 621

622

I. GRABEC

and J. PETRISIt

with our model. With these assumptions we still describe acoustic emission as a wave phenomenon in the field of elastomechanics, while the determination of material parameters describing the rheological properties of lattice elements are left to more profound physical study[9, 10, 111.

DYNAMICS OF A DISCRETE SEMIELASTIC

LATTICE

In technical practice acoustic emission is most often studied on objects composed of plates. The lateral dimensions of the plates are usually an order of magnitude or more greater than the plate thickness. For the sake of simplicity we therefore assume that the plate can be represented by a planar lattice of mass points, and we consider only the inplane movements. The interaction between lattice points will be represented by massless ligaments connecting only neighbouring masses. We generally assume that the ligaments can be represented by a nonlinear spring and a parallel dashpot, while corresponding parameters are specified by the formulation of a particular problem. The governing equations of lattice dynamics follow from the analysis of interaction between two points. For this purpose we consider two neighouring lattice points with equilibrium positions at Tj and Tj, which are displaced by the lattice movement to Tf and rl, as shown in Fig. 1. Due to this displacement the connecting ligament changes its length 1 = 1 tj - ?/I by

with V = (x, y), and ii = (u.~, z+) being the position and displacement vectors, while i” = I: + ij. Denoting the unit vector in the direction from the&h to ith points by (2)

2 = ( - Cos cp, - sin (s),

we can express the force, which is exerted on the ith point by the ligament due to its elongation by the equation of a spring: gi = - kiihll?.

(3)

The force on thejth point is then gj = - k;i. The parameter kji represents the stiffness of the ligament. We assume that it depends on material parameters, ligament length and time and is therefore generally not a constant. Besides the change of ligament length, we also assume that the rate of its elongation contributes to the interaction between interconnected points. In order to determine the cor-

Y

X

I

Fig.

I. Displacement

of points

due to lattice

movement.

Description

of acoustic

m

emission

d

F* C

k Fig. 2. Scheme

responding by

of a Voigt element

between

two lattice

points.

viscous force we first express the velocity of the ith point in the ligament direction

$‘l)

with & = G; being the displacement ith point is then given by

velocity.

D;

=

= (7;;.n)n,

The viscous force exerted by the ligament on the

(jii(@’

-

(4)

-

$“))

(

(5)

while the corresponding force on thejth point is bj = - 0;. Expressing eyn (4) by the components of the vectors, we obtain the expression D; = - d;j [(jr;., - iti.\) COScp + (iii,. - iti,.) sin cp] (COS

cp,

the scalar product in

sin cp).

(6)

The parameter d;j depends on the viscous properties of the material and is here assumed to be a constant. It mainly determines the internal damping of elastic waves in a free plate and also influences the rate of defect development. In technical practice acoustic emission is often analyzed on plates surrounded by liquids, as for example the wall of a tilled pressure vessel. In order to describe the influence of the surrounding liquid on the propagating AE signals, we assume that the corresponding force acting on the ith lattice point is proportional to the velocity of this point: 3; = -

(.;I;.

(7)

Here c’denotes the coefficient of viscous (Newton) damping due to contact between the surface and the surrounding fluid. The forces I$, bj and 3; represent the internal forces of the lattice surrounded by a liquid. In addition, outside forces can act on the lattice, which for the ith point are described by F;. The ligament between the two lattice points can be schematically represented by a Voigt element consisting of a nonlinear spring and a dashpot as shown in Fig. 2[9]. By various connections of such elements, models of viscoelastic materials can be constructed. In our case we represent a solid square element of side I and thickness I, by the construction given in Fig. 3. The element mass m = p/‘/r is represented by four equal masses m/4 in the corners of the construction. These masses are interconnected by marginal (I) and diagonal (II) Voigt ligaments. In equilibrium the diagonal ligament is g2 times longer than the marginal one. If the stiffness of all the elements is assumed to be equal to li and c = d = 0, the analysis at

,

m Fig. 3. Representation

of a solid element

by a construction

of Voigt ligaments

h24 low

1. GRABEC

deformation

and J. PETRISIC

shows that this model corresponds

to an elastic solid with modulus of elasticity

E = 8kl31,

(8)

and Poisson number I, = l/3. There are many materials with a Poisson number approximately equal to 0.3, so that the correspondence between this rheological model and the elastic properties of the material are already established if the stiffness is determined from eqn (8). The proposed rheological model is suitable for the description of the in-plane movement of a rectangular planar wall submerged in a liquid. For this purpose the wall is represented by N = t?.,:tz, square lattice cells, with (n, + l).(n,. + I) lattice points. The masses of these points are equal to m/4 in the corners, m/2 on the edges and ttl inside the lattice. Also, the marginal ligaments on the edges and the diagonal ones are single, while the interior marginal ligaments are double. It is also reasonable to assume that Newton’s damping due to surface interaction with the liquid is proportional to the surface of the lattice cell, which is further proportional to its mass. The equations governing the lattice dynamics can be most lucidly presented if we introduce generalized vectors of dimension 2(n, + I)@,. + 1). The force exerted by a ligament 1 between points i andj is then represented by a vector {K!$‘} having nonzero components at 2i - I, 2i and 2.j - I, 2j, corresponding to the x and y components of the force I?; given by eqn (3) or Kj = - i?;, respectively. The distribution of forces caused by deformation of ligaments in the lattice is then obtained by summing such vectors over all the ligaments: {K} =

c

{Kj;‘}.

Similarly the viscous damping due to the deformation

of ligaments can be expressed

by

where again the only nonzero components of LX\’ are equal to the component of Dj from eqn (5). or 6, = - fij at positions 2i - 1, 2i and 2j - I, 2j, respectively. Introducing further the generalized vectors of displacement,

{U) = {U,.Y,u,,... . . , I/n, u,., . .}, we can express the equation governing the lattice dynamics by m [M]

{ii}

+ {K} + {D} +

C’

[Ml

{Lq = {F}.

(I I)

Here {F} describes the external forces, and [M] denotes a diagonal matrix with elements equal to f, 8 or 1, according to corner, edge or internal lattice points. Expression (11) represents a system of second-order differential equations for the field of lattice displacements {U}. It is generally nonlinear due to the assumed nonlinear dependence of ligament stiffness on its length and also due to the nonlinear dependence of Al on the displacement vector as given by eqn (3). Therefore its solution must be found numerically. For this purpose it is advantageous to introduce the nondimensional vectors by the expressions

with W; = k/m, to = oo.t and U’ = dlJldto. The parameters k and d can be taken as arbitrary, but it is convenient if they represent the typical or average stiffness and damping coefficients

Description

of the corresponding as

of acoustic

625

emission

linear lattice. Equation (11) can then be written in a nondimensional

[WI/” 10 + {Wo + h(D)0 + b,[W{U’h = N”,

form

(12)

in which bi = d/moo and b,%= c/moo represent the nondimensional internal and surface damping coefficients. The damping is weak if b 4 1. Acoustic emission events represent transient phenomena. In such cases the system of differential equations given by expression (12) can be efficiently treated numerically by the application of the Runge-Kutta methods[ 121. For this purpose a discrete step in nondimensional time must be applied which is small in comparison with 1, and the initial conditions must be specified. In what follows we always assume that the lattice is at rest prior to the AE event. Particular examples resembling AE phenomena in real situations can then be treated by proper adjustment of the ligament parameters, by which also the boundary conditions can be accounted for.

EXAMPLES

Acoustic emission phenomena are often studied on stressed strips or beams. As a characteristic example we therefore study a transient phenomenon caused by the cracking of a stressed rectangular lattice. It consists of 50 x 6 points and is assumed to be fixed at one end and loaded by a constant force on the other end, as shown in Fig. 4. The lattice is assumed to correspond to homogeneous viscoelastic material (like PMMA) with constant stiffness and damping coefficients of all the ligaments except the marginal one on the upper edge at onethird of the strip length from the fixed end. The cracking is represented by changing to zero the stiffness of this ligament at t = 0. The resulting AE wave is characterized by the temporal dependence of displacement on the upper edge at two-thirds of the strip length. The calculated records of vertical 11~and horizontal II, displacements are shown in Fig. 5 for a nondamped lattice (hi = 6, = 0). The head of the transient signal exhibits a shock-like structure consisting of a weak forerunner and a more intensive jump. From the time of arrival it can be estimated that the forerunner propagates with the velocity of longitudinal waves and the jump with the velocity of flexural waves in a beam. High-frequency fluctuations in the transient signal are the consequence of the dispersion of elastic waves in a waveguide. The transient signals calculated with nonzero damping (Fig. 6) show that weak internal (dashpot) damping mainly influences the high-frequency components, while the surface damping mainly influences the low-frequency components. With growing internal damping the effect of a breaking spring becomes less pronounced, which results in reduced amplitude and a smoother form of the AE transient signals, as shown in Fig. 7. In polymeric materials discontinuities can develop by the growth of crazes or cracking. Crazes develop due to the viscous flow of the polymer and do not produce acoustic emission. On the contrary, cracks develop due to the sudden unstable degradation of the fibrillar net in a craze which is accompanied by intensive AE[14]. According to our model a crack can be represented by excluding a complete ligament, while a craze corresponds only to a change of equilibrium spring length.

Fig. 4. Representation

of a cantilever beam by a lattice of 50 x 6 points for numerical “cr” denotes the place of cracked marginal ligament.

treatment.

626

I. GRABEC

Fig. 5. The calculated

and J. PETRI&c

of vertical (U,.) and horizontal the case without damping (h, = h, = 0).

time dependence

(ci,) displacements for

bs = 0

0

bi110-3)

_A

6.6

IO

--

Fig. h. The iniluence of viscous internal damping on the wavcbrm

/

33

of the vertical displacement.

Description

of acoustic

emission

‘65

f

Fig. 7. The influence

of viscous

surface

damping

on the waveform

of the vertical

displacement.

In order to compare numerical results with experimental ones, an experiment was made on a strip of PMMA with free length 100 mm, height 10 mm and thickness 5 mm. One end was clamped, and the other was pressed down by a certain deflection. The cracking was caused by wetting the upper edge with a drop of isoproponal 33 mm from the clamped end. Initial silent development of crazes was followed by jumpwise cracking. The vertical displacement was detected by a wideband NBS transducer[3] placed 66 mm from the clamped end. The signal captured by a transient recorder is shown in Fig. 8. In its main features the form of the experimental record resembles that of the numerically determined signal. The clearest discrep-

tips)

;

100

Ot

I

/

t

1 i L Fig. 8. A record

of vertical

displacement detected by a broadband cracking PMMA.

transducer

on a heam

of

I. GRABEC and J. PETRiSIt

6’8

;,/

~.. _+--_*_.-+_.-_--+__,.._ _-c__.-_c

.._ _1

ttps1

100 experfment

b, = bs q0

I Fig. 9. Experimentally

theory and theoretically determined records of vertical displacement by a step-like force on the end of beam.

excited

ancies appear with the arrival of waves reflected from the ends. This is understandable because our model does not correspond to real boundary conditions. Otherwise, more pronounced oscillations and a more quickly decreasing average amplitude in the region following the shocklike jump are characteristic of the experimentaf result, which is partly caused by the lower sensitivity of a transducer in the low-frequency region, A still better agreement between the numerical and experimental results is obtained if the AE event is simulated by breaking a pencil lead at the end of the beam, which is usually performed during the calibration of the AE transducer[l4]. The corresponding experimental record and the signat calculated for the case of step-like loading at the end is shown in Fig. 9. CONCLUSIONS The aim of our article is to present a physical model by which a step from the linear to nonlinear description of acoustic emission phenomena could be taken. For this purpose a phenomenological description of nonlinear material properties based on rheological parameters and a discrete lattice model of material are applied. The simple examples presented here demonstrate the applicability of this model for the description of real problems, but its true advantage will be revealed by our forthcoming treatment of essentially nonlinear phenomena. Only the influence of viscous damping on AE phenomenon is demonstrated by our examples, which is important for understanding acoustic emission caused by the crazing and cracking of polymeric materials, which has not yet been thoroughly explained[l4]. REFERENCES [I] T. F. Drouillard, Awrt.slic Emission. A Bihkqyrrphy with Abstrur~s. Plenum, New York (1979). [2] A. E. Lord. Jr.. Acoustic emission-An update. In Physic4 Acoustics. Vol. XV, pp. 295-360. Academic Press, New York (1981).

Description of acoustic emission

629

[31 D. G. Eitzen and H. N. G. Wadley, J. Res. NBS 89, 75 (1984). [41 A. N. Ceranoglu and Y. H. Pao, J. Appl. Mcc,h. 48, 125 (1981); 48, 133 (1981); 48, 139 (1981). In Plr.v.\icm/ 151 W. Sachse and N. N. Hsu, Ultrasonic transducers for materials testing and their characterization. Acousfics, Vol. XIV, pp. 277-406. Academic Press, New York (1979). [61 W. Sachse and A. N. Ceranaglu, U/rrc~.sonic.rIntemutioncrl 79, Graz. p. 138. IPC Science & Technology Press. Guildford, England (1979). [71 Y. H. Pao. R.-R. Gajewski and A. N. Ceranoglu. J. Aw~ts~. SW. Am. 65, 96 (1979). [81 R. L. Weaver and Y. H. Pao, J. Appl. Mech. 49, 821 (1982). [91 B. Rosen, ed.. Fructure Processes in Polymeric Solids. Interscience, New York (1964). I M. K. Kanninen et al., ed., fne/us/ic Behat,ior c$So/ids. McGraw-Hill. New York (1969) 1101 I I I I J. A. Simmons and H. N. G. Wadlev, J. Rrs. NBS 89. 55 (1984). ii2j G. A. Korn and T. M. Korn. Mathemrtticrrl Handbook. McGraw-Hill, New York (1968). 1131 1. Grabec and M. Platte. Srtzsors cnrd Acrrrtrrors 5, 275 (1984). [I41 I. Grabec and J. PetriSit, Proc. 5. Kolloy. S~,1zcl//e,,lis.tion, Zittau. 1984. To be published. (ReceiL’ed 28 January

1985)