A single elastic fiber embedded in an infinite elastic matrix is considered. An incident plane SH wave is assumed in the infinite matrix, and an expression is derived for the total energy in the fiber due to the incident SH wave. A nondimensional form of the fiber energy is plotted as a function of the nondimensional wavenumber of the SH wave. It is shown that the fiber energy attains maximum values at specific values of the wavenumber of the incident wave. The results obtained here are interpreted in the context of phenomena observed in acousto-ultrasonic experiments on fiber reinforced composite materials

Nagem, Raymond J.

Williams, James H., Jr.

English

NASA Technical Reports Server

I 
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NASA Contractor Report 4205 
Energy in Elastic Fiber 
Embedded in Elastic Matrix 
Containing Incident SH Wave 
James H. Williams, Jr., and Raymond J. Nagem 
Massachusetts Institute of Technology 
Cambridge, Massachusetts 
Prepared for 
Lewis Research Center 
under Grant NAG3-328 
National Aeronautics 
and Space Administration 
Scientific and Technical 
Information Division 
1989 
https://ntrs.nasa.gov/search.jsp?R=19890009323 2020-03-20T03:59:42+00:00Z
Energy in Elastic Fiber Embedded in Elastic Matrix 
Containing Incident SH Wave 
James H.  Williams, Jr. and Raymond J .  Nagem 
Massachusetts Institute of Technology 
Department of Mechanical Engineering 
Cambridge, Massachusetts 021 39 
SUMMARY 
A single elastic fiber embedded in an infinite elastic matrix is considered. An incident 
plane SH wave is assumed in the infinite matrix, and an expression is derived for the total 
energy in the fiber due to the incident SH wave. A nondimensional form of the fiber energy 
is plotted as a function of the nondimensional wavenumber of the SH wave. It is shown 
that the fiber energy attains maximum values at specific values of the wavenumber of the 
incident wave. The results obtained here are interpreted in the context of phenomena 
observed in acousto-ultrasonic experiments on fiber reinforced composite materials. 
INTRODUCTION 
Quantitative ultrasonic nondestructive evaluation methods require analyses of wave 
propagation in the material or structure which is to be evaluated. In fiber reinforced com- 
posite materials, wave propagation characteristics depend on properties and arrangement 
of the constituent materials, as well as on the overall geometry and both the microscopic 
and macroscopic defect state of the composite. 
Analyses of wave propagation in fiber reinforced composites are reviewed in [I] and 
[Z]. According to these reviews, wave propagation analyses may be divided into two broad 
categories. In the first category, the composite is treated as an equivalent homogeneous 
medium, and mechanical responses such as stress and strain are described in terms of 
averages over the constitutive elements. It has been stated that the homogeneous or 
equivalent continuum approach is valid “when the scale of the changes in stress level (rise 
distance, wavelength, etc.) is much larger than the sizes of the constituents of the composite 
(fiber or particle diameter, fiber spacing, ply spacing, etc.)” [l]. In the second category 
of wave propagation analyses, the composite is modeled as a heterogeneous mixture of 
constituents, and the dynamic interaction between the constituent materials is considered. 
The heterogeneous analyses are complex, but are necessary if the scale of changes in stress 
level is smaller than or of the same order as the size of the composite constituents, or if 
small-scale phenomena such as interface bond failure are of interest. 
Recent acousto-ultrasonic experiments by Kautz [3] show some interesting wave prop- 
agation behavior in graphite fiber reinforced epoxy composite panels. Kautz introduces a 
broadband signal into a composite panel, and observes that the response of the composite 
appears to be divided into a high frequency component traveling at a relatively fast wave 
speed and a low frequency component traveling at a relatively low wave speed. When the 
same broadband signal is introduced into a geometrically similar panel consisting only of 
the epoxy matrix, the response is observed to consist only of the low frequency component. 
Kautz hypothesizes that the composite panel acts as a heterogeneous medium, with wave 
propagation along the fibers at the higher wave speed and wave propagation through the 
matrix at the lower wave speed. 
In this report, an infinite elastic matrix containing a single cylindrical elastic fiber 
is considered. An incident plane SH wave is assumed to exist in the elastic matrix, and 
2 
an expression is derived for the total energy of the fiber due to the incident SH wave. 
It is shown that the total fiber energy attains maximum values at specific values of the 
wavelength of the incident wave. The purpose of this work is to investigate, in a simple 
setting, the issue of energy transfer between the matrix and the fibers in a composite 
material. Recommendations are given for future research which may give further insight 
into the experimentally observed phenomena in [3]. 
ANALYSIS 
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A cylindrical fiber having circular cross section is shown in Fig. 1. Rectangular coordi- 
nates (5, y, z )  and cylindrical coordinates (t, 6 ,  z )  are defined as shown in Fig. 1. Displace- 
ments uz,  uy and u, are defined in the x ,  y and z directions, respectively, and displacements 
u,, ue and u, are defined in the r ,  6 and z directions, respectively. The fiber is assumed 
to have radius a and to be of infinite extent in the z direction. The fiber is assumed to be 
elastic, with mass density pf and Lam6 constants pf and Xf. It is assumed that the fiber 
is embedded in an infinite elastic matrix with mass density p and Lam6 constants p and 
A. Continuity of relevant stresses and displacements is assumed at the materials interface 
r = a. 
The infinite elastic matrix is assumed to contain an incident harmonic plane wave of 
the form 
where the superscript ( i )  denotes the incident wave, wo is the amplitude of the incident 
wave, k is the wavenumber of the incident wave, w is radian frequency, t is time and 
j = e. The wavenumber k and frequency w are related by 
w = c,k (4) 
where 
3 
Eqns. (l), (2) and (3) define a wave which propagates in the positive z direction with 
particle motion parallel to the fiber in the z direction. Such a wave, in this context, is 
denoted as an SH wave. 
Because of the cylindrical geometry of the fiber, it is convenient to write the incident 
SH wave in the cylindrical coordinates ( r ,  8, z )  as [4] 
u p ) ( r ,  e, Z ,  t )  = o 
u f ) ( r ,  e, z ,  t )  = o 
u p ) ( r ,  e, z ,  t )  = Re 
00 
c n j n J n ( k t )  cos 
where Jn is a Bessel function of the first kind and of order n, and where 
1, n=O 
2, n=1,2, ... e n =  { (9) 
It is shown in [4] that eqns. (1) through (3) and eqns. (6) through (8) are mathematically 
identical representations of the same incident wave. 
The interface conditions at  r = a and the three-dimensional equations of elasticity 
which govern the infinite medium and the fiber are satisfied by the total displacement field 
141 
where 
4 
The quantities A ,  and Cn are dimensionless complex coefficients. H i ' )  is a Hankel function 
of the first kind and of order n, and the prime denotes the derivative of a function with 
respect to its argument. 
Eqn. (16) represents a scattered wave which propagates outward from the fiber. As 
r --f 00, the amplitude of the scattered wave approaches zero, so that far from the fiber 
the total displacement field approaches the displacement field of the incident SH wave. 
Eqn. (17) represents a refracted or transmitted wave in the fiber. Eqn. (17) is a 
standing wave with time-harmonic particle motion in the z direction. Over any fiber cross 
section z = constant, the fiber displacement in the z direction varies as a function of r and 
fl according to eqn. (17). 
The nonzero fiber strains associated with the displacements given by eqns. (13) through 
(15) are 
and the nonzero stresses in the fiber are 
dUlf) 
Qrz = Pf 
The elastic strain energy Ut per unit length of fiber in the z direction is 
5 
The kinetic energy Kf per unit length of fiber in the z direction is 
The total energy Ef per unit length of fiber is, therefore, 
The energy E f  given by eqn. (29) is a function of time. In order to obtain a single 
numerical measure of energy, it is customary to average the energy Ef over one time period 
of the harmonic oscillation. The time-averaged energy < E > f  is defined by 
where 
27r T =  - 
W 
When eqn. (17) and eqns. (23) through (29) are substituted into eqn. (30), a long 
algebraic calculation gives the following result for the time-averaged fiber energy: 
where 
n = O  
P n =  { i!29 n = 1,2,. . . (33) 
and where the asterisk denotes the complex conjugate. The derivation of eqn. (32) requires 
the complex variable identity 
1 
Re{%} = - ( z  + z * )  , 
2 (34) 
the trigonometric identies [5 ]  
+. 
x. 6 
sin ne sin me 
and the Bessel function identities (61 
for n=O or m=O 
r6mn otherwise 
The quantity bmn in eqns. ( 3 5 )  and (36)  is the Kronecker delta. 
Eqn. ( 3 2 )  is an expression for the total fiber energy per fiber length in the z direction. 
In order to nondimensionalize the fiber energy < E >f, consider an infinite uniform elastic 
matrix containing only the incident SH wave given by eqns. (6) through ( 8 ) ,  and consider 
the time-averaged energy < E > m  contained, per unit length in the z direction, in the 
region 0 < r < a. By a calculation similar to the calculation leading to eqn. ( 3 2 ) ,  the 
time-averaged energy < E >m  is 
where 
A nondimensional fiber energy < e > f  is now defined by 
< E > f  
C e > f =  < E > m  (43) 
The dimensionless quantity < e > f  gives a measure of the fiber energy relative to the 
energy the matrix would contain if there were no fiber present. 
7 
By introducing the nondimensional parameters 
Pf r p  = - 
P 
r,, = !!L
c1 
(44) 
(45) 
eqns. (43) and (19) can be written as 
< E > f  
< E > m  <e >f= 
respectively, where 
From eqns. (46), (47) and (48) it can be seen that the nondimensional fiber energy < e >f 
depends only on the nondimensional parameters ka, rp and r,,. The parameter ka is a 
nondimensional wavenumber of the incident wave, and can be written in terms of the 
incident wavelength A; as 
2na 
ka= - 
A i  
w a  
k a =  - 
The nondimensional fiber energy < e > f  is plotted as a function of the nondimensional 
or in terms of frequency as 
C 
wavenumber ka in Fig. 2 for the numerical values r p  = 2.0 and r p  = 25.0. These numerical 
values correspond to an E-glass fiber in a PMR (polyimide monomeric reactant) matrix 
[7]. In Fig. 3, the nondimensional fiber energy < e > f  is plotted as a function of ka for the 
numerical values rp = 1.5 and rp = 7.5. These numerical values correspond to a Celion 
6000 fiber in a PMR matrix [7]. The numerical values used to generate Fig. 3 are estimates 
of the material properties of the graphite/epoxy composite used in [3]. The infinite series 
in eqn. (46) are computed by truncating when the absolute value of the individual terms 
becomes less than times the absolute value of the accumulated sum. 
8 
DISCUSSION 
In both Fig. 2 and Fig. 3, the nondimensional fiber energy < e > f  is approximately 
equal to one when ka is close to zero, or, equivalently, when the wavelength of the incident 
wave is long compared to the fiber diameter. This means that for long wavelengths, the 
energy in the fiber is approximately the same as the energy which would exist in a uniform 
matrix. In other words, the presence of the fiber does not significantly affect the energy 
distribution for long wavelengths. As ka becomes larger (or, equivalently, as the wavelength 
becomes smaller) the energy in the scattered wave becomes larger, and the energy in the 
fiber becomes smaller than the energy which would exist in the region 0 < r < a in the 
uniform matrix. 
The most interesting features of Fig. 2 and Fig. 3 are the local maxima in the values 
of < e >f. For an incident wave containing a broad band of wavenumber, the fiber energy 
will be greatest at frequencies corresponding to the values of ka where the local maxima 
occur. If the fiber response could be measured, the local maxima would appear in a 
frequency spectrum of the measured response. Thus it is sensible, at least in this simple 
case, to suppose that certain frequency components in a measured response may contain 
information about the fiber. 
The incident SH wave assumed here is independent of the spatial coordinate z and so, 
therefore, is the displacement field in the fiber. Therefore, it is impossible, in this analysis, 
to discuss the issue of wave propagation along the fiber in the z direction. The analysis of 
an incident wave which has a z component may allow consideration of wave propagation 
along the fiber. Equations which describe the scattering and refraction of such an obliquely 
incident wave are given in [4]. 
CONCLUSIONS AND RECOMMENDATIONS 
The purpose of this report is to investigate the transfer of energy to a single fiber in 
an infinite elastic matrix containing an incident SH wave. An expression is derived for the 
energy in the fiber, and it is shown that the fiber energy attains local maxima at specific 
values of the wavenumber of the incident wave. 
Future research should consider the problems of incident P and SV waves. As men- 
tioned above, an obliquely incident wave may allow analysis of wave propagation along 
9 
, 
the fiber. Also, properties of the scattered wave given by eqn. (16) should be investigated, 
I since the scattered wave may be available for experimental measurement. It would be in- 
teresting to see if the energy flux of the scattered wave has wavelength dependence similar 
to the wavelength dependence of the energy of the fiber. 
I REFERENCES 
[ 1) J.D. Achenbach, “Waves and Vibrations in Directionally Reinforced Composites”, 
[2] F.C. Moon, “Wave Propagation and Impact in Composite Materialsn, Composite 
Composite Materials 2, 1974, pp. 309-351. 
Materials 7, 1975, pp. 259-332. 
[3] H.E. Kautz, “Ray Propagation Path Analysis of Acousto-Ultrasonic Signals in Com- 
[4] Y.-H. Pao and C.-C. Mow, Diffraction of Elastic Waves and Dynamic Stress Concen- 
[5] S.M. Selby, ed., Standard Mathematical Tables, 21et ed., Chemical Rubber Co., 1973. 
[6] Y.L. Luke, Integrals of Bessel Functions, McGraw-Hill, 1962. 
[7] C.C. Chamis, “Predicting Microstresses in RP Composites”, SPI 41et RP/C Institute 
posites”, NASA Technical Memorandum 100148, July 1987. 
trations, Crane Russak and Co., 1973. 
Conference, Atlanta, GA, January 1986. 
10 
Incident 
SH Wave Fiber \ 
Direction \ 
of particle 
motion 
\ 
Fig. 1 Elastic fiber embedded in infinite elastic matrix containing incident SH wave. 
11 
I . 2  
I .I 
I 
0.9 
0.8 
0.7 
0.6 
0.5 
0.4 
0.3 
0.2 
0. I
0 2 4 6  8 IO 12 14 16 I8 20 
Nondimensional Wavenumber, ka 
Fig. 2 Nondimensional fiber energy, < e > f  as a function of nondimensional wavenumber ka 
for r p  = 2.0, r,, = 25.0. 
12 
I .2  
1 . 1  
I 
0.9 
0.8 
0.7 
0.6 
0.5  
0.4 
0.3 
0.2 
0.1 
0 2 4 6  8 IO I2 14 16 I8 20 
Nondimsnsional Wovenumber, ko 
Fig. 3 Nondimensional fiber energy, < e >f as a function of nondimensional wavenumber ka 
for r p  = 1.5, and r p  = 7.5. 
13 
National Aeronautics and 
Space Administration 
1. Report No. 
NASA CR-4205 
Report Documentation Page 
2. Government Accession No. 
17. Key Words (Suggested by Author(s)) 
7. Author(s) 
James H. Williams, Jr., and Raymond J. Nagem 
18. Distribution Statement 
~ 
9. Performing Organization Name and Address 
Massachusetts Institute of Technology 
Department of Mechanical Engineering 
Cambridge, Massachusetts 02 139 
12. Sponsoring Agency Name and Address 
National Aeronautics and Space Administration 
Lewis Research Center 
Cleveland, Ohio 44135-3191 
15. Supplementary Notes 
19. Security Classif. (of this report) 20. Security Classif. (of this page) 
Unclassified Unclassified 
3. Recipient's Catalog No. 
22. Price' 21. No of pages 
16 
5. Report Date 
January 1989  
6.  Performing Organization Code 
8. Performing Organization Report No. 
None (E-4489) 
10. Work Unit No. 
506-43- 1 1 
11. Contract or Grant No. 
NAG3-328 
13. Type of Report and Period Covered 
Contractor Report 
Final 
14. Sponsoring Agency Code 
Project Manager, Harold E. Kautz, Structures Division, NASA Lewis Research Center. 
16. Abstract 
A single elastic fiber embedded in an infinite elastic matrix is considered. An incident plane SH wave is assumed 
in the infinite matrix, and an expression is derived for the total energy in the fiber due to the incident SH wave. 
A nondimensional form of the fiber energy is plotted as a function of the nondimensional wavenumber of the SH 
wave. It is shown that the fiber energy attains maximum values at specific values of the wavenumber of the inci- 
dent wave. The results obtained here are interpreted in the context of phenomena observed in acousto-ultrasonic 
experiments on fiber reinforced composite materials. 
Nondestructive testing; Nondestructive evaluation; 
SH waves; Composites; Acousto-ultrasonics 
Unclassified - Unlimited 
Subject Category 38 
'For sale by the National Technical Information Service, Springfield, Virginia 22161 NASA FORM 1626 OCT 86 
NASA-Langley, 1989 


Energy in elastic fiber embedded in elastic matrix containing incident SH wave

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