- Email: [email protected]
A comparison of two theories of acoustoelasticity
A comparison of two theories acoustoelasticity
of
A.V. CLARK and R.B. MIGNOGNA The presence of a plane stress field causes small changes in the phase velocities of orthogonally polarized SH waves. The (small) difference in phase velocities (birefringence) can be used for non-destructive stress measurement. However, material anisotropy can affect phase velocity to the same extent as stress. Two theories have been developed which account for the effect of both stress and anisotropy. The theory of lwashimizu and Kubomura assumes isotropy in the third-order elastic moduli and anisotropy in second-order moduli. A different approach was taken by Okada, who assumed the existence of a matrix analogous to the index of the refraction matrix in optics. In this paper, we generalize the theory of lwashimizu and Kubomura by retaining anisotropy in third-order moduli. We show how Okada’s theory can be made to agree with this more general theory. We also compare the predictions of the various theories with birefringence data obtained from uniaxial tension tests on 2024-T351 aluminium specimens. Both the Okada theory and the theory of lwashimizu and Kubomura gave good agreement with experiment. KEYWORDS: ultrasonics, acoustoelasticity, acoustic birefringence
Introduction It has been known for some time that the presence of stress causes small changes in the phase velocities of sound waves propagating through solids. The idea of exploiting this change as a stress measurement tool was put forth over twenty years ago’, and its potential application investigated by several researchers*-” I*. As pointed out by Crecraft2, orthogonally polarized SH waves propagating through a solid in a state of plane stress can be considered the acoustic analogue of the photoelastic effect in optical materials. That is, a birefringence effect occurs so that for a homogeneous. isotropic solid in a state of plane stress the birefringence B is related to the difference between the principal stresses u1 and a2
B = A?,(u,- a,) Here d is the acoustoelastic
B=
v2
-
(1) constant,
and
VI
(2)
!h(V,+ V2) where V, and V, are phase velocities
of SH waves polarized along the (T, and u2 directions, respectively. Note that a2 - u, is the magnitude of the principal shear stress. The authors are in the Nondestructive Evaluation Section, Structural integrity Branch, Naval Research Laboratory, Washington, DC 20375, USA Paper rceived 16 May 1983.
0041-524>(/83/0502
ULTRASONICS. SEPTEMBER 1983
1749/$03.00
Equation (1) is derived from the equations of Hughes and Kelly3,who calculated the phase velocities of free waves propagating through homogeneous, isotropic materials under stress. For such materials, the acoustoelastic constant I@ is given by 4~ +
n
8P2
where p is the shear modulus and n is one of the three third-order elastic moduli of an isotropic solid. The birefringence technique was used by Crecraft in an attempt to measure the residual stresses in a nickelsteel bar subjected to bending forces large enough to cause plastic deformation in some portions of the bar. He recognized that the bar was not isotropic but had an initial anisotropy ‘evidently due to preferential grain alignment’*. He measured the initial birefringence (prior to deformation of the bar) and also found that this anisotropy had a principal axis parallel to the bar axis. The initial birefringence was subtracted from that measured when the bar was deformed, in an attempt to compensate for the initial anisotropy. Unfortunately, this did not give satisfactory results for measurement of the residual stress. The issue is further complicated by the fact that some of the birefringence measurements were made in regions where the bar was plastically deformed. Hsu4 used the birefringence technique to measure principal shear stress in a disc in diametral
the
0 1983 Butter-worthEt Co (Publishers)Ltd 217
compression. The disc was cut from a rolled plate of 2024 T-4 aluminium, and the loads were applied parallel to the rolling direction. The birefringence was measured along a diameter orthogonal to the load axis. By subtracting the initial birefringence (measured when the material was in the unloaded state) and attributing the remainder to the principal shear stress, good agreement with theory was obtained. It should be noted that the applied shear stress along this diameter vanishes due to symmetry. More will be said about this point later. Blinka and Sachsej used a different approach in relating birefringence to applied uniaxial stress. They propagated SH waves through a stressed aluminium specimen with a broad-band transducer orientated at 45” to the load direction. Due to the birefringence effect, two SH waves propagate through the specimen. If the specimen is isotropic, one wave will be polarized along the load axis, and the other polarized orthogonal to the load axis. The two waves will be out of phase when they return to the transducer, and the spectrum of the echo received will have minima which can be related to the time-of-flight difference of the waves and hence tothe applied stress. The authors of Ref. 5 also included a correction term in their birefringence against stress equation to account for the initial (unstressed) birefringence due to material anisotropy. It should be noted that in all the works referenced above, the effect of material anisotropy (‘texture’) was either ignored or accounted for by subtracting the initial unstressed birefringence, B,, from the birefringence B in the presence of applied stress and relating the remainder to the principal shear stress B -
B, = if (a2 -
a,)
We have recently measured the stresses around a single-ended crack in Mode I crack-opening displacement using Okada’s theory and obtained excellent agreement between our acoustoelastic measurements and the theoretical stress field in our specimens. It is curious, however, that Okada’s theory works as well as it does. Okada postulates the existence of a matrix analogous to the refraction index matrix in optics. However, using analogies between optics and acoustics requires some care, since there are obvious differences between the two fields; for example, the existence of only two optical phase velocities in anisotropic crystals, as opposed to three acoustic phase velocities. Also. the slownesses in optics are not calculated from the eigenvalues of the refraction index matrix. The eigenvalues of the refraction index tensor are the square roots of the principal values of the dielectric tensor. If a wave normal is along one of these principal axes, then the slownesses for wave propagation in that direction are given by the eigenvalues. For an arbitrary orientation of the wave normal, the slownesses are given by a geometrical construction involving the ‘ellipsoid of wave normals’
3. (3)
This method works for the special cases mentioned in Ref. 4; that is, either uniaxial stress or (locally) biaxial stresses, that are parallel and perpendicular to the rolling direction. Recently, two theories have been developed to account for the effect of material anisotropy on acoustic birefringence. The theory of Iwashimizu and Kubomura treats the case of free waves propagating through the thickness of a plate which is orthotropiP. In their equations, the material is treated as a nonlinear elastic solid with the effects of orthotropy retained in the second-order elastic moduli but ignored in the third-order moduli. They derived an equation relating birefringence and stress similar to (3) in that their equation contains the initial birefringence B, and a single acoustoelastic constant. However, they found that in the most general case, there was a non-linear relationship between birefringence and stress. Okada also derived a relation between birefringence and stress’. He made the assumption that there exists an index of refraction matrix for the solid and that the components of the matrix are linear functions of stress. The coefficients relating the matrix components to stress were assumed to be nine in number when referenced to axes of material symmetry in the plate (parallel and perpendicular to the rolling direction). This is analogous to the behaviour of the second-order elastic moduli for an orthotropic solid. Okada assumed that the inverse velocities (slownesses) of free waves propagating in an orthotropic solid are the eigenvalues of the index of refraction matrix. Okada then derived a
218
non-linear birefringence-stress relation involving B0 and three acoustoelastic constants. His equation reduces to that of Iwashimizu and Kubomura6 if one of the constants vanishes and the other two are equal, as will be discussed more fully later. His relation also reduces to (1) for an isotropic material.
e1
0: + -0: -52
=
constant
(4)
E3
where the E, are the principal values of the dielectric tensor and the Di are components of the electric displacement. The slownesses are proportional to the principal semi-axes of the ellipse generated by the intersection of the ellipsoid of wave normals with a plane whose normal is in the direction of free wave propagation. For more details, the reader is referred to standard optics texts, such as Ref. 9. For an unstressed anisotropic solid, the (squares of) acoustic phase velocities are the eigenvalues of a symmetric second-order tensor CVkinin, where the Cij,, are the second-order elastic moduli and nj are the direction cosines of the wave normallO. For a stressed solid. it has been shown” that the velocities are the squares of the eigenvalues of a second-order ‘acoustical tensor’, which is a function of the applied stress and of the second- and third-order elastic moduli. The theory of Iwashimizu and Kubomura is a special case of this more general theory, since they ignored anisotropy in the third-order moduli. Okada tested his theory and found that it gave good agreement with birefringence measurements for uniaxial tension specimens made of various aluminium alloys’.‘*. He noted in particular that for 5052 aluminium his three-parameter model was able to fit the data much better than the theory of Iwashimizu and Kubomura. We are thus left with somewhat of a paradox, in that the Okada theory, which appears to be the less rigorous, seems to give better agreement with
ULTRASONICS.
SEPTEMBER 1983
where the superscript ‘0’ refers to the isotropic values and the primes denote small perturbations from the isotropic values. For example,
experiment than the Iwashimizu and Kubomura theory, which is based on a seemingly more rigorous theoretical foundation. The purpose of the present paper is two-fold. First, we wish to resolve this (apparent) paradox by extending the theory of Iwashimizu and Kubomura to account for the anisotropy in third-order moduli. We will show that this gives a birefringence equation whose functional form is the same as Okada’s and contains three acoustoelastic constants. Secondly, we present some experimental results for 2024-T351 aluminium tensile specimens and compare the agreement between the two theories and the data.
Theoretical Rigorous
(11) where 8, is the Kronecker delta function, are the iamt constants for the material. Iwashimizu skim
=
study
derivation
of birefringence
equation
We begin the analysis by considering the case of an infinitesimal stress wave propagating through an elastic solid subjected to finite applied stresses tis. The stresses and strains are referenced to the deformed and undeformed states of the solid, respectively. In the undeformed state, the coordinates of a material point are given byXi; after deformation, the material point has coordinates Xi. The material densities in the undeformed and deformed states are denoted by p,, and p respectively. The passage of the stress wave causes an additional infinitesimal displacement Wi superposed on the finite displacement Ui = Xi - Xi. The equation of motion for the stress wave is given by” (tis 6kkr +
Skirs)
where p dxk -----
dxi
r
aw,
azwk
s1
dx,
=
(5)
ax,
a’@ (6)
where @ is the strain energy density for the elastic solid. Summation over repeated indices will be implied, unless otherwise stated. In the notation of Iwashimizu and Kubomura6, strain energy density is given by 1 1 Cijkl Eli Ekl + 3! Cijklmn Eii Eki &n “=F
the
(7)
where the and C$!+,,,, are the second and third order elastic moduli evaluated at zero strain; consequently, cijk,
and Kubomura ck$s
In what follows, terms that are second order in both the displacement gradients au&Xi and aWdaXi will be neglected, so that the strain tensor assumes the linear form
derive
skim
(124
+
2X(akiErs
+
&sEki)
’
2E.L (dkkrEis
+
sir Eks
+
Ckirsmn Emn
+
sks Eir
+
&is Ekr) (12b)
They retained the effects of anisotropy in second order moduli, but ignored it in third order moduli by setting the Ckirsmn = Ckh:smn in the above. We will consider effect of retaining the anisotropic terms in the third order moduli. Furthermore, we specialize to the case of a slightly orthotropic material, such as a rolled plate. Due to rolling a differential volume of the plate has one (local) two-fold symmetry axis, orientated normal to the plane of the plate. Also, if the texture is uniform throughout the plate thickness, then the rolling direction is also a two-fold axis, as is the in-plane axis perpendicular to the rolling direction. (The case of a small texture gradient is considered in Appendix A). Note that these axes are ‘local crystallographic axes’ embedded in material volumes which are considered large relative to a grain size. but small relative to an acoustic wavelength. On this length scale it is possible to consider the effect of the average orientations of all the grains in the volume, rather than of individual grains. Assume that there exists a triad of two-fold axes, so that a differential volume of the plate material behaves as a slightly orthotropic single crystal. Consider also the case of waves propagating through the thickness of the plate (along the xj axis). The other axes of material symmetry (parallel and perpendicular to the rolling direction) will be donated as the x2 and xi axes. The plate is assumed to be in a state of applied plane stress. We assume negligible residual stresses. The displacement
(8)
+
and X and p
Wi =
in the stress wave is of the form
Wi exp[i(fi,
-
of)]
(13)
where K is the wavenumber (w/v) associated with frequency w and phase velocity V. Since the plate is in a plane stress state, t,, = t,, = r23 = 0. For slight anisotropy, the usual stress-strain relations hold to a high degree of accuracy
(9) tij = For a slightly cijkl
=
anisotropic
c,;k,
+
material,
let
Ci;.k,
(104
XEkkdij
+
2PEEij
(14)
so that E,, = Ez3 = 0. Since there is no applied stress in the direction of propagation of the wave, (5) simplifies to
and Cgklmn
=
C;klmn
ULTRASONICS.
+
C;klmn
SEPTEMBER 1983
(lob)
(15)
219
Recall that the Skirs depend upon the elastic moduli and the strains, as shown by (12b). In general, the strains will vary on a length scale which is much larger than an acoustic wavelength (the scale on which W, varies). Consequently, gradients of strains can be neglected in (15). Also, it has been assumed that the elastic moduli are constant over a differential volume having sides much larger than an acoustic wavelength. The moduli vary on the scale of the plate thickness, which is much larger than a wavelength, and will be replaced in (15) by their values averaged through the plate thickness (see Appendix A). Consequently. the values of the moduli used hereafter are the thicknessaveraged values. Using the relation” (13) into (15) gives
p = pO (1 -
[email protected] = A,, + A,, f
[(A,, -AJ2
+ 4A22]1’2 (19)
where V, = V, + AL’. The angle @ between the x, axis and the eigenvector associated with the SH-wave having velocity V1 is given by
(20) The acoustic v2
BE---
birefringence
-
B is deGned by
v,
C-21)
vo
where V,= ‘/z (V, + V2). Since V, = V, f AV/2 and V, = V, - AV/2 and AV<< V, for small birefringence.
Ekk) and substituting [(A,,
-A,,)*
+
%1”2 (22)
a-)ov,*
(Akr -
PO v2 Sk,)w, =
where the acoustical
(16)
0
tensorAkr
is defined
From (17) (12b) and (18)
by An-A22
Akr = C& (I + E/d + S;krs The pop are the eigenvalues of the acoustical tensor, and the eigenvectors are polarization directions for particle motion in free waves propagating in the xj direction.
(G-
=
(17)
G4)
+
+
(Cl55-
Cl,,)
tll
+
(C,,,
G44)
tl3
-
2P
(71
+
-
(C25,
t/2)
-
C244)
(23)
where C,, = p + C;, and C,, = p + C;;
Consider terms such as A,,. If such terms are non-zero, there will be an eigenvector corresponding to a quasilongitudinal wave (rather th In pure longitudinal). Presently we have A:, = C&; expressed in the Voigt notation, C,;,, = C,“,. For an orthotropic material C,O, vanishes identically, as does C;,. Consequently A,, is non-zero only if Sir,, is non-zero. From (12b). the only non-zero contribution to S;,,, is the term C3133mnE,,. since E,, = E,, = 0. For an orthotropic material, the contribution to the strain-energy density function due to terms that are third-order in strains is13
A12
=
%(P
+
t/2
likewise, (24)
C,,,)
For an isostropic material, the following between the third-order modulir4:
relations
hold
c
144
=
c2,,
(W
c
155
=
c244
Wb)
c
344
=
c35,
(25c)
Since the material &,
=
and
cl44
+
is assumed
c;,,,
c244
=
slightly orthotropic,
c,,,
+
let
c;44.
G,, - Cj4, = CG,,
where Cl,, << C,,, etc. For plane +
6C,,,
tll
+
6712 L,
t/z
~3
+
d +.
671
.I
[G,,
d
+
Cm
d
+
CM
stress, the isotropic
tl3
+ 6~3KG,,vt: + .I
q1
=
-
2
‘12
(0,
=
Here the Voigt notation is used for strains as well as third-order moduli; for example, nr = E,,, q4 = E,,, etc. From the above equation, it is seen that C3133mnEmn is non-zero only when C3,333,E3, is non-zero, since all other third-order moduli vanish. However, E,, = 0, so s;,,, = 0. In like fashion, it can be shown that A,, = 0, so there is no coupling between longitudinal and transverse motion for waves propagating in the xj direction, even when the effect of orthotropy in third order moduli is accounted for. Consequently, the velocities and polarizations of pure mode SH waves are calculated from the eigenvalues and eigenvectors of the acoustical tensorA,s ((Y,p = 1, 2). The eigenvalues are given by
J-
+
fJ2)
relations
give
(0,
-
(2W
u*)
cos
2p (a, - 0,) sin 28
(18)
220
stress-strain
a:1
28
(26b)
qs = r (26~) 4p with v amd E being Poisson’s ratio and Young’s modulus, respectively. The principal stresses are u, and u,, and 6 is the angle between the x, axis and the u, direction. Substituting (26a) and (26b) into (23) gives A,, +
M,(~,
A22
= -
(G u2)
-
~0~
Ci4)
+
MAa,
+
28
~2)
(27)
where M, = (-v/E) C;,, and M2 = 1 i- (C,,, - C144)/2p. A term (C;,, - C;,,) t/2 has been neglected in the above.* Substitution of (26~) into (24) gives *We show that neglecting this term allows the birefringence equation to be written in a pleasing form In terms of the sum and difference of principal stresses. More will be said later about neglecting this term.
ULTRASONICS.
SEPTEMBER 1983
24,,
= MJ (a, - a*) sin 28
(28)
where M3 = 1 + C,,,lp. When (27) and (28) are substituted into the birefringence equation (22) there results B = ([B,+ m,(a,+ q) + m,(o,- uJcos 2fl]* (29)
+ [ms(o, - uJ sin 28]*)“*
where use has been made of the relation poV,’ = CL, and the m are defined by mi = A4il2E.Lfor i = 1,2,3. The initial birefringence B, is equal to (Cj, - CiJ/2p. In like fashion, the angle $ is given by (see (19)): m&J, tan 2@ =
uJ sin 28
B, + m,(a,+ UJ + m2(u,-
~~2) cos
28 (30)
u12
(31)
= 2m3
so that the shear stress can be determined birefringence and the angle @.
(32)
is recovered ifB, = m, = 0.m2 = m3 = G = (4~ -I- n)/ 8$ where n is the third order Murnaghan constant14. It is straightforward to show that these relationships hold by obtaining the appropriate forms of the third-order moduli from the expression given by Iwashimizu and Kubomura
and using v, = n/4.6 It was shown before that to obtain (29) and (30) it was necessary to assume that the term (C;,, - C&s)t/2 was negligible as compared to, for example, (C,,, - C344)tlj. In general, nZ and Q will be the same order of magnitude for plane stress, so the assumption is valid if C;,, if CS,, << C,,, - CS4,, or equivalently, Cl44
=
cm
Cl,,
=
Cl44
c,,,
#
c344
Note that there is in general a non-linear relationship between the birefringence B and the initial (unstressed) birefringence B. .Consequently, it is not generally possible to compensate for the effects of initial anisotropy by merely subtracting B, from B. Exceptions are the case of uniaxial stress, or a state of biaxial stress with the principal stress axes parallel to the local crystallographic axes. The experiment described in Hsu’s pape? falls into the latter category. The diametral loads were applied to the disc in directions parallel to the rolling direction and the birefringence measured along the diameter perpendicular to the rolling direction. In this case, the local shear stress vanishes due to symmetry. Assuming m, << m, for this material (2024-T4 aluminium),** the birefringence equation reduces to
This explains measurements
(34)
the excellent agreement and theoretical values.
between
and
It is of some interest to determine why Okada’s approach is successful, given the fact that he begins his analysis by seeking the eigenvalues and eigenvectors of a tensor which is supposed to be the acoustical analogue of the optical refraction index tensor. This tensor is not the same as the acoustical tensor of (16) which is derived from a rigorous analysis of the passage of a sound wave through a stressed solid. We also note that the symmetric (second-order) acoustical tensor Aik is found by contracting the fourth-order tensor sjj,L[ twice with the direction cosines n; for the wave vector K; that is, Ai/q = Sgkpin,. Okada seeks the eigenvalues of the symmetric tensor N,, = N,,T+ ANii, where N;,?is a diagonal matrix whose elements he assumed are’the slownesses of free waves propagating in the unstressed state. The AN, are (small) changes which are linearly related to stress ANii = %jkl ukl. We note that the units ofNji are inverse velocity and the units OfAik are density X velocity squared. This motivates us to formally define N;i by the relationship (in matrix form)
For more details,
SEPTEMBER 1983
his
We have thus far shown that solving for the eigenvalues of the acoustical tensor leads to a nonlinear birefringence equation (29) which involves three acoustoelastic constants m,, m2. and m3, when the effects of anisotropy are accounted for. This equation is formally equivalent to the birefringence equation derived by Okada and experimentally verified by him’.‘*.
WI PII-’ = Plho
It is interesting to note that the first two relationships above hold for a tetragonal material; that is, C,,, = C,,, and C,,, = C,,,. They do not hold in general for an orthotropic material. However, we have for a tetragonal material6 C,,, = C344. which does not satisfy the third relationship. Consequently, a material whose acoustic birefringence obeys the Okada equations has some properties of a tetragonal material and some of an orthotropic material.
ULTRASONICS.
equation
Comparison between the rigorous derivation Okada’s theory
from the
Equation (29) is the same as that derived by Okada, and reduces to the form of Iwashimizu and Kubomura form, = 0,m2 = m3. Furthermore the birefringence equation for the isotropic case B = ii@, -uJ
cases of the birefringence
B -B, = mz(u,- uJ
For the case of plane stress, (a, - a*) sin 28 = 2u,,. From (29) and (30) we have B sin 21$
Special
Performing convention)
(35)
see Appendix
the inversion
+
[A4
B.
gives (no summation
(36)
**We measured m, and m2 for 2024-T351 aluminium and found that mz * 300 ml. It seems reasonable to assume that similar results hold for 2024-T4 aluminium.
221
where
(374
that the rolling direction was at various angles to the specimen axis. The specimens were subjected to uniaxial tension, and the birefringence measured for various stress levels. For the special case of uniaxial
tension,
(29) becomes
(37b)
Mij
0
=
for either i orj = 3.
(37c)
B = B,+(m,+m,)a
(8 = 0")
(414
B = Be-l-(m,-m2)o
(8 = 90")
(41b)
B = [(B,+ m,u)2 f (m,u)2]"2 (8 = 45”)
Here (l/N;) are the phase velocities for free wave propagation in the x3 direction in the unstressed orthotropic solid; for example, (1/NF1)2 = (Z-L-I- C&)/P, etc. Following Okada, we assume
(4lc)
(38)
A straight line lit to the B against u curve for 8 = 0” and 90” will give the values of m, and m2. The value of m, can be calculated from a least-squares fit ofB against u for 0 = 45”.
where the summation convention applies and the Voigt notation is used.* Note that Z = 1, 2. . 6. whereas i = 1, 2, 3.
The birefringence was measured using the pulse-echo overlap technique and the instrumentation used in our experiments is described in Ref. 8. Fitting the data to (41 a)-(41c) gave values of:
mI =
&IJ UJ
With these definitions, becomes
the birefringence
equation
(22)
+411;~22)’ + (44’_
B
=
+
ml
+
v,
K%,
-
%I)
Ull
+
(a22
-
a121
u221J2
where the Voigt above equation Using the plane agrees with the that the Okada relationship.
notation is used only for the CYS.The is the same as that derived by Okada’. stress relation (26) we find that (39) birefringence equation (29) provided stress-acoustic constants satisfy the
-v,(%
-
m, =
a221
-vc;,
=
2
5
2@
(404
1 1 1
1+ [
(c
a12
-
;
(a,,
+
a221
155- c,‘A 2iJ
=
i%
(4Ob) I (4Oc)
m3 =
It is easy to show that (30) for the polarization directions is also recovered when the above relations are satisfied. Thus, we have shown that if Okada’s Nij tensor is related to the acoustical tensor Aij using (35), the Okada approach gives the same results as the rigorous theory based on non-linear elasticity, provided that the Okada stress-acoustic constants are related to the second- and third-order elastic moduli as above.
Experimental
study
To compare the theories of Refs 6 and 7, we performed a series of experiments using uniaxial tension specimens made of 2024-T351 aluminium. The specimens were cut from 6.3 mm thick rolled plate so *We note that Okada’ defines N, 2 = ageal *~‘2.
222
(42a)
rn2 = 3.64 X 10e5MPa-’
(42b)
m3 = 3.32 X 10-j MPa-’
(42~)
(39)
m%6~,2)2~“2
m2 = VCI [
ml = 0.01 X 10d5MPa-’
The quality of the fit is evident in Fig. 1 for 0 = 0”. 45”. and 90”. Note that we have also shown data from a specimen for which 8 = 60”. Reasonably good agreement is also obtained for this specimen for the above values of the acoustoelastic constants. The equivalent forms of the birefringence equations of Iwashimizu and Kubomura are obtained by setting m, = 0,m2 = m3 = m in equations (41a) - 41(c). The value of m obtained by a straight line lit of the data for 8 = 0” is m = 3.68 X 10-S MPa-I. This of course is very nearly equal to the value of rn2 in the Okada model. since fitting the data by the Okada theory gives m, z O,m, s m,; see(42a)-(42c). The quality of the fit between the birefringence data and the predictions of the Iwashimizu and Kubomura theory is shown in Fig. 2. The agreement is excellent for fl = 0” and 90”. This is to be expected. since both theorie?’ predict the same slope dB/du for 0 = 0” and 90” when m, = 0 and m2 = m. Since the Iwashimizu and Kubomura theory has only one acoustoelastic constant, it does not give as good a fit to the 0 = 45” specimen as the Okada theory: however, the agreement is still quite good. We also measured the angle @ for our uniaxial tension specimens, as a function of stress a; our experimental technique is described in Ref. 8. For the stress levels used in our experiments, we found that @ = 0” for 8 = 0” and 0 = 90”; that is, the slow acoustic axis remained normal to the rolling direction. This is in accordance with the Okada angle equation (30). The theory of Iwashimizu and Kubomura also predicts the same result. For specimens with 0 = 45” and B = 60”, we obtained the results shown in Figs 3 and 4. We estimate our experimental error in measuring C#Ito be + 2”, so the agreement between the theoretical predictions and the data is reasonably good.
ULTRASONICS.
SEPTEMBER
1983
developed by Okada. He assumed that the inverse phase velocities (slownesses) can be calculated as the eigenvalues of an index of refraction matrix. whose elements are assumed linearly related to stress.
7-
The phase velocities in the rigorous theory are proportional to the squares of the eigenvalues of the (second-order) acoustical tensor. By formally relating the acoustical tensor [A] to the Okada index of refraction [N] using [[Nj [N]]-* = [Al/p,, we found that the rigorous theory and the Okada theory give the same results, provided that the relationships (40) between the Okada stress-acoustic constants aIJ and the acoustoelastic constants rn;(of the rigorous theory) are satisfied.
6-
“0
40
20
60 Stress
Fig. 1
80
100
120
CMPol
The birefringence fit for 0 = 0’. 45’. 60’ and 90’
We compared the agreement between experimentally obtained values of birefringence B and polarization angle r$, and the predictions of the theory of Okada, for uniaxial tension specimens made from 2024-T351 aluminium plate. The agreement was excellent. We also compared the same data to the predictions of the theory of Iwashimizu and Kubomura. The agreement was also quite good. In fact, we found that for our specimens, m, e 0 and m2 r m,, so that the Okada theory reduces (approximately) to the theory of Iwashimizu and Kubomura for this case.
40-
30B 3 8 20-
IO
1
OO
I 20
I
40
,
I
60 Stress
80
Stress CMPal
x\/
I
Fig. 3
Angle @against stress for 6’ = 45” and 60”
Fig. 4
Angle @ against stress for 8 = 45” and 60”
CMPol
Fig. 2 The fit between the birefringence data and the predictions of the lwashimizu and Kubomura theory
Conclusions We have developed a rigorous theory which gives the phase velocities, birefringence. and polarization directions for free SH waves propagating through the thickness of a slightly orthotropic plate in a state of plane stress. The theory is an extension of that developed by Iwashimizu and Kubomura. In our derivation. we retained anisotropy in second- and third-order moduli. whereas they ignored anisotropy in third-order moduli. Our rigorous theory gave rise to a birefringence equation involving three acoustoelastic constants m,. m2. and m3, whereas the theory of Iwashimizu and Kubomura involves only one. The rigorous theory reduces to that of Iwashimizu and Kubomuraif m,= O,m, = m,. Our theory formally
ULTRASONICS.
gives the same results as that
SEPTEMBER 1983
Stress CMPol
223
where the ei are principal values of Ed. In this same coordinate system, the relations between D and E become D; = E;Ei (no sum), and the relation between electric energy density and the electric displacement is expressed as
Acknowledgement The research reported here was conducted in the Nondestructive Evaluation Section of the Naval Research Laboratory under the sponsorship of the Office of Naval Research.
Appendix
(B4)
A
Here we show that the elastic moduli in (15) can be replaced by their thickness-averaged values. Assume small variations in the moduli, SO that Cljk/ = + (d C,j~,/dx,)s, where is the thicknessaveraged value of C;;,,, .Y~is measured from the plate centre,* and < Ci.k, > > (8 C+JdXJ d and d = plate thickness. Then f’or a plane wave, (15) will contain terms such as
t-41) Comparing the magnitudes of the two terms on the right-hand side of the above. K Kd < C,kn > g
aC3kr3
C3L,.3(x3 = d) -
Cjkr, (x3 = 0)
ax,
>> 1
(Ml
Energy density
Here we use some analogies between optics and acoustics to explain how (35) is derived. The optical derivation follows directly from Ref. 9. In optics, the electric energy density G, in a light wave is given by the product (E.D)I&r where ,!? is the electric field vector, and D is the electric displacement. Since the electric displacement is related to the electric field through
G, =
% E;j U’ I/
036)
(Bl)
with E;, the dielectric tensor, the electric energy density becomes Ei Eii E, A = G, = constant. (B2) 8a Because of the symmetry of the dielectric tensor, (B2) can be reduced to a form in which only squares of the field components enter the expression, and not the products. Since the electric energy density is positive definite. the above expression can be thought of as an ellipsoid in a space with coordinate axes E,, EJ2 and E,. The ellipsoid has principal axes E,. E,. and E, and in this coordinate system. (B3)
*Actually, x3 should be measured from the centroid 2, of Ciifi rather than the plate centre. However, this merely entails replacing x3 by x3 - I, in what follows; the results of the analvsis are still the same.
strain by the
% Cijkl W, i Wk, ,
(B7)
with Wi the particle displacement in the stress wave. For a plane wave with direction cosines n, and wavenumber K, we have
G =
224
in stress waves
or, since stress is related to (linearized) second-order elastic moduli.
in light waves
E, E; + .q E; + E&E: = 87rG,
(B3
for a light wave in an optical crystal; here x denotes proportionality, and (Ni.*>-l is the ij,, element of the inverse of the matrix dij.
G, =
Di = Eii E,
ot 81~Ge
The acoustical analogue is the passage of an infinitesimal stress wave through an (unstressed) anisotropic solid. The strain energy density is given by
B
Energy density
of wave normals’9.
Consider the case of a light wave propagating in the 5, direction. which is parallel to a principal axis of the dielectric tensor. For a non-magnetic crystal, the two possible phase velocities in the xi direction are given by K = C/(Ei)“’ where C is the phase velocity of light in vacua, and i #_j. ** Since the optical index of refraction is defined by N = C/Vi, we see that it is possible to generalize to an index of refraction tensor nji = &,il/z having principal values N * = e;. Following Okada, we define Nil = i”,llC, SO that Nil has units of inverse velocity. Consequently. the eneigy density can be written in the form DlDj (N,‘)-’
since the moduli vary on length scaled, and the quantity Kd >> 1 for the frequencies used in ultrasonics.
Appendix
which is called the ‘ellipsoid
T
Aik
038)
wwk
where the symmetric second-order tensor A& = Cu,knjni. The above expression also can be written terms of particle velocities Ui as Aik Vi vk Gs = 2 v,2
in
(B9)
where slight anisotropy is assumed so that K2 s 02/Vzo Rewriting the above in terms of the principal values p,,vF of the tensor Aik gives G, =
L/zp.
@lo)
or defining
an acoustical
index of refraction
Ni = VdVi,
**The units will be correct if ci is replaced by rci where the permeability r = 1.
ULTRASONICS.
SEPTEMBER 1983
3
(Bll)
G, = ‘/2p. which is analogous
Analogy
to (B4).
between
optical
and acoustical
cases
If we make an analogy between electrical displacement D; and particle velocity Ui, there is an analogy between Aik and Nij; (compare (B5) and (B9)). Since& is the contraction of a fourth-order tensor, we form the fourth-order tensor Nij Nk,, and then contract to obtain a second-order tensor Nij Njk. TO obtain the correct units, we let
which is just (35), remembering are pO Vj’.
that the eigenvalues
of
d4jk
References I
2
Benson, R.W., Raelson, V.J., Acoustoelasticity.
Prod. Eng.
30 (1959) 56-59 Crecraft, D.I., The Measurement of Applied and Residual Stresses in Metals Using Ultrasonic Waves.1 Sound Vibr., 5, (1) (1967) 173-192
ULTRASONICS.
SEPTEMBER 1983
Hughes, D.S., Kelly, J.L., Second-order Elastic Deformation of Solids, Phys. Rev., 92, (5) (1953) 1145-l 149 4 Hsu, N.N., Acoustical Birefringence and Use of Ultrasonic Waves for Experimental Stress Analysis, Exp. Mach.. 14, (5) (1975) 147-152 5 Blinka, J., Sachse, W., Application of Ultrasonic-pulsespectroscopy Measurements to Experimental Stress Analysis, Exp. Mech.. 16, (12) (1976) 448-453 6 Iwashimizu, Y., Kubomura, K., Stress-induced Rotation of Polarization Directions of Elastic Waves in Slightly Anisotropic Materials. Znr.J. Solids Srructures, 9 1(1973) 99114 7 Okada, K., Stress-Acoustic Relations for Stress Measurement by Ultrasonic Techniques, J. Acoust. Sot. Jpn (E). 1, (3) (1980) 193-200 8 Clark, A,.V., Mignogna, R.B., Sanford, R.J., Acoustoelastic Measurement of Stress and Stress Intensity Factors Around Crack Tips, Ultrasonics 21 (1983) 57-64 9 Born. M.. Wolf. E.. Princiules of Ontics. 6th ed.. Pergamon L Press, New Yo;k, XIV. (IdsO) 10 Auld, B., Acoustic Fields and Waves in Solids, Wiley, New York, 6A_ (7D) (1973) 11 Tburston, R.N., Wave Propagation in Fluids and Normal Solids, Physical Acoustics. Vol 1. (A), Academic Press, New York (1964) 12 Okada, K., Acoustoelastic Determination of Stress in Slightly Orthotropic Materials, Exp. Mech., 21, (1981) 461-466 13 Mumaghan, F.D., Finite Deformation of an Elastic Solid, Wiley, New York, 4. (5) (1951) 14 Hearmon, R.F.S., ‘Third-Order’ Elastic Coefficients, Am. Ctymzl.. 6, (1953). 331-340
225
of
A.V. CLARK and R.B. MIGNOGNA The presence of a plane stress field causes small changes in the phase velocities of orthogonally polarized SH waves. The (small) difference in phase velocities (birefringence) can be used for non-destructive stress measurement. However, material anisotropy can affect phase velocity to the same extent as stress. Two theories have been developed which account for the effect of both stress and anisotropy. The theory of lwashimizu and Kubomura assumes isotropy in the third-order elastic moduli and anisotropy in second-order moduli. A different approach was taken by Okada, who assumed the existence of a matrix analogous to the index of the refraction matrix in optics. In this paper, we generalize the theory of lwashimizu and Kubomura by retaining anisotropy in third-order moduli. We show how Okada’s theory can be made to agree with this more general theory. We also compare the predictions of the various theories with birefringence data obtained from uniaxial tension tests on 2024-T351 aluminium specimens. Both the Okada theory and the theory of lwashimizu and Kubomura gave good agreement with experiment. KEYWORDS: ultrasonics, acoustoelasticity, acoustic birefringence
Introduction It has been known for some time that the presence of stress causes small changes in the phase velocities of sound waves propagating through solids. The idea of exploiting this change as a stress measurement tool was put forth over twenty years ago’, and its potential application investigated by several researchers*-” I*. As pointed out by Crecraft2, orthogonally polarized SH waves propagating through a solid in a state of plane stress can be considered the acoustic analogue of the photoelastic effect in optical materials. That is, a birefringence effect occurs so that for a homogeneous. isotropic solid in a state of plane stress the birefringence B is related to the difference between the principal stresses u1 and a2
B = A?,(u,- a,) Here d is the acoustoelastic
B=
v2
-
(1) constant,
and
VI
(2)
!h(V,+ V2) where V, and V, are phase velocities
of SH waves polarized along the (T, and u2 directions, respectively. Note that a2 - u, is the magnitude of the principal shear stress. The authors are in the Nondestructive Evaluation Section, Structural integrity Branch, Naval Research Laboratory, Washington, DC 20375, USA Paper rceived 16 May 1983.
0041-524>(/83/0502
ULTRASONICS. SEPTEMBER 1983
1749/$03.00
Equation (1) is derived from the equations of Hughes and Kelly3,who calculated the phase velocities of free waves propagating through homogeneous, isotropic materials under stress. For such materials, the acoustoelastic constant I@ is given by 4~ +
n
8P2
where p is the shear modulus and n is one of the three third-order elastic moduli of an isotropic solid. The birefringence technique was used by Crecraft in an attempt to measure the residual stresses in a nickelsteel bar subjected to bending forces large enough to cause plastic deformation in some portions of the bar. He recognized that the bar was not isotropic but had an initial anisotropy ‘evidently due to preferential grain alignment’*. He measured the initial birefringence (prior to deformation of the bar) and also found that this anisotropy had a principal axis parallel to the bar axis. The initial birefringence was subtracted from that measured when the bar was deformed, in an attempt to compensate for the initial anisotropy. Unfortunately, this did not give satisfactory results for measurement of the residual stress. The issue is further complicated by the fact that some of the birefringence measurements were made in regions where the bar was plastically deformed. Hsu4 used the birefringence technique to measure principal shear stress in a disc in diametral
the
0 1983 Butter-worthEt Co (Publishers)Ltd 217
compression. The disc was cut from a rolled plate of 2024 T-4 aluminium, and the loads were applied parallel to the rolling direction. The birefringence was measured along a diameter orthogonal to the load axis. By subtracting the initial birefringence (measured when the material was in the unloaded state) and attributing the remainder to the principal shear stress, good agreement with theory was obtained. It should be noted that the applied shear stress along this diameter vanishes due to symmetry. More will be said about this point later. Blinka and Sachsej used a different approach in relating birefringence to applied uniaxial stress. They propagated SH waves through a stressed aluminium specimen with a broad-band transducer orientated at 45” to the load direction. Due to the birefringence effect, two SH waves propagate through the specimen. If the specimen is isotropic, one wave will be polarized along the load axis, and the other polarized orthogonal to the load axis. The two waves will be out of phase when they return to the transducer, and the spectrum of the echo received will have minima which can be related to the time-of-flight difference of the waves and hence tothe applied stress. The authors of Ref. 5 also included a correction term in their birefringence against stress equation to account for the initial (unstressed) birefringence due to material anisotropy. It should be noted that in all the works referenced above, the effect of material anisotropy (‘texture’) was either ignored or accounted for by subtracting the initial unstressed birefringence, B,, from the birefringence B in the presence of applied stress and relating the remainder to the principal shear stress B -
B, = if (a2 -
a,)
We have recently measured the stresses around a single-ended crack in Mode I crack-opening displacement using Okada’s theory and obtained excellent agreement between our acoustoelastic measurements and the theoretical stress field in our specimens. It is curious, however, that Okada’s theory works as well as it does. Okada postulates the existence of a matrix analogous to the refraction index matrix in optics. However, using analogies between optics and acoustics requires some care, since there are obvious differences between the two fields; for example, the existence of only two optical phase velocities in anisotropic crystals, as opposed to three acoustic phase velocities. Also. the slownesses in optics are not calculated from the eigenvalues of the refraction index matrix. The eigenvalues of the refraction index tensor are the square roots of the principal values of the dielectric tensor. If a wave normal is along one of these principal axes, then the slownesses for wave propagation in that direction are given by the eigenvalues. For an arbitrary orientation of the wave normal, the slownesses are given by a geometrical construction involving the ‘ellipsoid of wave normals’
3. (3)
This method works for the special cases mentioned in Ref. 4; that is, either uniaxial stress or (locally) biaxial stresses, that are parallel and perpendicular to the rolling direction. Recently, two theories have been developed to account for the effect of material anisotropy on acoustic birefringence. The theory of Iwashimizu and Kubomura treats the case of free waves propagating through the thickness of a plate which is orthotropiP. In their equations, the material is treated as a nonlinear elastic solid with the effects of orthotropy retained in the second-order elastic moduli but ignored in the third-order moduli. They derived an equation relating birefringence and stress similar to (3) in that their equation contains the initial birefringence B, and a single acoustoelastic constant. However, they found that in the most general case, there was a non-linear relationship between birefringence and stress. Okada also derived a relation between birefringence and stress’. He made the assumption that there exists an index of refraction matrix for the solid and that the components of the matrix are linear functions of stress. The coefficients relating the matrix components to stress were assumed to be nine in number when referenced to axes of material symmetry in the plate (parallel and perpendicular to the rolling direction). This is analogous to the behaviour of the second-order elastic moduli for an orthotropic solid. Okada assumed that the inverse velocities (slownesses) of free waves propagating in an orthotropic solid are the eigenvalues of the index of refraction matrix. Okada then derived a
218
non-linear birefringence-stress relation involving B0 and three acoustoelastic constants. His equation reduces to that of Iwashimizu and Kubomura6 if one of the constants vanishes and the other two are equal, as will be discussed more fully later. His relation also reduces to (1) for an isotropic material.
e1
0: + -0: -52
=
constant
(4)
E3
where the E, are the principal values of the dielectric tensor and the Di are components of the electric displacement. The slownesses are proportional to the principal semi-axes of the ellipse generated by the intersection of the ellipsoid of wave normals with a plane whose normal is in the direction of free wave propagation. For more details, the reader is referred to standard optics texts, such as Ref. 9. For an unstressed anisotropic solid, the (squares of) acoustic phase velocities are the eigenvalues of a symmetric second-order tensor CVkinin, where the Cij,, are the second-order elastic moduli and nj are the direction cosines of the wave normallO. For a stressed solid. it has been shown” that the velocities are the squares of the eigenvalues of a second-order ‘acoustical tensor’, which is a function of the applied stress and of the second- and third-order elastic moduli. The theory of Iwashimizu and Kubomura is a special case of this more general theory, since they ignored anisotropy in the third-order moduli. Okada tested his theory and found that it gave good agreement with birefringence measurements for uniaxial tension specimens made of various aluminium alloys’.‘*. He noted in particular that for 5052 aluminium his three-parameter model was able to fit the data much better than the theory of Iwashimizu and Kubomura. We are thus left with somewhat of a paradox, in that the Okada theory, which appears to be the less rigorous, seems to give better agreement with
ULTRASONICS.
SEPTEMBER 1983
where the superscript ‘0’ refers to the isotropic values and the primes denote small perturbations from the isotropic values. For example,
experiment than the Iwashimizu and Kubomura theory, which is based on a seemingly more rigorous theoretical foundation. The purpose of the present paper is two-fold. First, we wish to resolve this (apparent) paradox by extending the theory of Iwashimizu and Kubomura to account for the anisotropy in third-order moduli. We will show that this gives a birefringence equation whose functional form is the same as Okada’s and contains three acoustoelastic constants. Secondly, we present some experimental results for 2024-T351 aluminium tensile specimens and compare the agreement between the two theories and the data.
Theoretical Rigorous
(11) where 8, is the Kronecker delta function, are the iamt constants for the material. Iwashimizu skim
=
study
derivation
of birefringence
equation
We begin the analysis by considering the case of an infinitesimal stress wave propagating through an elastic solid subjected to finite applied stresses tis. The stresses and strains are referenced to the deformed and undeformed states of the solid, respectively. In the undeformed state, the coordinates of a material point are given byXi; after deformation, the material point has coordinates Xi. The material densities in the undeformed and deformed states are denoted by p,, and p respectively. The passage of the stress wave causes an additional infinitesimal displacement Wi superposed on the finite displacement Ui = Xi - Xi. The equation of motion for the stress wave is given by” (tis 6kkr +
Skirs)
where p dxk -----
dxi
r
aw,
azwk
s1
dx,
=
(5)
ax,
a’@ (6)
where @ is the strain energy density for the elastic solid. Summation over repeated indices will be implied, unless otherwise stated. In the notation of Iwashimizu and Kubomura6, strain energy density is given by 1 1 Cijkl Eli Ekl + 3! Cijklmn Eii Eki &n “=F
the
(7)
where the and C$!+,,,, are the second and third order elastic moduli evaluated at zero strain; consequently, cijk,
and Kubomura ck$s
In what follows, terms that are second order in both the displacement gradients au&Xi and aWdaXi will be neglected, so that the strain tensor assumes the linear form
derive
skim
(124
+
2X(akiErs
+
&sEki)
’
2E.L (dkkrEis
+
sir Eks
+
Ckirsmn Emn
+
sks Eir
+
&is Ekr) (12b)
They retained the effects of anisotropy in second order moduli, but ignored it in third order moduli by setting the Ckirsmn = Ckh:smn in the above. We will consider effect of retaining the anisotropic terms in the third order moduli. Furthermore, we specialize to the case of a slightly orthotropic material, such as a rolled plate. Due to rolling a differential volume of the plate has one (local) two-fold symmetry axis, orientated normal to the plane of the plate. Also, if the texture is uniform throughout the plate thickness, then the rolling direction is also a two-fold axis, as is the in-plane axis perpendicular to the rolling direction. (The case of a small texture gradient is considered in Appendix A). Note that these axes are ‘local crystallographic axes’ embedded in material volumes which are considered large relative to a grain size. but small relative to an acoustic wavelength. On this length scale it is possible to consider the effect of the average orientations of all the grains in the volume, rather than of individual grains. Assume that there exists a triad of two-fold axes, so that a differential volume of the plate material behaves as a slightly orthotropic single crystal. Consider also the case of waves propagating through the thickness of the plate (along the xj axis). The other axes of material symmetry (parallel and perpendicular to the rolling direction) will be donated as the x2 and xi axes. The plate is assumed to be in a state of applied plane stress. We assume negligible residual stresses. The displacement
(8)
+
and X and p
Wi =
in the stress wave is of the form
Wi exp[i(fi,
-
of)]
(13)
where K is the wavenumber (w/v) associated with frequency w and phase velocity V. Since the plate is in a plane stress state, t,, = t,, = r23 = 0. For slight anisotropy, the usual stress-strain relations hold to a high degree of accuracy
(9) tij = For a slightly cijkl
=
anisotropic
c,;k,
+
material,
let
Ci;.k,
(104
XEkkdij
+
2PEEij
(14)
so that E,, = Ez3 = 0. Since there is no applied stress in the direction of propagation of the wave, (5) simplifies to
and Cgklmn
=
C;klmn
ULTRASONICS.
+
C;klmn
SEPTEMBER 1983
(lob)
(15)
219
Recall that the Skirs depend upon the elastic moduli and the strains, as shown by (12b). In general, the strains will vary on a length scale which is much larger than an acoustic wavelength (the scale on which W, varies). Consequently, gradients of strains can be neglected in (15). Also, it has been assumed that the elastic moduli are constant over a differential volume having sides much larger than an acoustic wavelength. The moduli vary on the scale of the plate thickness, which is much larger than a wavelength, and will be replaced in (15) by their values averaged through the plate thickness (see Appendix A). Consequently. the values of the moduli used hereafter are the thicknessaveraged values. Using the relation” (13) into (15) gives
p = pO (1 -
[email protected] = A,, + A,, f
[(A,, -AJ2
+ 4A22]1’2 (19)
where V, = V, + AL’. The angle @ between the x, axis and the eigenvector associated with the SH-wave having velocity V1 is given by
(20) The acoustic v2
BE---
birefringence
-
B is deGned by
v,
C-21)
vo
where V,= ‘/z (V, + V2). Since V, = V, f AV/2 and V, = V, - AV/2 and AV<< V, for small birefringence.
Ekk) and substituting [(A,,
-A,,)*
+
%1”2 (22)
a-)ov,*
(Akr -
PO v2 Sk,)w, =
where the acoustical
(16)
0
tensorAkr
is defined
From (17) (12b) and (18)
by An-A22
Akr = C& (I + E/d + S;krs The pop are the eigenvalues of the acoustical tensor, and the eigenvectors are polarization directions for particle motion in free waves propagating in the xj direction.
(G-
=
(17)
G4)
+
+
(Cl55-
Cl,,)
tll
+
(C,,,
G44)
tl3
-
2P
(71
+
-
(C25,
t/2)
-
C244)
(23)
where C,, = p + C;, and C,, = p + C;;
Consider terms such as A,,. If such terms are non-zero, there will be an eigenvector corresponding to a quasilongitudinal wave (rather th In pure longitudinal). Presently we have A:, = C&; expressed in the Voigt notation, C,;,, = C,“,. For an orthotropic material C,O, vanishes identically, as does C;,. Consequently A,, is non-zero only if Sir,, is non-zero. From (12b). the only non-zero contribution to S;,,, is the term C3133mnE,,. since E,, = E,, = 0. For an orthotropic material, the contribution to the strain-energy density function due to terms that are third-order in strains is13
A12
=
%(P
+
t/2
likewise, (24)
C,,,)
For an isostropic material, the following between the third-order modulir4:
relations
hold
c
144
=
c2,,
(W
c
155
=
c244
Wb)
c
344
=
c35,
(25c)
Since the material &,
=
and
cl44
+
is assumed
c;,,,
c244
=
slightly orthotropic,
c,,,
+
let
c;44.
G,, - Cj4, = CG,,
where Cl,, << C,,, etc. For plane +
6C,,,
tll
+
6712 L,
t/z
~3
+
d +.
671
.I
[G,,
d
+
Cm
d
+
CM
stress, the isotropic
tl3
+ 6~3KG,,vt: + .I
q1
=
-
2
‘12
(0,
=
Here the Voigt notation is used for strains as well as third-order moduli; for example, nr = E,,, q4 = E,,, etc. From the above equation, it is seen that C3133mnEmn is non-zero only when C3,333,E3, is non-zero, since all other third-order moduli vanish. However, E,, = 0, so s;,,, = 0. In like fashion, it can be shown that A,, = 0, so there is no coupling between longitudinal and transverse motion for waves propagating in the xj direction, even when the effect of orthotropy in third order moduli is accounted for. Consequently, the velocities and polarizations of pure mode SH waves are calculated from the eigenvalues and eigenvectors of the acoustical tensorA,s ((Y,p = 1, 2). The eigenvalues are given by
J-
+
fJ2)
relations
give
(0,
-
(2W
u*)
cos
2p (a, - 0,) sin 28
(18)
220
stress-strain
a:1
28
(26b)
qs = r (26~) 4p with v amd E being Poisson’s ratio and Young’s modulus, respectively. The principal stresses are u, and u,, and 6 is the angle between the x, axis and the u, direction. Substituting (26a) and (26b) into (23) gives A,, +
M,(~,
A22
= -
(G u2)
-
~0~
Ci4)
+
MAa,
+
28
~2)
(27)
where M, = (-v/E) C;,, and M2 = 1 i- (C,,, - C144)/2p. A term (C;,, - C;,,) t/2 has been neglected in the above.* Substitution of (26~) into (24) gives *We show that neglecting this term allows the birefringence equation to be written in a pleasing form In terms of the sum and difference of principal stresses. More will be said later about neglecting this term.
ULTRASONICS.
SEPTEMBER 1983
24,,
= MJ (a, - a*) sin 28
(28)
where M3 = 1 + C,,,lp. When (27) and (28) are substituted into the birefringence equation (22) there results B = ([B,+ m,(a,+ q) + m,(o,- uJcos 2fl]* (29)
+ [ms(o, - uJ sin 28]*)“*
where use has been made of the relation poV,’ = CL, and the m are defined by mi = A4il2E.Lfor i = 1,2,3. The initial birefringence B, is equal to (Cj, - CiJ/2p. In like fashion, the angle $ is given by (see (19)): m&J, tan 2@ =
uJ sin 28
B, + m,(a,+ UJ + m2(u,-
~~2) cos
28 (30)
u12
(31)
= 2m3
so that the shear stress can be determined birefringence and the angle @.
(32)
is recovered ifB, = m, = 0.m2 = m3 = G = (4~ -I- n)/ 8$ where n is the third order Murnaghan constant14. It is straightforward to show that these relationships hold by obtaining the appropriate forms of the third-order moduli from the expression given by Iwashimizu and Kubomura
and using v, = n/4.6 It was shown before that to obtain (29) and (30) it was necessary to assume that the term (C;,, - C&s)t/2 was negligible as compared to, for example, (C,,, - C344)tlj. In general, nZ and Q will be the same order of magnitude for plane stress, so the assumption is valid if C;,, if CS,, << C,,, - CS4,, or equivalently, Cl44
=
cm
Cl,,
=
Cl44
c,,,
#
c344
Note that there is in general a non-linear relationship between the birefringence B and the initial (unstressed) birefringence B. .Consequently, it is not generally possible to compensate for the effects of initial anisotropy by merely subtracting B, from B. Exceptions are the case of uniaxial stress, or a state of biaxial stress with the principal stress axes parallel to the local crystallographic axes. The experiment described in Hsu’s pape? falls into the latter category. The diametral loads were applied to the disc in directions parallel to the rolling direction and the birefringence measured along the diameter perpendicular to the rolling direction. In this case, the local shear stress vanishes due to symmetry. Assuming m, << m, for this material (2024-T4 aluminium),** the birefringence equation reduces to
This explains measurements
(34)
the excellent agreement and theoretical values.
between
and
It is of some interest to determine why Okada’s approach is successful, given the fact that he begins his analysis by seeking the eigenvalues and eigenvectors of a tensor which is supposed to be the acoustical analogue of the optical refraction index tensor. This tensor is not the same as the acoustical tensor of (16) which is derived from a rigorous analysis of the passage of a sound wave through a stressed solid. We also note that the symmetric (second-order) acoustical tensor Aik is found by contracting the fourth-order tensor sjj,L[ twice with the direction cosines n; for the wave vector K; that is, Ai/q = Sgkpin,. Okada seeks the eigenvalues of the symmetric tensor N,, = N,,T+ ANii, where N;,?is a diagonal matrix whose elements he assumed are’the slownesses of free waves propagating in the unstressed state. The AN, are (small) changes which are linearly related to stress ANii = %jkl ukl. We note that the units ofNji are inverse velocity and the units OfAik are density X velocity squared. This motivates us to formally define N;i by the relationship (in matrix form)
For more details,
SEPTEMBER 1983
his
We have thus far shown that solving for the eigenvalues of the acoustical tensor leads to a nonlinear birefringence equation (29) which involves three acoustoelastic constants m,, m2. and m3, when the effects of anisotropy are accounted for. This equation is formally equivalent to the birefringence equation derived by Okada and experimentally verified by him’.‘*.
WI PII-’ = Plho
It is interesting to note that the first two relationships above hold for a tetragonal material; that is, C,,, = C,,, and C,,, = C,,,. They do not hold in general for an orthotropic material. However, we have for a tetragonal material6 C,,, = C344. which does not satisfy the third relationship. Consequently, a material whose acoustic birefringence obeys the Okada equations has some properties of a tetragonal material and some of an orthotropic material.
ULTRASONICS.
equation
Comparison between the rigorous derivation Okada’s theory
from the
Equation (29) is the same as that derived by Okada, and reduces to the form of Iwashimizu and Kubomura form, = 0,m2 = m3. Furthermore the birefringence equation for the isotropic case B = ii@, -uJ
cases of the birefringence
B -B, = mz(u,- uJ
For the case of plane stress, (a, - a*) sin 28 = 2u,,. From (29) and (30) we have B sin 21$
Special
Performing convention)
(35)
see Appendix
the inversion
+
[A4
B.
gives (no summation
(36)
**We measured m, and m2 for 2024-T351 aluminium and found that mz * 300 ml. It seems reasonable to assume that similar results hold for 2024-T4 aluminium.
221
where
(374
that the rolling direction was at various angles to the specimen axis. The specimens were subjected to uniaxial tension, and the birefringence measured for various stress levels. For the special case of uniaxial
tension,
(29) becomes
(37b)
Mij
0
=
for either i orj = 3.
(37c)
B = B,+(m,+m,)a
(8 = 0")
(414
B = Be-l-(m,-m2)o
(8 = 90")
(41b)
B = [(B,+ m,u)2 f (m,u)2]"2 (8 = 45”)
Here (l/N;) are the phase velocities for free wave propagation in the x3 direction in the unstressed orthotropic solid; for example, (1/NF1)2 = (Z-L-I- C&)/P, etc. Following Okada, we assume
(4lc)
(38)
A straight line lit to the B against u curve for 8 = 0” and 90” will give the values of m, and m2. The value of m, can be calculated from a least-squares fit ofB against u for 0 = 45”.
where the summation convention applies and the Voigt notation is used.* Note that Z = 1, 2. . 6. whereas i = 1, 2, 3.
The birefringence was measured using the pulse-echo overlap technique and the instrumentation used in our experiments is described in Ref. 8. Fitting the data to (41 a)-(41c) gave values of:
mI =
&IJ UJ
With these definitions, becomes
the birefringence
equation
(22)
+411;~22)’ + (44’_
B
=
+
ml
+
v,
K%,
-
%I)
Ull
+
(a22
-
a121
u221J2
where the Voigt above equation Using the plane agrees with the that the Okada relationship.
notation is used only for the CYS.The is the same as that derived by Okada’. stress relation (26) we find that (39) birefringence equation (29) provided stress-acoustic constants satisfy the
-v,(%
-
m, =
a221
-vc;,
=
2
5
2@
(404
1 1 1
1+ [
(c
a12
-
;
(a,,
+
a221
155- c,‘A 2iJ
=
i%
(4Ob) I (4Oc)
m3 =
It is easy to show that (30) for the polarization directions is also recovered when the above relations are satisfied. Thus, we have shown that if Okada’s Nij tensor is related to the acoustical tensor Aij using (35), the Okada approach gives the same results as the rigorous theory based on non-linear elasticity, provided that the Okada stress-acoustic constants are related to the second- and third-order elastic moduli as above.
Experimental
study
To compare the theories of Refs 6 and 7, we performed a series of experiments using uniaxial tension specimens made of 2024-T351 aluminium. The specimens were cut from 6.3 mm thick rolled plate so *We note that Okada’ defines N, 2 = ageal *~‘2.
222
(42a)
rn2 = 3.64 X 10e5MPa-’
(42b)
m3 = 3.32 X 10-j MPa-’
(42~)
(39)
m%6~,2)2~“2
m2 = VCI [
ml = 0.01 X 10d5MPa-’
The quality of the fit is evident in Fig. 1 for 0 = 0”. 45”. and 90”. Note that we have also shown data from a specimen for which 8 = 60”. Reasonably good agreement is also obtained for this specimen for the above values of the acoustoelastic constants. The equivalent forms of the birefringence equations of Iwashimizu and Kubomura are obtained by setting m, = 0,m2 = m3 = m in equations (41a) - 41(c). The value of m obtained by a straight line lit of the data for 8 = 0” is m = 3.68 X 10-S MPa-I. This of course is very nearly equal to the value of rn2 in the Okada model. since fitting the data by the Okada theory gives m, z O,m, s m,; see(42a)-(42c). The quality of the fit between the birefringence data and the predictions of the Iwashimizu and Kubomura theory is shown in Fig. 2. The agreement is excellent for fl = 0” and 90”. This is to be expected. since both theorie?’ predict the same slope dB/du for 0 = 0” and 90” when m, = 0 and m2 = m. Since the Iwashimizu and Kubomura theory has only one acoustoelastic constant, it does not give as good a fit to the 0 = 45” specimen as the Okada theory: however, the agreement is still quite good. We also measured the angle @ for our uniaxial tension specimens, as a function of stress a; our experimental technique is described in Ref. 8. For the stress levels used in our experiments, we found that @ = 0” for 8 = 0” and 0 = 90”; that is, the slow acoustic axis remained normal to the rolling direction. This is in accordance with the Okada angle equation (30). The theory of Iwashimizu and Kubomura also predicts the same result. For specimens with 0 = 45” and B = 60”, we obtained the results shown in Figs 3 and 4. We estimate our experimental error in measuring C#Ito be + 2”, so the agreement between the theoretical predictions and the data is reasonably good.
ULTRASONICS.
SEPTEMBER
1983
developed by Okada. He assumed that the inverse phase velocities (slownesses) can be calculated as the eigenvalues of an index of refraction matrix. whose elements are assumed linearly related to stress.
7-
The phase velocities in the rigorous theory are proportional to the squares of the eigenvalues of the (second-order) acoustical tensor. By formally relating the acoustical tensor [A] to the Okada index of refraction [N] using [[Nj [N]]-* = [Al/p,, we found that the rigorous theory and the Okada theory give the same results, provided that the relationships (40) between the Okada stress-acoustic constants aIJ and the acoustoelastic constants rn;(of the rigorous theory) are satisfied.
6-
“0
40
20
60 Stress
Fig. 1
80
100
120
CMPol
The birefringence fit for 0 = 0’. 45’. 60’ and 90’
We compared the agreement between experimentally obtained values of birefringence B and polarization angle r$, and the predictions of the theory of Okada, for uniaxial tension specimens made from 2024-T351 aluminium plate. The agreement was excellent. We also compared the same data to the predictions of the theory of Iwashimizu and Kubomura. The agreement was also quite good. In fact, we found that for our specimens, m, e 0 and m2 r m,, so that the Okada theory reduces (approximately) to the theory of Iwashimizu and Kubomura for this case.
40-
30B 3 8 20-
IO
1
OO
I 20
I
40
,
I
60 Stress
80
Stress CMPal
x\/
I
Fig. 3
Angle @against stress for 6’ = 45” and 60”
Fig. 4
Angle @ against stress for 8 = 45” and 60”
CMPol
Fig. 2 The fit between the birefringence data and the predictions of the lwashimizu and Kubomura theory
Conclusions We have developed a rigorous theory which gives the phase velocities, birefringence. and polarization directions for free SH waves propagating through the thickness of a slightly orthotropic plate in a state of plane stress. The theory is an extension of that developed by Iwashimizu and Kubomura. In our derivation. we retained anisotropy in second- and third-order moduli. whereas they ignored anisotropy in third-order moduli. Our rigorous theory gave rise to a birefringence equation involving three acoustoelastic constants m,. m2. and m3, whereas the theory of Iwashimizu and Kubomura involves only one. The rigorous theory reduces to that of Iwashimizu and Kubomuraif m,= O,m, = m,. Our theory formally
ULTRASONICS.
gives the same results as that
SEPTEMBER 1983
Stress CMPol
223
where the ei are principal values of Ed. In this same coordinate system, the relations between D and E become D; = E;Ei (no sum), and the relation between electric energy density and the electric displacement is expressed as
Acknowledgement The research reported here was conducted in the Nondestructive Evaluation Section of the Naval Research Laboratory under the sponsorship of the Office of Naval Research.
Appendix
(B4)
A
Here we show that the elastic moduli in (15) can be replaced by their thickness-averaged values. Assume small variations in the moduli, SO that Cljk/ =
t-41) Comparing the magnitudes of the two terms on the right-hand side of the above. K
aC3kr3
C3L,.3(x3 = d) -
Cjkr, (x3 = 0)
ax,
>> 1
(Ml
Energy density
Here we use some analogies between optics and acoustics to explain how (35) is derived. The optical derivation follows directly from Ref. 9. In optics, the electric energy density G, in a light wave is given by the product (E.D)I&r where ,!? is the electric field vector, and D is the electric displacement. Since the electric displacement is related to the electric field through
G, =
% E;j U’ I/
036)
(Bl)
with E;, the dielectric tensor, the electric energy density becomes Ei Eii E, A = G, = constant. (B2) 8a Because of the symmetry of the dielectric tensor, (B2) can be reduced to a form in which only squares of the field components enter the expression, and not the products. Since the electric energy density is positive definite. the above expression can be thought of as an ellipsoid in a space with coordinate axes E,, EJ2 and E,. The ellipsoid has principal axes E,. E,. and E, and in this coordinate system. (B3)
*Actually, x3 should be measured from the centroid 2, of Ciifi rather than the plate centre. However, this merely entails replacing x3 by x3 - I, in what follows; the results of the analvsis are still the same.
strain by the
% Cijkl W, i Wk, ,
(B7)
with Wi the particle displacement in the stress wave. For a plane wave with direction cosines n, and wavenumber K, we have
G =
224
in stress waves
or, since stress is related to (linearized) second-order elastic moduli.
in light waves
E, E; + .q E; + E&E: = 87rG,
(B3
for a light wave in an optical crystal; here x denotes proportionality, and (Ni.*>-l is the ij,, element of the inverse of the matrix dij.
G, =
Di = Eii E,
ot 81~Ge
The acoustical analogue is the passage of an infinitesimal stress wave through an (unstressed) anisotropic solid. The strain energy density is given by
B
Energy density
of wave normals’9.
Consider the case of a light wave propagating in the 5, direction. which is parallel to a principal axis of the dielectric tensor. For a non-magnetic crystal, the two possible phase velocities in the xi direction are given by K = C/(Ei)“’ where C is the phase velocity of light in vacua, and i #_j. ** Since the optical index of refraction is defined by N = C/Vi, we see that it is possible to generalize to an index of refraction tensor nji = &,il/z having principal values N * = e;. Following Okada, we define Nil = i”,llC, SO that Nil has units of inverse velocity. Consequently. the eneigy density can be written in the form DlDj (N,‘)-’
since the moduli vary on length scaled, and the quantity Kd >> 1 for the frequencies used in ultrasonics.
Appendix
which is called the ‘ellipsoid
T
Aik
038)
wwk
where the symmetric second-order tensor A& = Cu,knjni. The above expression also can be written terms of particle velocities Ui as Aik Vi vk Gs = 2 v,2
in
(B9)
where slight anisotropy is assumed so that K2 s 02/Vzo Rewriting the above in terms of the principal values p,,vF of the tensor Aik gives G, =
L/zp.
@lo)
or defining
an acoustical
index of refraction
Ni = VdVi,
**The units will be correct if ci is replaced by rci where the permeability r = 1.
ULTRASONICS.
SEPTEMBER 1983
3
(Bll)
G, = ‘/2p. which is analogous
Analogy
to (B4).
between
optical
and acoustical
cases
If we make an analogy between electrical displacement D; and particle velocity Ui, there is an analogy between Aik and Nij; (compare (B5) and (B9)). Since& is the contraction of a fourth-order tensor, we form the fourth-order tensor Nij Nk,, and then contract to obtain a second-order tensor Nij Njk. TO obtain the correct units, we let
which is just (35), remembering are pO Vj’.
that the eigenvalues
of
d4jk
References I
2
Benson, R.W., Raelson, V.J., Acoustoelasticity.
Prod. Eng.
30 (1959) 56-59 Crecraft, D.I., The Measurement of Applied and Residual Stresses in Metals Using Ultrasonic Waves.1 Sound Vibr., 5, (1) (1967) 173-192
ULTRASONICS.
SEPTEMBER 1983
Hughes, D.S., Kelly, J.L., Second-order Elastic Deformation of Solids, Phys. Rev., 92, (5) (1953) 1145-l 149 4 Hsu, N.N., Acoustical Birefringence and Use of Ultrasonic Waves for Experimental Stress Analysis, Exp. Mach.. 14, (5) (1975) 147-152 5 Blinka, J., Sachse, W., Application of Ultrasonic-pulsespectroscopy Measurements to Experimental Stress Analysis, Exp. Mech.. 16, (12) (1976) 448-453 6 Iwashimizu, Y., Kubomura, K., Stress-induced Rotation of Polarization Directions of Elastic Waves in Slightly Anisotropic Materials. Znr.J. Solids Srructures, 9 1(1973) 99114 7 Okada, K., Stress-Acoustic Relations for Stress Measurement by Ultrasonic Techniques, J. Acoust. Sot. Jpn (E). 1, (3) (1980) 193-200 8 Clark, A,.V., Mignogna, R.B., Sanford, R.J., Acoustoelastic Measurement of Stress and Stress Intensity Factors Around Crack Tips, Ultrasonics 21 (1983) 57-64 9 Born. M.. Wolf. E.. Princiules of Ontics. 6th ed.. Pergamon L Press, New Yo;k, XIV. (IdsO) 10 Auld, B., Acoustic Fields and Waves in Solids, Wiley, New York, 6A_ (7D) (1973) 11 Tburston, R.N., Wave Propagation in Fluids and Normal Solids, Physical Acoustics. Vol 1. (A), Academic Press, New York (1964) 12 Okada, K., Acoustoelastic Determination of Stress in Slightly Orthotropic Materials, Exp. Mech., 21, (1981) 461-466 13 Mumaghan, F.D., Finite Deformation of an Elastic Solid, Wiley, New York, 4. (5) (1951) 14 Hearmon, R.F.S., ‘Third-Order’ Elastic Coefficients, Am. Ctymzl.. 6, (1953). 331-340
225