ht. 1. Engng Sci. Vol. 26, No. 1, pp. 13-26, 1988 OE?O-7225/B $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright @ 1988 Pergamon Journals Ltd

A THEORETICAL MODEL OF EDDY CURRENT NONDESTRUCTIVE TEST FOR

ELECTROMAGNETOELASTIC MATERIALS

S. A. ZHOU and R. K. T. HSIEH Department of Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden

AIrstract-This paper investigates the problem of theoretical modelling of eddy current non- destructive tests for electromagnetoelastic materials. With the aid of an adjustable static magnetic field acting on test electromagnetoelastic solids,.an eddy current NDT approach, which can account for the effect of electromagnetomechanical interaction, is proposed and a fundamental system of linearized field equations governing this approach is then derived. A perturbation analytical solution for the ECT systems with circular symmetry is also obtained. The results are being used to study quantitatively the effects of elastic deformation fields coupled with electromagnetic fields in eddy current NDT. It is shown that the proposed approach may provide more flexibilities and information than conventional eddy current method which is based on a rigid-body model.

1. INTRODUCTION During recent years an increasing amount of research has been conducted to develop methods and procedures for improving interpretations in nondestructive testings. In the area of electric and magnetic testings, the eddy current non-destructive test has been used to solve a variety of material and product evaluation problems in metal industry for hundreds of years and studied by many researchers (see, for instance, Hughes [l], Fiirster and Stambke [2], Burrows [3], Dodd et al. [4], Libby [5], Wait [6], Chari and Kincaid [7], Sabbagh [f?] etc.). However, the state of the art of the traditional eddy current NDT field today seems to be still in its infancy as stated in early 1959 by Hochschild [9] and lately in 1981 by Lord and Palanisamy [lo]. The limitations of the traditional eddy current NDT model for rigid body are encountered, for instance, in trying to relate electromagnetic properties to mechanical, metallurgical and chemical properties, though changes in temperature, heat treatment, hardness, strength, purity, chemical or alloy composition, internal stresses, etc. do alter electromagnetic properties. In addition, the eddy current testing (ECT) for magnetic materials also presents special problems (see Drunen and Cecco [ll]). The relatively high value of its magnetic permeability drastically reduces the penetration depth as compared to nonmagnetic materials. Though the skin effect may be offset by reducing test frequency, the inability to distinguish between defects and random permeability variation still remains since the reduction to lower test frequencies would in general also yield reduced sensitivity and poor discrimination. It is therefore needed to further explore certain ECT models in which not only electromagnetic parameters but also some other potential variables such as mechanical parameters could be used, measured and analysed directly. The recent increasing knowledge on electromagneto- mechanical interaction phenomena allows to improve this state of the art by deeper investigations on the conventional eddy current NDT and on the theoretical modellings (see, for instance, Zhou and Hsieh [12]). In this paper, an eddy current NDT approach which can take into account the effects of elastic deformation coupled with electromagnetic fields will be introduced and studied.

2. AN ADJUSTABLE STATIC MAGNETIC FIELD ON

ELECTROMAGNETOELASTIC SOLIDS

It is, so far, known that if an electrically conducting magnetoelastic solid is subjected to electromagnetic forces while immersed in a magnetic field, the electromagnetic fields are still governed by Maxwell’s equations with, however, a modified Ohm’s law, while the elastic deformation is determined by the Euler’s equation with a modified Hooke’s law. The governing

13

14 S. A. ZHOU and R. K. T. HSIEH

field equations are then, in general, given by a set of nonlinear coupled partial differential equations (see e.g. Moon [13]), which may be summarized as follows.

Field equations:

VXE+$=O, V*B=O, VXH=J,

V a J = 0 in conductive solids, V.E=Oinvacuum

(2.1)

(2.2)

(2.3)

Constitutive equations:

J=a E+ (

SXB dt and M=xH, (B=,u,(H+M)) (2.4)

T = A(V l U)S + G(VU + (VU)r) + poxHH (2.5)

where u is the electric conductivity, x the magnetic susceptibility, p. the magnetic permeability in vacuum, A and G the Lame constants. E, H, J, B, U and T are respectively the electric field, the magnetic field, the electric current density, the magnetic induction field, the elastic displacement field and the elastic stress field. It is noticed that this set of equations is valid under the following hypothesis: (i) the elastic deformation is small so that a linear strain-displacement relation is hold and magnetostrictive and piezostrictive effects may be neglected; (ii) considered electromagnetoelastic materials are good conductors so that electric free charges and electric displacement current may be neglected for low excitation frequencies (less than 1O’Hz); (iii) the matter velocity field (W/at) is small compared with the light velocity (c = l/G) so that relativity effects are negligible (theoretical treatments of electromagnetic deformable solids from the point of view of Einstein’s special theory of relativity are referred to Grot and Eringen [14], Dunkin and Eringen 1151, Penfield and Haus [16], Hutter 1171, Hutter and Pao [18], etc.); (iv) the physical properties of the electromag- netoelastic solids are linear and isotropic; (v) no mechanical body forces are acting on the solid.

Suppose now that the electromagnetoelastic solid is under the exertion of an adjustable static magnetic field He as well as a time-varying electromagnetic field, the total electromagnetoelas- tic fields in the solid can thus be written as

H(x, t) = w(x) + h(x, t) (2&a)

B(x, t) = B’(x) + b(x, t) (2.6b)

E(x, t) = E(x, t) (2.fjc) U(x, t) = v”(x) + u(x, f) (2-W T(x, t) = P(x) + t(x, t) (2.~1

where @, B”, ti and ‘J? are respectively the static magnetic field, the static magnetic induction field, the static elastic displacement field and the static elastic stress field caused by the applied static magnetic field in the absence of external time-var~ng electromagnetic field sources. Here, it has been noticed that the static magnetic field does not produce a static electric field according to the Faraday’s law of induction. In the absence of time-varying sources, the static electromagnetoelastic fields in the solid are determined by the following field equations.

VXH’=O and V-B’=0 (2.7) V~T”+~o~(HoV)~=O (2.8)

in which B”=/& (where cc = iuo(I + x)) (2.9)

p = A(V l U”)6 + G(VU’ + (VU“)=) + /_A~~J-J%J~ (2.10)

with prescribed boundary conditions.

Eddy current nondestructive test 15

The fields II, b, E, II and t generated by a time-varying excitation electromagnetic field can thus be determined by the following field equations:

VXE+;=O, V-b=O, VXh=J, (2.11)

V - J = 0 in conductive solids, V - E = 0 in vacuum (2.12)

V*t+JX13°+po~((~d’)h+(h4’)H”)+po~(h4’)h+JXb=p$ (2.13)

where

E+$xB’+ dt %b ) and b=ph (2.14)

t = A(V l u)6 + G(Vu + (Vu)‘) + ,u,,x(H% + h@) + ,uoXhh (2.15)

It is shown that because of the nonlinearity of the electromagnetoelastic coupling phenomenon, the electromagnetic fields due to certain time-varying excitation sources can be influenced by the applied static magnetic field. Such a phenomenon, however, disappears in an electromag- netic theory on rigid bodies. In the next section, an eddy current nondestructive testing approach will be proposed with the aid of the electromagnetoelastic coupling effect.

3. AN EDDY CURRENT NDT MODEL ON ELECTROMAGNETOELASTIC MATERIALS

It is known that in the case of conventional eddy current NDT, the electromagnetoelastic coupling effect, resulting from the elastic displacement velocity term in the modified Ohm’s law and the electromagnetic body forces in Eulerian motion equation, are in general too small to have significantly detectable effect on the ECT results because of the low excitation levels in the probes. The electromagnetoelastic coupling effect is, therefore, neglected in a conventional ECT model for rigid bodies, which has been used widely and successfully in many applications, However, as the rigid body ECT model uses only the electromagnetic parameters, such as electric conductivity and magnetic permeability, it may fail to study a variety of NDT problems in which other variables such as mechanical parameters rather than the electromagnetic parameters play an important role [9]. In addition, as excitation frequency is often the only test variable over which an inspector can exercise any control, the applicability of the traditional rigid body model is again limited by the well-known skin effect in eddy current NDT especially for ferromagnetic materials [ 111.

To offset the deficiencies of the conventional rigid body ECT model and to dig out the effects of mechanical parameters in eddy current NDT, an eddy current NDT approach for electromagnetoelastic materials is introduced here. In this approach, an adjustable static magnetic field is suggested to be applied on the testing electromagnetoelastic solid (see Fig. l), which is used to provide the necessary impetus to generate sufficient and detectable signals of electromagnetoelastic coupling effects as we can see in eqns (2.13)-(2.15). As additional test variables, the direction and strength of the applied static magnetic field may be used to exercise certain controls in the eddy current NDT besides the excitation frequency of the current in the probe.

Fig. 1. An eddy current NDT approach for electromagnetoelastic materials.

16 S. A. ZHOU and R. K. T. HSIEH

To dig out the ele~tromagnetoelasti~ coupling effect, one can adjust the applied static magnetic field so that the relationship lhi] << /HP1 (i = 1, 2, 3) is hold in the electromagneto- elastic solid during the ECT test. Thus, a set of linearized coupled field equations for the eddy current NDT problems of electromagnetoelastic solids under the exertion of the applied static magnetic field can be obtained by dropping the products of u, h and J, i.e. (&/at) X b, poy,hh and J X b. The linearized field equations write

vxE+g=o, V.b=O, Vxh=J, (3.1)

V l J = 0 in conductive solids, V.E=Oinvacuum

V~t+JXB”+po~((~d’)h+(h*V)@)=p$

(3.2)

(3.3)

where

J=a E+zXB’ ( >

and b=ph

t = A(V - u)6 + G(Vu + (Vu)‘) + pox(@h + hti)

The interface conditions read

(3.4)

(3.5)

nX [ E+$XB’ =O, 1 n l [J] = 0, n X [h] = K, n l [b] = 0, (3.6a)

[uf=O and [t+t”]+n=O (3.6b)

where [e] denotes the jump of a certain quantity on the two sides of the interface, K the surface current density and tM the electromagnetic stress which is defined by

It is to be noticed that in the second terms of the RHS of eqn (3.8), use has been made of the approximation p = pox for ferromagnetic materials (x >> 1). For diamagnetic and paramagnetic materials (Ix]<< l), this term as well as the third term on the RHS of eqn (3.5) and the magnetic body force, the third term on the LHS of eqn (3.3) may be neglected. In this paper, a linear approximation for the magnetic permeability of ferromagnetic materials, will be used, i.e. b=ph where p = constant. As eddy current NDT probe excitation levels are relatively low, a linear initial permeability value may be adequate for modelling the test ferromagnetic materials provided that the applied static magnetic field is not too strong. However, for applied static magnetic field strong enough to magnetically saturate the test material, the linear appro~mation may still be used, but the remanence effect has to be taken into account. Initial conditions are ignored since we shall only consider time-harmonic steady state cases in this paper. In addition, the static elastic fields caused by static magnetic body forces have no contribution to the eddy current NDT problems within the range of the considered approximation, and, therefore, will not be considered in the following sections.

4. ECT FORMULATION OF INHOMOGENEITIES IN MAGNETOELASTIC SOLIDS

Now, let us consider the ECT problem of an electromagnetoelastic solid with in- homogeneities. For simplicity, the material of the probe is supposed to be a nonmagnetic rigid conductor with electric conductivity CT o. The excitation current in the probe is considered as time-harmonic with the angular frequency w. The excitation electromagnetic fields, which exist in the absence of the test magnetoelastic solid will be denoted by E” and h”. The total electromagnetic fields in the presence of the electromagnetoelastic solid will be the sum of the excitation fields and the induced fields, i.e.

E = R” + Rind, J =Js +Jind, h = h" + hind, b = b" + bind (4.1)

Eddy current nondestructive test 17

in which Js = a,,fW + y”J” and Jind = (a - aoYo)ES + o(Eind + Y&X B”) (4.2)

b” = p& and bind = (p - po)hs + phind (4.3)

where J’ is the external source current density which is supposed to be only in the probe, y” and y are the indicative functions of the probe and of the test solid defined respectively by

1, yO(x) = ()

1,

x in Probe and y(x) =

( 1, x in test solid

otherwise o (4.4)

, otherwise

For harmonic excitation, the time dependence of any field function F(x, t), due to the linearity of the considered field equations, can be expressed by the complex exponential notation F(x, t) = Re{F*(x)exp(iot)}. Introducing now the magnetic vector potential A(x) defined by

b*i”d(x) = V x A(x) (4.5)

with the Coulomb gauge condition V. A = 0 (A may be complex), one can get from the linearized Maxwell’s equations (3.1) that

E*ind(x) = -ioA(x) - V+(x) (4.6) and

J *ind = (o _ ooyO)E*s _ aV# - ioa(A - yu* X B”)

where $(x) is a scalar electric potential. For N discrete inhomogeneities in the test solid, one may write that

(4.7)

I (4.8) 44 = uor"w + %YW( 1+ g 2 YW)

~=yl-(l-~(l-~l~y”(x)))y(x)) P(X) PO

(4.9)

(4.10)

C’,,(x) = C$l+ $. AC$)Y”(x) (4.11) LY=l

in which Au, = a, - a,,,, Apa = ,u~ - p,, Apn = pn - pm, and AC = C(“) - C” (which, for the isotropic materials, reads AC@ = Ai2’“‘6,6,, + AG’“‘(6,Sj,+ S&jk)) are respectively the perturbation values of the material properties between the cv-th inhomogeneity and the matrix solid. y”(x) is the indicative function characterizing the position, size and shape of the a-th inhomogeneity. a,, pm, pm and C” are respectively the electric conductivity, the magnetic permeability, the mass density and the elastic moduli of the matrix solid.

After some manipulations, a set of field equations for the determination of the magnetic vector potential A, the electric scalar potential $ and the mechanical displacement vector u* may be derived as follows:

V*A = iopou,,,yA + po(uoyo + u,,,y)V# - ~~u,yE*~

- iwpou,yu* X B” - V x ((1 - po/,QyV X A) - poJ*def in V” (4.12)

V*# = ioBo - (V X u*) + iwp,*d”f/um inV+VO (4.13)

V2@=0 inV”-V-V’ (4.14)

C$U~,[j + PmW2Ui* + /lO(x(I$hj + hiHy)),j +fFdefce) +fr’” = 0 in V (4.15)

Here, Jdef is the fictitious current density modelling the N inhomogeneities in a magnetoelastic solid defined by

1 J*def=-V X

c1tn x N * y”(V x A + pa*“)) a=1 Pm

+ 2 Au,y”(E *S - V# - iwA + iou* X B”) (4.16) a=1

18 S. A. ZHOU and R. K. T. HSIEH

Pt? def is the fictitious electrical charge density defined by

P V . (y”(E*” - V$ + io(u* x B” - A))) (4.17)

and fcdef@) is the fictitious body force (elastic part) defined by

f;d”f’e’ = -$ Ap,02y”u,* + $ AC$,,y‘%:,,, (

(4.18) o(=l Ir=l .i

and V and V” denote respectively the volume of the test body and the volume of the probe. The mathematical formulation (4.12)-(4.15) constitutes a set of differential equations to

describe the eddy current NDT problem for magnetoelastic solids with inhomogeneities under the exertion of a static magnetic field.

5. ECT PROBLEMS OF MAGNETOELASTIC SOLIDS WITH CIRCULAR SYMMETRY

In this section, a set of non-dimensional coupled ordinary differential equations with interface conditions is derived explicitly for the study of eddy current NDT problems of electromagnetoelastic cylinders. It is assumed that the test material is a cylinder of infinite length with radius R2, electric conductivity u2, magnetic permeability p2, elastic Lame’s constants AZ, G,, and mass density p2. It is also supposed that a cylindrical inhomogeneity with radius RI and with material properties ul, pl, Al, Cl, pl, may be embedded in the test electromag- netoelastic body. The probe with excitation current is supposed to be a very thin cylindrical shell with radius R (R > R2 > RI > 0) such that it may be approximately modelled by a surface current in a vacuum but its material properties are retained, i.e. its conductivity and rigidity. All considered material properties are supposed to be uniform along the direction of the axis of the cylinder (see Fig. 2). An external static magnetic field H” is supposed to be applied on the test cylinder, which is uniform and is along the direction of the axis of the cylinder. Circular symmetry of all properties is finally assumed.

Thus, the field quantities will only vary in the radial r-direction. The magnetic vector potential A(x) is then along the e-direction while the mechanical displacement u along the r-direction. A set of coupled ordinary differential field equations in a non-dimensional form can then be derived as

(R,IR<y<landl<y<m)

A(y)=ip2a + q&ii(y), cu=1,2

--A(Y) -Y, a=l,2

(5.1)

(5.4

(5.3)

Fig. 2. An eddy current NDT for an electromagnetoelastic cylinder with an inhomogeneity.

Eddy current nondestructive test 19

in which the following dimensionless quantities have been introduced

Rae A(y) = - 4V) iEZ”(R) and WY) = R25,&“(R) (5.4)

where y = rlR, and

E;“(R) = _ iWP;RrzS and & = U,B!“’

h, + 2G, (5.5)

where I:” is the total excitation surface current density and BP) = pa& is the static magnetic field in the cu-th area r, (i.e. r1 = (0 < y <RJR} and r2 = (RJR <y < R,JR) and r, = {RJR < y < l}). The dimensionless parameters pa, qLy and r, ((Y = 1, 2) are respectively defined by

Pa = Re, rll, = R2/3?,ou,(B$?,“))2

A, + 2G, (5.6)

(5.7)

It may be seen in eqns (5.3) and (5.7) that the part a((&~/%) X B”) X B” of the electromagnetic body force plays a role of a damping force. This implies that when elastic deformation in a test solid is considered, this force cannot be ignored because a resonance phenomenon would occur otherwise. The interface conditions in eqn (3.6) may be deduced here as follows:

A(3) = /jc2) _ & C(2) at y = R2fR

kC2) - (82)2 u Q2 -w =A(l) __u h -cl)

@I>” aty = RJR

(5.8)

(5.9)

[ii] = ii (a+l) _ jp = 0 aty = RJR (5.10)

and at y = RJR (a = 1,2)

where

W,=i Id(yA("+'))-ld ( ydy y dy (Ym) (5.13)

It is assumed here that no dynamic mechanical surface forces are acting on the outer cylinder surface of the test body. The first term on the LHS of eqn (5.12) therefore vanishes for (Y = 2. In addition, the physical conditions with ii(O) = 0, A(O) and A(w) finite, have to be satisfied. At y = 1, one also has

; (yp3’) = ?g (5.14) e

where AC3)(y) is defined in the vacuum region r3 and the induced current Z$‘” in the probe is caused by the presence of the test body. It is shown that the terms on the RHS of eqns (5.11) and (5.12) are due to the effect of ferromagnetic inhomogeneity, which vanishes for non-magnetic materials.

6. AN ECT SOLUTION OF MAGNETOELASTIC CYLINDER WITH INHOMOGENEITY

In this section, an analytical perturbation solution is derived for the eddy current NDT of an electromagnetoelastic cylinder with an inhomogeneous inclusion. The size effect of the

20 S. A. ZHOU and R. K. T. HSIEH

inhomogeneity on detecting electromagnetic fields can be used to study the important eddy current NDT problem of the measurement of thickness and cladding thickness of magnetoelas- tic materials. Differing from conventional rigid body results, this solution can account for the effect of elastic properties of testing materials, which may not only improve measuring accuracy but also may offset the skin effect in eddy current NDT especially for ferromagnetic materials.

To solve the eqns (5.1)-(5.3), the decomposition A(y) =A’(y) + A&y) is first considered. Here, A0 denotes the corresponding rigid-body solution determined by the equations

( 2

b+ld - +o(y) = 0, dy2 ydy Y’

in vacuum

( ((u=l,2)

(6.1)

(6.2)

ti is the perturbation magnetic potential due to electromagnetoelastic coupling effects. At its first approximation, the magnetic potential A, taking into account the electromagnetoelastic coupling effects, may be determined by the following equations:

( in vacuum

( ~+ld-(iSi,+;))A(y)=iB~+t),liO(y), ((u=1,2) dY2 YdY

in which the elastic displacement c”(y) is determined by the equation

( $+;$+ +$)tiyy) = -/qy) _Y, ((u = 1, 2)

with prescribed interface conditions. After some manipulations, one can obtain at its first approximation that

COY + Cl/Y, R,IR~ycl

A(Y 1 = C2W2yi1’*) + GK1(P2Yi1’2) - Y + F,(Y ), RJR < y -C RJR

GW~yi1’2) -Y + F,(Y), OsyCRR,IR

(6.3)

(6.4)

(6.5)

(6.6)

where ZI and K1 are respectively the modified Bessel function of the first kind of the first-order and the modified Bessel function of the second kind of the first-order. By using the interface conditions, one can derive that Co = Z’e”d/Z$, and C1, C2, C3, C, are determined by the following linear system of algebraic equations:

R -- R2

~l(X22) K&22) 0 \ /Cl\

0 ~22~0(~22) -~22K0(~22) 0

0 4(x12) K1h2) -~lhl)

0 x12~o(x12) -~12Koh2) -x11; Ioh 1)

(6.7)

where IO and K. are respectively the modified Bessel function of the first kind of the zero-order and the modified Bessel function of the second kind of the zero-order, and

xl1 = PI 2 il”, x12= 2 fi %i”‘, xz2 = /3* 2 il”,

and

Eddy current nondestructive test 21

Functions F’ and F2 are respectively the particular solution of the equation:

( 2 A+__ ’ d - (iSl+$))W) = rlS”(r>, dy2 ydy (a = L2) (6.8)

and can be found by a general method of the variation of parameters for the nonhomogeneous equation, an explicit solution of which will be given in the next section. Here, the elastic displacement ii”(y) is found at its zero-order approximation as

CgZI(p2yP2) + C!K,(fi2yPn)

ti + i@ > (RJR <y <RJR)

ii”(y) =

’ QsyCR,/R

(6.9)

where 51 and YI are respectively the Bessel function of the first kind of the first-order and the Bessel function of the second kind of the first-order, and the constants C$ C;, 6: and G are determined by eqn (6.7) in which one lets q, = 0 (i.e. F, = Fk= 0) (a = 1, 2) and Ck = c”, (k = 1,2, 3, 4). The constants D1, D2, and D3 are determined by the interface conditions (5.10) and (5.12), which lead to the following linear system of algebraic equations:

(3 S,t Z)(4)-(;;) (6.10)

where the elements of the coefficient matrix Q read respectively

Qll = 522JX~22) + M(E22>/(& + W, where E22 = r2R2/R Q12 = E22WE22) + ~2Y,(E22)/@2 + 2G2), Qn = 0, Q21= 51(512), Q22 = KG,,), Q23 = -JI@~II,) where 652 = t2R1/R and gll = t,R,IR,

Q31= E12JXE12) + ~24(512)/@2 + X2),

Q32 = h2Y;&2) + ~2Y,(t12)/(~2 + 2G2),

Q33 = - ~131//32)2(511Ji&) + WI(~II)/(& + X&J),

and

r~ = W%x22W22) + A2&(~22)/(A, + X2)) + C$22KI(~22)

+ ~2Kdx22)/(3c2 + 2G2)))/(6 + iSI) - %/Bf

r2 = (C%l,(x12) + CXI(~~))/(G + $4) - CZ,(~~J/(~~ + &),

r3 = (C%IZG(XIZ) + A24(~12)/@2 + 2G2)) + C%~,,G(x,,)

+ A2K&12)/(~2 + 2G2)))/(6 + $2)

- C~(B~/,S~)‘(~,,Z;(X,,> + ~I~IW/& + 2G))/(6 + b%) + %/P$

in which W”, = i _$ (yAO("+')) _ $ (YA ))I -o(n) (a = 1, 2)

y=R,IR

(6.11)

the rigid-body solution of the magnetic potential A0 can be found as

COY + C/Y> R,/R<y<l

AO(y) = C~ZI(~2yi1’2) + CzK1(p2yP2) - y, RJR <y < RJR

C!ZI(&yi”*) - y, Osy <RJR

(6.12)

22 S. A. ZHOU and R. K. T. HSIEH

The obtainment of the magnetic potential A(y) and the elastic displacement field ii”(y) provides the solutions of calculating a variety of electromagnetic induction and electromag- netoelastic coupling phenomena. A concrete application will be illustrated in the following section.

7. ELASTIC DEFORMATION EFFECT IN EDDY CURRENT NONDESTRUCTIVE TESTS

A quantitative study of the effect of elastic deformation fields coupled with electromagnetic fields in the eddy current nondestructive test is performed based on an impedance analysis (Hochschild [14]) in this section. For simplicity, the testing material sample is supposed to be a homogeneous nonmagnetic elastic conductive cylinder with radius R2 = b (and RI = 0). With the aid of the results given in the last section, one can easily calculate the solenoids impedance which is the ratio of the coil voltage to the current flowing in the coil. After some manipulations, an analytical expression for the normalized solenoid impedance may be obtained. In its first approximation, it writes

in which 2’ = ionpoR2 is the empty solenoid impedance and the functions S(y) and F(y) are defined respectively by

(7.2)

F(Y) ‘4% -~JI(zY)

’ + ” ( r2 + iB’)pi’” I,( p g i112)

+ S(y)WYi”2) + P2WWW2)

/%“210( /3 f iln) (7.3)

where Pi and P2 are respectively the functions defined by

PI(y) = +$ (1 + x’(z;(x)Kl(x) - (1+ $*(x)K’(x)))

P2(Y) = - $5 (r;20 - (1+ $)m)

in which x = pyi”2. The g-factor is given by

where the denominator reads

(7.4)

(7.5)

(7.6)

(7.7)

It can be seen that if 77 = 0, the classical result of the normalized impedance predicted by a rigid body model (see Libby [5]) is recovered.

The total eddy current (complex amplitude) in the test magnetoelastic body can be written as

J;ind(r) = J;rigid(r) + J;ehStiC(,)

(7.8)

where, by means of eqn (4.2), the eddy current Jzrigid predicted by a conventional eddy current model for a rigid body reads

J grigid(r) = aE~“(r) - iouAz(r) (7.9)

Eddy current nondestructive test 23

and the eddy current JZ;e’astic generated by the elastic deformation coupled with electromagnetic fields in the test body is

JPtiC (r) = -ioaAA,(r) + iwauf0Z3~ (7.10)

where u is the electric conductivity of the test cylinder. The amplitude of the eddy current induced by elastic deformations at any radius inside the

nonmagnetic elastic conductive cylinder as compared with the conventional rigid body eddy current at depth b - Y’ below the surface of the test body can then be found as

IJ$‘stic(r)( = iaoR2(B:)*(gJl(ty) - Z,(/?yi”*)) lJggid(ro)l (A + 2G)(z* + ifi2)Zl(/3yoi1’*)

where y” = r”/R. To study the dependence of the part of eddy current induced by elastic deformation on the excitation current frequency, on the applied static magnetic field and on the material properties of the test body, let us first look at the g-factor and plot the value of its denominator lg*I in Fig. 3. It is shown that when an excitation current frequency is close to characteristic frequencies of the test system, the value of lg*I becomes very small and lgl, therefore, becomes very large. It would go to infinite if we ignored the part of the electromagnetic body force induced by the velocity of the elastic displacement i.e. f = a((h/&) x B”) x B”. This damping effect is shown to increase as the strength of the applied static magnetic field and the conductivity increase. The characteristic frequences can be found

3.0

2.5

p = 2.7 x. IO3 kg /m3

Y : 0.35

15cmoo 250000 350000

Excltotion frequency w ( rod / s)

Fig. 3. Characteristic frequencies of the testing system, damping phenomenon and their dependance on the applied static magnetic field and material properties of the test body.

(1) B,=OSTesla, E=7x10gN/mZ, a=3.4~10’1/61.m; (2) B, = 2.0 Tesla, E = 7 x 10” N/m*, u = 3.4 x 10’ l/Q.m; (3) B, = 2.0 Tesla, E = 7 x 10” N/m’, u = 3.4 x lo6 l/Q.m; (4) B, = 0.5 Tesla, E = 7 x 10” N/m2, u = 3.4 x 10’ l/Q.m.

24 S. A. ZHOU and R. K. T. HSIEH

as the roots of the equation given by

J;(x) +&x-V&) = 0 (7.12)

where x = bo(pl(A + 2G))“‘. Figure 4 gives the quantitative distribution of the eddy current in the test cylinder as given in

eqn (7;ll). It is shown that if the excitation current frequency is close to the characteristic frequencies of the system, the eddy current generated by elastic deformations could amount up to 30% of the value of conventional rigid body eddy current at depth (b - r”) = 36 below the surface of the test body, depending on the strength of the applied magnetic field and the material properties of the test body. Here, S = (2/(~c~))~‘~ is the conventional skin depth. The quantitative results also show that as the depth below the surface increases, the mechanically generated eddy current decreases very slowly rather than exponentially. This phenomenon thus provides a possibility to reduce the skin effect which is one of the main ECT inspectability

factors ([ll]). Optimal choice of the excitation current frequency is, however, found to be necessary to dig out the electromagnetoelastic coupling effects.

The dependence of the normalized impedence on the material properties of testing cylinders under the exertion of a static magnetic field are quantitatively studied in Figs 5a and b. It is found that the effects of the elastic deformations and the applied static magnetic field on the impedence are in general small. The effect of the elastic deformations on the impedence increases as the Young’s modulus of the test solid decreases. Figure 5b also shows the possibility of detecting the Young’s moduls E by the proposed ECT model provided that enough measuring accuracy can be reached by high sensitive ECT instruments. In all figures, relevant quantities and their unit will be p (kg/m3), Y = 0.35, p = p. = 4n x lo-’ (H/m), b/R=0.9 (R=O.lm), a(ll(Q. m)), B = Bz (Tesla), E (N/m2) and o (rad/s), unless specification is made. Here, Y and E are respectively the Poisson’s ratio and the Young’s modulus of the test material.

100 c p--2.7x10' kg/m', ~~0.35, u=3.4xlO' 1Ifi.m

3 eo- (II RigId body modal 1.22 X?O'

c 12) 05 7 xlOS 1.22 I IO' =: "> (3) 20 7 110'0 1.60 x.10'

$D (41 0.5 7 I 10'0 1.60 x IO' 60-

\ (5) 0.5 7 r10' 1.60 x10'

s :

:a L

; 40-

?

:

(1 1

I I I I I I I I I I I eo 81 92 93 94 85 96 87 69 99

r - Radius coordinate (mm)

Fig. 4. Distribution of eddy current induced by elastic deformation in test body and its dependence on the applied static magnetic field, the excitation frequency and the Young’s modulus.

Eddy current nondestructive test

0.245r (a)

0.235

c- err 0.225

i

2 0.215 Y

1 W’i.22X

cl005 0.015 QO25 0.035 0.045

Re (Z/I Z’l)

Fig. Sa. Dependence of the normalized impedence on material properties of the testing cylinder.

0.2195 - 0 Rtqld model

. E=7x101°, 812

n E=7x10g. B-2

0 E=7x10g, B=4

r r\,

i $ 0.2192 -

Re (Z/lZ’l)

Fig. 5b. Effects of elastic modulus and static magnetic field on impedence in the eddy current NDT.

8. CONCLUSIONS

A theoretical model for the eddy current NDT of magnetoelastic materials has been introduced in this paper. The introduction of the applied static magnetic field provides the impetus to dig out the electromagnetoelastic coupling effect, as well as the additional test variables over which one can exercise certain controls during the eddy current testing. An A - qb - u formulation for the analysis of eddy current NDT problems of magnetoelastic solids with inhomogeneities has also been developed, in which the inhomogeneities are modelled as fictitious multipole electric current, fictitious multipole electric charges and fictitious multipole body forces. A set of non-dimensional coupled ordinary differential equations for the determination of the ECT problem of magnetoelastic cylinder with inhomogeneity is then derived and an analytical perturbation solution is obtained. The results on the quantitative study of the effect of interaction between elastic deformation field and electromagnetic fields in eddy current NDT are as follows: (1) There exist characteristic frequencies at which the effect of elastic deformation is amplified and may become important in ECT. Optimal choice of the excitation frequency is, therefore, necessary to dig out electromagnetoelastic coupling effects. (2) The part of electromagnetic body force (a(&~/&) X B) X B in magnetoelastic solids under the exertion of harmonic loads may be not negligible due to resonance phenomena. (3) The softer the material is, the larger the effect of elastic deformation on ECT becomes. (4) The “elastic” eddy current in the test body under permanent exertion of a static magnetic field has almost no skin effect.

26 S. A. ZHOU and R. K. T. HSIEH

REFERENCES

[l] D. E,: HUGHES, Phil. Mug., Series 5 8, 50 (1879). [2] F. FORSTER and K. STAMBKE, Z. Mefal. 45, 166 (1954). [3] M. L. BURROWS, Theory of eddy current flaw detection. Ph.D. Dissertation, University of Michigen.

University Microfilms, Ann Arbor, Mich. (1964). [4] C. V. DOOD, W. E. DEEDS and J. W. LUQUIRE, Inf. J. NDT 1,29 (1969/70). [5] t9kj, LIBBY, Introductron to Elecfromagnetrc Nondestructive Test Methods. Wiley-Interscience, New York

[6] J. R. WAIT, IEEE Inst. Meas. IM-27, 235 (1978). [7] M. V. K. CHARI and T. G. KINCAID, In Eddy Current Characterization of Materials and Sbuctures, ASTM

STP 722 (edited by G. Birnbaum and G. Free), Philadelphia, pp. 59-75 (1981). [8] H. A. SABBAGH and L. D. SABBAGH, Znr. Adu. NDT 10, 267 (1984). [9] H. HOCHSCHILD, Prog. NDT 1, 59 (1958).

[lo] W. LORD and R. PALANISAMY, In Eddy Current Characterirarion of Marerials and Structures, ASTM STP 722 (edited by G. Birnbaum and G. Free), Philadelphia, pp. 5-21 (1981).

[ll] G. VAN DRUNEN and V. S. CECCO, NDT Int. 17,9 (1984). [12] S. A. ZHOU and R. K. T. HSIEH, In Electromagnetomechanical Interactions in Deformable Solids and

Sbuctures, Proc. IUTAM Symp. (edited by Y. Yamomoto and K. Miya), pp. 221-226. North-Holland, Amsterdam (1987).

[13] F. C. MOON, Magneto-Solid Mechanics. John Wiley & Sons, New York, (1984). [14] R. A. GROT and A. C. ERINGEN, Inr. J. Engng Sci. 4,639 (1966). [15] J. W. DUNKIN and A. C. ERINGEN, Znt. J. Engng Sci. 1,461 (1963). [16] P. PENFIELD and H. A. HAUS, Electrodynamics of Moving Media, Research Monograph No. 40. MIT Press,

Cambridge, Mass. (1967). [17] K. HUTTER, Electrodynamics of deformable continua. Ph.D. thesis, Cornell University, Ithaca, N.Y. (1973). [18] K. HU’ITER and Y.-H. PAO, J. Elasticity 4, 89 (1974). [19] R. HOCHSCHILD, Nondestructive Testing 12, 35 (1954).

(Received 28 February 1987)