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Dynamic elastic stiffness with service generated inhomogeneity for fibrous composites
Following the work on the determination of dynamic elastic constants for the state of affairs a~-ound a single inhomogeneous ellipsoid. this work considers a fibrous composite whose inhomogeneity i: ixrraed on acco!mt ofchaogr ix3 porosity or moisture content after the composite has been in service for some time. The added inhomogeneous phase. measured in terms of volume fraction of uniformly suspended particks in composite. is related to v&city measurrment!si. The analysis is useful when the volume fraction of one type of inciusion phase is not lino\\n a pnori.
1. Introductory remarks The use of velocity measurements to detect linea r elastic moduli is weii developed [Ij. Elastic moduii determined in this manner are usually called dynamic elastic constants. For a Linear elastic isotropic material, the Lame constants A and I_Lcan be obtained as /L =pi.;,
h + 2p =pr.f
(1)
~r+udinal wave speed. respre~ivciy. where p, ~‘r and r’r are mass density. transverse wave speed. and I,.,,: When mass density is given. the moduli h and g can be found by the measured t4oci:ie$ 1’7 and li. For anisotropic materials, elastic waves rdn stif! be propa_nated in any direcrion i-scent nilrely compressive or transverse waves can be obsained on@ along specific dirxtioms. Wirh reference io Fig. I 4-1 111. the eIastic modrili C,,,, (i. j.k.I= !. 2. 3t wouid he for a transversely isotropic elastic maicr:,A determined by the mass density and wave x,elocity as C 35; =p~i
(along
x,-axis).
Cz,,, = pt.+ (polarized: %Crrr,
- C,rl,)
CIlir=pl.i
{sionp
x1 -axis)
x3 - axis)
= pi,:- f polarized:
The subscripts L and T stand for the iongitudinat in the work to follow.
I?) x2 - axis) and transvet~ e direction.
The above uiii be understood
2. Problem statement Consider a fibrous composite consisted of malrls and chopped short fibres. both linear elastic and isotropic. Let all the fibres be distributed uniformly an d aligned parallel to themsefves, Fig. 1. The composite may then be treated as transversely isotropic and with the foliowing constirutive relations i2j:
O~67-S~~~,~OZ/SO.i.OO B 1992 - Elsevier
Science Publishers
B.V. Ali rights resrnrd
When a composite of this type is placed in service and is subjected to service conditions that may generate an additional inclusion phase such as porosity or moisture absorption, change in elastic moduti and mass density will result. The purpose of the current investigation is to present a method to nondestructively detect these changes by employing plane wave velocity analysis. The study is theoretical in nature. The analysis is based upon the solution of the elastodynamic fields around an ellipsoidal inhomogeneity [3], a self-consistency scheme and an averaging theorem over energy considerations [4.53. The basic equations are recapitulated in a more convenient form for the present application. Certain derivations that are not entirely obvious are given in the Appendix.
3. Theory Consider an incident plane harmonic wave passing through the composite medium shown in Fig. I. Due to the presence of the fibres and the second inclusion phase, assumed to be in the form of spheres. there are generated strain disturbance and inertia disturbance. Following the approach in [3]. the total &spiacemeni fiezg.3is taken to be the superposition of that of the matrix without inclusions plus the disturbance due to the presence of inclusions. Let their average values due to all inhomogeneities be Eij and p”Gj, then the total stress field and inertia force fieId in the matrix are given by c$’ + (u;~)~ and p’“‘a~) + ( pn, >%*,where
and
=poaj
(3
where the superscript “0” indicated zeroth phase or matrix. When an additional “single” inhomogeneity of type 0, is introduced, the additiona! disturbance in stress, strain and inertia force can be denoted by v;), .+;) and p”‘ay’. The equivalent inclusion
inD
(7)
where crij= aij + &r and a, = t!ij+ alf) are the disturbed stress and inertia, and E$‘) and 17Tir’ are the uniform eigenstraik’and e&enforces due to the presence of the additional rth phase in reg:on R, and from [3j
in which f,,, Fjkl are defined on p. 392 in j3j and p. 3 in [Sj rvhers constants A, p were associated with the matrix material or A = hO, J.C= $_ The tensor S&i. is derived from the voiume integral in the second term of eq. (16) in [3] by using eq. (12). The macroscopic average of the total disturbance in stresses over th..e volume. ~LT:,dV. and th2: in inertia forces. .!pai dV, can be shown to vanish i9] nnd lead to the condition as folEo&:
where only the rea! part should be used in computing any physical quantities and the super bar indicates time average over one cycle. It is seen that when the eigenstrains and eigenforces are known, the properties of the composite can be found from eqs. (14) and (15). The eigenfields can first be solved from eqs. (6) and (7) in terms of the averaged fields E,, and ij. Using eqs. (10) to (131, the averaged fields can be determined. The computation is based upon known volume fraction for each type of embedded inhomogeneities.
& Ik~lr~aj~lijl;rlir~rr ~$1’ c~ig&klrfs
fzi:i :rrtd l/$’
t,ct t)tc incrc&:ttt w;rye tivlds IX f~lanc Lima*harmonic
and frrc)pagatcrl along the S,-axis, (Fig. I):
u:‘) = fl,,ijjj cxp i( k-o t )
t&re
C16)
i. i = I, k and w are the longitudinal
respectively.
The
time
harmonic
~~(7, t) =nj(i:)
displaccmcnt
wave number and frequency of the incident and eigenfields can be written as:
cxp( -id)
wave field.
(17)
and E:( r:, f) = E:( ?) exp( -iwt)
(IS)
17,*(?, t) =flT(f)
(19)
exp(-iwt)
from which the total field CL;is the superposition of the inhomogeneities, i.e.,
of the incident
uf( r, t) = uJ.“( F, t) + Uj( F, t) = Lfj”( r, t) + tij(‘,
wave field plus that due to the presence
f) i-
LLI”(
r,
t)
(20)
For the composite as depicted in Fig. 1, it is observed that ,$I’= E$), and E*(‘ .i3 ) are the only eigenforces. They are non-vanishing eigenstrains and 17:“’ = IT,*“‘, and 17cCr’ are the non-vanishing determined in terms of the average strains and inertia in the matrix by expanding eqs. (6) and (7): [2(Ah +A/.#
-Si;‘,,
+ [ AA(1 -S&)
-S$,)
-AA(S$;‘,,
tS&j]~$)
- 2( AA + A@j‘I;$a]~;*:~r
=AAu:“+2(AA+Ap)E,,
+AAZ-,,
i21)
[2Ah(I-S~;~,-~~;~,)-(Ah+2A~)(S~;~,+S~’,~,)]~,L,“’ + [(AA + 2Ap)( = (AA + 2Ay)u:”
1 - S&,)
+ 2AAC,, + (Ah + 2A/.~)Eaa
Apf f,, +flz + l/(p”o2Ap)] = -Ap(F,,,&” Adf;,
+fd~P”‘+
- 2AAS;.;;,] .&@’
c F,&” [fs
= -Apu; i”--Ap(F3,,q,
(22)
IX,*“’ ‘f,JZ;” + F,33~j*:r)) -A&
(23
+ b’(~~~~A~)]fl:“’ *W + F,,,E,*,“’ + F,&“‘j
- Apii,
(24)
where u3(it = LL,
exp ikz
(25)
Note that AA = A(‘) - ~07
Ay =#r
_
0 FL,
Ap = #” _ PO
(26)
in which (r) = 1 denotes the short fibre phase and (r) = 2 denotes the additional inclusion phase particles. The quantities St!) r,k.r f,$;‘, F,$$ are volume integrals associated with each inhomogeneity geometries. Solving for e1*1,E& IIF and ri: rrom c eqs. (21) to (24) in terms of Err, E33, L7, and ii, and substituting in eqs. (10) to (13) will yield their values. The eigenfieids are thus determined. Employing eqs. (14) and (15): the composite material moduli and mass density are determined if all volume fractions are given. Tn the case of porosity or moisture absorption problems, the volume fraction f2 is not known,
eqs. (14) and (15) determination.
5. Longitudinal
musk
hc combined
with velocity measurements
such as those in eqs. (2) to complete
the
wave velocity of the composite along x,-direction
From eq. (161, ail the strains induced E$’= iku,
by the incident
wave field vanish except
exp i( kz - wt)
(271
Using or& the real part of the complex expressions for the strains, averaging timewise over one cycle, eqs. (14) and (1% are reduced to
pc = po+ (2/u$Y)
velocities
and eigenfields
i: f,@Ep r=
and
(29)
1
??
When the eigenfields :(r) and nFcr’, r = 1, 2, are determined by the procedures as described in Section 4, eqs. (28) and (29) contain thre: unknowns, C&,,. pf and _f?. They can be solved when the veioci~ cL along the x,-axis is measured. Using eqs. (21, (281 and (291, the longitudinal wave velocity of the composite along the x-a-direction can be obtained as
The trend of the ratio between the longitudinal velocity in &he compesirs with edditiona0 homogeneity phase and that without it is shown in Fig. 2. A band calcuiator, I-If’ 44. was used. The additional inhomogeneity is taken to be pores. The aspect ratio of the short &es and the matrix are assumed to be v0 = ~(‘r = l/3 and @)/p
01 0
I 0.1
0.2
0.3
0.4
0.5
Total Volume Fraction. f = f+ f2 Fig. 2. Longitudinal
wave velocity along x,-direction.
cylindrical
tibres, ilj /a,
= 10, n: = oz.
;h ~,~r~~~k~~klI~~~l~~~ wiih lmgix I’ ,. Ttrc dr’oplGtg of lri,$cr order terms in Lhc cvalualion of the integrals can IX ;i major source ot’error aI higher end ol’Wil\;UnL~lllbCr. ‘I’hc eigunstr;k xict eigenkmes in the short
fitm: trhasc of vi~imc fractiw Etep~constml on the right-hand
ibssurnirlg IW presence of pores. These are then j, art first dewwined sicle of cqs. (211 to 124) for computing the eigenstrains and eigenforces in
f1,. Without writing a ra!her extensive computer program for evaluating the volume integrals, a Tayior series expansion is used where only terms up to (ka)’ are kept. It is noted that the terms in eqs. (14) and (15) are functions of material properties and geometric dimensions after time averaging and taking only the real parts in computations.
6. Discussion of results
The analysis given allows the estimation of the changes in dynamic elastic stiffnesses znd mass density due to the generation of an additiona! inbomogeneous phase in a fibres composite. To determine the eigenstrains and eigenforces by the procedures described in Section 4, the vo!ume integrals i,j, F,,zij, Sijk, must be evaluated for the given geometries of each type of inhomogeneities. Since these integrals are spatially dependent, volume average for these need to be performed. This can be, however, a relatively complicated task in computation. Using the multinomial theorem and series representation, Fu and Mura [6] gave the results for regions inside and outside the inhomogeneity in a series form. Numerical results presented in [3,7] are valid to about wave number less than two, ka G 2. The computations required computer usage to a certain degree. Recently, Mikata and Nemat-Nasser gave an exact closed-form solution for a spherical inclusion and a truncation approximation for an eilipsoidal inclusion [8]. Various spatial dependence of the stress amplitudes is displayed for low wavenumbers. It can be observed that the stress amplitudes change very little within the inclusion for wavenumbers less than one half. This indicates that the volume integrais are essentially spatial independent at the low end of the wavenumber and computation may be somewhat simplified.
Acknowledgement
The author is grateful to the comments and editing assistance of the reviewers and the editor-in-chief.
Appendix
The total field in the composite is the linear superposition of that from the incident wave field pins the disturbed field that due to the presence of the inhomogeneities. The latter vanishes identicaIIy in the .._f+‘_”_il _c absence of the inhomogeneities. For examp!e, the displacements and the sircsses can 1.. UC.wIILctiil u;(r, t) =lp(F:
r) +u,(F, r)
(A-1)
$(F,
t) +ajj(r,
(A-2)
t) =up(r,
f)
For a given composite with an additional inhomogeneity of the rth phase, the total disturbed field can be written as Ui(‘, t) = z$( r, t> + Zf!“( r, t)
(A.31
gij( F, t) = C$i’(?, r) + a$)( r, t)
(A.9
where the superscript “-” indicates the average value.
Its time rate of change can be writter: as [I]: 3E - = /p’a,‘~‘C dV + ~LT,~F;~ dV at
(A.6j
where the superscript “ .” den&es partial differentiation with respect to time. Combining eqs. (A.l), (A-21, (A.51 and (A.6); there results jpca;r;’
I
The integral
qfi,;
dV
dl/=l(p”ap+pn,)[c:“$_i;;~~“] dV= _/(q;
+ qi)
I
;;j) + iij + ij;j
df’
=
(T
.,l.~‘(~)dS -
r,
1 I
g. L’(i! ‘I
Id
dV=
0
whcrc $) can bc any prescribed virtual velocity. Similarly, the integral
vanishes by the fact that the rate of work done by surface traction is equivalent to the rate of increase of the strain energy. The final result from eq. (A.7) is therefore (A.10) in which the eigenforces 7i;*(‘) vanish identically outside a,.
References [l] R. Truell. C. Elbaum and B.B. Chick: (iltrasorzic Methods in Solid State Physics (Academic Press: New York, 1969). [2] M. Taya and T.W. Chou, On two kinds of ellipsoidal inhomogeneities in an infinite elastic body: an application to a hybrid composite, In?. .I. Solids and Structures 17 (1981) 553-563. [3] L.S. Fu and T. Mura, The determination of the elastodynamic fields of an ellipsoidal inhomogeneity. J. .IppJ. itleciz.Xl (1983) 390-396. [4] T. Mura, Micromechanics of Defects in Solids (Martinus Nijhoff, Dordrecht. The Netherlands, 1982). [5] L.S. Fu, A theory of dynamic elastic constants of heterogeneom media, Phibsophical Mugmine A 56(l) (19871 149-159. [6] L.S. Fu and T. Mura, Volume integrals associated with the inhomogeneous Helmholtz equation: 1. Ellipsoidal region. NASPl Contractor Report 3749, 1083: a!so: Warr Marion 4 (1982) 141-149. [7] L.S. Fu and Y.C. Sheu, Ultrasonic wave propagation in two-phase media: spherical inclusions, Composite Str-z~tures 2 (1954) 289-303. [8] Y. Mikata and S. Nemat-Nasser, First-order dynamic Eshelby tensor. 25th P.cc-z! TV -i-‘^-’ “--*‘ -- ,LE;. ICCLII.L~Y. I.“C._L’ *‘ociety of Engineering Science, ESP25.88014, University of California, Berkeley, CA, June, 1988. [9] L.S. Fu, On the macroscopic average of the disturbed stress and inertial fields in an inhomogeneous elastic media. submitted for publication.
1. Introductory remarks The use of velocity measurements to detect linea r elastic moduli is weii developed [Ij. Elastic moduii determined in this manner are usually called dynamic elastic constants. For a Linear elastic isotropic material, the Lame constants A and I_Lcan be obtained as /L =pi.;,
h + 2p =pr.f
(1)
~r+udinal wave speed. respre~ivciy. where p, ~‘r and r’r are mass density. transverse wave speed. and I,.,,: When mass density is given. the moduli h and g can be found by the measured t4oci:ie$ 1’7 and li. For anisotropic materials, elastic waves rdn stif! be propa_nated in any direcrion i-scent nilrely compressive or transverse waves can be obsained on@ along specific dirxtioms. Wirh reference io Fig. I 4-1 111. the eIastic modrili C,,,, (i. j.k.I= !. 2. 3t wouid he for a transversely isotropic elastic maicr:,A determined by the mass density and wave x,elocity as C 35; =p~i
(along
x,-axis).
Cz,,, = pt.+ (polarized: %Crrr,
- C,rl,)
CIlir=pl.i
{sionp
x1 -axis)
x3 - axis)
= pi,:- f polarized:
The subscripts L and T stand for the iongitudinat in the work to follow.
I?) x2 - axis) and transvet~ e direction.
The above uiii be understood
2. Problem statement Consider a fibrous composite consisted of malrls and chopped short fibres. both linear elastic and isotropic. Let all the fibres be distributed uniformly an d aligned parallel to themsefves, Fig. 1. The composite may then be treated as transversely isotropic and with the foliowing constirutive relations i2j:
O~67-S~~~,~OZ/SO.i.OO B 1992 - Elsevier
Science Publishers
B.V. Ali rights resrnrd
When a composite of this type is placed in service and is subjected to service conditions that may generate an additional inclusion phase such as porosity or moisture absorption, change in elastic moduti and mass density will result. The purpose of the current investigation is to present a method to nondestructively detect these changes by employing plane wave velocity analysis. The study is theoretical in nature. The analysis is based upon the solution of the elastodynamic fields around an ellipsoidal inhomogeneity [3], a self-consistency scheme and an averaging theorem over energy considerations [4.53. The basic equations are recapitulated in a more convenient form for the present application. Certain derivations that are not entirely obvious are given in the Appendix.
3. Theory Consider an incident plane harmonic wave passing through the composite medium shown in Fig. I. Due to the presence of the fibres and the second inclusion phase, assumed to be in the form of spheres. there are generated strain disturbance and inertia disturbance. Following the approach in [3]. the total &spiacemeni fiezg.3is taken to be the superposition of that of the matrix without inclusions plus the disturbance due to the presence of inclusions. Let their average values due to all inhomogeneities be Eij and p”Gj, then the total stress field and inertia force fieId in the matrix are given by c$’ + (u;~)~ and p’“‘a~) + ( pn, >%*,where
and
=poaj
(3
where the superscript “0” indicated zeroth phase or matrix. When an additional “single” inhomogeneity of type 0, is introduced, the additiona! disturbance in stress, strain and inertia force can be denoted by v;), .+;) and p”‘ay’. The equivalent inclusion
inD
(7)
where crij= aij + &r and a, = t!ij+ alf) are the disturbed stress and inertia, and E$‘) and 17Tir’ are the uniform eigenstraik’and e&enforces due to the presence of the additional rth phase in reg:on R, and from [3j
in which f,,, Fjkl are defined on p. 392 in j3j and p. 3 in [Sj rvhers constants A, p were associated with the matrix material or A = hO, J.C= $_ The tensor S&i. is derived from the voiume integral in the second term of eq. (16) in [3] by using eq. (12). The macroscopic average of the total disturbance in stresses over th..e volume. ~LT:,dV. and th2: in inertia forces. .!pai dV, can be shown to vanish i9] nnd lead to the condition as folEo&:
where only the rea! part should be used in computing any physical quantities and the super bar indicates time average over one cycle. It is seen that when the eigenstrains and eigenforces are known, the properties of the composite can be found from eqs. (14) and (15). The eigenfields can first be solved from eqs. (6) and (7) in terms of the averaged fields E,, and ij. Using eqs. (10) to (131, the averaged fields can be determined. The computation is based upon known volume fraction for each type of embedded inhomogeneities.
& Ik~lr~aj~lijl;rlir~rr ~$1’ c~ig&klrfs
fzi:i :rrtd l/$’
t,ct t)tc incrc&:ttt w;rye tivlds IX f~lanc Lima*harmonic
and frrc)pagatcrl along the S,-axis, (Fig. I):
u:‘) = fl,,ijjj cxp i( k-o t )
t&re
C16)
i. i = I, k and w are the longitudinal
respectively.
The
time
harmonic
~~(7, t) =nj(i:)
displaccmcnt
wave number and frequency of the incident and eigenfields can be written as:
cxp( -id)
wave field.
(17)
and E:( r:, f) = E:( ?) exp( -iwt)
(IS)
17,*(?, t) =flT(f)
(19)
exp(-iwt)
from which the total field CL;is the superposition of the inhomogeneities, i.e.,
of the incident
uf( r, t) = uJ.“( F, t) + Uj( F, t) = Lfj”( r, t) + tij(‘,
wave field plus that due to the presence
f) i-
LLI”(
r,
t)
(20)
For the composite as depicted in Fig. 1, it is observed that ,$I’= E$), and E*(‘ .i3 ) are the only eigenforces. They are non-vanishing eigenstrains and 17:“’ = IT,*“‘, and 17cCr’ are the non-vanishing determined in terms of the average strains and inertia in the matrix by expanding eqs. (6) and (7): [2(Ah +A/.#
-Si;‘,,
+ [ AA(1 -S&)
-S$,)
-AA(S$;‘,,
tS&j]~$)
- 2( AA + A@j‘I;$a]~;*:~r
=AAu:“+2(AA+Ap)E,,
+AAZ-,,
i21)
[2Ah(I-S~;~,-~~;~,)-(Ah+2A~)(S~;~,+S~’,~,)]~,L,“’ + [(AA + 2Ap)( = (AA + 2Ay)u:”
1 - S&,)
+ 2AAC,, + (Ah + 2A/.~)Eaa
Apf f,, +flz + l/(p”o2Ap)] = -Ap(F,,,&” Adf;,
+fd~P”‘+
- 2AAS;.;;,] .&@’
c F,&” [fs
= -Apu; i”--Ap(F3,,q,
(22)
IX,*“’ ‘f,JZ;” + F,33~j*:r)) -A&
(23
+ b’(~~~~A~)]fl:“’ *W + F,,,E,*,“’ + F,&“‘j
- Apii,
(24)
where u3(it = LL,
exp ikz
(25)
Note that AA = A(‘) - ~07
Ay =#r
_
0 FL,
Ap = #” _ PO
(26)
in which (r) = 1 denotes the short fibre phase and (r) = 2 denotes the additional inclusion phase particles. The quantities St!) r,k.r f,$;‘, F,$$ are volume integrals associated with each inhomogeneity geometries. Solving for e1*1,E& IIF and ri: rrom c eqs. (21) to (24) in terms of Err, E33, L7, and ii, and substituting in eqs. (10) to (13) will yield their values. The eigenfieids are thus determined. Employing eqs. (14) and (15): the composite material moduli and mass density are determined if all volume fractions are given. Tn the case of porosity or moisture absorption problems, the volume fraction f2 is not known,
eqs. (14) and (15) determination.
5. Longitudinal
musk
hc combined
with velocity measurements
such as those in eqs. (2) to complete
the
wave velocity of the composite along x,-direction
From eq. (161, ail the strains induced E$’= iku,
by the incident
wave field vanish except
exp i( kz - wt)
(271
Using or& the real part of the complex expressions for the strains, averaging timewise over one cycle, eqs. (14) and (1% are reduced to
pc = po+ (2/u$Y)
velocities
and eigenfields
i: f,@Ep r=
and
(29)
1
??
When the eigenfields :(r) and nFcr’, r = 1, 2, are determined by the procedures as described in Section 4, eqs. (28) and (29) contain thre: unknowns, C&,,. pf and _f?. They can be solved when the veioci~ cL along the x,-axis is measured. Using eqs. (21, (281 and (291, the longitudinal wave velocity of the composite along the x-a-direction can be obtained as
The trend of the ratio between the longitudinal velocity in &he compesirs with edditiona0 homogeneity phase and that without it is shown in Fig. 2. A band calcuiator, I-If’ 44. was used. The additional inhomogeneity is taken to be pores. The aspect ratio of the short &es and the matrix are assumed to be v0 = ~(‘r = l/3 and @)/p
01 0
I 0.1
0.2
0.3
0.4
0.5
Total Volume Fraction. f = f+ f2 Fig. 2. Longitudinal
wave velocity along x,-direction.
cylindrical
tibres, ilj /a,
= 10, n: = oz.
;h ~,~r~~~k~~klI~~~l~~~ wiih lmgix I’ ,. Ttrc dr’oplGtg of lri,$cr order terms in Lhc cvalualion of the integrals can IX ;i major source ot’error aI higher end ol’Wil\;UnL~lllbCr. ‘I’hc eigunstr;k xict eigenkmes in the short
fitm: trhasc of vi~imc fractiw Etep~constml on the right-hand
ibssurnirlg IW presence of pores. These are then j, art first dewwined sicle of cqs. (211 to 124) for computing the eigenstrains and eigenforces in
f1,. Without writing a ra!her extensive computer program for evaluating the volume integrals, a Tayior series expansion is used where only terms up to (ka)’ are kept. It is noted that the terms in eqs. (14) and (15) are functions of material properties and geometric dimensions after time averaging and taking only the real parts in computations.
6. Discussion of results
The analysis given allows the estimation of the changes in dynamic elastic stiffnesses znd mass density due to the generation of an additiona! inbomogeneous phase in a fibres composite. To determine the eigenstrains and eigenforces by the procedures described in Section 4, the vo!ume integrals i,j, F,,zij, Sijk, must be evaluated for the given geometries of each type of inhomogeneities. Since these integrals are spatially dependent, volume average for these need to be performed. This can be, however, a relatively complicated task in computation. Using the multinomial theorem and series representation, Fu and Mura [6] gave the results for regions inside and outside the inhomogeneity in a series form. Numerical results presented in [3,7] are valid to about wave number less than two, ka G 2. The computations required computer usage to a certain degree. Recently, Mikata and Nemat-Nasser gave an exact closed-form solution for a spherical inclusion and a truncation approximation for an eilipsoidal inclusion [8]. Various spatial dependence of the stress amplitudes is displayed for low wavenumbers. It can be observed that the stress amplitudes change very little within the inclusion for wavenumbers less than one half. This indicates that the volume integrais are essentially spatial independent at the low end of the wavenumber and computation may be somewhat simplified.
Acknowledgement
The author is grateful to the comments and editing assistance of the reviewers and the editor-in-chief.
Appendix
The total field in the composite is the linear superposition of that from the incident wave field pins the disturbed field that due to the presence of the inhomogeneities. The latter vanishes identicaIIy in the .._f+‘_”_il _c absence of the inhomogeneities. For examp!e, the displacements and the sircsses can 1.. UC.wIILctiil u;(r, t) =lp(F:
r) +u,(F, r)
(A-1)
$(F,
t) +ajj(r,
(A-2)
t) =up(r,
f)
For a given composite with an additional inhomogeneity of the rth phase, the total disturbed field can be written as Ui(‘, t) = z$( r, t> + Zf!“( r, t)
(A.31
gij( F, t) = C$i’(?, r) + a$)( r, t)
(A.9
where the superscript “-” indicates the average value.
Its time rate of change can be writter: as [I]: 3E - = /p’a,‘~‘C dV + ~LT,~F;~ dV at
(A.6j
where the superscript “ .” den&es partial differentiation with respect to time. Combining eqs. (A.l), (A-21, (A.51 and (A.6); there results jpca;r;’
I
The integral
qfi,;
dV
dl/=l(p”ap+pn,)[c:“$_i;;~~“] dV= _/(q;
+ qi)
I
;;j) + iij + ij;j
df’
=
(T
.,l.~‘(~)dS -
r,
1 I
g. L’(i! ‘I
Id
dV=
0
whcrc $) can bc any prescribed virtual velocity. Similarly, the integral
vanishes by the fact that the rate of work done by surface traction is equivalent to the rate of increase of the strain energy. The final result from eq. (A.7) is therefore (A.10) in which the eigenforces 7i;*(‘) vanish identically outside a,.
References [l] R. Truell. C. Elbaum and B.B. Chick: (iltrasorzic Methods in Solid State Physics (Academic Press: New York, 1969). [2] M. Taya and T.W. Chou, On two kinds of ellipsoidal inhomogeneities in an infinite elastic body: an application to a hybrid composite, In?. .I. Solids and Structures 17 (1981) 553-563. [3] L.S. Fu and T. Mura, The determination of the elastodynamic fields of an ellipsoidal inhomogeneity. J. .IppJ. itleciz.Xl (1983) 390-396. [4] T. Mura, Micromechanics of Defects in Solids (Martinus Nijhoff, Dordrecht. The Netherlands, 1982). [5] L.S. Fu, A theory of dynamic elastic constants of heterogeneom media, Phibsophical Mugmine A 56(l) (19871 149-159. [6] L.S. Fu and T. Mura, Volume integrals associated with the inhomogeneous Helmholtz equation: 1. Ellipsoidal region. NASPl Contractor Report 3749, 1083: a!so: Warr Marion 4 (1982) 141-149. [7] L.S. Fu and Y.C. Sheu, Ultrasonic wave propagation in two-phase media: spherical inclusions, Composite Str-z~tures 2 (1954) 289-303. [8] Y. Mikata and S. Nemat-Nasser, First-order dynamic Eshelby tensor. 25th P.cc-z! TV -i-‘^-’ “--*‘ -- ,LE;. ICCLII.L~Y. I.“C._L’ *‘ociety of Engineering Science, ESP25.88014, University of California, Berkeley, CA, June, 1988. [9] L.S. Fu, On the macroscopic average of the disturbed stress and inertial fields in an inhomogeneous elastic media. submitted for publication.