REFERENCE

NBSIR 84-3002

RESIDUAL-STRESS MEASUREMENTS

USING SHEAR-HORIZONTAL WAVES

FROM ELECTROMAGNETIC ACOUSTIC

TRANSDUCERS

NBS

Publications

ill ffliS iiiIIiii AlllOb 43713^

B^

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303

March 1984

•oc 101

. 155

34-3112

1934

RATIO. hi. 5:\ J OF STANDARDS

UEKARY

NBSIR 84-3002

RESIDUAL-STRESS MEASUREMENTS

USING SHEAR-HORIZONTAL WAVES

FROM ELECTROMAGNETIC ACOUSTIC

TRANSDUCERS

R. B. King

C. M. Fortunko

Fracture and Deformation Division

National Bureau of Standards

Boulder, Colorado 80303

March 1984

Funded by: U.S. Navy, David Taylor Naval Ship Research and

Development Center, Annapolis, Maryland 21402

Office of Nondestructive Evaluation, National Bureau of Standards,

Washington, D.C. 20234

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Residual -Stress Measurements Using

Shear-Horizontal Waves from Electromagnetic

Acoustic Transducers

R. B. King* and C. M. Fortunko**

Fracture and Deformation Division

National Bureau of Standards

Boulder, Colorado 80303

This collection of technical papers covers a two-year effort on a novel

approach to measuring residual stresses using changes in ultrasonic wave

velocities of horizontally polarized shear waves produced by electromagnetic

acoustic transducers. The initial three papers deal with the theoretical

developments pointing to the measurement of in-plane residual stress as well

as preliminary experimental verification. The later two papers generalize

this theory and extend it to include surface residual stress measurement in an

arbitrary plane; again, empirical validation is included.

Major support for this work came from the U.S. Navy, David Taylor Ship

Research and Development Laboratory. Support also came from the Office of

Nondestructive Evaluation of the National Bureau of Standards.

Key words: acoustic; electromagnetic; nondestructive; residual stress;

shear horizontal; velocity,

* Present Address: IBM, 5600 Cottle Rd., San Jose, California, 95193

**Present Address: Aerojet Ordnance, 2521 Michelle Dr., Tustin, California,

92680
TABLE OF CONTENTS

PAGE

In-Plane Residual -Stress Measurements (1982) . 1

1. Evaluation of Residual States of Stress and Material Texture

Using Ultrasonic Velocity Measurements with Electromagnetic

Acoustic Transducers, 1982 IEEE Ultrasonics Symposium, ed. by

B. R. McAvoy (IEEE, New York, 1982) pp. 885-888 3

2. Evaluation of Residual Stress States Using Horizontally

Polarized Shear Waves, Review of Progress in Quantitative

Nondestructive Evaluation, Vol . 2B, ed. by D. 0. Thompson

and D. E. Chimenti (Plenum Press, New York, 1983) pp. 1327

1338. ' 7

3. Determination of In-Plane Residual Stress States in Plates

Using Horizontally Polarized Shear-Waves, Journal of Applied

Physics, 54(6), pp. 3027-3035, 1983 19

Arbitrary Plane Residual -Stress Measurements (1983) 29

4. Acoustoelastic Evaluation of Arbitrary Plane Residual Stress

States in Nonhomogeneous , Anisotropic Plates, Ultrasonics,

21(6) pp. 256-258, 1983 31

5. Evaluation of Surface Residual Stresses Using Horizontally

Polarized Shear Waves at Grazing Incidence..^ " 35

IV
IN-PLANE RESIDUAL-STRESS

MEASUREMENTS (1982)

EVALUATION OF RESIDUAL STATES OF STRESS AND MATERIAL TEXTURE USING ULTRASONIC VELOCITY MEASUREMENTS WITH ELECTROMAGNETIC ACOUSTIC TRANSDUCERS

R. B. King and C. M. Fortunko

National Bureau of Standards

Fracture and Deformation Division

Boulder, Colorado 80303

In this paper a new approach for using acoustic

measurements to evaluate residual stresses in the presence of unknown material property variation is presented. Procedures previously applied to the evaluation of stress with acoustic measurements are reviewed, and it is shown that these involve using measurements with bulk waves propagating along the

normal to the surface of a plate and do not provide

sufficient information to separate the influences of stress and material property variations. To overcome

this fundamental limitation, an alternative theory is developed that governs the propagation of shear waves

polarized horizontally with respect to the surface of

a plate (SH-waves), but propagating at oblique angles

with respect to the surface normal. The question of separating the effects of residual stress and material

properties on acoustic velocity is addressed in detail. A practical experimental procedure is developed

that permits the evaluation of the in-plane components

of the principal stresses in a plate exhibiting an unknown inhomogeneous initial anisotropy caused by

material texture or microstructure. The procedure is then verified experimentally using an aluminum specimen

with a known residual stress state, but unknown

anisotropy.

Introduction

initial

In recent years considerable attention has been

focused on the possibility of evaluating residual

stress states in practical engineering materials by

exploiting the fact that the presence of stress influences the velocities of elastic waves in a solid.

However, progress has been slow because most practical

engineering materials exhibit unknown material property

variations, which also influence the elastic wave

velocities. In practice, the problem can only be solved by developing a procedure permitting the unambiguous separation of the influences of the main

factors determining the elastic wave velocities:

metal 1 urgical properties, stress, and temperature.

In this paper the problem of separating the effects of stress and metallurgical variations on the velocities of elastic waves in weakly anisotropic

polycrystalline aggregates is addressed. Restricting

measurements to elastic waves propagating along the normal to a plate does not provide sufficient information to separate the effects of stress from those of metallurgical variations because while several unknown

stress components and material properties are present,

only three independent measurements can be made. To overcome this limitation it is proposed that additional

information be obtained from measurements of elastic

wave velocities along directions that do not coincide

with one of the principal stress directions. In particular, it is shown that sufficient information

can be obtained using measurements of shear waves propagating in planes of symmetry of the anisotropic,

polycrystalline aggregate.

To permit evaluation of stress and anisotropy

from the velocity measurements, a special theory of acoustoelasticity is developed. A closed form

theory is developed that describes the propagation

of pure shear waves propagating in the planes of

symmetry of the polycrystalline aggregate. Based on this theory, an experimental procedure is developed

that permits the evaluation of biaxial stress states

in plates when the principal stress directions lie

in the planes of symmetry of the anisotropic,

polycrystal 1 ine aggregate. The proposed experimental

procedure has important technological applications,

particularly in the nondestructive evaluation of residual stresses in butt weldments.

The proposed experimental procedure is verified

using an aluminum plate specimen containing a cylindrical plug that produces a known stress state.

A new electromagnetic-acoustic transducer (EMAT)

configuration is used that permits the generation

and detection of horizontally polarized shear waves (SH-waves) from a very wide range of oblique propagation directions, and significantly increases the

reproducibility of the measurements. The measured

stress distributions are then compared with strain

gauge data and calculated values.

Theory of Acoustoelasticity - Media of Orthorhombic

Symmetry

The theory of acoustoelasticity for media of orthorhombic symmetry can be developed from the equations of motion for a solid in the presence of stress. Following Tokuoka and Iwashimizu [1], the expression governing propagation of plane waves in a stressed anisotropic medium is found to be:

^ijk£ pi

( °i£ P

i

2 , - pv ) 6 , ] Jk j d

k = 0 (1)

where p. is the direction of the wave normal, 0 the unii vector in the direction of particle motion,

o. is the stress tensor, p is the density, v is the 'wa9e phase speed, and C... is a function of tie

second and third order elastic tensors of the material , C . and C . i j

k £ ijk£mn

U S Government work not protected by U S. copyright

3

1982 ULTRASONICS SYMPOSIUM 88f,
Closed form solutions of the eigenvalue problem

of Eq. 1 are obtainable when the propagation direction

lies in a plane of material symmetry. The details of the derivation are presented in Ref. [2]. The key

results for weakly anisotropic media are:

the same angle 0 . Both measurements must then be repeated for a different value of 4> . The difference between the two measurements is then used in Eq. 2 to evaluate (o- - a.) and the quantity (C.r - C, ). (In many technological applications the kr?owlr J --

SH 23 - SH

13 cos

SH 2p(SH°

T ( C 55

' C 44 ) + a(<^ 1°<B

0,1

edge of

) is sufficient because additional information refated

1 to the specimen geometry is usually available to estimate the relative magnitudes of o

7 and o

In principle, the separate values of o.

1"

and c, and

SH,,-, + SH, ,

- 2SH '23 13

b( <t> ) (c. 2SH

where SH .

.

propagatilr^

(3)

denotes the velocity

in the X_.-X^ pi ane, o. of an and o SH-waveare the

ill pi I

' I V_ I p I t. > cut Jtpuiuct V u I u t -> t amj \j . can be determined experimental ly by using Etfs. 2 ana

3. However, in this case a set of five independent measurements must be performed at different values of 0 . The five measurements are needed to determine the values of c^, o_, C.., C,,, and C--. In practice, an additional measurement may

D be required to compen- sate for any path length variations.

I UpQ L I I I III tile '< ^

' '

• p

i U i I t j U o UIIVJ U

| tilt tilt in-plane principal stresrs components, a (t>) and b(<S) are functions of components of the second and third

order elastic tensors, and SH° denotes the mean unstressed velocity, which is explicitly given by:

SH° = [(

C 44 cos "0 + C 66

'55 cos

sin 0 1/2

)

+ Cgg sin 0 1/2

] (4)

estimate of SH may be used in the denominator

2 and 3, because a small error in the denominator

in a higher order error in the calculated

_..n ^

error

An

of Eqs. resul ts stress states. However, the exact value of SH needed in the numerator of Eq. 3 because a small

in the numerator results

calculated stress states.

in a large error in the

A Practical Experimental Configuration

To obtain experimental data for utilizing

Eqs. 2 and 3, transducers of SH-waves are required

that produce ultrasonic beam patterns with wide angular distributions. An electromagnetic acoustic

transducer which meets this requirement is described

in Ref. [2]. A representati ve ultrasonic radiation

pattern of this EMAT is shown in Fig. 1. In Fig. 1, the transmitted ultrasonic amplitude (at 520 kHz) is plotted as a function of the angles ( 0 ) in the plane normal to the surface of a plate and bisecting the transducer between the magnets (the sagittal plane).

It is evident that the radiation pattern is very broad and, in particular, does not vanish at any angle in the sagittal plane. By reciprocity, the same directivity pattern applies upon reception.

This property of the transducer is needed to study the velocities of ultrasonic waves as a function of propagation direction.

(d»gr •••

)

The importance of Eqs. 2 and 3 is that the same unknowns are present regardless of the value of 0 . As

a consequence, measurements at different values of 0 provide additional information without increasing the number of unknowns. Equation 2 is particularly

important because only two unknowns are present: (C-r

- C^) and (oo - 0 ,). Thus, measurements at only tvvo° values of 0 are required to evaluate the quantity

( 0 - oi).

An Experimental Procedure

It follows from the preceding discussion that the residual stress state and the relevant elastic constants

can be uniquely determined by making precise velocity measurements with off-axis SH-waves in the planes of material symmetry. To separate the effects of texture

and stress, measurements must be made in mutually

orthogonal material symmetry planes.

Figure 1.

—Th#or»l»c»l M> 1 • = O* D 3 2 mm

520 kHi Ultrasonic radiation pattern of electromagnetic -acoustic transducer.

It is straightforward to determine the difference

between two principal stresses by making relative measurements and using Eq. 2. In this case, the

procedure involves first making a velocity measurement

of an SH-wave propagating in the x ?

- x, plane at an angle 0 with respect to x^ axis, hrext, the measurement

is repeated in the x^ - x^ plane for

A practical experimental configuration for determining stress states in anisotropic plates is shown in Fig. 2. In the configuration of Fig. 2 two SH-wave EMATs are used. One of the EMATs is used as

a generator and the other as a receiver of the probing ultrasonic signals. Because the two EMATs are aligned along the same sagittal plane, the receiver EMAT is sensitive only to ultrasonic

886 - 1982 ULTRASONICS SYMPOSIUM

4
signals with a displacement component parallel to the spaces of the plate and normal to the sagittal plane

(SH-waves

)

.

Transmitter Receiver

Figure 2. Transducer configuration for stress measurement . In the special case when one of the principal

stress directions is parallel to a crystal axis and

the sagittal plane is also a plane of crystal symmetry,

the received signals are pure SH-waves. Since SH-waves

are decoupled from longitudinal (L) and vertically

polarized shear (SV) waves upon reflection, they are

not phase shifted upon reflection from the surfaces of

the plate. This important observation results in a significant simplification in the inversion of the experimental data and forms the basis for the experimental determination of residual stress states in planar geometries.

An examination of Fig. 2 reveals that many

independent ultrasonic signal paths (rays) exist that

connect the centers of the generator and receiver

EMATs. In fact, useful experimental information is available over a range of angles $, from grazing U = 90°) to nearly normal U ^ 0°). As a consequence, by

measuring the phases of a number of the ultrasonic

signals in planes of symmetry, sufficient information

is provided to permit the unambiguous determination of

the in-plane stress components, elastic constants and

adjustment for plate thickness variations, if needed.

X2

Figure 3. Specimen containing a known residual stress state.

Experimental Verfication

To verify the experimental procedure, measurements were carried out on an aluminum plate specimen,

25-mm thick, which contained a known biaxial stress state and unknown initial anisotropy. Aluminum was chosen for this study because it is typical of engineering materials that although nominally elastically isotropic, exhibit a small amount of anisotropy. The anisotropy causes an effect on velocity

that is not negligible in comparison to the effect

of stress. The design of the experimental specimen

is shown in Fig. 3. The test plate was square,

686 mm on each side, and made of type 6061-T6

aluminum. In the center of the plate is a hole into

which was inserted a circular plug, 102 mm in diameter, made of type 304 stainless steel. Prior

to insertion, the stainless steel plug was cooled to

77 K and the aluminum test plate heated to 423 K. After insertion, the resulting strains were monitored

with strain gauges placed along radial lines until

the whole test specimen was returned to the ambient

temperature. The strain gage readings confirmed the

theoretically predicted stress distribution.

To obtain ultrasonic data, the specimen shown

in Fig. 3 was scanned using the experimental SH-wave

EMAT configuration, as shown in Fig. 2. The scanning

was carried out along one of the radial lines that

coincide with the axis of plate symmetry, as shown

in Fig. 3. To measure the phases of the ultrasonic

signals, a modified system of Kino et al. [3] was

used.

The relative velocity differences (SH„,-SH.. ,)/SH : were calculated for several positions along the scan line, with the x~ axis taken to coincide with the plate rolling direction. The ultrasonic data were obtained from the first and third arrival signals,

as shown in Fig. 2, with the SH-wave EMAT pair aimed

along the scan line and, then, at 90 c with respect

to the scan line. The first and third arrivals were selected because of the large difference in the angle defined by their rays and the plate surface

normal, 33.8° and 12.6°. The difference in the principal stresses, o~ - a., was then evaluated from

the velocity measurements using Eq. 2. As discussed

b

b

i i

eg o Experimental results b using new procedure

- 0.4

. * Experimental results CO - \ using acoustical -

CO

111 cr \ birefringence \ 0

CO NP ^ Exact stress value 4 V 0 o 0.2 A ° o

111 N

_J< * x, A

oc n I 1 O 50 100 150 z

DISTANCE FROM PLUG BOUNDARY, mm

Figure 4. PrinciDal stress difference evaluated in SDecimen of figure 3.

5

1982 ULTRASONICS SYMPOSIUM R87
in the preceding section, the constant c($) was neglected because of the weak anisotropy. The values

of the constants aU.) and a(<t>„) were determined

independently using tensile calibration specimens

obtained from the ends of the test-block shown in Fig. 3. The calibration values determined were

a ( 33 . 8° ) = 3.7 x 10" b /MPa, a( 12.6°) = 4.4 x 10" 5 /MPa.

The calibration constants were found to vary with

position, but the variations were less than 1%. The experimental results for the ul trasonically measured

values of o~ - o, are shown in Fig. 4. For comparison,

the calculated values of (o^ - Oj) from the elasticity

solution are also shown in Fig/ 4. The calculated

distribution is parabolic, and the experimental

results are seen to follow this shape.

Summary

Closed-form expressions relating the velocities

of elastic waves propagating at arbitrary angles in the planes of symmetry of weakly anisotropic, polycrystalline solids have been presented. In particular,

it has been shown that using measurements of the velocities of pure shear waves as a function of propaga ti on , di rect ion , sufficient information is obtained to permit the evaluation of residual stress

states in a practical plate geometry. In fact, measurements at only two independent propagation

directions are required to obtain the difference in the two in-plane principal stress components. Furthermore, an experimental procedure, based on the new

theory, has been proposed and evaluated using a new SH-wave electromagnetic-acoustic transducer (EMAT)

design

.

Acknowl edgments

The authors are indebted to S. A. Padget,

J. Arvidson, Jr., and R. A. Schramm, who participated

in the construction of the experimental apparatus and

in the experiments described. The authors are also

indebted to J. D. McColskey who prepared the specimens

used in the experiments. This work was supported in part by the U.S. Navy, David Taylor Naval Ship Research

and Development Center, and the Office of Nondestructive

Evaluation of the .National Bureau of Standards.

References

1. T. Tokuoka and V. Iwashimizu, Int. J. Solids and Structures 9, 383 (1968).

2. R. B. King and C. M. Fortunko, "Determination of In-Plane Residual Stress States in Plates Using

Horizontally Polarized Shear Waves", submitted to Journal of Applied Physics.

3. G. S. Kino, J. B. Hunter, G. C. Johnson, A. R. Selfridge, D. M. Barnett, G. Hermann, and C. R. Steele, J. Appl . Phys. 50, 2607 (1979).

888 - 1982 ULTRASONICS SYMPOSIUM

6
EVALUATION OF RESIDUAL STRESS STATES USING

HORIZONTALLY POLARIZED SHEAR WAVES

R. B. King and C. M. Fortunko

Fracture and Deformation Division

National Bureau of Standards

Boulder, Colorado 80303

ABSTRACT

A new approach for using acoustic measurements to evaluate

residual stresses in the presence of unknown material property

variations is presented. It is shown that measurements using shear waves propagating along the normal to the surface of a plate do not

provide sufficient information to separate the influences of stress

and material property variations. To overcome this fundamental

limitation, an alternative theory is developed that governs the

propagation of shear waves polarized horizontally with respect to the surface of a plate (SH-waves), but propagating at oblique

angles with respect to the surface normal. The question of

separating the effects of residual stress and material properties

on acoustic velocity is addressed in detail. In addition, a practical experimental procedure is developed that permits the

evalution of the in-plane components of the principal stresses in a plate exhibiting an unknown inhomogeneous initial anistropy caused

by material texture or microstructure. The procedure is then

verified experimentally using an aluminum specimen with a known

residual stress state, but unknown initial anisotropy.

INTRODUCTION

The application of acoustic velocity measurements to the

evaluation of residual stresses has received considerable attention

in recent years. A major problem has been the fact that practical

engineering alloys exhibit considerable material property variations. These variations affect acoustic velocity and may mask the

effects of stress.

1327

7
1328 R. B. KING AND C. M. FORTUNKO

In this paper, the problem of evaluating residual stresses

from velocity measurements in alloys with unknown material property

variations is addressed. In particular, it is demonstrated that

experimental procedures, in which velocity measurements of acoustic waves at normal incidence to the surface of a solid are made, do

not provide sufficient information to determine the stresses

unambiguously. Specifically, it is shown that the commonly used

technique based on acoustical birefringence is hampered by the

influence of material anisotropy.

A new technique is proposed which is based on measurements

with horizontally polarized shear waves (SH-waves) propagating in a plate at arbitrary angles to the normal to the free surfaces. The new technique is a generalization of the acoustical birefringence

method. It is shown theoretically that the technique provides

enough information to permit the evaluation of in-plane stress

components in the presence of unknown material anisotropy. An

experimental procedure is then described that uses electromagneticacoustic-transducers (EMATs) to generate the probing SH-wave signals. Results of a verification experiment using a plate specimen

containing a known residual stress state are presented.

The basis for evaluation of stress with measurements of

acoustic velocity is provided by the theory of acoustoelasticity,

which relates changes in the velocity of elastic waves to the

stress state in a solid. The most common application involves the use of bulk waves propagating at normal incidence to the surface of

a plate. The velocity measurements with these waves are used to characterize the average through-thickness values of the in-plane

stress components. Alloys are polycrystalline aggregates which can exhibit macroscopic anisotropic elastic properties due to microstructure or texture. The polycrystalline material is elastically

equivalent to an anisotropic crystal. The acoustoelasticity

relations for waves propagating along the normal to a plate made of

anisotropic material are.

THEORY

V„ “ v „°

£ £

«= A^ + o

2

) + D(o

2 - Oj)

£

( 1 )

V,S2

+ B(o

2 - o

j

) + E(o

1 + o2

) ( 2 )

= C(o

1 + o

2

) + F(o

2 - oj

) (3)

8
EVALUATION OF RESIDUAL STRESS STATES 1329

where V^, and are the velocities in the presence of stress

of a longitudinal wave, a shear wave polarized parallel to the

direction, and a shear wave polarized parallel to the X^ direction,

respectively; V^° is the velocity of a longitudinal wave in the

unstressed state, Vg° is the average shear wave velocity in the

unstressed state, and and are the in-plane principal stress

components. The constants A through F in Eqs. (1-3) are functionally dependent on the second and third order elastic constants of

the solid, and are commonly referred to as acoustoelastic constants

or stress-acoustic constants. It is shown 1 that for solids with

weak anisotropy and with principal directions of stress parallel to

crystal axes, the constants D, E, and F are small and may be

neglected for practical measurement purposes.

Equations (1-3) have been applied in three ways. In the

acoustical birefringence method, Eq. (2) is employed to determine

the difference of the principal stresses from the relative difference of the velocities of shear waves polarized in the mutually

orthogonal principal directions. 2-4 Longitudinal wave measurements

have been used by Kino and co-workers 5 ’ 6 to evaluate the quantity

(°2 + us in g Eq. (1). Guscha, et al. and Guz, et al. 7 * 8 have

used both shear and longitudinal wave measurements in an attempt to

determine the separate values of o and a in nominally isotropic

steels.

It is relatively straightforward to apply acoustic

measurements to the characterization of applied stress states if

independent measurements can be made in the unstressed state for

in this case, the values of V^°, V^ 0

, and V ° can be measured

throughout the region of interest. However, the evaluation of

residual stress states using acoustic velocity measurements is

seriously complicated by the fact that constants characteristic of

the unstressed state are not available. The nominal value of the

velocities in the absence of stress can be measured using a separate reference sample of the same materials. However, most

alloys of engineering importance are not perfectly homogeneous. As

a consequence, variations in the velocities in the unstressed state

can be present that are of the same order of magnitude as the

velocity variations caused by stress. In particular, in the

commonly used acoustical birefringence method, the presence of

initial anisotropy can result in an unknown, and not negligible,

value of the quantity ( V

S 2°~V

S 1

°

^ ^ V

S°

’ even * n nomina Hy isotropic

materials. The importance of this ^initial anisotropy effect is

widely recognized. Various procedures have been proposed for

providing information using other measurements in addition to acoustic velocity measurements to permit evaluation of stress in

the presence of the unknown initial velocity term. 9 However, none

of these procedures has been satisfactorily experimentally verified

to date.

9
1330 R. B. KING AND C. M. FORTUNKO

In this paper, a new technique is proposed for overcoming the

problem of unknown initial anisotropy. The conventional birefringence method is limited by the fact that, with measurements con- fined to elastic waves propagating along the normal to the surface

of a plate, insufficient data is obtained to separate the effects

of stress and anisotropy. Equation (2) involves two unknowns,

( v

s2

e -v

S i°)/vs

6 and (o -o ), only one quantity, (V -V

gl ) /V 0

, is measured. Additional Information would in principle De provided

by measuring the velocity of waves propagating at an angle with

respect to the plate normal, but in general this approach would be

difficult because a closed-form theory relating velocity to stress

is not available and the waves are not pure modes. However, it is

shown 1 that for the special case of horizontally polarized shear waves (SH-waves) propagating in planes of crystal symmetry, in

solids whose directions of principal stress are parallel to crystal

axes, a closed-form theory can be derived. In particular, the

following equation is shown to be valid for weakly anisotropic

solids

SH - SH C - C 23

sr

- 2

Is - * a(e)(0

2 44

' V (A)

where SH and SH are the velocities of SH-waves propagating in

the X^-X^ and X.-X.T planes of a plate, X^ is parallel to the normal

to the pT-ate, X^ and X^ are parallel to crystal axes, 0 is the

angle between the propagation direction and the plate normal. C

,55

and are elastic constants, SH is the average velocity in the

unstressed state, given by

SH° = |(SH

23

° + SH

13

°) (5)

The acoustoelastic constant a(0) is a function of the second- and

third-order elastic constants of the material.

The first term on the right-hand-side of Eq. (4) is equal to

. SV - SH 13° . ' SH° ; 0

'

Thus, Eq. (4) is a generalization of the usual acoustical birefringence relation, Eq. (2) and, in fact, reduces to that relation for

the special case 0=0°. The main advantage of using Eq. (4) is

that measurements can be made at various values of the angle 0,

thereby providing additional information without increasing the

number of unknowns. Measurements at two values of 0 provide two

equations, which can be solved for the unknowns (C._-C,,)/C^ and

(o

2 ~c^). (Measurements at more than two angles of § will produce

an overdetermined system of equations, which can be solved using

least squares fitting.)

10
EVALUATION OF RESIDUAL STRESS STATES 1331

EXPERIMENTAL PROCEDURE

To permit implementation of Eq. (4), transducers capable of

generating SH-waves propagating at arbitrary angles with respect to

the plate normal are required. A practical electromagnetic-acoustic-transducer (EMAT) capable of fulfilling this requirement has

been developed. A schematic representation of this transducer is

shown in Fig. 1. A typical acoustic radiation pattern for the

transducer in Fig. 1 is shown in Fig. 2. This pattern can be

regarded as an angular spectrum of plane waves. Thus, by properly

locating transmitting and receiving transducers appropriate rays

representing plane waves at desired angles may be utilized. A

practical experimental configuration for applying the transducer of

Fig. 1 to stress characterization is shown in Fig. 3. The

transmitting transducer and receiving transducer are aligned so that the plane passing between the magnets of the transducer and

normal to the plate (the sagittal plane) is shared by the two

transducers. SH-waves are generated by the transmitter which

propagate in this plane and whose polarization direction is normal

to it. Upon examining Fig. 3, it is evident that because of

multiple reflections from the back surface of the plate, many rays

connect the two transducers. Measurements with plane waves at

various angles with respect to the plate normal thus can be made by

utilizing different rays. The procedure for implementing the

configuration in Fig. 4 is as follows. In a plate for which the

crystal axes are known (e.g., a rolled plate) the transducers are oriented so that the sagittal plane is parallel to a crystal axis.

A single echo is isolated by time-gating so that velocity of a plane wave at an angle 0, with respect to the plate normal can be

measured. The transducers are now rotated 90° and the velocity

measurement is repeated. The quantity

SH

23

- Sli 13

SH°

is now determined. This procedure is then repeated using a different echo, so that the relative velocity of plane waves propagating

at a different angle, is now measured. Two equations are then

available for evaluating the unknowns

, C55

" C44 . ( j- ) and (Oj-Oj).

44

A precision velocity measuring system is required to implement

the experimental procedure described. Since only relative measurements are needed to apply Eq. (4), a modified version of the system

developed by Kino et al. 6 was utilized in the present work. In this system, the relative frequency change required to maintain a

11
1332 R. B. KING AND C. M. FORTUNKO

Sm-Co

Magnet

Fig. 1. A practical electromagnetic -acoustic transducer design.

^ (degree*)

Fig. 2. Ultrasonic radiation pattern of EMAT.

12
EVALUATION OF RESIDUAL STRESS STATES 1333

Transmitter Receiver

constant phase difference between an acoustic signal and a reference is measured. Relative velocity changes can be evaluated using

the relation

AV

= Af A£

V f +

£

( 6 )

where f denotes frequency and £ denotes acoustic path length. In using this system for making velocity measurements on two orthogonal

planes as described above, the path length remains unchanged so that the term A £/

£

vanishes.

13
1334 R. B. KING AND C. M. FORTUNKO

EXPERIMENTAL RESULTS

To verify the theory and experimental procedure presented

above, measurements were carried out on an aluminum plate specimen,

25 mm thick, which contained a known, biaxial, residual stress

state and unknown initial anisotropy. The specimen design is shown

in Fig. 4. A 100 mm hole was machined in the aluminum plate. The

plate was heated to 423 K and an oversized plug of 304 stainless

steel which had been cooled to 77 K was inserted in the hole.

Strain gages had been placed on the aluminum plate so that the

resulting residual stress state could be evaluated after the

specimen returned to ambient temperature. The measured strains

agreed with the predictions of an exact theory for this specimen. 10 The resulting values of (°

2 ~a p

as a function of position along one

of the plate symmetry axes (normal to an edge) are shown in Fig. 5. Before the ultrasonic experimental procedure described above

could be used to evaluate the residual stresses in the specimen,

the acoustoelastic effect had to be calibrated to determine the

constants a(0) in Eq. (4). Two values of 6 were chosen, 33.8° and

12.6°. These correspond to the first and third echoes from the

back surface of the plate, respectively. A calibration specimen in

the form of a tensile bar was cut from the end of the aluminum

plate from which the specimen containing residual stress was made.

The tensile axis was parallel to the rolling direction. Tension

Fig. 5. Principal stress difference evaluated in specimen of

Fig. 4.

14
EVALUATION OF RESIDUAL STRESS STATES 1335

Fig. 6. Typical calibration curves for acoustoelastic constants.

was applied in a servo-hydraulic testing machine. Ultrasonic data were taken with the SH-waves polarized parallel to and perpendicular to the rolling direction. Results for relative frequency change

from the unstressed state as a function of applied stress are shown

in Fig. 6 for the first and third echoes. The constant a(6) is

given by the difference of the slopes of the lines from the two

orthogonal polarization directions. The calibration values ^determined were a(33.8°) = 3.7 x 10 /MPa, a(12.6°) *= 4.4 x 10 /MPa.

The calibration constants were found to vary with position by

approximately 1%.

After calibrating the acoustoelastic constants, velocity

measurements were made in the aluminum plate containing a known

residual stress state. Measurements of relative velocity change,

(SH^-SH^) /SH° as a function of position, were made along a scan

line perpendicular to the rolling direction of the plate and along

a radial line from the plug. The difference of the principal

stresses, (o„-o.), was evaluated from the relative velocity measurements ana the values of the constants a(33.8°) and a(12.6°)

using Eq. (4). This evaluation was possible using this technique

despite the fact that an unknown amount of initial anisotropy was

present and no measurements were made in the unstressed state. The

results for (o^-o ) are contraste d with the exact theoretical

values in Fig. o

.

15
1336 R. B. KING AND C. M. FORTUNKO

For comparison with the experimental results obtained with the new technique, results obtained using a more conventional application of acoustical birefringence are also plotted in Fig. 5. It was assumed that the anisotropy in the specimen was homogeneous and

could be characterized from measurements in an unstressed reference

sample. Thus the value of ( v

S 2 °“V c i °)

/

v

s

° in E<3* ( 2 ) was evaluated

on a separate specimen and assumed to De constant. A single

relative velocity measurement was then used to infer the value of

Erom Fi 8- it is evident that the assumption of homogeneous anisotropy results in significantly poorer agreement with

the exact values of (o^-o^).

DISCUSSION

A new experimental procedure has been proposed for using

velocity measurements with SH-waves for characterizing residual

stress states in anisotropic materials. The theoretical basis for

the method has been presented in the form of closed-form expressions for the velocity of SH-waves propagating at arbitrary angles

in planes of crystal symmetry. The new procedure is subject to

certain limitations, some of which are also present in existing

techniques, and some of which are peculiar to this technique:

1. A principal assumption of the theory upon which the proposed

procedure is based is that the principal directions of stress

are parallel to the crystal axes of the unstressed solid. In

many applications of engineering importance this assumption

will be valid, for example, the case of butt weldments where

the rolling direction is either perpendicular or parallel to the welding direction.

2. Measurements are needed using SH-wave signals propagating at two different angles with respect to the plate normal in order

to evaluate (°

2_0 p

i-n t *ie presence of unknown anisotropy. An error propagation analysis was conducted 1 and it was found

that the accuracy of the evaluation of (c^-op decreases as the angular difference between the rays corresponding to the

two ultrasonic signals decreases. There is thus an interesting trade-off in the proposed technique between precision and

spatial resolution. For the measurement precision obtainable

using the current experimental apparatus, acceptable precision

(M0%) in the evaluation of stress can be attained if the

angular difference between the two SH-waves rays is 10° or greater

.

3. The error propagation analysis mentioned above also showed

that errors in the values of the relative velocity measurements or the acoustoelastic constants would lead to magnified

errors in the values of (o -o ). For an angle difference of

10° between the rays used in the measurement it was found that

16
EVALUATION OF RESIDUAL STRESS STATES 1337

±1% errors in the measurement would cause approximately ±10% errors in the value of (o^-o.). Precision velocity measurement systems are readily capable of making relative velocity

measurement to a precision of greater than 1%. However, the use of a separate reference sample to evaluate the acoustoelastic constants can lead to greater than 1% errors in their

values if significant material inhomogeneity is present. If

this is found to be a serious problem then it may be necessary

to use the in-situ calibration technique proposed by Scott. 11

In conclusion, an experimental procedure has been demonstrated

which shows promise for evaluating residual stress states in plate

geometries with unknown material property variations. This is an important step in extending acoustoelastic stress measurements to applications of practical engineering importance.

ACKNOWLEDGMENTS

The authors are indebted to S. A. Padget, J. Arvidson, Jr.,

and R. A. Schramm, who took much of the experimental data and

participated in the construction of the experimental apparatus.

The authors are also indebted to J. D. McColskey who prepared the

specimens used in the experiments. This work was supported in part

by the U.S. Navy, David Taylor Naval Ship Research and Development

Center, and the Office of Nondestructive Evaluation of the National

Bureau of Standards.

REFERENCES

1. R.B. King and C.M. Fortunko, "Determination of In-Plane Residual

Stress States in Plates Using Horizontally Polarized Shear

Waves," submitted to the Journal of Applied Physics.

2. H. Fukuoka, H. Toda and T. Yamone , "Acoustoelastic Stress Analysis of Residual Stress in a Patch-Welded Disk," Exp. Mech. 18,

227, 1978.

3. N.N. Hsu, "Acoustic Birefringence and the Use of Ultrasonic

Waves for Experimental Stress Analysis," Exp. Mech. 14, 169,

1974.

4. W.H. Cooper, D.H. Saunderson, C.M. Sayers and M.G. Silk,"Ultrasonics and the Residual Stress Measurement Problem," A.E.R.E.

Harwell, Report No. AERE-R 9588, United Kingdom, 1980.

5. G.S. Kino, J.B. Hunter, G.C. Johnson, A.R. Selfridge, D.M.

Barnett, G. Herrmann and C.R. Steele, "Acoustoelastic Imaging

of Stress Fields," J. Appl. Phys., 50, 2607, 1979.

6. M.P. Scott, D.M. Barnett and D.B. Ilic, "The Nondestructive

Determination of Residual Stress in Extruded Billets from

Acoustoelastic Measurements," in 1979 Ultrasonics Symp. Proc .

,

J. de Klerk and B.R. McAvoy (eds.), IEEE, New York, p. 265, 1980.

7. O.I. Guscha and F.G. Makhort, "Acoustic Method of Determining

Biaxial Residual Stresses," Prikl. Mech. 12, 32, 1976.

17
1338 R. B. KING AND C. M. FORTUNKO

8. A.N. Guz, F.G. Makhort, 0.1. Guscha and V.K. Lebedev, "Theory

Underlying the Determination of Initial Stresses from the Results of Ultrasonic Measurements," Prikl. Mech. 7, 1205, 1971.

9. M. James and 0. Buck, "Quantitative Nondestructive Measurements

of Residual Stress," CRC Critical Reviews in Solid State Physics

and Materials Science 9, 61, 1980.

10. C.M. Kachanov, Foundations of the Theory of Plasticity, MIR

Press, Moscow, USSR, p. 127, 1974.

11. M.P. Scott, Doctoral Dissertation, Department of Materials

Science and Engineering, Stanford University, Stanford, CA, 1981.

18
Determination of in-plane residual stress states in plates using horizontally

polarized shear waves

R. B. King and C. M. Fortunko

Fracture and Deformation Division, National Bureau ofStandards, Boulder, Colorado 80303

I Received 10 September 1982; accepted for publication 18 January 1983)

In this paper a new approach for using acoustic measurements to evaluate residual stresses in the presence of unknown material property variation is presented. Procedures previously applied to the evaluation of stress with acoustic measurements are reviewed, and it is shown that these

involve using measurements with bulk waves propagating along the normal to the surface of a

plate and do not provide sufficient information to separate the influences of stress and material

property variations. To overcome this fundamental limitation, an alternative theory is developed

that governs the propagation of shear waves polarized horizontally with respect to the surface of a

plate (SH waves), but propagating at oblique angles with respect to the surface normal. The

question of separating the effects of residual stress and material properties on acoustic velocity is addressed in detail. A practical experimental procedure is developed that permits the evaluation

of the in-plane components of the principal stresses in a plate exhibiting an unknown

inhomogeneous initial anisotropy caused by material texture or microstructure. The procedure is then verified experimentally using an aluminum specimen with a known residual stress state, but

unknown initial anisotropy.

PACS numbers: 46.30.Rc, 43.88.Dv, 43.35.Cg

INTRODUCTION

In recent years considerable attention has been focused on the possibility of evaluating residual stress states in practical engineering materials by exploiting the fact that the presence of stress influences the velocities of elastic waves in a solid. However, progress has been slow because most practical engineering materials exhibit unknown material property variations, which also influence the elastic wave velocities. In practice, the problem can only be solved by

developing a procedure permitting the unambiguous separation of the influences of the main factors determining the

elastic wave velocities: metallurgical properties, stress, and

temperature.

In this paper the problem of separating the effects of

Stress and metallurgical variations on the velocities of elastic waves in weakly anisotropic polycrystalline aggregates is addressed. First, using the theory of acoustoelasticity, the ef- fects of stress on elastic waves propagating along the principal stress directions are reviewed. It is shown that restricting

measurements to elastic wave propagating along these special directions does not provide sufficient information to sep- arate the effects of stress from those of metallurgical variations. In particular, it is shown that the usual experimental

procedure, based on acoustic birefringence, cannot be used

to separate the effects of stress and initial anisotropy, caused

by microstructure or nonrandom grain orientations in a polycrystalline aggregate (texture). Second, to overcome the above limitation it is proposed that additional information

be obtained from measurements of elastic wave velocities along directions that do not coincide with one of the principal stress directions. In particular, it is shown that sufficient information can be obtained using measurements of shear waves propagating in planes of symmetry of the anisotropic,

polycrystalline aggregate.

To permit evaluation of stress and anisotropy from the

velocity measurements, a special theory of acoustoelasticity

is developed. It is shown that the system of equations governing elastic wave propagation in media of orthorhombic symmetry in the presence of stress can be solved numerically. A

closed-form theory is also developed that describes the propagation of pure shear waves propagating in the planes of

symmetry of the polycrystalline aggregate. Based on this theory, an experimental procedure is developed that permits

the evaluation of biaxial stress states in plates when the principal stress directions lie in the planes of symmetry of the

anisotropic, polycrystalline aggregate. The proposed experimental procedure has important technological applications,

particularly in the nondestructive evaluation of residual

stresses in butt weldments.

The proposed experimental procedure is verified using an aluminum plate specimen containing a cylindrical plug

that produces a known stress state. A new electromagnetic

acoustic transducer (EMAT) configuration is used that per- mits the generation and detection of horizontally polarized

shear waves (SH-waves) from a very wide range of oblique

propagation directions, and significantly increases the re producibility of the measurements. The measured stress dis- tributions are then compared with strain gauge data and cal- culated values.

THEORY OF ACOUSTOELASTICITY: GENERAL

BACKGROUND

The theory of acoustoelasticity relates measurable

changes in the velocities of elastic waves to the stress state in a solid. The theory has been studied by many authors

Earlier work was motivated by the need to determine third order elastic constants, which play a significant role in mans

branches of solid-state physics. More recently . greater atten

3027 J. Appl. Phys. 54 (6), June 1983 19 3027
tion has been focused on the specific application of the the- ory of acoustoelasticity to the problems ofdetermining stress states in solids. In applying the theory to the determination

of residual stress states, it has been commonly assumed that residual stress states are equivalent to applied stress states in terms of their effect on the velocities of elastic waves. This

assumption is also used in the present work.

In the past, explicit formulae have been derived that

relate the velocities of elastic waves in solids to multiaxial

stress states when the principal stress directions and the propagation directions coincide with the crystal axes.

3,5,6 Most technological applications involve a need to evaluate

residual stress states in polycrystalline materials. Microstructure or texture (nonrandom orientation of the grains)

result in elastic anisotropy of the polycrystalline aggregates.

For the special case of a plate subjected to a biaxial plane stress state, with the propagation direction along the

plate normal, the following relations can be obtained for iso- tropic solids6

:

( vi — v°i)/v° — A[ax + a2 ), (1)

{vS2 =B(a2 -ax ), (2)

(vS2 + vS i - v°sV2v°s = C W\ + ( 3 ) where v°, and v°s are the longitudinal and shear wave velocities in the absence of stress; the subscript / denotes the longitudinal wave component, and the subscripts S 2 and 5 1 refer to the two shear waves whose polarization directions coin- cide with the two principal in-plane stress directions (X{ and

X2 ).A,B, and C are commonly referred to as the acoustoelastic constants, which are functionally related to the secondand third-order elastic constants. The principal stress components are denoted by cr, and a2

. In anisotropic solids, Eqs. ( 1 )—(3) can be generalized as

follows:

(v, — v°,)/v°, = A (cr, + a2 ) + D(cr2 — a,), (4) (%2 - ” v 1 )/«& = v°s 2 ~ v°sl /v°s + B (a2 — cr.) + E (a t + cr2 ), (5)

(uS2 + us , — 2v°s)/2

v°s = C(ai + a2 ) + F(cr2 - a,), (6) where D, E, and F are additional acoustoelastic constants and v°s

is the mean shear wave velocity in the absence of

stress:

v°s =2(^2 +Usi)- ( 7 ) Equations (4)—(6) are derived in the Appendix for solids exhibiting orthorhombic symmetry. Orthorhombic symmetry

has been assumed in the present theory because it is believed

that it is sufficiently general to represent most technologically important solids. (It is shown in the Appendix that for weak anisotropies the constants D, E, and Fare small).

Equation (2) is known as the acoustical birefringence

equation and is analogous to the optical birefringence equation used in the theory of photoelasticity. The distinctions between acoustoelasticity and photoelasticity are discussed

in detail by Henneke and Green. 7 The acoustical birefrin- gence effect has been widely used in experimental evaluation

of residual stress states.

8" 10 Its main advantages are that only

relative measurements of the velocities need be made and

that initial specimen inhomogeneities are canceled. However, weak anisotropy can result in large absolute errors in the determination of the quantity a2 — a,. The errors are caused by the indeterminacy of the (i£2 — v°s , )/v°s term in Eq. (5). The importance of this initial anisotropy effect is widely recognized. Various techniques have been proposed

for overcoming this difficulty." To date, none of the above

techniques have led to a fully successful experimental de- monstration.

As an alternative approach, measurements of the longitudinal wave velocities have been used. 12,13 An examination

of Eq. (4) reveals that the velocity of longitudinal waves is not

strongly sensitive ’ ' initial anisotropy. However, initial specimen inhomog cities can cause large errors in the de- termination of th uantity a2 + cr,. Also, absolute velocity

measurements r ceded to determine the absolute magnitude of the quality a2 + cr,. Scott, Barnett, and Ilic

1

3

have

shown that by supplementing the available experimental

data by theoretical information, such as the fact that residual

stresses must self-equilibrate, the need to obtain absolute velocity data can be eliminated. Another approach has been proposed by Guscha et

o/.

6,14 for isotropic solids. It was shown theoretically using

Eqs. ( 1 )—( 3 ) that absolute shear and longitudinal velocity

measurements can be used to determine cr, and a2 , separately. In addition, the effects of the initial material inhomogeneity can be eliminated using the above approach. However,

in the derivation in Ref. 14, it is assumed that the ratio of v°, and v°s remains unchanged throughout the specimen. This

assumption appears questionable in view of the fact that v°s would be more strongly affected than u° by any variations in material anisotropy with position. In all of the work described above, use is made of bulk

elastic waves that propagate along the direction normal to the surface of the plate. This approach is fundamentally li- mited. In particular, it is not possible to determine uniquely

the five unknowns cr,, a2 , v

°S2 , vs 1 > v°i from Eqs. (4)—(6). In the present work, a procedure is proposed that over- comes the above limitations. The procedure is based on making measurements with bulk waves propagating at an angle

with respect to the plate normal. In plates, the use of longitu- dinal (L), and vertically polarized shear (SV) waves in this context would lead to significant difficulties, because such waves are coupled upon reflection and exhibit substantial phase shifts as a function of the angle of incidence. 15 In addition, an adequate theory for L and SV waves in the presence of stress is very complicated. However, the difficulties outlined above can be successfully overcome by using shear waves that are polarized in the plane of the plate (SH waves).

THEORY OF ACOUSTOELASTICITY: MEDIA OF

ORTHORHOMBIC SYMMETRY

The theory of acoustoelasticity for media of ortho- rhombic symmetry can be developed from the equations of

motion for a solid in the presence of stress. In this paper, the approach used in the derivation of the relevant equations of

motion follows that of Tokuoka and Iwashimizu. 16 In this approach, the wave motion is treated as an infinitesimal state

3028 J. Appl. Phys., Vol. 54, No. 6, June 1983 20

R. B. King and C. M. Fortunko 3028
of deformation that is superimposed on a large initial, static

state of deformation.

Let V, x and x' denote the position vectors of a point in an undeformed reference state, the initial deformed state, and the final deformed state, respectively (Fig. 1). The subscripts / and j are defined using the same basis vectors for all three states. In terms of the Cauchy stress, the equations of

motion in the final state are given by

dx7 K) = P

d

2 u,

dt 2 d

2 u)

dt 2 = P' d

2 u)

dt 2

( 8 )

However, this interpretation is misleading, because the ef- fective stiffness tensor does not exhibit the familiar symmetry properties.4 For example, in general

eff / eff ^ 1212 ~f U 2121 * Thus, in the presence of stress, a shear wave propagating

along the x t direction and polarized along the x2 direction

will not have the same velocity as a shear wave propagating

in the x2 direction and polarized along the x t direction. Therefore, only Eq. (14) will be used in the subsequent dis- cussion.

where p, p\ u,, u), and oq

, a), denote density, displace- ment. and stress, respectively, in the initial and final de- formed states. The usual implicit summation convention on

repeated indices is used throughout this paper. The nonlinear-elastic constitutive law relating the Cauchy stress to the Lagrangian strain is given by

The strain energy density W can be expressed as Eq.

(17): w= {Cijk ,EijEkl 4 \Cl]klmn Etj Ekl Emn , (17) where CiJklmn is the third-order stiffness tensor. Substituting

into Eq. (12), the following expression for Cljkl is obtained:

dW 1 dx) dx)

Po dE I; dXr dXs

state. (The thermodynamic aspects of Eq. (9) are considered

in Ref. 3.

)

Making use of the strain-displacement relationship

- 1 dxl

(9) ^riikl — £- icy^mnpq +T c^ mnpqrs Ers ) Po

dx, dxj dxk dx.

the La- X

dXm dXn dXp

,J 2 \ dx, dX, — 5, ( 10)

and substituting for aq in Eq. (8), the following expression is obtained:

(18)

Equation (18) can be linearized at this point. Thus

E>j ~Sjjki ak„ (19) where Sijkl is the elastic compliance tensor, and to first order

in dp,/dXj.

dx,

dX,

Ejj 4- Piij + 8ijt ( 20)

( C,tkj 4 aij&,k ) dxjdx,

( 11 )

In the derivation of the above expression only the lowest

order terms in u, and its derivatives have been retained. Cijk ,

is defined by

p d

2 W dx, dxj dxk dx,

7o dErs dEpq ~dX~r ~dX~s ~dX~p ~dX~q

' ( 12 )

(In the derivation of Eq. (11) it has been assumed that Cijkl and <j

q are homogeneous.)

Next, a plane wave solution is assumed for u):

ujr= di em(x,p, - vt), (13) where /? is the wave number, p, the direction cosine of the wave normal, and d, the unit vector in the direction of particle motion. A substitution of Eq. (13) into Eq. (11) leads to the generalized Christoffel’s equations:

[ Cljklp,p, + [aaPiP, - pv

2 )6jk ]dk = 0. (14) The eigenvalues of Eq. (14) are pv

2 - crilPiPl , and the eigenvectors are the particle displacement vectors dk

. It is tempting to rewrite Eq. (14) in the form

[[Cijki + tra $jk)PiPi ~ pv

2 ]d, = 0, (15) and define the effective stiffness tensor

Cfki = CiJkl -(- a„8Jk . (16)

where 12

y

is the local rigid body rotation. In general, for propagation directions that do not lie in the planes of symmetry, the eigenvalue problem of Eq. 14' cannot be solved in closed form. However, numerical solu- tions can be obtained quite easily. Once the stress tensor ar is specified, the tensor Ci]kl can be calculated. Then, for each

propagation direction p„ the product C,jklp,p, can be obtained, resulting in a real three by three matrix whose eigenvalues and eigenvectors can be found. Computational sub- procedures for solving this type of eigenvalue problem are readily available. 18 Closed-form solutions of the eigenvalue problem can be

obtained when the propagation direction is parallel to the material symmetry axes and thex,, x2 andx, axes are parallel to the principal stress directions. The solutions are de- rived in the Appendix and the results are given by Eqs. 4i

( 6). Closed-form solutions of the eigenvalue problem arc

also obtainable when the propagation direction lies in i plane of material symmetry. This is a consequence of the fact that in the planes of material symmetry, the wave whose

particle displacements are normal to the plane of material

symmetry is a pure mode. Planes of material symmetrs in the absence of stress will remain planes of material symmetry in the presence of stress when the principal stress dire^ tions are parallel to the material symmetry axes Consider a plane wave propagating in the x x . plane in a medium of orthorhombic symmetry where v , . s and s ire the principal stress directions. In this case the eigenvalue

3029 J. Appl. Phys., Vol. 54, No. 6, June 1983 21 R B King and C M Fortunko 3029
problem Eq. (14) simplifies to

{.{CijkiPi + 2Cljk iP2Pi + Cijk 3 p\

)

+ [ ^2lPl + <?3lPl - pv

1

] }dk = 0. ( 21 ) The elements of the matrix enclosed by the parentheses vanish when K — 2,j^ 1, orj = 1, k ^ 1. This can be verified by

using Eq. (18) and examining the symmetry of the tensors Cljk , and Cijklmn . The symmetry properties of the tensor Cijkimn are given in Ref. 19. The SH wave, which is denoted

by = 1 , d2 — 0, d3 = 0, is then a pure mode that can easily

be split off and treated separately. The velocity of this mode

is given by

pv~ = C2X x2p\ 4 2C2i \ 1P2Pi 4- C31 1 3/^3 4- &22P2 4- &13P1 >(22)

where C21 13 vanishes as a result of the assumed symmetry.

Next, making use of Eq. ( 1 8) and noting that in the principal coordinates x lt x2 , and x3 dx

—7- = Slk ( 1 +Ea ) (no sum on /), (23) dXk

it can be shown that

SH223 [ C'66( 1 +2E\ X + 2E22 ) Po

+ C-66\(E\ I + •E'22 )] + E66iE33 + ^*22 ] +

cos

2

Po x [C55 (l + 2£',| + 2E33 ) + ^66 1(^11 ~b E33 ) + C44X E22 4 <J33 ]

,

(24)

where d> is defined as the angle between the propagation di- rection and thex, axis, and the contracted (Voigt’s) notation

is used. For further convenience, a new notation is intro- duced: SH,y is taken to denote the velocity of the SH wave

propagating in thex,-xy plane.

The eigenvalue problem then reduces to a system of two

equations for j, k = 2, 3. Upon expanding, a quadratic is obtained for the eigenvalue [pv

2 — cr xxp\-a33p\), which can

be solved for the velocity in closed form. However, obtaining

a solution is tedious and not required for subsequent devel- opment.

In a manner analogous to that used in deriving Eq. (24),

it can be shown that SH2

3 = ^ [Q6

(l + 2£'

11 4- 2is 22 ) Po + C66l (E, , 4- E22 ) 4- C663E33 -f <7, , ] + [C44( 1 + 2E22 + 2E33 ) + C44XEu

Po

+ C(>(,\(E22 + £"33) + cr33 ] . (25)

For experimental purposes, it is convenient to express

Eqs. (24) and (25) in terms of differences in the velocities. Only terms to first order in the difference between the velocities in the absence and presence of stress are retained. Velocities in the unstressed state are denoted by the superscript 0. The full derivation is given in the Appendix and only the

results are repeated below:

SH23 - SH 13 . SH° U

+ c{<i> )(a2 + £7,), and

( 26)

SH23 + SH| 3 - 2 SH"

2 SH°

*=b(d> )(cr

2 + a,)

+ d (<f> )(cr2 — a,). (27) For weak anisotropies, c and d are the first order in e, where e is a parameter characterizing the anisotropy [e.g., C55 = C44 (l 4 e)]. Thus, c and d can be neglected.

In the above discussion SH° denotes the mean unstressed velocity, which is explicitly given by

sho = 1 [7 C44 cos

2 & 4 QftSin

2

(j> V

/2

2 IA po J + ( C55cos

2

4> 4 c66sin

2

<j)

b po

An estimate of SH" may be used in the denominator of

Eqs. (26) and (27), because a small error in the denominator

results in a higher order error in the calculated stress states However, the exact value of SH° is needed in the numerator

of Eq. (27) because a small error in the numerator results in a

large error in the calculated stress states. Equation (26) is a generalization of the acoustical bire- fringence relation [Eq. (5)] and, in fact, reduces to it for the

special case d) = 0. Similarly, Eq. (27) is a generalization of Eq. (6). The importance of Eqs. (26) and (27) is that the same unknowns are present regardless of the value of <p. As a con- sequence, measurements at different values of d provide additional information without increasing the number of unk- nowns. Equation (26) is particularly important because only

two unknowns are present: (C55 — C44 ) and [a2 — £7,). Thus,

measurements at only two values of <f> are required to evaluate the quantity (£72 — £7,).

EXPERIMENTAL PROCEDURE

It follows from the preceding discussion that the residual stress state and the relevant elastic constants can be

uniquely determined by making precise velocity measure- ments with off-axis SH waves in the planes of material symmetry. To separate the effects of texture and stress, measure- ments must be made in mutually orthogonal material

symmetry planes.

It is straightforward to determine the difference between two principal stresses by making relative measure- ments and using Eq. (26). In this case, the procedure involves

first making a velocity measurement of an SH wave propagating in the x2 -x 3 plane at an angle 4> with respect to x3 axis. Next, the measurement is repeated in thex,-x3 plane for the same angle Both measurements must then be repeated for a different value of <t>. The difference between the two measurements is then used in Eq. (26) to evaluate (cr-, — £7, ) and

the quantity (C55 — C44 ). (In many technological applications the knowledge of (a2 — £7,) is sufficient because addi3030 J. Appl. Phys., Vol. 54, No. 6, June 1 983 22

R. B. King and C. M. Fortunko 3030
tional information related to the specimen geometry is usually as ailahle to estimate the relative magnitudes of a2 and cr, ). In principle, the separate values of a2 and cr, can be

determined experimentally by using Eqs. (26) and (27). However. m this case a set of five independent measurements

must be performed at different values of 6. The five measure- ments are needed to determine the values of <7,, a2 , C^, C55 , and C. , In practice, an additional measurement may be re- quired to compensate for any path length variations.

In practical applications, small errors e,, e2 can be present in the \alues of SH; , — SH I3/SH" measured at angles <b x and 6 . respectively, and errors e„ e4 can be present in a[d>\) and a\d> i in Eq. 26|. Inverting the two equations obtained

from the measurements at and d2 to evaluate a2 — cr, re- sults in

( a2 - a,) = |(7-> — <7, 1 +

S] C2 -S2Cl e4 £7-,U, e^d , U-,

£7 , fl.C |

(29)

where (cr

: — cr, ) cx

is the exact value of <7-, — cr,; <5,, 52 are the

values of iSH; 3 — SH,,)/SH" at 6\ and 62, a, = a[6t ), and

C = cos

2 i<£, ). The term in the parentheses represents the rel- ative error in stress caused by the accumulation of errors in velocity measurements and calibrations. The most severe

case of error magnification occurs when e, = — e2 and = e4

. In this case, Eq. (29) becomes

(cr2 — <7,

)

= (cr2 — <7,) 1 +

g,C2 + g2 C1

L 8,C,-8,C,

q2 C

i

“l-

£7

| C->

£7 , C, £7 •> C1

(30)

To obtain realistic numbers for insertion in Eq. (30), anisotropic second-order elastic constant values for steel and

aluminum were obtained by slightly perturbing nominal iso- tropic values. The third-order elastic tensor was taken to be

isotropic, and the third-order constants /, m, and n were

taken from Ref. 24. The acoustoelastic constants a[6 ) were

calculated from the second-and third-order elastic constants

using the expression in the Appendix. Using the values for aluminum, for 6 X = 10° and 62 = 15°, the error term then

becomes — 6 If, — 7 ley. It follows that errors in the relative velocity measurements or in the calibration of the constants

a\tb ) cause errors of two orders of magnitude larger in a2 — ay

. If d ] = 10° and 62 = 20°, the error term is — 9e, — lle2

. Thus, the error magnification is greatly re- duced as the angular difference between two successive measurements is increased. (Similar results were obtained using

the elastic constants for steel.) Measurement imprecision of less than 1 % can be obtained with the relative velocity measurement system used in the present work. From the above, it follows that (£72 — cr,) can be determined with an error of less than 10%. To apply

Eqs. (26) and ( 27), the functional forms ofa[6 I and b \<b I must

be known in advance. As shown in the Appendix, the knowledgeofo(<£ land& \6 iat<£ = 0° and 90° is all that is required to

determine the unique functional forms for a{6 land/? \d> 1 . Unfortunately, published values of third-order elastic constants

to evaluate a{6 ) and b (d> ) at 6 — 0° and 90° are not applicable

because nominally similar engineering alloys can exhibit large variations of the acoustoelastic constants.

10 20 As a consequence, the values of the acoustoelastic constants must be

calibrated directly on the material under study. In principle, the required calibration can be performed on a separate reference sample subjected to known applied

stresses, but nominally of the same composition as the speci- men whose residual stress state is to be determined. However, slight variations in the values of the acoustoelastic constants may still be present in the specimen to be examined.

This uncertainty would normally be expected to result in errors of the same order in the calculated values of cr, and <7:

. However, as discussed below, magnification of error can oc- cur if measurements at different angles are used to evaluate

stresses. If errors are expected to be of such a magnitude that they cannot be tolerated, then an in situ calibration tech- nique of Scott can be used.

22

PRACTICAL EXPERIMENTAL CONFIGURATION

To obtain experimental data for utilizing Eqs. l26| and

(27), transducers of SH waves are required that produce ul- trasonic beam patterns with wide angular distributions. Recent technological advances have made it feasible to construct such transducers. A practical transducer design is shown in Fig. 1. The transducer configuration as shown in Fig. 1 is de- rived from the periodic permanent magnet EMAT that has

been developed for nondestructive evaluation NDE appli- cations. 23,24 However, to produce a compact impulse re- sponse and ultrasonic beams with a wide angular pattern, the transducer of Fig. 1 employs only one \M = 1 pair of permanent magnets. The principles of operation and design details are described in detail elsewhere. 24 25 Instead, in this paper,

attention is focused on the ability of the new EMAT configuration to produce wide angular beam patterns and. by reci- procity, to receive ultrasonic signals from a w ide range of

angles.

Initial Deformed

FIG. 1. Geometry used for acousloclasliciiy derivaii< n

3031 J. Appl. Phys., Vol. 54, No. 6, June 1983 23

R B King and C M Fortunk 3031
Sm-Co

Magnet

FIG. 2. Practical electromagnetic-acoustic transducer design.

A representative ultrasonic radiation pattern of the EMAT depicted in Fig. 1 is shown in Fig. 2. In Fig. 2, the transmitted ultrasonic amplitude (at 520 kHz) is plotted as a function of the angles [4> ) in the plane normal to the surface of

a plate and bisecting the transducer between the magnets

(the sagittal plane). It is evident that the radiation pattern is very broad and, in particular, does not vanish at any angle in the sagittal plane. By reciprocity, the same directivity pat- tern applies upon reception. As a consequence, in the sagittal

plane, the simple EMAT of Fig. 1 acts in a manner similar to that of a finite line-source of SH waves that is positioned on

the surface of the plate. This property of the transducer is needed to study the velocities of ultrasonic waves as a function of propagation direction for two reasons. First, it per- mits the collection of ultrasonic velocity data from a wide

range of angles. Second, it reduces the influence of diffrac- tion effects, which must often be taken into account in accurate velocity measurements.

25-27 (The diffraction properties

of elementary and practical SH-wave EMATs are discussed

in detail by Pardee and Thompson. 28

) A practical experimental configuration for determining

stress states in anisotropic plates is shown in Fig. 3. In the configuration of Fig. 3 two SH-wave EMATs are used. One

4

* (degrees)

FIG. 3. Ultrasonic radiation pattern of electromagnetic-acoustic trans- ducer.

of the EMATs is used as a generator and the other as a re- ceiver of the probing ultrasonic signals. Because the two

EMATs are aligned along the same sagittal plane, the receiv- er EMAT is principally sensitive to ultrasonic signals with a displacement component parallel to the surfaces of the plate

and normal to the sagittal plane (SH waves). In the special case when one of the principal stress di- rections is parallel to a crystal axis and the sagittal plane is also a plane of crystal symmetry, the SH-wave signals propagate are pure modes. Since SH-waves are decoupled from

longitudinal (L) and vertically polarized shear (SV) waves upon reflection, they are not phase shifted upon reflection from the surfaces of the plate. This important observation

results in a significant simplification in the inversion of the experimental data and forms the basis for the experimental

determination of residual stress states in planar geometries.

An examination of Fig. 3 reveals that many independent ultransonic signal paths (rays) exist that connect the

centers of the generator and receiver EMATs. In fact, useful experimental information is available over a range of angles

<J), from grazing (^ = 90°) to nearly normal (<6 — 0°). As a con- sequence, by measuring the phases of a number of the ultra- sonic signals in planes of symmetry, sufficient information is provided to permit the unambiguous determination of the

in-plane stress components, elastic constants and adjustment for plate thickness variations, if needed.

EXPERIMENTAL VERIFICATION

To verify the. experimental procedure, measurements

were carried out on an aluminum plate specimen, 25 mm

thick, which contained a known biaxial stress state and un- known initial anisotropy. Aluminum was chosen for this study because it is typical of engineering materials that al- though nominally elastically isotropic, exhibit a small amount of anisotropy. The anisotropy causes an effect on

velocity that is not negligible in comparison to the effect of

stress. The design of the experimental specimen is shown in Fig. 5. The test plate was square, 686 mm on each side, and

made*of type 6061-T6 aluminum. In the center of the plate is a hole into which was inserted a circular plug, 1 02 mm diameter, made of type 304 stainless steel. Prior to insertion, the

stainless steel plug was cooled to 77 K and the aluminum test plate heated to 423 K. After insertion, the resulting strains were monitored with strain gauges placed along radial lines until the whole test specimen was returned to the ambient

temperature. The strain gauge readings confirmed the theoretically predicted stress distribution. To obtain ultrasonic data, the specimen shown in Fig. 5 was scanned using the experimental SH-wave EMAT configuration, as shown in Fig. 4. The scanning was carried out along one of the radial lines that coincide with the axis of

plate symmetry, as shown in Fig. 5. To measure the phases of

the ultrasonic signals, a modified system of Kino et al.

1

1

was

used. In this system the relative frequency change required

to maintain a constant phase difference between an acoustic

signal and a reference signal is measured. Relative velocity

changes can then be evaluated from the relation

3032 J. Appl. Rhys., Vol. 54, No. 6, June 1983 24

R. B. King and C. M. Fortunko 3032
Transmitter Receiver 20

FIG. 4. Transducer configuration for stress measurement.

AV _ Af At

V~f t

’ (31)

where f denotes frequency and t plate thickness. Measurements were made using 850-kHz SH-wave probing signals

with a 10% fractional bandwidth. It should be added that the scanning was performed manually, using a grid laid out on the surface of the test specimen without the aid of mechanical positioning mechanisms.

The relative velocity differences (SH23 — SH 13 )/SH°

were calculated for several positions along the scan line, with

the x2 axis taken to coincide with the plate rolling direction. The ultrasonic data were obtained from the first and third

arrival signals, as shown in Fig. 4, with the SH-wave EMAT

pair aimed along the scan line, and, then, at 90° with respect

to the scan line. The first and third arrivals were selected because of the large difference in the angle defined by their rays and the plate surface normal, 33.8° and 12.6°. The differ- ence in the principal stresses, cr2 — av was then evaluated

from the velocity measurements using Eq. (26). As discussed

in the preceding section, the constant c[<t> ) was neglected be- cause of the weak anisotropy. The values of the constants

u(<^,) and a[ct) 2 ) were determined independently using tensile calibration specimens obtained from the ends of the test block shown in Fig. 5. Tension was applied parallel to the

rolling direction in a servohydraulic testing machine. Measurements of relative frequency change as a function of applied stress for an SH-wave polarized parallel and perpendicular to the rolling direction were made. The results are plotted in Fig. 6. The values of the constant a# ) is given by

FIG. 5. Specimen containing a known residual stress state.

FIG. 6. Typical calibration curves for acoustoelastic constants.

the differences of the slopes of the line plotted. Measure- ments were taken using the first and third echoes from the back surface of the specimen. The calibration values deter- mined were c(33.8°) = 3.7X 10~5 MPa,

a(12.6°) = 4.4 x 10 5 MPa. The calibration constants were found to vary with position, but the variations were less than

1%. Therefore, they were judged acceptable and in situ call bration was not needed. The experimental results for the ultrasonically measured values of a2 — cr, are shown in Fig.

7. For comparison, the calculated values of {a:

- a,i from

the elasticity solution are also shown in Fig. 7. The calculat- ed distribution is parabolic, and the experimental results are seen to follow this shape.

As an additional comparison, the experimental results obtained with obliquely propagating SH waves were related to results that would have been obtained using one of the conventional procedures, based on acoustical birefringence

which are also plotted in Fig. 7. It was assumed that the texture was homogeneous and could be characterized using

measurements on unstressed reference specimens. Corre

spondingly, the value of (F52 °— ys] °)/Vs

° in Eq 5 obtained using an unstressed specimen. Then, the values !

( VS2 — VS] )/Vs

° were measured on thetest specimen shown

in Fig. 4. It is clear that the assumption of homogeneous

texture results in poor agreement between the ultras in, measurements and the calculated stress state I his result was expected because the texture was measured ult ras, >mcal

ly in an unstressed plate of specimen material and bund t< be inhomogeneous. The values obtained using the emeu

tional procedure were lower than the value' ’blamed u n theory and strain gauges. Therefore, it is belies d that tie

R B King and C M Fortunko 3033 3033 J. Appl. Phys., Vol. 54, No. 6, June 1983 25
b

DISTANCE FROM PLUG BOUNDARY.mm

FIG. 7. Principal difference evaluated in specimen of Fig. 5.

new experimental procedure would be inherently more ac- curate than the conventional procedure in practical engineering applications. In many such applications (e.g., welding and forming) significant variations of texture with

position are expected.

DISCUSSION

Closed-form expressions relating the velocities of elastic waves propagating at arbitrary angles in the planes of

symmetry of weakly anisotropic, polycrystalline solids have

been presented. In particular, it has been shown that using

measurements of the velocities of pure shear waves as a function of propagation direction, sufficient information is obtained to permit the evaluation of residual stress states in a

practical plate geometry. In fact, measurements at only two

independent propagation directions are required to obtain

the difference in the two in-plane principal stress components. Furthermore, an experimental procedure, based on

the new theory, has been proposed and evaluated using a new SH-wave EMAT design. However, the new experimental procedure is subject to certain limitations.

( 1 ) One of the principal assumptions of the new theory is that the principal stress directions are parallel to the crystal axes of the unstressed anisotropic solid. If this condition is violated the SH waves in a plate are no longer pure shear waves and the theory cannot be assumed to be correct. However, in many applications of technological importance the assumption of coincidence of the symmetry planes with the

principal stress directions is valid. Prominent examples in- volve most butt weldments in which the rolling direction is either perpendicular or parallel to the welding direction. In

addition, this assumption is also present in the commonly

used acoustical birefringence method, and thus does not limit the proposed procedure more than conventional procedures.

(2) Measurements using SH-wave signals propagating

at two different angles with respect to the plate normal are required to obtain the difference of the two principal stresses

cr, and cr2 in the presence of unknown anisotropy. The accu- racy of such measurements increases in proportion to the angular difference between the rays corresponding to the two ultrasonic signals, because the matrix relating (a2 — a,) and anisotropic effects becomes singular for A<p — 0. However, it has been shown that an acceptable accuracy ( ~ 10%)

can be attained if the angular difference between the two SHwave rays is 20° or greater. As a consequence, the spatial resolution of the new experimental system is directly deter- mined by the thickness of the test specimen.

In conclusion, an experimental procedure has been

demonstrated that can be used to determine the residual

stress states in plate geometries. This is an important step in extending the theory of acoustoelasticity to practical applications of major technological importance.

ACKNOWLEDGMENTS

The authors are indebted to S. A. Padget, J. Arvidson,

Jr., and R. A. Schramm, who participated in the construction of the experimental apparatus and in the experiments

described. The authors are also indebted to J. D. McColskey

who prepared the specimens used in the experiments. This

work was supported in part by the U. S. Navy, David Taylor

Naval Ship Research and Development Center, and the Office of Nondestructive Evaluation of the National Bureau of

Standards.

APPENDIX: DERIVATION OF ACOUSTOELASTIC

RELATIONS FOR A MEDIUM OF ORTHORHOMBIC

SYMMETRY

Explicit formulae for velocities of waves propagating in specific directions are obtained by substituting for p, in Eq.

(14) and making use of Eq. (18). For the case of a wave propagating parallel to the x3 crystal axis, Eq. (14) reduces to

[ c3jk 3 + (<733 - pv

2 )8Jk ] dk = 0. (32) Furthermore, by considering the symmetry of the tensors Cjjk , and Cijklmn it can be shown that the elements of the matrix in brackets vanish for j^k.2A Thus, there are three uncoupled modes whose velocities are given by

v

2,= - [C33( 1 + 4E33 ) + <733 + C33] Eu

Po+ C332^22 + C333E33],

Vl 2 — [£44(1 + 2E33 + 2E22 ) + cr33 + C44i Eu

PoT C442E22 + C443E33 \, (33) V

2s 1 = [ C55 ( 1 + 2E33 + 2EX ! ) + <733 + C55 X E,

,

Po+ C552E22 + ^553^33 ]

• The relative velocity difference expressions [Eqs. (4)—(6) in the text) are derived by using Eq. (33) to obtain linearized expressions for V,, VS2 , and FS 3

. Thus, for example,

3034 J. Appl. Phys., Vol. 54, No. 6, June 1 983 26R. B. King and C. M. Fortunko 3034
—' ' = — 1 4C33(5, 3 4 S23 ) + C33 ,(5, | + 5, 2 ) v, 4C3 j + C + 522 ) 4 <Z333 (5, 3 + *5*23) ] (^1 + ^2)

+ —— 1 4C33 (5, 3 — 523 ) + C33 ,(5, , — 5, 2 ) 4C33 4- C33 2(*S |2 — ‘S'22 ) 4 C333(5, 3 — 523 )](<7, <+>)> (34)

where StJ is the contracted matrix form of the compliance

tensor Sijk ,

. The coefficients offer, + a2 ) and (cr, — a2 ) are the constants A and D in Eq. (4). For an isotropic material

5*12 = 5,3 = 523 , 5,, =512 , and C33I = C332 so D = 0. For

I

weak anisotropy, D is of the order e, where e is a small pa- rameter characterizing the degree anisotropy (e.g., e = C55 — C44). Similarly, Eqs. (5) and (6) in the text are obtained with E and F negligible for weak anisotropy.

Expressions for the square of velocities of SH waves

propagating in tht Xj — X3 and X2 — X3 planes are presented in the text [Eqs. (24) and (25)]. Velocity difference expressions are obtained from these by deriving linearized expressions for the velocities and introducing the compliance

tensor to relate strain to stress. The result for the difference between the SH-wave velocities in the X2 -X3 and X,-X3 planes is

SH23 -SH, 3 =

SH23°-SH,3

°

. _ , ab

SH° * SH°

* 2 (a + b)

X [S(a - b )(2C6658 + C66 ,55 + C66256 + C6635 7) -(a + b)]

4 C {a [2C44(5 ,

1

] 4 5, 3 — 5,2 — + C55 \S 5 4 C552S 6 + C553S 7] — b [ 2C44(s, 2 + 5, 3 522 -*23 ) 4 C44 ,5 5 C4425 6 4- C443S 7 ] (cr2 <j,

)

4 2 — [5 (a — b ){ 1 + 2C66(5, , 4 522 + 2s 12 ) (a + b)

4 C66 ,52 + C6625 3 + C6635 4} + C {a[2C55 (5, , +51)

+ C55 ,52 + C552S 3 + C553S 4] - b [2 C44 (s22 + 5 1) + C44 ,52 + C4425 3 + C4435 4]}](ct2 + a,), (35)

where 5 = sin

2

</), C = cos

7't, a — \

>[ Aj(Q(A 4-C55 C) _ 12’ b = |[po(Q^4C44 C)]-i, 51 = 5 12 + 523 + *13> 52 = 5, | + ^tzi 53 = *12 + **22’ 54 = 5 )3 4 •*23’ 55 = s \ 1 5 12> 56 = S 12 S22< 57 = 5 13 — •*23> 58 = 5,

,

— 522 . The coefficients of cr2 - and 1j2 + <7, in Eq. (35) are theconstantsa(<^ )andc(<^ (presented in the text. For isotropic

materials, C55 = C44, 1 — &22> C55 I — ^442’ C44 I — ^552> and C553 = C443, so c(<f> ) = 0. For weak anisotropy, c(4> ) is of order (e) and may be

neglected and Eq. (35) reduces to SH,3 — SH, 3 —— , C44 — C55 2 ± 11 \ L = — cos- <p + a(4> ; SH° '* 2 C44

a[(f))— — [sin 2 (p — cos

2 (5,, — 5, 2 ) C44 X(2C55 + C551 -C552 )] +0(6). (36)

In a similar manner Eq. (37) in the text is derived, and it is found d (tf> ) is negligible for weak anisotropy.

'D. S. Hughes and J. L. Kelly, Phys. Rev. 92, 1 145 (1953).

2 A. Seeger and O. Buck, Z. Naturforscher A15, 1056 (1960).

3 R. N. Thurston and K. Brugger, Phys. Rev. A 133, 1604 (1965).

J R. N. Thruston, J. Acoust. Soc. Am. 37, 348 (1965).

5 T. Tokuoka and M. Saito, J. Acoust. Soc. Am. 5, 1241 (1971).

60. I. Gushcha and F. G. Makhort, Prikl. Mekh. 12, 32 (1976).

r

7 E. G. Henneke II and R. E. Green, Jr., J. Acoust. Soc. Am 45, 1367 1 1969

"H. Fukuoka, H. Toda, and T. Yamane, Exp. Mech. 18. 227 ( 1 978 1

. g N. N. Hsu, Exp. Mech. 14, 169 (1974).

I0 E. Schneider, K. Goebbels, and G. Hubschen, Rev. Cethedec 17. 155

(1980). "M. James and O. Buck, CRC Crit. Rev. Solid State Mater. Sci 9. 61

(1980).

I2 G. S. Kino, J. B. Hunter, G. C. Johnson, A. R Selfridge. D. M. Barnett. G. Hermann, and C. R. Steele, J. Appl. Phys. 50, 2607 1 1 979)

l 3 M. P. Scott, D. M. Barnett, and D. B. Ilic, in 1979 Ultrasonics Symposium

Proceedings, edited by J. de Klerk and B. R. McAvoy (IEEE, New York.

1980), p. 265.

I4 A. N. Guz, F. G. Makhort, O. I. Gushcha, and V K. Lebedev, Prikl Mekh. 7, 1 20S (1971).

I5 L. Niklas, Material Prufung 7, 281 (1965)

lhT. Tokuoka and Y. Iwashimizu, Int. J. Solids Struct 9. 383 1 1 968 1 l7 K. Brugger, Phys. Rev. A133, 1611 (1964). '"Computer Code EIGRS (International Mathematical and Statistical Li- braries, Houston, Texas, 1980).

I9 R. Bechmann, R. F. Hearmon, and S. K. Kurtz, Elastic, Piezoelectric and

Related Constants ofCrystals [ Springer, Berlin, 1969), Vol. 2. pp 102-123

20D. I. Crecraft, J. Sound Vib. 5, 173 (1967).

2I G. C. Johnson, Doctoral Dissertation, Stanford University . 1 1 979

22 M. P. Scott, Doctoral Dissertation, Stanford University, 1981

21 C. M. Fortunko, R. B. King, and N. Tan, J. Appl. Phys 53. 345( 1 24C. F. Vasile and R. B. Thompson, J. Appl. Phys. 50, 2853 1 1979|.

25 D. B. Ilic, G. S. Kino, A. R. Selfridge, and F. R. Stanke. in 1979 Ultra

ics Symposium Proceedings, edited by J. deKerk and B R M, \\ > (IEEE, New York, 1980), p. 269.

26 H. J. McSkimin, J. Acoust. Soc. Am. 35, 490 1 1963

1

27 E. P. Papadakis, J. Acoust. Soc. Am 40. 863 1 1966i

2XW. F. Pardee and R. B. Thompson, J Nondestr Eval 1 157(1980

3035 J. Appl. Phys., Vol. 54, No. 6, June 1983 27

R B King and C M Fortunko 3035

ARBITRARY PLANE RESIDUAL-STRESS

MEASUREMENTS (1983)

29

Acoustoelastic evaluation of arbitrary

plane residual stress states in nonhomogeneous, anisotropic plates

R.B. KING and C.M. FORTUNKO

In this paper it is shown that relative ultrasonic velocity measurements can be

used to determine the difference of normal stress components (T22 ~ Tu ) in non-homogeneous, anisotropic plates containing arbitrary residual stress states.

It has previously been demonstrated that measurements with horizontally

polarized shear waves (SH-waves) can be used to determine {T22 - Tu ) unambiguously in plates with unknown material property variations. However,

previously the theory relating the velocity of SH-waves to stress and material

anisotropy was limited to the case where the principal directions of stress are

parallel to the axes of material symmetry. In this paper, we remove this

restriction by extending the theory. We also suggest possible experimental

procedures for validating the new theory.

KEYWORDS: ultrasonics, stress measurement, anisotropic plates

Introduction

In a previous publication, we have addressed the problem of measuring two-dimensional residual stress distributions in non-homogeneous. anisotropic metal

polycrystals 1. By non-homogeneous we mean that the

material properties vary with position, resulting in a variation of the wave speeds in the unstressed state. Specifically, we have developed a special theory of

acoustoelasticity to describe the effects of in-plane

residual stress states in plates on the velocities of shear waves polarized parallel to the surfaces and propagating obliquely with respect to the surface normal

(SH-waves). To develop the new theory, we have

assumed that the axes of symmetry of the textured,

polycrystalline aggregate coincide with the principal

stress directions. This assumption was needed to obtain

simple, closed-form expressions describing the effects of stress and elastic anisotropy on the velocities of SHwaves in the planes of symmetry.

We verified the theory using a new experimental

procedure for evaluating the difference of the two in- plane principal stress components. (T22 — Tn ), in the coordinate system of Fig. 1. (Tn

is the stress tensor.)

In this paper, we extend the new theory to the case of a more general stress state in a plate. In particular, we

treat the case where the in-plane principal stress directions are not coincident with the in-plane material

coordinates, as indicated in Fig. 1. Method

In Ref. 1. we derived the following expression for the

relative velocity of SH-waves

The authors are in the Fracture and Deformation Division. National Bureau

of Standards. Boulder, Colorado 80303, USA. Paper received 1 3 June

1983. Contribution of the National Bureau of Standards, not subiect to copyright.

SH23 - SH; SH^~

cos

2 0 +a(0)(T22 - Tn ) (1)

where SH,, denotes the velocity of a plane SH-wave

that propagates in the x, - x, plane, as defined in Fig. I: the angle d is the acute angle between the direction of

propagation and the normal to the surface; C„ is the

elastic constant tensor in contracted (Voigt) notation:

and a(6) is a calibration constant that is determined b\ the second and third-order elastic constants.

In deriving (1) we made the following assumptions: I the principal stress directions and material symmetry

axes coincide, and 2. only plane stress is present. It should be noted that coincidence of material symmetry

axes and principal stress directions occurs only when

Fig. 1 Geometry of plate defining material a«et « . . <

,

C55 - C44

Q ^44

0041 -624X/83/060256-03/$03.00 © 1983 Butterworth 8- Co (Publishers) Ltd 256 ULTRASONICS NOVEMBER 1 983

31
Tn is zero. As a consequence of the second

assumption, the stress components Tu, T2y and T}J vanish.

For the special case of grazing incidence (8 — rr/2). the anisotropy term in (1) vanishes. As a result, it can be

rewritten as

= a(7r/2) {Tu - Tn ) (2)

n/2

where

a(n/2) = 1/(2 C66 )

The special result of (2), which holds for plane SHwaves propagating at grazing incidence, has been

proposed as the basis for evaluating residual stress

states in plate-like geometries by Thompson et al2

. However, the usefulness of the result is restricted by the assumption that the material symmetry axes and

principal stress directions coincide. In this section, we

extend the theory to include more general cases of

stress distributions in plate-like geometries.

We follow the general theory of acoustoelasticity

developed by Tokuoka and Iwashimizu3

. Thus, the propagation of a plane wave in a stressed continuum is described by the following eigenvalue relationships

[SijkiPiPi + (Tu Pi Pi ~ pv*)$jk}dk = 0 (3)

where p, is the unit normal to the wavefront. dk

is a unit vector in the direction of the particle motion, p is the material density, v is the wave phase velocity, and

Sijkl ~

P/Po (Cmnpq ^-mnpqn^rs)

dx,- dx, dx k dx, — —— (4)

where p0 denotes the material density in the unstressed

state; x, and X, are the position vectors in the stressed and unstressed states respectively; £,; is the strain tensor; and Q*/ and Cjjkimn are the second and third order stiffness tensors. We assume the tensors Cl/ki and

Cjjkimn exhibit orthorhombic symmetry. (The general

symmetry properties of C,jkimn are described in Ref. 4.) A summation from 1 to 3 is implied when the indices

are repeated. To evaluate (4), we find it convenient to introduce the relation

~

(5 y + Eij + !V,y) (5)

where fV

q

is the local rigid body rotation tensor. We first consider the case of a plane SH-wave propagation in thex2 — x, plane. In this case, p, = 0, Pi = sin#, and p 3 = cos#. We rewrite (3) using a simplified matrix notation

[Ajk ~

where

- p0 v

2 S,k] dk = 0 (6)

Ajk = (Sjki Pi Pi "b Tp pi pi 8jk ) (7)

From an expansion of (6) we obtain

(/In ~ Pov

1

) l(A u ~ p0 v

2

) (A }} - p0 v

2

) - A

2 n | ~A l 2 [A l 2(A }} —

p0 v

2

) A 23A l} ]

~b An [A ij A 2} (A 22 Po v

2 Mn] = 0 ( 8 )

Next, from (3). (4), and (7) we obtain the following

A 12 [(C 12 "b C22 + 2C56 4- C266 )El2 + (C22 - C,2 - 2C66 ) IF.21 sin2 0 + [(C44 — C55 + C456 ) El2 + (C44 — CSJ ) IFn] cos

2 9 (9)

and

A 13 2 sin# cos^[(C 1J 4- C23 + CJ66 ) El2 + (C23 - c13 ) (F12 ] (10)

Noting that T 12 and A n are of one order higher than

A U,A 2 2

, and/ljj in the displacement gradients, we

obtain the following approximation to (8)

(A u -

Po v

2

) (A 22 - Po v

2

) (A }2 - Po v’ 2

) = 0 (11)

The solution corresponding to the SH-wave is

P0 v

2 = A n (12)

Now. we introduce the notation 57/,, to denote the

velocity of an SH-wave in the x, — xf plane. From (3). (4) and (7) we obtain in the.tj — x2 plane

Po S//2 23 = [C66(l + 2En + 2E22 ) + CfaEj + r2 2] sin2 9 + [C3J (

1 + 2

£

n + 2£33 ) + C55, Ej + T3 3 ] cos

2 9 (13)

Similarly, in the.v, — x } plane we obtain

p^13 = [C66(l + 2En 2Eu) + C66i Ej + r„] sin2 ^ + (C44(

1 + 2 £33 + 2 £22) + C44,£, + r33 ] cos

2 9

(14)

Conclusion

Equations 13 and 14 are identical to those presented in

Ref. 1 for the special case Tn = 0. Thus, to first order

in displacement gradients, the presence of non-zero Tu

produces no effect on the velocities of SH-waves

propagating in the planes of material symmetry.

To obtain an expression for experimental evaluation, we developed a linearized velocity difference equation

from (13) and (14). In addition we assumed weak

anisotropy and expressed strain in terms of stresses using Hooke's law. After algebraic simplification we

obtained

(SH2i - SHi} )/SH0 = cos

2 0(CSS - C44 )/2C66 + (Tn Tn ) [sin2 # cos

2 #(C441 C551 )/2C64 ]/2C66

(15)

SH23 — SH!3 SH~o

ULTRASONICS . NOVEMBER 1983

32

257
For weak anisotropies, the quantity ii, M1 — 331j» 13 approximately equal to (-n/2), where n is an isotropic

Mumaghan constant Equation (15) is identical to (1), with a(d) explicitly given by

a(6) = [sinJ 0 - cos

J 0 n/4C66 ]/2C66 (16)

Furthermore, an error magnification effect that limits the precision of stress evaluation with obliquely

incident waves' is not present for grazing SH-waves.

However, to our knowledge an accurate experimental

procedure based on the use of grazing SH-waves has

not yet been experimentally demonstrated.

The authors have learned by private communication

that a similar theory to that presented in this note has

been derived by A.V. Clark. He has investigated the more general case where the stress components T13 and

1 23 are non-vanishing and has also derived an

interesting restriction on the theory leading to (15) for SH-waves approaching normal incidence.

Acknowledgements

This work was supported by the National Bureau of

Standards. Office of Nondestructive Evaluation. The

authors wish to express their appreciation to JoAnne

Wilken for typing the manuscript

References

1 King, R.B., Fortunko, C.M. J. AppL Phys. in press

2 Thompson, R.B., Smith, J.F., Lee, S.S. unpublished.

3 Tokuoka, T., Iwashimizu, Y. Ini J. Solids Structures. 9 (1968) 383

4 Bechmann, R., Hearmon, R.F., Kurtz, S.K. Elastic

Piezoelectric, and Related Constants of Crystals SpringerVerlag. Berlin (1969) 2. 102-123

258

33

ULTRASONICS . NOVEMBER 1983
-
Evaluation of Surface Residual Stresses Using

Horizontally Polarized Shear Waves At Grazing Incidence

R.B. King and C.M. Fortunko

Fracture and Deformation Division

National Bureau of Standards

Boulder, Colorado 80303

to be published by: Journal of Applied Physics

Key words: acoustoelasticity; anisotropy; elastic waves; geometrical

nondestructive evaluation; residual stress.

acoustics;

Contributions of National

the U.S.

Bureau of Standards are not subject to copyright in

35
ABSTRACT

In this paper a new theory and experimental method are described for

evaluation of surface residual stresses in inhomogeneous, anisotropic

materials. The method is based on the use of horizontally polarized shear

waves (SH-waves) that propagate at a grazing angle with respect to the surface

of a sample. In addition, a new theory is presented for grazing SH-waves

propagating through a body in which the stress distribution varies with depth.

It is shown that to first order the average velocity of the grazing SH-waves

is dependent only on the surface values of residual stress. Based on the use

of electromagnetic-acoustic transducers (EMATs) to generate and detect grazing

SH-waves, preliminary experimental verification of the theory is presented.

36
INTRODUCTION

It is well known that ultrasonic determination of residual stress is

fundamentally complicated by texture and mi crostructure effects. Texture and

microstructure can cause the material to be anisotropic, and variations in

texture and microstructure lead to material inhomogeneity. Anisotropy and

inhomogeneity influence ultrasonic wave propagation and, often, mask the

stress related effects. In a previous paper [1], the authors have demonstrated

that conventional ultrasonic methods, which rely on bulk shear or longitudinal

waves at normal incidence with respect to a free surface, cannot provide

enough information to separate the effects of stress and material properties

on ultrasonic wave velocities. Techniques based on changes in the Rayleigh

wave velocities are also sensitive to limitations caused by material property

variations. In Ref. 1, the authors have described a new ultrasonic measurement

technique that can be used to overcome the above limitations. To verify the

new technique, the authors have compared ul trasoni cal ly-determi ned stress

distributions in a plate with theoretical predictions and strain gage measurements. Good quantitative agreement has been demonstrated.

In contrast to the conventional ultrasonic techniques, the technique

proposed by the authors in Ref. 1 is based on the use of shear waves that are

polarized parallel to and propagate obliquely with respect to the free

surface. These special ultrasonic probing waves are SH-waves, which can now

be efficiently generated and detected using electromagnetic- acoustic

transducers (EMATs) [1].

The new ultrasonic technique described by the authors in Ref. 1 is

limited in two respects. First, the theoretical basis for the technique was

limited to cases when the stresses coincided with the material coordinate

axes. This limitation was later removed, however, by extending the theory. Tf-c

37
modified theory is presented in Ref. 2. A second limitation involves variations

of stress in the direction normal to the free surfaces of a sample. When such

variations are present, the average through-thickness value of stress is

determined. Thus, stress distributions which have a vanishing average through

thickness value (for example, bending stresses) are not detected. This limitation is shared by all ultrasonic techniques that involve through-thickness

propagation of the ultrasonic probing signals.

In situations where the residual stress varies in the thickness direction,

the value of stress at the free surface is often of greatest importance.

Consequently, this paper specifically addresses the problem of determining the

surface values of residual stresses. In particular, a practical

experimental procedure is described that permits the determination of surface

residual stresses in weakly inhomogeneous and anisotropic materials.

The new experimental procedure is based on the use of SH-waves that propagate at near-grazing angles with respect to the free surfaces. Thus, the new

procedure is different from the Rayleigh wave procedures that have also been

used to evaluate surface stresses [3,4]. The Rayleigh wave procedures are

sensitive to material property variations. On the other hand, the new SH-wave

procedure appears to be inherently insensitive to material property

variations.

THEORETICAL BACKGROUND

Before describing the theory underlying the new experimental for evaluating

surface residual stresses, it is useful to outline the shortcomings of the

Rayleigh wave technique. The theoretical basis for the Rayleigh wave technique

was first proposed by Hayes and Rivlin [5]. In Ref. 5, the following relationship was derived:

(v - v

0

)/v

0

= aTj + bT

2

38

( 1 )
where v and v q denote the velocity of a Rayleigh wave in the unstressed and

stressed states, respectively, a and b are functions of the second- and

third-order elastic constants of the material, and and are the two

nonvanishing principal stress components at the free surface. In Eq. 1 the

axes of material symmetry and principal stress directions are taken to be

parallel. The coordinate system relevant to Eq. 1 is shown in Fig. 1.

A linear combination of and can be inferred by making the relative

velocity measurement needed to evaluate the left-hand-side of Eq. 1. However

for many engineering materials, the value of v

q is not constant with position.

This effect causes a large uncertainty in the ul trasoni cally determined value

of (aTj + bT^). This particular problem can be overcome by making use of

SH-waves that propagate at near-grazing incidence.

The advantages of using SH-wave for evaluating stress in the presence of

material property variations were first demonstrated in Ref. 1. In particular,

based on the theory presented in Ref. 1, the following relation can be derived

for SH-wave propagating at near-grazing incidence:

(v

2

" v

l )/v

o

= [1/(2C

66 )](T

22

' 1U

) (2)

where, v^, v^ are the velocities of grazing SH-waves propagating in the x^,

Xj, directions, respectively, v q is the mean unstressed shear wave velocity,

T ^2and T^ are normal stress components referred to the coordinate system of

Fig. 1, and Cgg is an elastic constant. Equation 2 was independently proposed

as the basis for evaluating bulk residual stresses by Thompson, Smith, and

Lee [6].

It is important to note that no unstressed velocity term appears in Eq.2.

In particular, there is no difference between v^ and v^ in the unstressed

state even though the theory is valid for orthorhombic materials. Thus, from

a relative measurement of velocities in the unstressed state alone, the

39
left-hand-side of Eq. 2 can be evaluated, and the quantity ( T ^

- T^) can be

inferred. The theory used in deriving Eq. 2 is valid only for the case where

the material properties (elastic constants and density) and stresses are

homogeneous. In addition, in deriving Eq. 2, the assumption was made that the

principal directions of stress and the material symmetry axes are parallel.

This is equivalent to the assumption that T^ vanishes in the coordinate

system of Fig. 1 [1]. This restriction was later removed by the authors in

Ref. 2, where the theory was extended to the case of arbitrary values of T.^

In this paper, the theory of Ref. [1] is extended to the case of neargrazing SH-wave propagating through inhomogeneous stress distributions. The

elastic constants and the density of the material are taken to be constant in

the region of interest, while the stress components are functions of x^ only.

This permits the study of a situation of practical importance. The problem of

evaluating residual stresses in plates can be considered as an example. In

plates, the stress distribution can vary over a short length scale in the

thickness direction. The variation of stress with in-plane dimensions is over

a longer distance, and can be neglected in the region of interest (the region

interrogated experimentally).

Following Tokuoka and Iwashimizu [7], the equation of motion for a small

displacement superimposed on a large elastic initial strain state is derived:

(3)

where

S + C E ) mnpq mnpqrs rs (4)

o dX mnpq dX dX dX

40
p , p are the densities in the unstressed and stressed states .respectively

,

C and C are the second- and third-order elastic tensors of the

mnpq mnpqrs

material, E.. is the strain tensor, u. is the displacement vector describing

ij l r

the small superimposed motion, and represent the position vectors of a

particle in the unstressed and stressed states, respectively. The deformation

gradients can be replaced using the relation (valid to first order in

displacement gradients)

5x

^v1 (6. . + E. . + W. .) 5X . = ij ij ij (5)

where W.. . is the local rigid body rotation. Equations 3 and 4 are valid for

any small elastic displacements superimposed on an initial elastic strain

state. The special case of an SH-wave propagating in the x^-x

3 plane is

treated first. In this case:

u.j = u 3

= 0 , u 2

= u

2

( x

^ , x 3, t)

Substituting the above in Eq. 3, the following result is obtained:

2 a du 0 . du 0 5 u 0 d

( ril \r v ) 4- ^ (8(X 9 X ) 2 2 _ p. 5x

1 a X

1 ,x

3 5 Xl dx

3

1 3 dxj

' p o “St7 (5)

Henceforth the subscript "2" will be omitted. The functions a and 6 depend on

the elastic constants and strains and are presented in the Appendix.

It is noted that in Eq. 5 the functions a and e are only slightly

perturbed from homogeneous, isotropic values. Thus, they can be expressed as

ot - Egg ef(x

1 ,x

3

) , B - x

i ,x3

)

where e is a small parameter. (Strain and degree of anisotropy are assured to

be of the same order of magnitude in this analysis. The value of c is of the

41
order of strain). Thus, a direct perturbation solution in terms of e is

possible. It can be shown, however, that this approach does not lead to a

simple closed-form expression for u. In this paper, a geometrical acoustics

approach is used instead. Following Kara! and Keller [8], let

u = A(Xj,Xg) exp [iu)(S(Xj jX^-t)] (7)

The amplitude A is expanded in reciprocal powers of frequency,

°° n A = A

n (l/<i) (8)

n=0

Substituting Eqs. 7 and 8 into Eq. 5, the zero'th order solution (valid in the

high-frequency limit) is

A° [-“<§/ + P„] - 0 Ol

For arbitraryJ A , there results

Q

/dS \2 , . ,dS >,2 /

,

n

\ al^

+ B ^ = p o (10)

Equation 10 is the Eikonal equation for SH-wave in an inhomogeneous,

anisotropic medium. Solutions to Eq. 10 are cylindrical surfaces along which

the phase, S, is constant (wavefronts). The family of curves that is

orthogonal to the wavefronts is the family of rays. The phase, S, can be found

by integrating forward along a ray from a known starting point. Equation 10 is

not in convenient form because the functions a and 6 are in general not equal.

However, by a pertubation approach Eq. 10 can be converted to an isotropic

42
equation with inhomogeneous right-hand-side. By substituting Eqs. 6 into

Eq. 10 and moving terms involving e to the right-hand-side, one obtains:

(S

\ + / 5 S ^ o - e1% E9(H’ ;/c

66

= n (ID

where n is the effective index of refraction of the medium. Equation 11 can be

evaluated using an iterative procedure: First, e is set equal to zero to

obtain S , the solution to zero'th order in e. Then, S is substituted into

o o the right-hand-side of Eq. 11 and the first-order solution, S. , is found.

This procedure is used to study the case of a line source on the surface of a

half-space (Fig. 2). The wavefronts in a homogeneous, anisotropic medium are

circles defined by

S o

‘ ( VC 66 )1/2r < 12 >

where R is the radius measured from the line source. Substituting Eq. 12 into

Eq. 11, for points in the vicinity if the surface, the approximate result is

n = (p

o/C66

)1/2 (l-ef/2C66 ) (13)

If the stresses vary only in the x- direction, then f=f(x,), and from Eq. 13 n

is a function of x~ only. For isotropic Eikonal equations of the form of Eq.

11, the rays are defined by [8]:

d

dl (n

dx . dn

i' = dT

-

' dx

T (14)

where i takes on the values 1 and 3 and JL denotes arc length alone a ray.

43
The special case where the stresses vary linearly with x^ is considered

next. (This can be thought of as the first segment of a multilinear fit to an

arbitrary stress distribution). In this case

f = f + f. Xo

o 13

where f and f^ are constants. Equation 14 is evaluated to find the rays.

Following Officer [9], the rays can be shown to be arcs of circles (Fig. 3)

with centers along a line a distance

[l-ef

0

/(2C

66 )] / [ef1 /(2C66 )]

above the surface. Along a ray, the difference in phase, S, between two

points Jl and l is [8]

If the source and observation points, both located along the free surface, are

separated by a distance, 2x, as shown in Fig. 3, then to first order in e the

phase difference between the two points is given by

AS = 2iu(C

66 /p

0

) 1/2 [l-ef

0

/(2C66

)]x (15)

The phase difference in the unstressed medium is given by

AS

0

= 2w(C

66/p0

) 1/2

x (16)

44
From Eqs 15 and 6 and the definition of a in the Appendix, it is seen that f

depends only on the values of stress at the surface. Thus, to first order in

strain the phase at the surface is only dependent on the stress values at the

surface.

Equation 15 was derived for an SH-wave propagating in the x^-Xg plane.

If a similar calculation is performed for a near-grazing SH-wave in the Xg-Xg

plane, an equivalent result to Eq. 15 is obtained. Calculating the relative

phase difference between the near-grazing SH-waves in the x^-Xg and Xg-Xg

planes results in:

(AS

2

-AS

1 )/AS

0

= -1/(2C

66

)(T

22

(X

3

=0)-T

11

(x

3

=0) ) (17)

2 1 where AS , AS denote the phase difference between the source and observation

points for the near-grazing SH-waves in the Xg-Xg and x^-Xg planes. It is

convenient to introduce the concept of average velocity along the arc connecting

the source and observation points. If the frequency of the wave and the arc

length between source and observation points are held constant while the plane

of propagation is changed from the x^-Xg to the Xg-Xg plane, then

(AS

2

-AS

1 )/AS

0

= -(v^vp/Vg (18)

When Eq. 18 is substituted in Eq. 17, the result is

( Vv

l )/v

o

= 1/(2C

66 } [T

22

(x

3

=0) ' T

11

(x

3

=0)] (19)

Thus, the relative velocity difference between a near-grazing SH-wave in the

x^-Xg and Xg-Xg planes is dependent only on the value of the quantity ( Tg

^ -T

]

^

)

at the surface. This result forms the theoretical basis for a new experimental

procedure for evaluating the surface values of residual stress in weakly

inhomogeneous and anisotropic materials.

45
Several comments are in order concerning the theory leading to Eq. 19.

Equation 19 is valid to first order in strain. By keeping the next higher

order term, the effects of stress gradients can be quantified. This is carried

out in the Appendix, and it is shown that for realistic stress gradients, Eq.

19 provides results that are sufficiently accurate for engineering purposes.

It is interesting to note that the construction of Fig. 3, with the line

of centers of the rays above the surface, is valid only when the stress

increases with depth. If the stress decreases with depth, the line of centers

will be below the surface. Because the curvature of the rays is very slight,

for the latter case, the source and observation points would have to be

slightly below the surface. In practice, this is not a limitation since the

transducers used to generate and receive SH-waves operate by producing and

detecting body forces in the sample in a region slightly below the surface [1].

As a final note on the theory, Eq. 19 was derived using a geometrical

acoustics approximation and thus is valid only in the high-frequency limit.

However, keeping higher order terms in the asymptotic expansion (Eq. 8)

affects only the amplitude and not the phase. Therefore, Eq. 19 is valid for

lower frequencies as well. From physical considerations, the theory is

likely to become invalid when the length scale over which significant stress

variations occurs becomes comparable to or less than the ultrasonic wavelength.

EXPERIMENTAL PROCEDURE AND RESULTS

Recent developments have made it possible to construct efficient transducers

of SH-waves, that employ an electromagnetic-acoustic transduction principle.

The principle and operation of electromagnetic-acoustic transducers (EMATs) is

described in Ref. 10. A typical transducer construction is shown in Fig. 4.

46
The radiation pattern of the EMAT in Fig. 4 in the plane normal to the

surface and containing the source and observation points is shown in Fig. 5.

It is seen from Fig. 5 that indeed a grazing SH-wave can be generated by this

EMAT.

In the present work, the velocity measurement system described in Ref. 1

was used to make relative velocity measurements with grazing SH-waves. The

transducer arrangement shown schematically in Fig. 6 was used in the experiments,

with the two transducers separated by a distance, 2x, equal to 35 mm.

Preliminary experiments were conducted, to investigate the sensitivity of

SFI-waves to anisotropy, and to validate the near-grazing SH-waves that propagate

in an inhomogeneous stress field. The plate specimen shown in Fig. 7 was used

in the experiments. The plate was made of aluminum 6061-T6 and was 25 mm

thick.

In the first experiment, the anisotropy was characterized by making

measurements with shear waves at a steep angle of incidence along the line

shown in Fig. 7. The average value of velocity anisotropy obtained from

several measurements in the unstressed sample was 0.005 ± 12 percent. On the

other hand, the velocity difference ( v 2” v i^/

v 0 9 raz ^ n 9 SH-waves in the same

unstressed sample averaged 0.00011 from several measurements. The effect of

anisotropy is seen to be substantially smaller along a grazing path than

along a steep path, by a factor of almost 50. This result confirms the

prediction of the theory (Eq. 1).

A calibration test was next performed by placing the specimen shown in

Fig. 7 in a tensile testing machine. A maximum tensile stress of 210 MPa was

applied. Relative velocity measurements, ( v 2~ v l^ v o’ were macie at various

stress levels. From the calibration test and Eq. 2, the result for was

g 2.63 x 10 MPa. As expected, this does not differ significantly from the

47
nominal value of the shear modulus for aluminum 6061-76 of 2.66 x 10^ MPa [11]

The specimen shown in Fig. 7 was placed in a four point bending apparatus. A

load of 30 kN was applied, giving rise to a maximum stress of 226 MPa at the

surface and a stress distribution that varies linearly through the thickness

in accordance with beam theory. (The stress was constant along the beam in

the vicinity of the middle of the specimen). Several relative velocity

measurements were made at different points in the region near the center of

the specimen. The resulting average value for ( v 2“ v i^/

v 0 was 0.0045 ± 15°/o.

Using the calibration result for C^, the estimate of the surface value of

(

22 “Ti 1

) from the velocity measurements using Eq. 19 was 236 MPa ± 15°/o.

The discrepancy of the average value from the correct value of 226 MPa is only

5°/o. (It is expected that better precision and reproducibility can be

obtained with improved transducers) . The above experimental results are in

agreement with the new theory.

Discussion

Based on Eq. 19, relative velocity measurements with grazing SH-waves can

be used to evaluate the surface value of (T22~^ll^ ^homogeneous,

anisotropic materials subjected to residual stress. This technique has two

distinct advantages over the use of other ultrasonic procedures such as

measurements with Rayleigh waves. The first is that no unstressed velocity

term appears in Eq. 19, the basis for the new procedure, while (v^-v^)/^ does

appear in Eq. 1, the basis for the Rayleigh wave approach. Slight material

inhomogeneities can cause variations in v

q which are of the same order as the

small change in v caused by stress. Thus, severe errors would result in the

evaluation of stress from Rayleigh wave measurements and Eq. 1. This is not a

problem with the new procedure proposed in this paper. A second advantage is

48
that the coefficient relating stress to velocity (henceforth referred to as an

acoustoelastic constant) in Eq. 19 is only dependent on one second-order

elastic constant, C^. In contrast, the coefficients a and b in Eq. 1 are

functions of the second- and third-order elastic constants of the material.

Similarly, the coefficients relating velocity to stress for techniques based

on through-thickness propagation of bulk waves are also dependent on both

second-and third-order elastic constants. In typical engineering materials,

slight inhomogeneities are likely to cause a stronger effect on the secondorder elastic constants than on the third-order elastic constants. For

example, Hunter [12] reported results comparing properties of specimens, that

had nominally identical compositions but undergone different forming

operations. The Young's modulus and shear modulus are similar for the

specimens of the two materials. However, the acoustoelastic constant for bulk

longitudinal waves differed by almost 50°/o between the specimens made of the

two materials. Johnson [13] explained this result by showing that texture,

which is different in Aluminum 6061-T6 bars than in Aluminum 6061-T651 plates,

has a stronger effect on third-order elastic constants than on second-order

elastic constants.

The variation of the acoustoelastic constant with position will cause

errors in the evaluation of stress from velocity measurements if an average

value of the acoustoelastic constant has been used which has been calibrated

using a separate reference sample of the material under study. Scott [14]

proposed an in-situ calibration technique that permits calibration of the

acoustoelastic constant as a function of position directly in the residual

stress sample of interest. However, this technique requires a means of

superimposing an additional elastic stress state on the residual stress state

in the sample. Therefore this procedure may be inconvenient in practical

49
configurations. An in-situ calibration technique is less likely to be

necessary for the procedure presented in this paper because the calibration

constant is likely to vary less strongly with position.

The effective spatial resolution of the new procedure is determined by

the spacing between the transmitter and receiver transducers. The lower limit

of the spacing is controlled by the decay time of the initial electromagnetic

drive pulse (currently about 10 microseconds), corresponding to approximately

30 mm.

50
Appendi

Explicit expressions for the functions a and B discussed in the text are

as follows:

( 21 )

( 22 )

“ " C 66 * 1+E

1 +E2' E 3' + C 66i

E

i + C11

E

1 + C12

E

2 + C13

E

3

B ' c 44 ( 1+E

2

+E3" E i) + c 44iE

i + c 13

E

i + C23

E

2 + C33

E

3

In Eqs. 21 and 22, contracted notation has been used for both the stiffness

and strain tensors.

Keeping the next higher order term, the result for the phase difference

between source and observation points for a grazing SH-wave propagating in the

x^-x^ plane is

S = (p o/C

66

)1/2

[ 1

- ef°/( 2C

66

)]x ‘ [ef

l /(3C66 )](po/C66

)1/2,<2e (23)

A realistic residual stress distribution in a plate is shown in Fig. 8. For a

source-to-observation point spacing equal to h, the thickness of a plate,

based on the distribution in Fig. 8, the error introduced by neglecting the

second term in Eq. 23 is less than 8 percent. Because this technique

eliminates the potentially large errors in other approaches for evaluating

surface values as residual stress (e.g. the use of Rayleigh waves) caused by

unknown material property variations, the error level of 8 percent can be

considered acceptable for engineering purposes. (The error is reduced as the

source-to-observation point spacing is reduced).

51
Acknowledgments

This work was supported by the U. S. Navy, David Taylor Naval Ship

Research and Development Laboratory, and the National Bureau of Standards,

Office of Nondestructive Evaluation. The authors wish to express their

appreciation to JoAnne Wilken for typing the manuscript. The assistance of

John Moulder and John Arvidson, Jr. in constructing the special EMATs used and

obtaining experimental data is also gratefully acknowledged.

52
REFERENCES

[1]

. R.B. King and C.M. Fortunko, J. Appl . Rhys., in press.

[2]

, R.B. King and C.M. Fortunko, unpublished.

[3]

. M. Hirao, H. Fukuoka, and K. Hori , J. Appl. Mechanics 48, 119 (1981).

[4]

. D. Husson, S.D. Bennett, and G.S. Kino, Proceedings, 1982 IEEE Ultrasonics

Symposium, ed. by B. R. McAvoy (IEEE, New York 1982) pp. 889--892.

[5]

. M. Hayes and R.S. Rivlin, Arch. Rat. Mech. Anal. 8, 358 (1961).

[6]

, R.B. Thompson, J.F. Smith, and S.S. Lee, Proceedings, 1982 AF/DARPA

Review of Progress in Quantitative NDE, ed. by D. 0. Thompson and

D. L. Chimenti (Plenum, New York, 1983).

[7]

. T. Tokuoka and Y. Iwashimizu, Int. J. Solids and Structures 9,383 (1968).

[8]

, F.C. Karal and J.B. Keller, J. Acoust. Soc. Am. 31,694 (1959).

[9]

. C.B. Officer, Introduction to the Theory of Sound Transmission,

(McGraw-Hill, N.Y., 1958) pp. 59-61.

[10]

. C.M. Fortunko and R.B. King, J. Appl. Phys. 53,3450 (1982).

[11]

. A. Higdon, E. Ohlger, W. Stiles, J. Weese, Mechanics of Materials ,

(Wiley, N.Y., 1967) pp. 554-555.

[12]

. J. B. Hunter, Ph.D. Di ssertai ton , Mechanical Engineering Department,

Stanford University, 1979, unpublished.

[13]

. G. C. Johnson, Proceedings, 1982 AF/DARPA Review of Progressin

Quantitative NDE, ed. D. 0. Thompson and D. L. Chimenti (Plenum, NY,

1983) in press.

[14]

. M. P. Scott, Ph.D. Dissertation, Materials Science and Engineering

Department, Stanford University, 1980, unpublished.

53
Fig. 1. Coordinate system used in derivations.

Fig. 2. Line source on surface of half-space.

54
Source Point Observation Point

JLFig. 3. Rays in vicinity of surface for constant velocity gradient.

Sm-Co

/

I ///

I /

Fig. 4. Transducer construction.

55
$ (degrees)

o

Fig. 5. Radiation pattern of transducer in fig. 4.

2x

Transmitter Receiver

L

Surface of Sample

Fig. 6. Transducer configuration used in experiments.

56
Pin Holes for Gripping

in Tensile Experiment

Scanned Line in Unstressed

Velocity Experiment

J

o —1 o

>» M ouu mm.

t

90 mm.

I

25 mm.

Fig. 7. Plate specimen used in experiments.

X 2

Fig. 8. Schematic of through-thickness variation of residual stress

in a plate sample.

57
NBS-114A (REV. 2-80

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR 84-3002

2. Performing Organ. Report No. 3. Publication Date

March 1984'

4.

TITLE AND SUBTITLE

Residual -Stress Measurements Using Shear-Horizontal Waves from Electromagnetic

Acoustic Transducers

5.

AUTHOR(S)

R. B. King and C. M. Fortunko

6.

PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8 . Type of Report & Period Covered

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS ( Street . City. State. ZIP)

U.S. Navy, David Taylor Naval Ship Research and Development Center, Annapolis, MD 2140

Office of Nondestructive Evaluation, National Bureau of Standards, Wash., D.C. 20234

10.

SUPPLEMENTARY NOTES

[EJ Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bi bl iography or literature survey, mention it here) This collection of technical papers covers a two-year effort on a novel approach

to measuring residual stresses using changes in ultrasonic wave velocities of

horizontally polarized shear waves produced by electromagnetic-acoustic transducers.

The initial three papers deal with the theoretical developments pointing to the

measurement of in-plane residual stress as well as preliminary experimental

verification. The later two papers generalize this theory and extend it to include

surface residual stress measurement in an arbitrary plane; again, empirical

validation is included.

12. KEY WORDS (Six to twelve entries; alphabet! cal order; capitalize only proper names; and separate key words by semicolon s) acoustic; electromagnetic; nondestructive; residual stress; shear horizontal;

velocity

13. AVAILABILITY

EX] Unlimited

14. NO. OF

PRINTED PAGES

| | For Official Distribution. Do Not Release to NTIS

J Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

57

15. Price

|_XJ Order From National Technical Information Service (NTIS), Springfield, VA. 22161 $10.00

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