NASA Contractor Report 362 3

Phenomenological and Mechanics Aspects of Nondestructive Evaluation and Characterization by Sound and Ultrasound of Material and Fracture Properties

Li-Sheng W. Fu

GRANT NSG-3269 OCTOBER 1982

NASA Contractor Report 3 62 3

TECH LIBRARY KAFB, NM

llnlll~lllnm~~l~lllilRllll I?lb235Z

Phenomenological and Mechanics Aspects of Nondestructive Evaluation and Characterization by Sound and Ultrasound of Material and Fracture Properties

Li-Sheng W. Fu The Ohio State University Columbus, Ohio

Prepared for Lewis Research Center under Grant NSG- 3 2 69

National Aeronautics and Space Administration

Scientific and Technical Information Branch

1982

INTRODUCTION

The recent thrust in the study of .elastic wave scattering has been

highly motivated by its applications in various fields such as seismic

explorations, nondestructive testing, material property evaluation and

dynamic stress concentration. An excellent account of history and ,

fundamentals of elastic wave motions is given in [l-2]' and a‘comprehensive

discussion of applications in nondestructive evaluation from a.theoretical

viewpoint can be found in [3-41.

The scattering of a single ellipsoidal scatterer in an infinite

medium is of fundamental importance and is attacked by the method of

matched asymtotic expansions [S], the direct volume integral formulation [6],

the surface integral formulations [7], and also, recently, the polarization

approach [8].

It appears that Ma1 and Knopoff [6] were first in presenting a direct

volume integral formulation where they gave the scattered displacements in

terms of volume integrals involving the displacements and strains inside

the scatterer. Not knowing these fields, they used the solution when no

scatterer was present to.obtain .an approximate solution for a perfect

sphere. The- same approach was later taken by Gubernatis [9] for an

ellipsoidal inhomogeneity. These so-lutions are appropriate at longwave

scattering, ka < 1, where k is wave number and "a" is a typical geometric

dimension.

Using the equivalent inclusion method, Eshelby (10,11,12] studied

the static edastic fields "inside" and "outside" an ell.ipsoidal inclusion

1 Number in brackets designate References at the end of paper.

or inhomogeneity embedded in an infinite isotropic elastic medium under

applied tension. All through this paper, an "inhomogeneity is referred

to as a region of different elastic moduli and density compared with its

surrounding matrix and an "inclusionl' is referred to as a region with the

same elastic moduli and density as its surrounding matrix but include in

it a distribution of eigenstrains. Eigenstrains are strains that are not

derived from mechanical loading. As examples we note that thermal strains

and also the swelling strains due to the presence of moisture are special

types of eigenstrains.

The method of equivalent inclusion is a method where the inhomogeneity

is replaced by an inclusion such that solutions for the two problems are

exactly the same. the basic concept is sketched in Figure 1. Wheeler

and Mura [13] first developed but did not apply a complete eigenstrain

formulation to the dynamic case. The purpose of this study is to extend

the method of equivalent inclusion to fully develop the equivalence

conditions and to present a method for complete determination of the

eigenstrains and/or their derivatives as appropriate.

The equivalence conditions and the solutions to the scattering of an

inhomogeneity in terms of eigenstrains and/or their derivatives are first

developed. Agreement with other approaches is then easily seen. The

formulation is general and both the inhomogeneity and the host medium

can be anisotropic. The scattering of an ellipsoidal inhomogeneity in a

linear elastic isotropic whole space subjected to plane time-harmonic wave

is studied and the differential and total cross-sections for a uniformly

distributed eigenstrain are detailed and shown.

Mr. H.S. Jing and Mr. Y. Paul Hsu assisted in the computer

programming and graph plotting. Numerical work was conducted by using

the AMuU-IL 470 computing system and plotting was done on the MING System

of the Department of Engineering Mechanics, Ohio State University.

2

EQUIVALENCE CONDITIONS

In this section, we develop the equivalence conditions by requiring

that the displacement and stress fields obtained in the inclusion problem

be identical to those for the inhomogeneity problem, Fig l(a) and l(b).

The Inhomogeneity Problem (Problem I)

Consider the problem of a single inhomogeneity occupying the region n

in the infinitely extended region D-a subjected to applied incident wave

field u(~), Figure 1. j

Let the elastic moduli and the mass density be

denoted by Cijkn. and P 1 for the inhomogeneity, and by Cijkn. and P for the

host medium, respectively.

The governing equations for the displacement field are:

. . C jkrs Ur,sk + AC jkrs Ur,sk = p"j + Ap ii.

3

in which we used

'jk =c jkrs ur,s + AC jkrs ur,s

Ap = 0 in D-R

P ' -P in 52

in D (1)

(2)

(3)

IO in D-G AC jkrs = C! - Cjkrs in s2 (4)

JkrS

Let the superscripts (i) and (m) denote fields associated with the

incident wave and the mis-match in mass density and elastic moduli. It is

clear that

3

- -..... -- ._ . . _. - .-

= #I 5 j

+ ,(m) j

(ml as in the absence of mis-match,u. 3

disappears and the total field is

identical to the incident field.

The boundary conditions are that (1) the displacements and tractions

at the intersection of the regions Q and D-Q must be continuous, and (2)

the characteristics of out-going wave field and that the stresses die out

at infinity must be observed.

The Inclusion Problem (Problem II)

Consider next an infinite elastic solid of homogeneous moduli C ijka and density p with distributed eigenstrains, denoted by cf., in a region R,

II such that Q is identical in shape and size to that in Problem I, and

E* = 0 ij Ef .

II

The total strain field is

where

in D-Q in Q

E = rs (u +u

r,s &/2 = Ee + E* rs rs

= c e 'jk jkrs srs

(7)

(8)

Using Equations (6,7) in the equations of motion, we easily obtain the

governing equations for the total displacement field as follows:

C U jkrs r,sk =pii.+c jkrs , % k in D

J (91

4

It is, clear from Equation

= $1 + u* 5 j j

(9) that

(10)

where u'! J

is the displacement field due to the presence of E?. and it 1J

disappears when s?. vanish. 17

The only boundary conditions are those regular

conditions at infinity and the radiation condition.

Equivalence Conditions

For a complete equivalence between Problem I and Problem II, we

require that the displacement and stress fields in the two problems be

identical. Hence, for equivalence in stress field, we require, from

Equation (2) and Equations (7,8),

C jkrs ur,s + AC C jkrs CU 1111

For equivalence in displacement fields we require that the Equations (1)

and (9) be identical, hence

Ao ;; j

- AC C jkrs E* rs,k II (121

It is clear that Equations (11,12) are automatically satisfied in the region

D-Q by observing the definitions given in Equations (3,4,6). It is convenient

to split the RHS of Equations (12) into two parts such that

C jkrs E* (11 rs,k = -AC jkrs Ur,sk UW

C * (21 jkrs crs,k = Ap ii.

J

5

Cl=)

Employing Equations

conditions as:

(11-13) and Equations S,lO) we obtain the equivalence

AC uCm)(r) + Cjkrs E$'(') = jkrs r,s -AC Lw (r) , jkrs r,s in s2 UW

Ap tiim)(;, + Cjkrs &?)=- Ao iJ?)(:) in 52 (14b)

These conditions can be used to determine the eigenstrain distribution that

is necessary for the equivalence of Problems I and 11 provided that we can

write u(m) r in terms of the eigenstrains. One such method is given in [15].

It is of interest to note that Equations (14a) are identical in form as the

equivalence conditions in the static case and that only the jth components

of c =* (2) Cm> jkrs , -rs k are needed for determining u. . Further discussion on the

J determination of the eigenstrains will follow in the next section.

6

THE SCATTERING OF AN INHOMOGENEITY

Instead of finding the solution to the physical problem stated we

seek the solution to the equivalent inclusion problem. The governing

equations are Equations (6,7,8,9). Let the incident wave field be plane

time-harmonic then the time harmonic displacement and eigenstrain fields

can be written as, for example,

uj(m)(?,t) = u?)(f) exp (-iwt)

EGs(?,t) = E;~(?) exp (-id>

where w is the frequency of the incident wave field and i-i =-1. Using the

dynamic version of the Betti-Rayleigh reciprocal theorem and suppressing

the time dependence we obtain the displacement field as:

urn(?) = urn Ci) (r) - /I V Ic jkrs gjm(',r'

or, upon employing Gauss' theorem,

1 E* rs ,(:I dV ,

urn(+) = urn (i)(,q + I/( cjkrs gjm k(%?) E;$) dv V , (16)

where gjm(?,?l) is the spatial part of the solution to the associated

Green's function problem for Equation (9). Note that the use of the Green's

function preserves the characteristics of an outward propagating wave and

satisfies the boundary condition on stress at infinity. Since there are

two types of eigenstrains, we write the displacements as

u,(i') = urn ci)(i') - l// gjm(i,il) Cjkrs c;::;(i) dV cl

- .fI/ gjm kf(i,if) Cjkrs ~$~'(i) dV R '

(17)

where Equations (6,15,16) are used. It is clear that only the jth component

of c E* (2) jkrs , rs k are needed to determine the displacements. We can therefore

view this as a vector quantity, say OTT, where

Substituting Equation

J

(18) in Equation

(18)

we obtain the solution form

given as Equation (2.19) in Reference [8], p. 291, for the time-harmonic

case. If the Equations (14) are substituted in Equation (17), we find the

solution form Equation (12) in Reference [6], p. 379, or Equation (2.25)

in Reference [14], p. 2806.

The stress field that is arisen from the presence of mis-match or

equivalently by the presence of eigenstrains can be obtained by using Hooke's

law and Equations (5, 10, 17) as:

{y"rf) = - $ 'pqmn Il/[(gjm,n,(i~i') + gjn,m,(i,il)]T-r;(i) dv

n

lC l/l (g - 2 pqmn n jm,k'n' + gjn,k'm' 1 c jkrs C(i) dV (19)

Quantities of interest such as the differential cross section

dP(w)/dn defined as: [16,14]

dP(w) _ lim <r2 Iii ui. bq>

dQ r- <IO>

can be obtained in terms of the eigenstrains via Equations (17-ZO),

where I 0 is the incident power,

I0 =n Ci> ;li> , i u.. 13 j

<f(t)> denotes time averaging of a function

<f(t)> = +; f(t)dt 2 0

and 11., ni I. are direction cosines for i and n, respectively.

The differential dR is the differential element of a solid angle. The

total cross section is simply

(20)

(21)

In what follows we give examples for the scattering of an isotropic

smooth inhomogeneity in a linear elastic isotropic infinite medium subjected

to plane time harmonic incident wave field.

9

LINEAR ELASTIC ISOTROPIC MEDIUM

For such a medium the spatial part of the Green's function is

(exp iBR)/R

+ [(exp iBR)/R - (eq iaR) /RI , jml (221

where

2 2 2 2 R = 1;-cfl, a2 = q = z , 8' = s = 5

vL VT

and X, U, v L' vT are the Lame's constants, longitudinal wave speed, transverse

wave speed, respectively. Before we substitute Equation (22) in Equation (17),

we expand the unknown quantities associated with the eigenstrains in form

of a polynomial as [15,17]:

i7; (i) = A.+Ajk xk + Ajka xk xR + . . . J

~;$l)(i) ='Bij + Bijk xk + Bijkn. xk xR + . . .

(23)

(24)

where A., A. B B ijk' '.. are constants. 3 Jk' "'f ij' Substituting Equations

(22,23,24), with IT; defined by Equation (18), we obtain

u:)(i) = urn(i) - uii)(i) = fmj(i) Aj + fmjk(i) Ajk + . . .

+F mij (5) Bij + Fmijk(i) Bijk + . . . (25a)

10

where

and

4apw2 fmj(G) = + $9 mj - ~,mj

4apw2 f mjkC') = d2g 6 k mj + 'k,mj - 'k,mj . . .

47rpu2 F mij(‘l = - [Aa2 $9, 6ij + 2ll B2 $,i 6mj

- a $9 mij +b4, 1 mij

4npw2 F mijk(‘) = - lx a2 $k,m "ij + 2lJ 82 $k,i dmj

- 2lJ JI k,mij + 2lJ 0 k,mij 1

. . .

O(r) = I// (exp iBR)/R dV1 Q

. . .

'ka...sCr) n - = [I/ xk xL . . . xs (exp iBR)/R dV1

$J(?) = [A/ (exp iaR)/R dV'

Q,(r) = //I xk (exp iaR)/R dV' R

(2331

WC1

C2W

(2W

(26b)

(26~)

(26d)

(264

11

Jl,,,s(?) = lA/ xk x11 . . . xs (exp iaR)/R dV1 Wf)

The o- and $-integrals given in Equations (26) are the volume integrals

associated with the inhomogeneous Helmholtz equation. They can be carried

out for an ellipsoidal region by expanding (exp ikR)/R in Taylor series

expansions with respect to ?, for r > r' and with respect to ? for r < r'.

'Here k can be either a or B. Details are given in Reference [18]. This

type of expansion for the integrand is particularly useful in determining

the coefficients of a "polynomial" distribution of "3 and A?‘.. 1J

To determine the coefficients A., A. B B 3 ok' **" jk' jka, *--, we

substitute Equations (18,23,24,25) in Equations (14) and note that we are

*Cl) dealing with time-harmonic displacements. Since the ~7 and E..

in terms of polynomials we expand the u i"' (5) , up; (2) ,

and uifl(?)

are given

in Taylor

series expansions with respect to the coordinate origin, by matching the

coefficients of terms to the same power of XT, x., X.X., . . . . 1

we obtain a 1J

set of infinite number of algebraic equations for A., A. B B J J k' *" Jk' W' "'

The scattering cross sections are given in terms of these coefficients by

way of Equations (17,19,20).

12

EXAMPLES:

Uniformly Distributed Egenstrains

Let the region $2 be an ellipsoidal region of 2a1, 2a2 and 2a3 along

the x, Y, and z-axis, respectivelyi Figure 2. We first expand the integrals

and their derivatives in Taylor series for r < r' and obtain the Taylor

series for ujm)(?) from Equations (25). Substituting this series for u 5"' (r)

and its derivatives in Equations (14) with Equations (23,24) and the Taylor

series for usi)( we obtain the governing algebraic equations for A., A. 3 Jk'

. ..) B N ij' ijk' '*' by comparing the order in the power series i.e. xy, x., 1

x.x., etc. iJ

To save space these equations are not shown here. Once these

coefficients are determined, we go back to Equations (25) and find the

scattered displacement field which is um(?) when r + 0. 3

By using formulas

given in [lS] the scattered displacement field is given in terms of a triple

SUlll.

In what follows we consider the case for a plane time harmonic wave

propagating in the +z-direction, i.e.

uii)(?,t) = u. exp i(az - wt)

For a given (ka) enough terms must be taken in determining the coefficients

A's and B's in Equations (23,24) from

[f] (A1 + [F] {B} = 2; mXm mX1 mXm mX1

[d] (A} + [D] IB) = (El mXm mX1 m&n mX1 mX1

(27)

where [f], [F] are defined by the Taylor expansion of the f- and F- functions

13

at r = 0, and WI , @I are the average of the symmetric part of [f], [F].

Gil The RHS are obtained from the Taylor series expansion for u. and sci) 3 ij ,

respectively.

Using Equations (12,17) in [IS] and Equation (25), we find that, for

(11 the lowest order of a?. and or? or C E* (2) 13 J jkrs rs , i.e. keeping only the

constant terms in Equations (23,24), at a distance far away frori the

ellipsoid:

up&t) = Lp(G,t) 1 r+=

II k n-%-k 9, k n-e-k (-1)"4nala2a3 ala2a3 111R2R3 n=O ll=O k=O I (2n+3)(2n+l)n! (L/2)!(k/2)!(n-L-k)/21 l

l ( y [-a'(ia)" Em!ZjAj - Xia3(ia)ncm6kjBkj

- 2uia3' * (la)" LmLkllj Bkj I

+ =y= [62(Em'j-6mj) Aj - 2uiB3 L 6 B . k mj kJ

+ 2diB3 II II 11 B. ] (i31n] exp C-i&> mkj kj (28)

where the repeated subscripts must be summed from 1 to 3. Note that Bkj

is non-dimensional and is homogeneous in (auo) and A. is of dimension J

(Ap l w2 l uo). The first term in the expression, i.e. n = 0, gives the

following:

14

Lp (5) s (aa,) u.

= p G;(e,c$) + ex;riBr H;(e,$)

where (r,B,$) are spherical coordinates and

G;OWl = - (a2a3/3alall [ern’ljA3 (‘P/P>

+ (l-a2/B2) t m B3j + 2 em'lkLj Bij]

PaI

Wb)

H;le,e) = (a2a3/3alal) [ CB/aJ3 (k R. mJ - 6mj) “7 CAP/P)

4 - 2(6/a) kk Bcm + 2(B/a) 4 emakLjB;j] (29cl

in which

A? = Aj/(Aou2 uo) (29d)

Bitj = -Bkj/(iauo) (29e)

By using Equations (20,27,28) we obtain the differential scattering

cross section, after manipulation, as

dP (~1 dQ =aP(e,+) + (a/B) u’(e,U (30)

where

15

~22(e,~) = (aa,) l H;Iht4 Hf,(eAl CD n l $1

!L n-R-k R k n-L-k

cn = Y i c n-9. (-ilnlaalln(a2/a$ !a,/a,> %'2'3

n=O L=O k=O (2n+3)(2n+l) n! (L/2)! (k/2),! (k/2!: (n-L-k)/2!

C-i)“@ a,)“Ca,/a,l n-L-k 2 k n-R-k 5 R2 R3

Dn = n=O L=O k=O (2n+3)(2n+l)n!(%/2)!'(k/2)!(n-R-k)/2! -

The super bar here denotes complex conjugate. The total differential cross-

section can be easily obtained as

= (a/B I 3Ca alI 4/ c

Hz "; Dni$ dn . (31)

16

Discussion

An eigenstrain approach to the scattering of a single ellipsoidal

inhomogeneity is studied in detail. A complete formulation of the

equivalence between the inhomogeneity problem and the inclusion problem

is given while the case Ao=O was given in NASA contractor report #3445 [15].

It is shown that this approach is identical to other approaches such as

the direct volume integral formulation, Refs. [6,8,9].

The eigenstrains are expanded as a geometric series and the coefficients

are determined by the equivalence conditions derived here and also Ref. [19].

The scattered displacements and stresses are given in an analytic series

form in terms of these coefficients. The scattering of an ellipsoidal

inhomogeneity in an isotropic elastic medium under plane time-harmonic

incident wave is worked out as an example. The eigenstrains are assumed

to be uniformly distributed.

The advantages in using the approach taken is at least three-fold:

(i) the radiation condition for out-going waves and the continuity con-

ditions at the interface between the inhomogeneity and the matrix are

automatically satisfied, (ii) the solution for special inhomogeneity

geometric shape, such as sphere, cylinder, plate, disk, prolate and oblate

spheroids, can be easily obtained by setting the appropriate ratios between

al,a2,a3 in evaluating the volume integrals (181, (iii) with the assistance

of asymptotic expansion method the solution can be easily modified to

obtain solutions appropriate for different ranges of wavelengths that are

of interest.

17

Numerical results and graphical plots for an inhomogeneity of

spheroidal geometry, Fig. 2, are given in Figs. 3-9, for the case of

uniformly distributed eigenstrains. A comparison of the results obtained

with "exact" solution can only be done for simple cases. Such is done

for the cases of a planar or a spherical geometry. The comparison is

given in a subsequent report. Finally, it should be mentioned that a'

generalization of this work for investigating the attenuation and velocity

factors is being planned.

18

I

REFERENCES

[l] Y.H. Pao and C.C. Mow, Diffraction of Elastic Waves and Dynamic Stress Concentrations, Crane-Russek, N.Y. (1973).

[2] J.D. Achenbach, Wave Propagation in Elastic Solids, North-Holland Publishing Co. (1975).

[3] Elastic Waves and Non-destructive Testing of Materials, Ed. Y.H. Pao, ASME Div Appl. Mech., AMD-Vol. 29 (1978).

[4] Modern Problems in Elastic Wave Propagation, ed. J. Miklowitz and J. Achenbach, Proc. IUTAM (1977); J. Wiley, N.Y. (1978).

[S] S.K. Datta, "Diffraction of Plane Elastic Waves by Ellipsoidal Inclusions," J. Acoust. Sot. Am., 61, #6, 1432-1437 (1977). -

[6] A.K. Ma1 and L. Knopoff, "Elastic Wave Velocities in Two-component Systems," J. Inst. Maths. Applies., 3, 376-387 (1967).

[7] Yi-Hsing Pao, "The Transition Matrix for the Scattering of Acoustic Waves and for Elastic Waves," in Ref. (41, 123-144 (1978).

[S] J.R. Willis, "A Polarization Approach to the Scattering of Elastic Waves - I.‘ Scattering by a Single Inclusion," J. Mech. Phys. Solids, 28 (S/6), 287-306 (1980). -

[9] J.E. Gubernatis, "Longwave Approximations for the Scattering of Elastic Waves From Flaws With Applications to Ellipsoidal Voids and Inclusions," J. Appl. Phys., 50 (6), 4046-4058 (1979). -

[lo] J.D. Eshelby, "Elastic Inclusions and Inhomogeneities," in Progress in Solid Mechanics, 2, eds. I.N. Sneddon and R. Hill, North-Holland, Amsterdam, 89-140 (1961).

[ll] J.D. Eshelby, "The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems," Proc. Roy. Sot., A241, 376-396 (1957).

[12] J.D. Eshelby, "The Elastic Field Outside an Ellipsoidal Inclusion," Proc. Roy. Sot., A2S2, 561-569 (1959).

[13] P. Wheeler and T. Mura, "Dynamic Equivalence of Composite Material and Eigenstrain Problems," J. Appl. Mech., 40, 498-502 (1973). -

[14] J.E. Gubernatis, E. Domany and J.A. Krumhansl, "Formal Aspects of the Theory of the Scattering of Ultrasound by Flaws in Elastic Materials," J. Acoust. Sot. Am., 48, #7, 2804-2811 (1977). .

[15] L.S. Fu, "Fundamental Aspects in the Quantitative Ultrasonic Determination of Fracture Toughness: General Equations," NASA Contractor Report 3445 (July, 1981).

19

[16] R. Truell, C. Elbaum and B.B. Chick, Ultrasonic Methods in Solid State Physics, Academic Press, N.Y. (1969).

[17] G. Sendeckyj, Ellipsoidal Inhomogeneity Problem, Ph.D. Dissertation, Northwestern University (1967).

[18] L.S. Fu and T. Mura, "Volume Integrals of Ellipsoids Associated with the Inhomogeneous Helmholtz Equation," Wave Motion,A(2), 141-149 (1982).

[19] L.S. Fu, 'Method of Equivalent Inclusion in Dynamic Elasticity," Proceedings, AF/DARPA Progress Review on Quantitative Nondestructive Evaluation, Plenum Press (1982), Presented at Boulder, Colorado, (Aug., 1981).

20

D-n

C ijkl’ 4

D-i-2

C ijkl’ p

1

Fig. 1 (a) The inhomogeneity problem, (b) The inclusion problem.

x3 matrix: A, u ; p inhomogeneity : i\: P'; p0

Fig. 2 An ellipsoidal inhomogeneity under incident wave

22

I I I

*. . N=6 +: N=lO

N=20

2 3 4 5

ALPHA Al

Fig. 7 Convergence of Cn as a function of aal: a /a 2 1 = 2/3,

a3’al = 2/3.

23

10

9

8

7

6

5

4

3

2

1

I I I I

* *. . tungston in titanium . +: titanium in tungston

* 8: cavity in aluminum

uO .4 . 8 1. 2

ALPHA Al

Fig. 6 G(e,$) vs aal: a /a = 2 1 1, a3/a G . G = Gm * cm.

'1 = 112, 4 = 0, 8 = 0,

24

I- -

I I I I I

* *. . tungston in titanium

. +: titanium in tungston * 8: cavity in aluminum .

. % 1 t .

\ \ \ \ . \ \ * \ \ .

\b . I \

1 1 . I *

\ \ .

1 I

\ \

\ I \ 1 1

I 0 -.._ I I I -. - 0 .4 .8 1.2 1. 6 2

ALPHA Al

Fig. 5 G(8,4) vs &al: a2/al = 2/3, a3/al = 2/3, 9 = 0, 8 = 0

G . G = Gm - cm .

25

II

t *: 2/3, 2/3

t

\

+: l/3, l/3

0: 1, 2/3

y: 1, l/2

0 1

0

F .g. 4 G(e,$) vs. ual for different ratio of a .a a_nd a3/al: titanium in tungston,$ = 0, 8 = 0, G-G = G;G,.

.8 1.2

ALPHA Al

26

13

12

11 r( I

; 10

9

0

7

6

5

4 t3

3

2

1

\ *: 2/3, 2/3

\

+: l/3, l/3 l : 1, 2/3

\ g: 1, l/2

0 a’ .4 .8 1.2

ALPHA Al

Fig. 3 G(e,#) vs. aal for different ratios of a2/a, and

a3/al i tungston in titanium, 4 = 0, 8 = 0, G;G =

Gm Frn.

27

.8

.6

l-4 LA .4 x

. 2

0

-. 6

-. 8..

. : void in aluminum + : tungston.in Xitanium

'.- u 0

exp iaz-iwt t

.8 .6 .4 .2 0 .2 .4 .6 .8 1 x E-l

Fig. 8 Polar plots for the scattering amplitudes for the case aa = 2.0; I$ = 0, a2/al = a3/al = 2/3.

28

S

4

3

2

2 1 x

0

-1

-2

-3

-4

I I I I I I I I I

4 3 2 10 1223 4 x E-i

S

Fig. 9 Polar plots for the scattering amplitudes for the case aa 1 = 2.0, Ip = 0, a2/al = a3/al = l/3.

29

1. Report No.

NASA CR-3623 2. Government Accession No. 3. Recipient’s Catalog No.

4. Title and Subtitle PHENOMENOLOGICAL AND MECHANICS ASPECTS OF NONDESTRUCTIVE EVALUATION AND CHAR- ACTERIZATION BY SOUND AND ULTRASOUND OF MATERIAL AND FRACTURE PROPERTIES

5. Peport Date

October 1982 6. Performing Organization Code

7. Author(s)

Li-Sheng W. Fu

9. Performing Organization Name and Address

The Ohio State University Department of Engineering Mechanics Columbus, Ohio 43210

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration Washington, D. C. 20546

8. Performing Organization Report No. pYLo. I 11. Contract or Grant No.

NSG-3269

---I 13. Type of Report and Period Covered

Contractor Report

I 14. Sponsoring Agency Code

506- 52-62 (E-1324: 15. Supplementary Notes

Final report. Project Manager, Alex Vary, Materials Division, NASA Lewis Research Center, Cleveland, Ohio 44135.

16. Abstract

Advances have been made on abroad front in nondestructive testing (NDT) in terms of measurement methods, instrumentation, automation and computer-assisted signal acquisition and processing while recent developments in fracture mechanics and elastic wave theory have enabled the understanding of many physical phenomena in a mathematical context. The purpose of this review is to bring together the available literature in the material and fracture characterization by NDT, and the related mathematical methods in mechanics that provide fundamental underlying principles for its interpretation and evaluation. Information on the energy release mechanism of defects and the interaction of microstructures within the material is basic in the formulation of the mechanics problems that supply guidance for nondestructive evaluation (N-W.

7. Key Words (Suggested by Author(s))

Nondestructive testing/evaluation; Fracture ’ mechanics; Ultrasonics; Microstructure;

Elastic waves; Materials characterization; Properties characterization

L

18. Distribution Statement

Unclassified-unlimited STAR Category 38

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PO. Security Classif. (of this page)

I

21. No. of Pages

Unclassified 34 9. Security Classif. (of this report)

Unclassified I I

* For sale by the National Technical Information Service, Springfield, Virginia 22161 NASA-Lqngl ey , 1982