Acoustoelasticity in a weakly anisotropic monoclinic elastic material

WAVE MOTION NORTH-HOLLAND

65

8 (1986) 65-75

ACOUSTOELASTICITY MATERIAL Tatsuo TOKUOKA*

IN A WEAKLY ANISOTROPIC

20 March

ELASTIC

and Kuniaki MORIKAWA

Department of Aeronautical Engineering, Kyoto Unioersiiy, Received

MONOCLINIC

1985, Revised

Kyoto 606, Japan

27 June 1985

The acoustoelasticity in a stressed monoclinic elastic material is analyzed theoretically. It is assumed that the material weak anisotropy, such that the second-order elastic constants differ slightly from those of an isotropic material and third-order elastic constants retain general monoclinic anisotropy. The propagation velocities, the polarization directions the acoustoelastic effects for principal longitudinal and transverse waves are obtained and presented as functions of elastic constants, principal stresses and directions of principal axes of stress. The coefficients appearing in the formulas tabulated for Laue groups.

has the and the are

1. Introduction

Elastic waves in stressed elastic materials have been studied by many investigators. Hughes and Kelly [I] determined the elastic constants of an isotropic material by the method of ultrasonics in a uniaxial stress state. Toupin and Bernstein [2] and Thurston [3] studied theoretically the elastic waves propagating in stressed anisotropic elastic materials, and they obtained the relations between the propagation velocity and the second- and the third-order elastic constants of the materials. Acoustoelasticify was first proposed by Benson and Raelson [4] as a new experimental nondestructive stress analysis, and they studied experimentally the polarizations and the acoustical birefringence of two transverse waves propagating perpendicular to uniaxial stress direction. Tokuoka and Iwashimizu [5] analyzed the acoustoelastic effects in an isotropic material, and they established the stress-acoustical law, that is, the polarization directions are parallel to the secondary principal axes of stress and the acoustical birefringence is proportional to the difference of those principal stresses. But they did not take the third-order elastic constants into consideration. The acoustoelastic effects in stressed crystals were investigated by Tokuoka and Saito [6] by the perturbation method. They assumed that the inherent material anisotropy is dominant and that the induced stress anisotropy by deformation is supposed to be a small perturbation. On the other hand, usual metal plates and blocks usually have weak anisotropy induced by working processes, and we can assume that the material and the stress anisotropy of the materials are of the same order. From this point of view, Iwashimizu and Kubomura [7] treated wave propagation through the thickness of a plate which is slightly orthotropic, and they showed that the polarization directions of shear waves rotate largely as the stress increases. However, their theory is based on a nonlinear elastic material with the effects of orthotropy retained in the second-order elastic constants but ignored in the third-order ones. In order to generalize this theory Okada [8] applied the refraction index matrix in optics to the acoustoelasticity in a stressed orthotropic material, and he was able to obtain formally exact formulas of *

Deceased,

28 July 1985,

0165-2125/86/%3.50

@ 1986, Elsevier

Science

Publishers

B.V. (North-Holland)

66

T. Tokuoka, K. Morikawa

acoustical

birefringence

Iwashimizu Okada’s

and formula

orthotropic

agrees

materials

acoustoelastic

and polarization

Kubomura

directions.

by retaining formally

/ Acoustoelasticity

anisotropy

with their

propagating

at oblique

In this paper we analyze

results.

velocities,

results are completely

cubic, hexagonal

and isotropic

for a monoclinic

elastic

King

and

formula

is a special

[lo]

the theory

and they showed studied

polarization

directions

case of their results.

and Okada’s

in a stressed monoclinic

general,

[9] generalized moduli,

Fortunko

an article of acoustoelasticity

waves propagating

and derive the propagation The obtained

and Mignogna

in the third-order

shear

elastic material

and acoustoelastic

and we also show that formulas

elastic material

can be considered

formula

of that

waves

angle with respect to the crystal axes, and they obtained

effects for those waves. Okada’s

Sachse and Fukuoka [ 1 l] published other theories are reviewed in it.

Clark

in a monoclinic material

Recently,

in the

Pao,

along with many

with weak anisotropy

effects for principal for orthorhombic,

waves.

tetragonal,

as special cases of the obtained

formulas

material.

2. Infinitesimal waves in stressed anisotropic elastic material An elastic

material

is deformed

from a reference

state

K

to a state x by a static

deformation. The displacement vector is denoted by u(X, t), where X denotes point in K and t denotes the time. Then, the point vector in x is given by

and homogeneous

the point vector of a body

x=x+u. The Cauchy

(2.1)

stress tensor

T,=----,

8W

p

PO a&

can be given by

aXi

8X,

=k

(2.2)

8x1

where p. and p are mass densities and

au, Venant

K

and x, respectively,

W is the strain-energy

function

(2.3)

-

strain

tensor.

Here, we assumed

the strain-energy

function

is a function

We consider that an infinitesimal wave with the displacement vector w(x, t) propagates state x. Then, according to [ 51, from the equation of motion, we can obtain a fundamental governs

of the material,

au, au,

-+-+-

is the Green-St. strain tensor.

in

of the

on the deformed equation which

the wave propagation a2wk -=poz,

PO Sijkl +-

Tj8ik

ax,ax,

P

a*Wi

(2.4)

at

where Silk,=

We consider

a2 w

aE,,aE,,

axi axj -ax,,, ax,, axp

a sinusoidal

ax,

axk

ax,’

wave defined

wi = ai sin( knAx, - wt)

(2.51

by (2.6)

T. Tokuoka, K. Morikawa

/ Acoustoelasticity

67

in a monoclinic material

with amplitude a, wave number k, frequency o and propagation direction n. Then, substituting (2.6) into (2.4) we can obtain (&-PoV’&k)ak

(2.7)

=O,

where the symmetric tensor A.rk G&‘.. n.n +B T.n.na. tk ykl , 1 J,

J

(2.8)

,

P

is called the acoustical tensor, V=i

(2.9

denotes the propagation uelocity. Therefore, we have an eigenvalue problem. The eigenvalue gives pOVz and the eigenvector specifies the polarization direction a.

3. Weakly anisotropic elastic materials In studying elastic properties of crystals, instead of the thirty-two point groups, we can concern only with the eleven Luue groups: N: triclinic,

M: monoclinic,

0: orthorhombic,

C II 8z C I: cubic

T II & T I: tetragonal, R II & R I: rhombohedral

H II & H I: hexagonal.

(trigonal),

Also we indicate the isotropic group by the symbol I. We assume that the strain-energy function W can be expanded by strain components w = %j&ij&

=

Cklij

=

Cijklmn

Cjikb

=

Cklijmn

and we express (3.2)

+%jk,mn&j&&,n,

where CVk,and C#l,,,n are the second- and the third-order the following symmetric properties: cijkl

(3.1)

=

Cmnkl(j

=

elastic constants, respectively,

Cjiklmw

and they have

(3.3)

The independent

number of these elastic constants depend on the Laue groups. See, e.g., Brugger [12] and Thurston [13]. We assume that anisotropy of the concerned material is weak and then its second-order elastic constants can be expressed by c,k,

=

c;k,

+

(3.4)

C:j.k,,

where Cikl E A6,6,, + p

(8&jl+

$$jk)

(3.5)

is the second-order elastic constant of isotropic material, A and p are LamC’s constants, and by the assumption we can estimate that O( c:j.kJ = /.&E.

(3.6)

68

T. Tokuoka, K. Morikawa / Acoustoelasticity in a monoclinic material

Here &= O( Eij)

(3.7)

is the order of strain However, that

magnitude.

we adopt general

anisotropy

with respect

to the third-order

elastic constants.

We also assume

(3.8)

O( Cijklmn) = A which holds for usual Substituting and higher,

metals,

e.g., steel and aluminium.

(3.2) into (2.5), referring we can obtain

to (3.5), (3.6) and (3.8), and discarding

the terms of order

WE

that

sij,,= c&t + c ijk,+ ( qkh + Cgdrn”1&n,

(3.9)

where

Cyjklmn = A[(~irn~jn

+

+ CL[(

6im6kn

6insjm)6kt

+

+

(6km6tn + skn6/m)sijl

1sjt + ( Gimstn

6inskm

Sinslm)8,/c

+

+ (sjmskn + 6jn6km)6iI + (sjmsh+ 6jn6h)6ikl~ In this approximation

the acoustical

tensor

(3.10)

(2.8) is given by

Aik = A:, + A; + A$,

(3.11)

A$~(A+p)nink+p&,

(3.12)

Afk s C&njnl,

(3.13)

Af, = ~,njn,Sik + ( C$lrnn njn, + Cgklmnnjnl)E,,.

(3.14)

where

Here, Avk denotes the acoustical tensor of isotropic material in natural state, and A;k and Afk denote, respectively, the deviations of acoustical tensors due to material and stress anisotropy. The orders of Ayk, Alk and A$ are equal to CL,HUEand PF, respectively.

4. Principal waves We take the two-fold rotation axis of a monoclinic elastic material as the x,-axis and we concern the deformation state x of no shear strains in the planes x1x3 and x2x3, that is, we assume E,3 = E,, = 0.

It is convenient to adopt replaced in pair 11-1,

22-2,

The components E,=

of strain E,,,

only

(4.1) an abbreviated

33-3,

notation,

23=32-4,

are, then, expressed

E2= Ez2,

E3= EJ3,

the Voigt notation,

31=13-5,

12=21-6.

in which the tensor

subscripts

are

(4.2)

as

Ed= E5=0,

E,=

E,>.

(4.3)

T. Tokuoka, K Morikawa / Acoustoelasticity Thus,

69

in a monoclinic material

for example Cijk,mn&n = C,,,J& = Cm,,& + C,,,&+

(4.4)

Cup&~ + 2C,,,&,

where Caav is the Voigt notation of C+,,,,,,. From the approximation adopted in the above section and the estimation (3.6), the strain tensor appearing in (3.14) can be expressed by the stress tensor which is given by the isotropic linear stress-strain relation. Thus, we have

(4.5)

where ur (r = 1,2,3) are the principal stresses and 8 is the angle between the x,-axis and the first principal axis of stress. Now we consider the waves propagating along the x,-axis. Such a wave is called the principal waves. In this case we have (4.6)

-& = Si3k3+ a,Sik, Si3k3

=

(4.7)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

where from (3.10) we can obtain ~~,,=4h+a/_&,

CL, = Cj,$, = C&, = C&, = 2p, (4.8)

c:,,= cg6 = p,

others = 0.

According to the tables compiled in [ 121 and [ 131 for the second- and the third-order constants for Laue group M, we can easily ascertain that A,, = A,, = 0.

Therefore,

(4.9)

we have the characteristic

&-PcJ’~ A12 0

An 42 - PO V2 0

equation: 0 0

= 0.

(4.10)

A,, -PO V2

For the materials which belong to Laue groups 0, T II, T I, C II, C I, H II and H I, where two, four or six-fold rotation axes coincide with the x,-axis, the relations (4.9) hold. However, for the material which belong to Laue groups N, R II and RI, these relations do not hold. Then, a principal wave propagating along the x,-axis in the latter materials is neither pure longitudinal nor pure transverse. In fact, the strain state (4.1) in these materials produce, in general, non-vanishing stress components T,, and T23.

Table 1 shows the second-order constant C& (i, k = 1,2,3) and the third-order (i, k = 1,2,3; y = 1,2,3,6) for Laue groups M, 0, T II, T I, C II, C I, H II, H I and I.

constants

CiJk3y

70

T. Tokuoka, K. Morikawa

in a monoclinic material

/ Acoustoelasticity

Table 1 Second- and third-order constants for Laue groups M

0

T II

TI

CII

CI

H II

HI

I

33 44 45 55

33 44 0 55

33 44 0 44

33 44 0 44

11 44 0 44

11 44 0 44

33 44 0 44

33 44 0 44

0 0 0 0

133 233 333 336

133 233 333 0

133 133 333 0

133 133 333 0

112 113 111 0

112 112 111 0

133 133 333 0

133 133 333 0

112 112 111 0

C MY

144 244 344 446

144 244 344 0

144 155 344 446

144 155 344 0

144 166 155 0

144 155 155 0

144 155 344 145

144 155 344 0

b c c 0

c 5%

145 245 345 456

0 0 0 456

145 -145 0 456

0 0 0 456

0 0 0 456

0 0 0 456

145 -145 0 a

0 0 0 a

0 0 0 d

C 557

155 255 355 556

155 255 355 0

155 144 344 -446

155 144 344 0

155 144 166 0

155 144 155 0

155 144 344 -145

155 144 344 0

f, c 0

Chp

c 337

a=(155-144)/2,

c=(lll-112)/4,

b=(112-123)/2,

5. Acoustoleastic

d=(111L3.112+2.123)/8.

effects

5.1. Pure longitudinal

wave

The characteristic

equation

a,=a,=O

,

(4.10) gives a pure longitudinal

wave

4 # 0,

(5.1)

PO V’, = A,,.

(5.2)

Referring to (3.6), (4.6) and (4.7), retaining order higher than that, we have

the terms

of order

(p~/p~)“*

and discarding

the terms

of

(5.3) where PL”

d-h+2/.l PO

’

v;=v”,

1+ [

G ah

+2/-4)

1

(5.4)

T. Tokuoka, K Morihwa

/ Acoustoelasticity

71

in a monoclinic material

are, respectively, the propagation velocities of pure longitudinal waves in natural state of an isotropic material and a monoclinic material. The acoustoelastic effect for longitudinal wave can be expressed by the following quantities K-v”, -=AO+A,(a,+az)+(A,cos28+A3sin28)(a,-a2)+A4a3, v”,

(5.5)

v,- vl. -=AA,(cr,+a,)+(A,cos28+A,sin28)(a,-a,)+A,a,, vt

(5.6)

where, substituting (4.5) and (4.8) into (5.3), we have

Ao=

’

Ci3,

%A+2cL)

1

A,=

8~0 +2~)(3~ A,=

8p(A;2p)

+2~)

(Cm-

Cd

1

Ad=

[(A +2~)(G33+

~P(A +2~)(3~ +2~)

AJ =

4~0

Cm) -2A(4A +8~ + G~)l,

’

+2~)

Cm,

(5.7)

[2~(3h+2~~)-A(C133+C233)+2(A+~~)(4h+8~~+C333)1.

5.2. Pure transverse wave The characteristic

6 #O,

equation (4.1) gives two pure transverse waves a3=0 ,

a2#0,

(5.8)

P~(V,,)~=~[A~~+A~~*J(AII -Azz)‘+4A:,l

(r = L2),

tan 24 = A 2A11 11

(5.9) (5.10)

22

angle between the x,-axis and the polarization directions. Retaining the terms of order and discarding the terms of order higher than that, we have (P~IPOY2

where

4

is the

vTr= v”T l+~[c:,+CL+2~~+(C~,,+C~,+C,,,+C,,)E, { ~J[c;,-c~+(~,“,,-~~,,+c,,,-

c,,)~,12+4[c,,+(~~~,,+c,,,)~,12

(I-=1,2) (5.11)

where (5.12) is the propagation

velocity of pure transverse wave in natural state of an isotropic material.

72

T. Tokuoh,

Then,

the acoustoelastic

K. Morikawa

effect for transverse

/ Acousioelasticity

in a monoclinic material

waves can be expressed

by the following

quantities

I VT, - VT21= I VT, - b*l v”,

G-

={[B,+B,(cr,+a,)+(B,cos28+B,sin2B)(cr,-a,)+B,a,l2 +[C,+C,(a,+a2)+(C2~~~2~+Cjsin2~)(cr,-a2)+C,a,]2}“2, ;( v,, + VT21- v”,

(5.13)

=D,+D,(~,+a2)+(D2~0~20+Djsin2e)(a,-a2)+D,a,,

(5.14)

=D,(a,+a2)+(D2cos2~+D,sin20)(a,-a,)+D,cr,,

(5.15)

v”, 8 v,, + VT21- G v: where v;_

v”, l+C:5+Ck, 4P

is the mean value of the propagation material, and where we have

B,

5

(5.16)

> velocities

of two transverse

waves in natural

state of a monoclinic

SLL2(3~+2~)[(~+~P)(Clli+C2iS-C1~~-C211)-2h(C?~I-C341)lr

B, = &

(4/.~ + C,,, - C,,, - Gw + c2+1),

c, = 4p2c3;

+2p.) [(A +2~)(C145+

c3+‘+c”“).

(5.17)

C245)-2AC3451,

(5.18)

T. Tokuoka, K. Morikawa

The polarization

direction

/ Acoustoelasticity

(5.10) is, then, expressed

in a monoclinic

73

material

by

(5.20) Table 2 shows the coefficients A,,, Al,. . . , D4 for Laue groups I. From the table we have the following special cases. (i) For 0 groups we have

M, 0, T II, T I, C II, C I, H II, H I and

IVT,- &*I ={[B,+B,(~,+~,)+B,(~,-~,)COS~~+B,~,]*+[C,(~,-U,)~~~~~]~)‘~*.

(5.21)

V;.

tan 24 =

C,( U, - a,) sin 28

(5.22)

Bo+B,(~,+~,)+~,(~,-~,)c0s2e+B,u,’

Table 2 Coefficients

of acoustoelastic

effects for Laue groups

M

0

43 A, A2 4 A4

A0 A, A2 0 A4

A0 A, 0 0 A4

A0 A, 0 0 A4

Bo 4 B2 4 B4

BO 4 4 0 84

0 0

0 0

CO G c2 G G

0 0 0 G 0

DO Q D, D3 04

DO D, D2 0 04

TII

TI

c II

A0 A, A2

0 A4 0 B,

CI

H II

HI

I 0

A0 A, 0 0 4

A0 A, 0 0 4

A0 A, 0 0 4

0 0

0 0

0 0

B2

82

B2

B2

0 0

0 -2B,

0 0

0 0 0

0 0 0

0 0 0

G 0

G 0

B2

B2

0

G 0

0

0

DO Q 0 0

4 4 0 0

4 Q 8112 0

DO D, 0 0

D0 D, 0 0

D, 0 0

04

04

04

04

04

D4

B2 4

0 0 0 c2 G

4

0 0 0 -B,

B2

0 0 0 0 0

00

A, 0 0 A.4 0 0 B* 0 0 0 0 0 J-4 0 0 D, 0 0 04

14

T. Tokuoka, K. Morikawa

These formulas

B,=m,,

are identical

B2=mZ, C,=m,

/ Acoustoelasticity

with ones obtained and a,=&

(ii) For T II, T I, C I, H II, H I and I groups

by Okada

in a monoclinic material

[8] and Clark and Mignogna

[9] when we put

we have (5.23)

(5.24) (iii)

For T I and C I groups 1VT,;

vTd = JB;

we have

cos* 28 + C: sin2 28 ]ml - u2],

(5.25)

+ (5.26)

tan2+=9tan28. 2

(iv) For H II groups

we have (5.27)

(5.28)

4=w, where p

E

i

tan-’

2.

(5.29)

2

(v) For H I and I groups

we have

‘vT1;“-’ =IB21lul - ~~1,

(5.30)

+=e,

(5.31)

e+$lT.

(1953). [l] D.S. Hughes and J.L. Kelly, “Second-order elastic deformation of solids”, Phys. Rev. 9.2, 11451149 [2] R.A. Toupin and B. Bernstein, “Sound waves in deformed perfectly elastic materials. Acoustic effect”, J. Acoust. Sot. Amer. 33, 216-225 (1961). [3] R.N. Thurston, “Effective elastic coefficients for wave propagation in crystal under stress”, J. Acoust. Sot. Amer. 37, 348-356 (1965). [4] R.W. Benson and V.J. Raelson, “Acoustoelasticity”, Prod. Engrg. 30, 56-59 (1959). “Acoustical birefringence of ultrasonic waves in deformed isotropic elastic materials”, Internat. [5] T. Tokuoka and Y. Iwashimizu, J. Solids Structures 4, 383-389 (1968). [6] T. Tokuoka and M. Saito, “Elastic wave propagations and acoustical birefringence in stressed crystals”, J. Acoust. Sot. Amer. 45, 1241-1246 (1969). “Stress-induced rotation of polarization directions of elastic waves in slightly anisotropic [7] Y. Iwashimizu and K. Kubomura, materials”, Internat. J. Solids Srrucfures 9, 99-l 14 (1973). J. Acoust. Sot. Japan (E) I, 193-200 (1980). [8] K. Okada, “Stress-acoustic relations for stress measurement by ultrasonic technique”, Ultrasonics 21, 217-225 (1983). “A comparison of two theories of acoustoelasticity”, [9] A.V. Clark and R.B. Mignogna,

T. Tokuoka, K. Morikawa / Acoustoelasticity

in a monoclinic material

15

[lo] R.B. King and C.M. Fortunko, “Determination of in-plane residual stress states in plates using horizontally polarized shear waves”, J. AppL Phys. 54, 3027-3035 (1983). [11] Y.-H. Pao, W. Sachse and H. Fukuoka, Acoustoelasticity and Ultrasonic Measurements of Residual Stresses, Physical Acoustics XVII, Academic Press, New York (1984) Chapter 2. [12] K. Brugger, “Pure modes for elastic waves in crystals”, J. AppL Phys. 36, 759-768 (1965). [13] R.N. Thurston, Waues in Solids, Encyclopedia of Physics Via/4, Springer, Berlin (1974) Chapter E.