NBSIR 87-3630

Composite Materials Interface

Characterization

H. N. G. Wadley, J. A. Simmons, R, B Clough, F, Biancaniello,

E, Drescher-Krasicka, M. Rosen, T. Hsieh, and K, Hirschman

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Institute for Materials Science and Engineering

Gaithersburg, MD 20899

March 1988 ^ of

Stimulating America s Progress 1913-1988

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

30W

NBSIR 87-3630

COMPOSITE MATERIALS INTERFACE

CHARACTERIZATION

H. N. G. Wadley, J. A. Simmons, R. B Clough, F. Biancaniello,

E= Drescher-Krasicka, M. Rosen, T. Hsieh, and K. Hirschman

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Institute for Materials Science and Engineering

Gaithersburg, MD 20899

March 1988

U.S. DEPARTMENT OF COMMERCE, C. William Verity, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
'
This program supported by the Strategic Defense Initiative

Office/Innovative Science and Technology under

ONR Contract N00014-85-F-0087

Program Monitor Dr. S.G. Fishman

TABLE OF CONTENTS

Table of Contents. i

1. Executive Summary 1 2. Introduction 3

2.1 Interface Elastic Properties 5

2.2 Interface Adhesion 6

3. Calculation of Ultrasonic Scattering and Guided Interface

Waves from Near-interface Properties 8

3.1 General Theory for an Isolated Interphase ........... 9

3.1.1 Constitutive Relations 11

3.1.2 Conservation Relations . 13

3.1.3 The Boundary Conditions and the Propagation Vector 14

3.1.4 Equations of Motion as Propagation Equations 16

3.1.5 Energy Flow and Group Velocity. 17

3.1.6 Representation of Eigenwaves Outside the Interphase 19

3.1.7 The Scattering Matrix 22

3.1.8 Normal Modes and Leaky Waves 26 -

3.2

Cylindrical Interfaces ....... .... 29

3.2.1 Basic Equations 29

3.2.2 Propagation Equations in Cylindrical Coordinates ....... 31

3.2.3 Potential Functions and the Bessel Representation for

Eigenwaves in Regions I and III 33

3.2.4 Radial Mode Interface Waves . 36

3.3 Future Directions 40

3.4 References ................. 41

4. Experimental Study of Guided Interface Waves ......... 42

4.1 Planar Interfaces 42

4.1.1 Background Research 43

4.1.2 Measurement Methodology 46

4.1.3 Experimental Results ..... 48

4.1.4 Discussion 50

4.1.5 Summary of Results for Planar Interface 54

l
4.2 Cylindrical Interfaces ..... ......... 54

4.2.1 Wave Propagation of Embedded Cylinders . ...... 55

4.2.2 Aluminum-Steel Interfaces 61

4.2.3 Aluminum-SiC Interfaces. ........... 65

4.3 Future Directions. .......... 67

4.4 References 68

5. Acoustic Emission Assisted Measurement of Fiber and

Interface Strength 69

5.1 Microscopic Origins of Acoustic Emission 69

5.2 Fibrous Composite Fracture Micromechanisms .71

5.2.1 Load Transfer from the Matrix to the Fiber 71

5.2.2 Micromechanics of Fiber and Interface Fracture 73

5.2.3 Interphase Effects on Fiber Fracture . . 78

5.3 Model Experimental Approach 80

5.4 Discussion 84

5.5 Future Directions 89

5.6 References 91

Appendix I. . 93

i i
1 . Executive Summary

The designs for many of tomorrow’s defense systems cannot be implemented

with the materials available today. Lighter, stiffer, stronger materials

with higher temperature stability are required. A new generation of

advanced composites with metal and even ceramic matrices show the

greatest promise for satisfying many of these needs. Early attempts at

processing these materials met with mixed success. The properties needed

have in some cases come close to fulfillment, but other batches of

material have been found to exhibit very poor behavior. Micromechanical

modeling studies have directed attention to the microscopic interfaces

between matrix and reinforcement. Bulk behavior has been predicted to be

strongly influenced by the local elastic properties, residual stresses

and adhesion of the interface. Techniques to measure these newly

perceived quantities of importance do not exist. Thus it is not possible

experimentally to (i) confirm the micromechanical model predictions,

(ii) explore the relationships between interface properties and

processing variables and (iii) ensure acceptible interface properties in

materials destined for defense systems.

This research program is directed at developing experimental techniques

for characterizing interfaces in composite materials and coupling this

expertise to other ONR composite programs to enable optimum interfaces to

be designed for the next generation of advanced composites. We have

explored two approaches, guided interface waves and acoustic emission.

The former refers to a family of ultrasonic techniques with potential for

characterizing interface elastic and anelastic properties. The latter

utilizes acoustic emission to provide measurements of the adhesion of an

interface

1
This year, we have achieved the following substantial achievements:

o Formulation of the ultrasonic scattering/interface wave

propagation problem for a general interface.

o Theoretical prediction of the suspected existence of guided

interface waves at model Al-Fe, Al-SiC, Al-B, and Al-Graphite

cylindrical interfaces. The velocity was found to contain both

real and imaginary components. The values of both were calculated

as functions of frequency and cylinder radius.

o Experimental verification of the theoretically predicted real part

of the velocity for model Al-Fe and Al-SiC cylindrical interfaces

and detection of leaky radiation due to the non-zero imaginary

velocity component.

o Determination of the effect of microstructure and stress upon the

velocity of pure Stoneley waves at planar Fe-Ti interfaces.

o Development of an experimental approach using composite single

crystals to determine interface and fiber mechanical properties in

both metal and ceramic matrix composites.

o Determination of SiC fiber strength and Al-SiC interface shear

strengths in model single crystal composites as a function of

liquid metal-SiC contact time during processing.

These achievements have, we believe, significantly advanced us toward our goal of determining the elastic properties and adhesion of interfaces in

advanced composites. We have begun to devise schemes for implementing

these approaches on actual composites. In our future work, we propose to

evaluate these methods and develop data from the advanced composites

developed in the SDIO/IST Consortium.

2
2. Introduction

The materials needs of tomorrow’s defense systems cannot be met with the

materials available to us today. Greater specific stiffness, higher

strength, creep resistance at high temperature and better corrosion

resistance are needed for future materials. These properties are beyond

the reach of traditional metals and alloys, and ceramics suffer from high

mass density and low temperature brittleness. Advanced composites,

composed from either metal or ceramic matrices with graphite, Al^O^, B^C

and SiC reinforcement, show great promise of satisfying many of our future materials needs.

Because of the dissimilar chemical and physical properties (particularly

thermal expansion coefficient differences) of advanced composite

constituents, extensive chemical reactions and residual stresses can be

found at the internal interfaces between matrix and reinforcements. The

strong local stresses, due to elastic inhomogeneity, that result during

subsequent loading interact with the weakened interface to retard the

load bearing and fracture resisting mechanisms of advanced composites.

Micromechanical modeling of these processes is revealing the need for

optimized interface properties to realize the full potential of these

composites

.

Among the properties of greatest importance are the local elastic

properties at and near the interface, the adhesion between matrix and

reinforcement and the residual stress distribution in and around the

reinforcement. The measurement of these quantities with a spatial

resolution on the order of 1-10 pm is required if the interaction between

interface and bulk behavior is to be fully understood, and processing

schemes for interface optimization developed.

3
These measurement needs are a great challenge. No methods exist for

measuring these quantities at the microscopic level needed for

composites. Bulk methods of modulus measurement using either static or dynamic loads, yield a poorly defined "average" modulus in multiple fiber

composites, and the techniques of Broutman et al. that were developed for

adhesion measurements in transparent polymer-based composites fail in

opaque metal/ceramic matrices.

The purpose of the NBS/JHU program is to (i) develop methods for the

in-si tu characterization of interfaces in metal (and, perhaps later,

ceramic) matrix composites, (ii) use these in-situ techniques to better

understand the micromechanics of composites and ( i i i ) explore the

relationships between process variables and interface properties. It is our view that this approach will allow optimization of interface

properties and maximize exploitation of advanced composites.

The measurement of the quantities of import: local elastic properties,

adhesion and residual stress, were initially being addressed through

three techniques:

o Guided Interface Waves (GIWs) to determine the local elastic

properties

.

o Acoustic emission to determine the local adhesion.

o Energy dispersive x-ray dif f ractometry to determine local residual

stresses

.

Funding cut-backs have regrettably forced temporary curtailment of the

third technique, residual stress profiling, even though proof of concept

experiments were successful (see first Progress Report).

4
2.1 Interface Elastic Properties

The aim of the GIW study is to use the measured propagation velocity to

determine the elastic constants (moduli) of the interface. It is an

extension of the procedure utilized in solid state physics to determine

the single crystal elastic constants by measuring the velocities of

longitudinal and transverse bulk ultrasonic waves. The extension to

interfaces is a challenging topic not devoid of considerable risk.

Important milestones for this approach include:

o Confirmation of the existence of interface waves at the

metal-ceramic interfaces of importance in MMCs.

o Calculation and experimental confirmation of GIW characteristics

from the near-interface elastic (and anelastic) properties.

o Exploration of the effects of reinforcement geometry upon GIW

velocity expressions.

o Development of robust inverse methods for determination of

near-interface elastic properties from GIW measurements.

o Establishment of GIW measurement methods at the pm size range.

Each milestone carries with it the responsibility to apply the

information gained to MMCs being studied in other parts of the SDI/SDIO

Advanced Composites Consortium.

In sections 3 and 4 we report on our progress toward fulfilling the first

three of our milestones. While much still remains to be done, we have

found that interface waves do exist at metal-ceramic interfaces in all

composite systems studied. The effect of cylindrical fiber geometry has

5
been thoroughly explored and found to provide new opportunities for

ultrasonic characterization. We also have reason to suspect that a modified form of acoustic microscopy may facilitate velocity measurement

at the pm scale, and are beginning to explore the nature of the inverse

problem.

Our approach has deliberately been a general one. The additional

complexity introduced to the mathematical modeling is acceptable since

the resultant measurement methodologies promise to reveal anelastic

behavior and interface defects as well as elastic properties. This

approach has been carefully risk managed, however, by first solving

restrictive subsets of the general problem and experimentally confirming

their solutions. This coupling of theory and experiment is turning out

to be a productive approach to this complex research topic.

2.2 Interface Adhesion

Adhesion at interfaces in MMCs is being explored through the use of

acoustic emission techniques applied to specially designed test samples.

We are using acoustic emission as a means for detecting, locating and

characterizing individual interface or fiber failures in MMCs. The load

at which each emission is generated is related to the local stress at the

fracture site, thus allowing determination of the local strength,

provided the fiber/matrix geometry is simple and well defined.

Our research to date has utilized single SiC fibers embedded in aluminum

single crystal matrices to make tractable the stress analysis. The

aluminum single crystal matrix is extremely soft and provides a starting

point from which we may also investigate the effect of matrix properties

(by alloying) upon the micromechanisms of interface failure independent

of the interface state. The high purity of the matrix also provides a starting point from which we may investigate interface segregation and

interphase formation; and the system provides us with the opportunity to

6
explore the role of interphases upon adhesion by varying the temperature

and time of contact of either liquid aluminum with SiC or other

combinations of interest to ONR. All of these effects can be studied

independently of fiber-fiber stress interactions. Conversely, having

once established the behavior for single fibers, we can begin to study

the role of fiber-fiber interactions, again independently of other

effects

.

7
3. Calculation of Ultrasonic Scattering and Guided Interface Waves from

Near-interface Properties

In this part of our program, we want to determine the elastic properties

of the interphase by utilizing data gathered through propagating

ultrasonic waves across or along interfaces in MMCs which have an

interphase region between the reinforcement and matrix phases. Our

strategy has been to construct a general model of the interphase from

which ultrasonic wave behavior can be calculated while pursuing a

practically important subproblem in parallel.

The solution for the general model will then be used for solving specific

inverse problems to quantify interphase properties.

A model composite may be thought of as having either a layered, fibrous

or particulate configuration. In our program we are especially

interested in the physics near an interface. However, in models with

many layers, fibers or particles, multiple scattering between

reinforcements can obscure wave behavior associated to individual

interphases. Consequently our model will concentrate on a single

reinforcing element embedded in a matrix. In the idealized layered case,

this would involve two planar half-spaces separated by an interphase

layer; the fibrous model would be that of a cylinder surrounded by an

interphase layer and embedded in a matrix, while the particulate case

would be of a similar nature but with a spheroidal particle.

For layered or fibrous models, the wavelength is not limited to the pm

range, because the reinforcing material can be of macroscopic size in at

least one direction. The particulate case does not have this feature and

so requires very high frequency waves to interrogate individual

particle/interphase configurations. While such experimental

8
measurements are possible, they are not yet at hand. So the formulation

as given, while encompassing the particulate geometry, will center on the

layered and f ibrous cases

.

In section 3.1 we develop the general theory and exemplify it using the

planar geometry of the layered model. (Section 4.1 deals with an

experimental application of Stoneley waves in the planar geometry.)

Section 3.2 examines the cylindrical geometry of the fibrous composite

and gives detailed expressions, including leaky wave predictions, for the

special case of perfect bonding (no interphase) involving isotropic

media. (Section 4.2 gives experimental confirmation to the leaky wave

calculations of section 3.2 and points toward exciting novel applications

of these waves.) Section 3.3 discusses possible directions for future

theoretical work.

3.1 General Theory for an Isolated Interphase

In this section we set forth the basic elements needed to determine both

ultrasonic scattering from guided wave motion along composite interphases

of thickness down to atomic dimensions and illustrate it for the case of

planar interfaces.

We first give the geometric model for the composite material with

interphase. A general set of constitutive relations from which

ultrasonic wave behavior can be computed is then set out. Conservation

relations, yielding three invariants of motion (frequency and two

slownesses) follow from the geometry of the model and the equations of

motion. They allow us to express the result of any scattering or interface wave experiment in terms of the superposition of monochromatic

eigenwaves characteristic of the composite structure.

9
An eigenwave is the simplest type of monochromatic wave which can travel

through the infinite composite medium without internal stimulus, the

energy being supplied at infinity. The name derives from the time

separated wave equation giving rise to a generalized type of Helmholtz

equation from which these waves arise as eigenfunctions. The remainder

of the section concentrates on calculating these eigenwaves, which are

themselves physically measurable. Each of these eigenwaves is shown to

be determined everywhere in the composite by specifying its three

displacements and three displacement derivatives (in a direction normal

to the interface) at any single point in the composite. The method for

calculating the energy flow, needed for many ultrasonic experiments, is

given for any eigenwave.

Each eigenwave has a fixed representation in either the matrix or the

reinforcement. This is shown in the plane interface example using the

natural plane waves for each region. The relationship between these two

representations defines a transformation matrix. Distinguishing the

incoming from the outgoing components in an eigenwave allows us to write

down the scattering matrix which, from a different perspective, shows how

each eigenwave may be thought of as a complete scattering experiment with

both inputs and outputs. Interface waves are shown to arise under

conditions where the scattering matrix fails to exist, because,

physically, such waves need no external energy source from the bulk

material, only from along the interface at infinity. Stoneley waves are

specifically discussed, but this general approach indicates that there is

a rich variety of other interface waves which might be experimentally

detectable, and therein provide a means for characterizing the

interphase. A number of such waves are discussed in the context of

experimental results in sections 4.1 and 4.2.

10
CO X

Fig.

3.1

Schematic

representation

of

a

planar

interfacial

region

(interphase)

sandwiched

between

two

materials.
3.1.1 Constitutive Relations

The I— I I— I I I sandwich configuration is shown in the planar geometry in

Figure 3.1. We assume that regions I and III are homogeneous linear

elastic matrix and reinforcement media with an interphase of material II.

We then seek to use as general a description of region II as possible

consistent with a one-parameter solution for scattering of elastic waves

entering II from either I or III. Such a "constitutive relation" will

have the same general form for the planar, fibrous and particulate

models, but each geometry will impose different symmetry constraints on

the coefficients of the relation. Here, regions I and III are assumed to

be linear anisotropic elastic solids, while region II may be anisotropic

elastic, viscoelastic or anelastic. ^ Region II may contain defects

provided that the interdefect spacing is small compared with wavelength.

The elastic properties in all regions will be assumed constant over each

of a one-parameter family of surfaces (e.g. , planes or cylinders) which

fill out the interfacial region. These surfaces must possess certain

symmetry properties with respect to the equations of elasticity, so that •• we have restricted ourselves to considering such surfaces to be planar,

cylindrical or spheroidal.

The equations of elasticity for either region I or III take the form:

d

2

u d

2

u

.

C

ijk£ dx Sx . = P "72 (la)

e j at

It may be of interest to include anelastic dislocation motion produced

by ultrasonic waves in a matrix stressed beyond its yield point by

differential CTE effects. This would involve a slight extension of the

matrix constitutive relation which does not affect the general features

of the theory at all and has only been excluded for the sake of

notational simplicity.

11
?? f? or, using ' " and " " to denote partial differentiation,

j t

^ijk£ Uk,j£ p u

i,tt

(lb)

Here represent the linear, generally anisotropic elastic constants

in either region I, , or III,

°ij Cijk£ £k£

fck£ " ( Uk,£

+ u

e,\J /2

( l c)

(l d)

C

ijk£ " S^ij " Ci j^k

(le)

with potential elastic energy density given by

2

£

ij

C

ijk« ek£

(If)

To permit the maximum latitude in describing the interfacial region II

within a linear formalism, we shall allow it to be a linear viscoelastic

material, so that it includes ordinary elasticity and possible

dislocation anelastici ty . The properties of the medium can vary in the

Xj direction, but are assumed independent of x^ and x^ (the inclusion of

local scattering centers will not be discussed in this report). The

constitutive relation for the interfacial region takes the form:

<7. .(x, t)

ij

00 II

. C

ijk£ (*r

t)uk^(x, t-T) dT ( 2a)

12
or o „ . ij

= C

II

ijk£*uk,£ (2b)

II

where * denotes convolution in time [1]. Here CL depends only on t and the space variable x^ with (x^.t) = 0 for t < 0. Note that

,11 C_k£ bas a delta function component at t = 0 which gives the ordinary

elastic response.

For t < 0 in the constitutive equation, no disturbance has occurred and u

is zero, so that the integral in equation (2a) actually only extends

between the limits 0 and t. However, equation (2b) can be Fourier

transformed and written in terms of frequency, gj:

cr. .(x,oj) ij v - ' = c

ijk«( x r

u ) Uk,£j^- U ) (2c)

thus removing the convolution in time and producing a simpler,, frequency

by frequency, constitutive relation. Note that this constitutive

relation reduces to ordinary linear homogeneous anisotropic elasticity if

C

11

is independent of x^ and w, i.e., C]f jk£^~l ,GJ ) = ^ijk#'

3.1.2 Conservation Relations

The conservation relations follow from well-known techniques used in wave

scattering theory. If we write down the elasticity operator

L(u) C

i jk2* (3a)

13
and the three homogeneous shift operators:

S

x°

2

u(x^ , » t

)

= u(x1> x

2

+ x^.x^t) (3b)

S o *3u(x

1 ,x2

,x3> t) = u(x

1 ,x2> x2

+ X^,t) (3c)

to

u(x^ ,

x

2

, x^ , t ) = u(x1> x2> x3> t + to) (3d)

then all four operators commute and, consequently, under quite general

conditions, can be simultaneously diagonalized. We can then use the

exponential eigenfunctions of the shift operators to represent the

harmonic eigenwaves of the entire I-II-III ensemble:

£(x, t) = Real exp icj(t-s2x

2

~s

3x3

) -fCXj) (4)

where the frequency (j and the "slowness" values s2

and s 3are fixed

constants for any eigenwave — that is, they are conserved during the

passage of the eigenwave <£ through the sandwich.

3.1.3 The Boundary Conditions and the Propagation Vector

The geometries with which we are dealing are characterized by a

one-parameter family of surfaces (planar, cylindrical or spheroidal) on each of which the material properties are assumed constant, only varying

in the direction of the interface normal. The boundary conditions for a

slab of such material are usually given by specifying a linear

combination of the normal tractions and displacements on each of two

bounding surfaces, thus giving three boundary conditions on each of two

14
surfaces, or six conditions in all. In the scattering formalism we allow

the material to be of infinite extent, ignore the boundary conditions at

infinity (which will be taken into account later), and instead determine

the wave function by specifying six conditions on one surface, i.e., the

three displacements and the three components of stress across the surface

(normal tractions) — the three in-plane stresses are not restricted, and

can even be discontinuous on occasion. The conservation relations are

then invoked to simplify the eigenwave, <£, and the equations of motion

are used to propagate it through the material.

Let us consider the plane x^ = x° in the planar geometry and write, using

equations (2) and (4), the displacements and normal tractions on that

surface. The displacements are given by

2u. =2 Real

i exp iw( t-s2x2

~s

3x3

) f i< x

i> exp iw( t-s

2x2

~s

3x3

) W

+ exp -icj(t-s2x

2-s3x3

) W (5)

when ” " denotes complex conjugate and the tractions are given by

2t . = C

.

1

. 0 *

i ilk£ lcjs^exp i(j( t-s

2x

2

~s

3x3

) f^+icos^exp -iu( t-s

2x2~s3x3

)

+ c - 11 i* llkl exp iu( t-s

2x2~s3x

3

) f k 1 + exp -iw(t-s

2x2

-s

3x3

) k, 1

= 2 Real

+ C

ilkl^ x r

w ^ f k,l^ x l^ exp iw(t-s

2x2

-s3x

3

) ( 6 )

Here the index £ = 2 or 3 and f. . = df. (x. )/dx,

k, 1 k

v \

J 1

15
Although these expressions appear complicated, they yield the remarkable

result that the displacements and surface tractions are determined over

the entire surface = x° if the values of the displacements and

tractions are known at any point on that surface. Thus, if only six

complex numbers f^(x°) and t\(x°) or, equivalently, f^(x°) and f i,i (x ) are known, the entire wave function <£ is determined, although not yet

explicitely calculated. We call the six-dimensional vector constructed

from either of these sets of six numbers the propagation vector.

The propagation vector, given in terms of displacements and tractions,

has been used in geophysics for calculating wave motion through

stratified media, and is called the Thompson-Haskell vector. Its

continuity across strata is used to calculate the propagating wave [2].

However, we shall find it more convenient to express the propagation

vector in terms of the complex vector f and its first derivative f. .. i. J Although these determine the Thompson-Haskell vector, they are not

strictly determined by it. The vector £ and its first derivative will

also be continuous throughout the medium.

3.1.4 Equations of Motion as Propagation Equations

The equations of motion o. . . = pu. may be written down from equations

ij, j Kl, tt

(5) and (6) by noticing that the expression for also contains the

expression for the full stress cr_ by replacing the index "l", occurring

in the elastic constants, with a "j”. These expressions only imply the

equality of the real parts of two complex expressions, but if we write

the more restrictive expressions requiring full complex equality, the

16
exponential dependence on cj, and drops out, producing the equation

of motion in the form of a one-dimensional propagation equation:

C

ilkl fk,ll

+

[ C

ilkl,l 1(JS ^C

ilk^]

f k,

w

[P<5 ik"S

j S Ajk^ ' iwS '‘ C - £~ilk£,lj f k

” 0

These are homogeneous ordinary differential equations which can be

directly integrated to determine <£ everywhere. Although they at first

appear formidable, they can be routinely integrated on today’s modern

computers provided that the constitutive functions, determining C. in

region II are reasonably well-behaved.

3.1.5 Energy Flow and Group Velocity

Sections 3.1.3 and' 3. 1.4 describe stress propagation for harmonic

eigenwaves. Stress is a linear function of the displacements. In this

section we describe the power flow, which is a quadratic function of

displacement, for such eigenwaves. We can write down the power flow for

a monochromatic elastic wave in terms of the energy change in an

arbitrary region V:

E - LA

2 dt

(C. v

.. n (x^ ,Gl>)u. .u. n + p6 . .u .u . )dx

l jk£ v 1

J i,j k, £ ij i J “J

p6

iju

j - ( c

ijk« (xr“)uk.« ) -jj u .dx +

l “ o

.

.u .ds . 1 J 1 —

J av

a. .u . ds . =

ij i ~J

(o

.

.u . ) , . dx

ij i J ~ ( 3 ) av

17
where the first expression in the second equation of (8) contains the

equilibrium equations and, consequently, vanishes. This expression

allows us to define the power flow, or Poynting vector, for harmonic

elastic waves. Such a vector with components P describes the energy per

unit area and unit time flowing through a surface with normal in the x^. 2 direction. To these three components of dimension 1/x t we can add a fourth component Po, which is the total energy density. This component

3 has dimension 1/x and can be thought of as the time component of the

space- time generalization of the Poynting vector:

P = E

o

P. = -a. .u. , j = 1,2,3 . (9)

J ij i

In a conservative system this four-dimensional energy flow vector, P, is

divergence free, P = 0. While the spatial component of P defines the

power flow or elastic Poynting vector, the vector with components Pj/P

q

describes the energy flow velocity field, and is, in fact, the

generalized group velocity [3,4].

By integrating P through time and over a volume, one can trace the

distribution of energy associated to any eigenwave. Usually, however, we

shall not be interested in the instantaneous ebb and flow of energy

during any cycle, but rather in the time averaged power flow, that is,

the power flow integrated over a period. This approach combines with the

complex field representation to produce results which have simple

physical interpretations. If one draws an arrow with components at

each point, these arrows not only point in the direction of maximum

energy flow, but no energy flows perpendicular to the arrows. Thus, in a

18
conservative system, starting from a surface at time t , one can use the

o arrows (vector field) to develop a four-dimensional tube inside of which

energy is conserved and from which one can follow the energy transfer.

3.1.6 Representation of Eigenwaves Outside the Interphase

In order to describe scattering across an interphase, we must first find

a simple representation for the harmonic eigenwaves in the outer regions

An eigenwave <£ takes a particularly simple form in such regions. In the

planar geometry, for example, «£ can be decomposed into the sum of six

plane waves.

To see this decomposition, consider a trial plane wave solution from the

infinite body, <£q = p exp( icj( t-s*x) ) , w / 0, with polarization vector p,

|p | = 1, and phase "slowness" vector s, each of whose components has

dimensions of 1/velocity [2]. Equation (1) then takes the form

C. „s .s „p, = pp., or Kp = pp, where K is the matrix given by

1 j c k i — —

K.. = C. .. „s .s n lk ijk£ j e

( 10 )

For <£q

to be a plane wave solution, then, the vector p must be an

eigenvector of the matrix K with eigenvalue p. Thus, it must satisfy the

determinant equation

S(s) |C. .. n s .s n -p8 . .

' i jk£ j £ ij = 0 ( 11 )

19
The set of real vectors in s space satisfying this equation is called the

2 slowness surface. To determine the eigenfunction <£ in region III

bounded by the plane at =0, for example, we seek those plane waves

which conserve the quantities w, s^ and s^- Such waves are those whose

s^ value is obtained by the intersection of the line in s space parallel

to the s^ axis and passing through the point (O.s^.s^) (Fig. 3.2). This

line hits the 6^ order slowness surface at six different values for s^

(some of which may be complex) and thus determines six possible "plane”

waves in III which satisfy the conservation relations. It can be shown

that all real roots for s^ occur in pairs such that the Poynting vector

for one root points towards the interface while that for the other root

points away from the interface. Complex roots appear in conjugate pairs

and are associated to "inhomogeneous plane waves" which decay or grow

Since (11) is a bicubic equation for s (+ s are each solutions), the

slowness surface is a three-sheeted surface symmetric with respect to

reflection through the origin. Positive definitiveness of the energy

expression guarantees three independent solutions in every direction in

slowness space. This surface determines both phase and group velocity

directions for all plane waves in any homogeneous elastic material. In

particular: 1) If x is some point in real space with magnitude x, and

if the vectors in s space are scaled in units of x, then the intersection

of the slowness surface with the plane perpendicular to x and lying at a distance t units from the origin, parameterizes all plane waves passing

through the point x at time t. 2) The direction of the normal to the

slowness surface at any point is parallel to the time-averaged Poynting

vector for the associated plane wave, and, therefore, indicates the

directions of energy flow for that plane wave [4].

20
Si

Fig. 3.2 Diagram showing six real plane waves possible in one material during a scattering reaction at an interface. Each

of the six points of intersection determines a different

possible plane wave. The vectors normal to the slowness

surface at each point of intersection give the Poynting

vector for that plane wave. Three of the plane waves carry

energy towards the interface, and three carry energy away

from the interface. Each material needs a separate slowness

diagram.
exponentially away from the interface. The time-averaged Poynting vector

for such waves is parallel to the interface [4].

The sum of six plane waves in region III can be represented

6 as 2

M=1

*11% exp [iu [t-s“x

1 -s2

x

2

-s3x3]

]

,

where the a^, . . . ,a^ are arbitrary

constants. On the plane x^ = 0, we have already seen that <£ is

completely determined by the six Values of f^ and f^

^ at, say, t = 0,

Xg = 0 and x^ = 0. We can solve the six linear equations giving the

quantities f^ and f^

^ in terms of the unknown constants a^ for the six

constants a

III

.a.III The function <£ represented as the sum of the

, " HI six plane waves, 2 a^. pM

exp

M=1

i(j t_S

l Xr

S

2

X2"S

3

X

3

, is an elastic

eigenwave in all of region III and matches the six boundary conditions

not only at t = 0, x^ = 0, x

the plane x^ = 0, and is, therefore, uniquely determined.

2

= 0 , x^ = 0, but for all t, and x^ on

The boundary conditions can also be expressed in terms of the

Thompson-Haskel 1 vector which gives the displacements and normal

tractions on x^ = 0. In that case, if the displacements and tractions

are known at X

2

= x^ = t = 0, then the equations to determine the a^^^’s

are

= u . and

1 Real

" III

2 P

M=1

Mi

Real . ® hi

M=1

5=1,2

C

ilk5 S 5

M

C

ilkl Sl Mk = T.1

( 12 )

21
Since the can be freely chosen as any complex numbers, and since u^

and are real numbers, we can remove the Real operators, leaving six

ordinary complex linear equations for the a^j^’s, completing the

description of the eigenfunction <£ in region III.

In a similar manner the values of the functions f^ and f^

^ (or those of

the Thompson-Haskel 1 vector) at one point on the plane x^ = x° contained

in or bounding region I are equivalent to a set of six complex numbers

a^ 8 ...,ag describing <£ in region I. Thus any eigenwave in the planar

geometry has a plane wave representation in each of the outer half spaces

I and III in terms of their own natural set of plane waves. (The

representation for eigenwaves in cylindrical geometry is carried out in

terms of potential functions. That representation is discussed for the

isotropic case in section 3.3.)

3.1.7 The Scattering Matrix

In section 3.2.6 we saw that any primitive monochromatic wave (eigenwave)

which can exist in a model composite with an interphase region has a

fixed representation in either the matrix or reinforcement phases. For

the planar geometry this representation was in terms of the sum of six

plane waves. From another perspective, each of these eigenwaves is a

self-contained solution to a scattering problem and all interphase

scattering solutions are included as combinations of these eigenwaves.

We need only extract that scattering information from the eigenwave

representation. This will be accomplished using the scattering matrix;

and this matrix is best introduced through what is called the

transformation matrix. The equation determining guided interface waves

can be found as a consequence of the relationship between these two

matrices

.

22
Continuing with our example of the planar geometry, any eigenwave in the

I-II-III sandwich has a plane wave representation in each of the outer

half spaces I and III in terms of the natural plane waves of that region.

The eigenwave can be integrated across region II using a one-dimensional

form of the equations of motion as described in section 3.2.4. Thus, for

any set of coefficients, a^,...,ag in region I, the eigenwave so obtained

gives rise to a corresponding set of coefficients a?j^ in region III.

Clearly this is a linear correspondence, since linear combinations of the

I III a^'s produce the same linear combinations of the a^ ’ s. Consequently,

we can catalog all harmonic eigenwaves for the I-II-III sandwich in terms

of a 6x6 invertible complex transformation matrix T, where

T I III

TaM

- (13)

However, the matrix T, while embodying the entire wave mechanics of the

medium and interface, does not allow us to directly solve scattering

problems. To do so, we need to derive a related quantity, the scattering

matrix.

If all the plane wave components of an eigenwave in an outer region have

real slowness, s^, then we have seen in section 3.2.6 that they break up

into two groups of three plane waves in each region: those whose

Poynting vector is oriented towards the interface (so that the energy

flow is towards the interface) and those whose Poynting vector faces away

from the interface. In this case three incoming waves are easily

identified in each of the regions I and III as those transmitting energy

toward the interphase, while the outgoing waves transport energy away

from the interphase. The scattering matrix S is that matrix which

23
connects the incoming and outgoing waves in both regions. However,

before giving the form of the scattering matrix, we also recall that if

the slowness s^ is not real, then there are two complex conjugate

slownesses associated to inhomogeneous plane waves [4]. Each of these

waves transmits energy parallel to the interphase, but one has

exponential growth away from the interphase, while the other has

exponential decay. In this case it’s reasonable to identify the

exponentially increasing waves with the incoming waves and the

exponentially decreasing waves with the outgoing waves. Nonetheless,

other identifications are possible, and as we shall see, are physically

meaningful in describing leaky waves.

To obtain the scattering matrix S from the transformation matrix T, let

us order the plane wave bases according to their type, incoming (I) or outgoing (0), and decompose the matrix T into a left half, T. , and a

right half, TD

: =K

III T I o

h-*

fl

’ h

'

o

to J

2

°3 J

3

r

l = II T=R °1

°2

, . °3 .

(14)

If region I is above region III, for example, the upward travelling waves

would be incoming on region III and outgoing in region I. We can think

of equation (14) in simple block form, where blocks of incoming or outgoing bases are grouped together:

24
0

m

‘

i

111 =

L

11

R

12

L

21

R

22

r T i

i

T i 0

or

0

m = Ljjl 1 + R^O1

I 111 = L

21

I

: + R^O1

(15a)

(15b)

Here is comprised of the blocks and while T^ is made up of the

blocks and •

Regrouping, we have

0

IH - R^O1 = L I1

R

22

q1 = -lIH + L

21

lX

or Ln

0

L

21

- 1

I.III1 -R

12

0 -R22

0

III

0'

(16a)

(16b)

where we have arranged the bases so as to diagonalize as much as possible

the block matrices on either side of equation (16b). This is done byputting the downward travelling waves first. Equation (14) then can be

regrouped as:

25
1 0 0

III 1

0 4

0 1 0 4

n

I I3

0 0 1 4

n

!l -1 0 0 T III = T

=R

0 -1 0 T III

2

0 <4

0 0 -1 HI

L 3 i°3 )

from which we derive the scattering matrix

( 17 )

(18)

It can be shown that, if the scattering process is reversible (no energy

loss, etc.), then the matrix S is unitary, i.e., S

* = S [2].

Equation (18) completes the scattering representation for the composite

sandwich with interfacial region. Knowledge of this matrix allows us to

predict how any monochromatic ultrasonic wave sent in towards the

interphase region will both reflect backwards and refract through the

interphase region. We have shown how this matrix can be computed if the

constitutive relation of the interphase region and the elastic constants

of the outer regions are known. It forms the basis for inverting

ultrasonic scattering data to determine local elastic properties.

3.1.8 Normal Modes and Leaky Waves

As seen from equation (17), the scattering matrix S, given by

equation (18), exists provided that the matrix S

p

is invertible. If

26
|SR

| = 0, then S does not exist, but then all the input waves can be set

equal to zero and a viable output wave can still be found. This is

exactly the condition for the existence of normal modes, so that the

equation

|SR

| = 0 (19)

becomes the normal mode equation in terms of the variables w, s^ and s^-

Guided interface waves is a relative term describing detectable normal

modes whose attenuation in the direction of motion along the interface is

sufficiently small that the waves can be detected at interface positions

remote from their point of injection.

6

For the planar case there are actually 2 =64 possible combinations of

incoming and outgoing waves in the regions I and III, and these give rise

to 64 variants or branches for equation (19). Of these branches only

one, which must have all s^ slowness roots complex, can have the signs of

all six slowness roots chosen so that exponential decay (rather than

growth) of the amplitude will occur away from the interface on both sides

3

for the outgoing waves and thus describe a normal mode of finite energy.

This "regular" branch contains the nondissipative Stoneley and Rayleigh

The representation of the function f^ in section 3.1.3 was shown in

section 3.1.6 to be a combination of exponentials outside the interphase

region. The real component of the exponent, which we call the growth

exponent, arises from the imaginary part of each s^ slowness root and

describes the asymptotic behavior away from the interface. In the

cylindrical geometry, the behavior outside the interphase also becomes

asymptotically exponential.

27
4

waves generalized to anisotropic media [5]. The other 63 '’irregular”

branches, however, can still give rise to "leaky" interface waves.

As we shall discuss in detail for the cylindrical geometry in the case of

isotropic elasticity, where there are only four possible branches, the

"irregular” branches of equation (19) can have complex s^ and roots

over a range of real frequency values. Such values, corresponding to a

normal mode with complex phase velocity, describe coherent but

attenuating "leaky" waves. These waves also include physically

detectable guided interface waves (see section 4).

The properties of the scattering matrix as well as the velocities and

other characteristics of both nondissipative and leaky modes, depend —

often strongly — on the constitutive relation of region II and the

elastic constants of regions I and III. If we can describe the entire

configuration in terms of a limited number of parameters, then the

formulation we have presented allows us to solve the forward problem and

predict both the scattering and normal mode properties as a function of

frequency. Usually the properties of regions I and III are well known so that one needs only parameterize the constants of the interfacial region

through its possibly unknown thickness. Then the experimentally

measurable scattering and/or dispersion relations serve as input data for

the inverse problem of interface characterization whose goal is to

extract the unknown parameters of the interfacial region.

^The above is true if one passes to the limit of zero density for one material while maintaining a shear velocity above the Rayleigh velocity

of the remaining material. Otherwise the Rayleigh wave will appear as

the limit of a leaky wave, as has been noted in the geophysics literature

[5],

28
Fig.

3.3

Schematic

representation

of

a

cylindrical

interfacial
3.2 Cylindrical Interfaces

In section 3.1 we showed how the scattering matrix and normal modes can

be obtained, with particular emphasis on the planar geometry. In that

geometry a material of any elastic anisotropy may be described in the

scattering formalism, since the elastic constants are invariant under

linear translation. In this section we wish to apply the scattering

formalism to the cylindrical geometry illustrated in Figure 3.3.

3.2.1 Basic Equations

Translational symmetry in the direction in the planar geometry

corresponds to polar rotational symmetry about the cylinder axis in the

cylindrical geometry. This symmetry restricts the elastic constants for

the cylindrical case to those of transverse isotropy (or hexagonal

symmetry) in which the plane of isotropy is perpendicular to the cylinder

axis. Under these conditions the strains in the cylindrical coordinate

system are given by:

e u

rr r , r

e u zz z , z

e „ = (ruQ - uQ + u Q )/2r

r0 v 0,r 0 r,0' ( 20 )

and the stresses by:

29
rr

CL 1 e + Cin eQQ + C10 e

11 rr 12 00 13 zz

00

C1 n e + C1

1

eoa + C1

0

e 12 rr 11 00 13 zz

zz C10 e + C10 eQQ + CL,. e 13 rr 13 00 33 zz

rz 2C/tA e 44 rz ( 21 )

0z

2CL . eQ 44 0

z

a

r0

2C

66

e0r

where the displacement vector u has components u^.Ug and in the r, 0

and z directions, respectively, and where is the force in the a

direction across a planar section of unit area whose normal is in the |3

direction. Here also CL.. - C, 0 = 20^ [6]. The five elastic constants

li LZ bo

for the transverse isotropic material are connected to X and p for the

isotropic material byJ Ciri = Ci0 = X, C. . = = p, C. , = = X + 2p. 12 13 44 66 ^ 11 33 ^

In cylindrical coordinates the equations of motion are

to

rr,r

+c7

r0,0 qq + ct rz,z

+ (cr v rr

- cjqo00 )7 = pru

r,tt^

rCT 0z,z

+ CT00,0

+ a

r0

+ CTr0,r = pru

0.tt

rCTzz,z

+ CT0Z,0

+ o + a = pru

rz rz,r z,tt ( 22 )

As in the planar case, we assume that in the interfacial region II the

elastic constants can only depend on r and gj. The elasticity operator,

obtained here from equation (22), commutes with both the axial shift

30
operator and polar rotation operator, giving rise to a representation for

eigenwaves in the cylindrical geometry as

<£(r , 9 , z , t) = Real f(r) exp i(L>( t v - — 0 - sz)

to

1

where n is an integer.

(23)

3.2.2 Propagation Equations in Cylindrical Coordinates

Equation (23) shows that the quantities u and n are conserved during

motion of a harmonic eigenwave in the cylindrical geometry. The variable

parametrizing the change of structure is the radius, r. The six

quantities making up the propagation vector in this case are £(r) and

£,^(r). The equations of motion are, as in the planar case, translatable

to ordinary second order differential equations for £(r) which can be

thought of also as linear algebraic equations for £* rr ( r ) on any

cylinder, given £(r) and £»

r (r) at some point on the cylinder. Knowledge

of £, rr (and £, ) allows one to increment the propagation vector through

a distance dr: £(r + dr) = £(r) + £>

r (r)dr and

f >r (r + dr) = £ >r (r) + £- rr (r)dr. As in the planar case, simple models

of interfacial regions (such as one, two, or three thin layers) can be

directly calculated without integrating the differential equations by

using a propagator matrix for each layer. Each transition between layers

will produce artificial scattering, but these effects should be

negligible unless the wavelengths become of the order of the interlayer

thickness

.

Taking the interface layer to be transversely isotropic viscoelastic, its

wave propagation equations take the form:

31
Cn (A + rf - f ) + pa

2

r2f - Cccn

2

f - i(C10 + Ccc )nrf Q ll v r,rr r,r r' r r 66 r

v 12 66' 0,r

+ + C-,,)nf Q - i(C1<:) + C..)kr2

f + C11 r2

f v 11 66' 0

v 13 44' z,r 11 ,r r,r

+ C

12,r^

rf

r ‘ inf 0> " iC13,r

kf

z = 0 (24a)

i(C10 + CL~)nrf - i(Cn + CL~)nf + CL,,(r f Q + rf Q - f v Q ) 12 66' r,r v 11 66' r 66 v 0,rr 0,r 0'

+ (pa

2

r2 - C

44k

2

r2 - C^n2)i Q - (C1Q + C^Jnkf,

11 '0 v 13 44' z

+ (“inf + rf Q - f Q ) = 0

66, r

v r 0,r 0' (24b)

i(C„ + C. Jkr(rf +f ) - (C10 + C.JnkrfQ + C. .(r

2

f + rf -n

2

f v ) 13 44' v r,r r' v 13 44' 0 44 v z,rr z,r z'

+ (pu

2

r2

- C

33k

2

r2

)£ z + C44r(rfzr - ikrf

r

) = 0 (24c)

Equations (24a-c) also include the propagation equations in the matrix

« o ~ and fiber when the appropriate relations between isotropy and transverse

isotropy are taken into account.

32
3.2.3 Potential Functions and the Bessel Representation for Eigenwaves

in Regions I and III

Although the potential function approach outlined here can be extended to

a transversely isotropic fiber and matrix, we shall restrict ourselves in

this report to isotropic media surrounding the anisotropic interfacial

region. The potential approach we present is the complex version of that

given by Gazis [7].

Rather than using a plane wave representation of the solution in the

outer regions, we attempt, for this cylindrical geometry, a

representation of the solution in the general Helmholtz form:

<p = v\ + v X ± (25a)

together with

= F(x.t) , (25b)

where x is a scalar potential, ^ is a vector potential, and F an

arbitrary selected gauge function.

Under these conditions, the equations of motion (22) separate in the

homogeneous outside regions into two equations:

a

2

v

2x = x. tt

(26a)

b

2vV = , (26b)

i l , tt v

where a is the longitudinal wave velocity of the appropriate region,

2 2

a = (A + 2p)/p, and b is the shear wave velocity, b = p/p. We assume

for x and each component of the eigenform (23):

33
K = f(r)

-in0 iu(t-sz)

e e

v 7

$

r

z

ig

0

(r) e-in0

gg(r)

-in0

e e

Sz

(r)

-in0

e e

ico( t-sz)

e

v 7

iw( t-sz)

iw( t-sz)

(27)

where we have used condition (24b) to set gg( r ) = -ig

r (r). Then

equations (22a-c) reduce to the 3 equations:

f

.

rr

gz , rr

S0,rr

2

2 r 2 1 1 , n — OJ S - + -rr r 2 2

„ a r

* 2 r 2 1 1 2]

, n

z , r

GJ s ^

+ “2 gz

L b J r J

2

CJ

' 2 1

, ( n+ i)

2

0 , r 2b

Z

J

1 2

r

= 0

g0 = 0

(28a)

(28b)

(28c)

In this case, the six components of the propagation vector can be thought

of as given by f ^ , df^/dr.gg, dg^/dr, g^ and dg^/dr. The displacement

fields are then given by the real part of:

=

r

*e

=

r in , . f - — g + l(JSgQ r r z 0

in

r

f - g + wsgQ z , r 0

-in0 iw(t-sz)

e e

v 7

-in0 iw(t-sz)

e e

v 7

(29a)

(29b)

= - iwsf + g

-g0 mg9

0 , r r -in0 iw(t-sz)

e e

v 7 (29c)

34
The important feature to notice about equations (28a-c) is that they are

all forms of Bessel’s equation (where we have chosen to use the modified

Bessel functions with notation more adaptable to the Lee and Corbly [8]

type example to be given shortly). The solutions to these equations can

be given as follows:

f = ClK

n

(Ar) + c2In(Ar) (30a)

gz

= c3

K

n

(Br) -h c4I n(Br) (30b)

gr = °5K

n+ l (Br) + c6

I n+l(Br) (30c)

where

A = ±

2 2 (j2

w s ' “2

n a

and

B = 2 2G) S

(31a)

(31b)

The square root functions here give rise to four branches (see

section 3.1.8), characteristic of the isotropic cylindrical problem,

which does not permit exponential growth inside the cylinder, and,

therefore, has only one representation inside the cylinder.

35
The boundary conditions on 1^ and imply that inside the fiber,

C- - c„ = = 0, while outside the fiber - c . - c„ = 0, so that the

1 3 5 2 4 6

solution inside the fiber is represented in terms of I and I r+ ^, while

the solution outside the interface can be represented in terms of K and

K

n+ ^. The six degrees of freedom are described by the three non-zero c coefficients and the two branches of the square root defining A and B.

3.2.4 Radial Mode Interface Waves

The wave structure in the cylindrical geometry is particularly rich due

to the curvature of the interface. In this report we want to investigate

the most common class of interface waves — those of axial-radial type —

that can occur in a fiber reinforced cylinder without an interfacial

region. This class of waves contains the Stoneley waves generalized to

the cylindrical geometry as well as leaky modes. The representation of

these special waves in an NDE context was first given by Lee and

Corbly [8], but not followed through by them in any detail, possibly

because of imprecise codes for calculating Bessel functions of a complex

argument or because of an error in the asymptotic representation of the I type modified Bessel functions.

If one sets n = 0 in equation (23) one obtains the ground state

eigenwaves for the cylindrical geometry. Without an interfacial region

present, the transformation matrix is obtained by matching boundary

conditions at the interface between the fiber and the matrix. The free

travelling normal modes requiring no inputs (c.f. equation (19) in

section 3.2 ) then produce a sixth order determinant which is the product

36
of two factors, one second order factor corresponding to waves with

displacements in the circumferential direction (torsional modes) and one

fourth order factor corresponding to waves whose circumferential

displacements are zero. In this section, we shall investigate these

latter, axial-radial, waves. Because of the radial symmetry of these

waves, the ordinary plane waves found in planar geometry will not be

included — they arise for n > 0.

To write down the fourth order equation giving the coefficient for the

axial-radial waves, we specialize equations (30) to n = 0. With no

tangential displacements, <Pg = 0 and g^ can be eliminated. Thus, we can write

X T = aK (AT G)r) exp[icj( t-sz)]

i o l

= j3K^(Bj(i)r) exp[i(j( t-sz)

]

X TTT = nrl (A TTT(or) exp[i(j( t-sz) ] 111 o ill

^III

= ^1 i j

(jr ) exP[fw ( t-sz) ] , (32)

and setting v = — and k = ur, we have:

s

37
in

cn

cno

ii ca «o

*fH *rH 1 +

'S' X ' *

HH

HHHH <

1—4 *

HH HH

HH HH - J- HH HH

/-

—sHH /

CQ * < * >— M CM HH r-H HH > HH HH M HH /-—X HH 1 /—x < HH * CQ / \ *

CM « HH ' ' HH

»—1

rO

N HH *—H HH HH HH CM HH HH HH HH

HH 1 t—t < HH CQ HH

HH / s t—t CM CQ >— HH

HH •—(

1-H > o HH < HH < HH 1 O HH CO > HH V ' HH HH HH HH

HH HH o HH HH HH HH r"H HH < —t

HH CM >-h HH HH HH

CM hh 1 < HH cq & o > CM CQ >

H-4HH i HH HH >

HH CM * HH HH a > HH HH

CM 1 a CM >-h h-»

rO

HH HH CM h-<

HH Q HH

CM CL CM

CL

HH

CQ X*

r-H cq

i—H + i—i /

*

>6 * HH * *— i—i CQ r— CM CQ CQ > '—‘ o 1 o r— CM ' HH S HH CQ CM CQ >

HH a >

CM

—'

X)) 1 CM

38

°This

notation

and

representation

differs

somewhat

from

that

of

Lee

and

Corbly,

who

use

the

wave

number

k,a

slightly

different

definition

of

the As

and B

s,

and

who

use

the

engineer

ing

representation

exp[ i

(kz-wt) ]

rather

than

exp[ i

(wt-kz) ].
The solution to these equations requires, of course, that the determinant

of the coefficients on the left hand side be zero, after which the

coefficients a, (3, nr and 8 can be found from the 3x3 minors of any row.

A difficulty with this determinant is that its value becomes

exponentially large for most of the useful range of the arguments. This

can be compensated by dividing out the asymptotic forms for K

m

(£) and

~r

I (D. e ** and cosh£, respectively. These asymptotic forms were chosen

to be analytic functions so that the determinant would remain an analytic

function of f and, hence, of v at every point of the v plane except

branch points of the square root functions A(v) or B(v)

.

The fact that I (£) is an even function of £ and I~(£) is an odd function

of £ shows, by inspection of equation (33) that the ± branches in region

III produce the same solutions. Hence, instead of the 64 possible

parameters in the anisotropic planar case;’ there are only 4 branches

here. Physically, this corresponds to the fact that the fiber is a

bounded region not admitting exponentially increasing solutions. Thus,

exponential leakage can only occur outside the fiber. Of course,

interchanging the two materials and thus reversing the curvature produces

another four modes, but the possibility of leaky modes with exponential

growth on both sides of the interface, as can happen in the planar

geometry, is foreclosed in the cylindrical geometry.

To solve equation (33), improved high accuracy computer codes for

calculating Bessel functions of a complex argument were developed. An

adaptive Newton’s technique was used to solve the equations and an

adaptive integral implementation of Rouche’s theorem (the integral of the

logarithmic derivative of an analytic function around any closed contour

equals 2iri times the number of zeroes minus the number of poles enclosed

39
by the contour) was used to guarantee that the approximate roots found

were actual roots, and that no roots were overlooked.

Because each mode is a harmonic wave and because there is no tangential

velocity, each particle traces out an elliptical trajectory in the (r,z)

plane. Graphical representations of particle motion have been developed

allowing one to visualize the interaction of each mode with the

interface. In addition, using the results of 3.2.5, the time-averaged

Poynting vector, generalized group velocity, energy flow curve, and

attenuation along the interface were calculated, permitting comparison

with experiments. For leaky modes, the displacement amplitude falls off

exponentially along the z direction while the Poynting vector components

and energy density drop off as the square of the amplitude. In contrast,

the group velocity, which is their ratio, is independent of z and depends

only on radius. Because of this independence of z, energy leaking away

from the interface follows parallel curves.

3.3 Future Directions

The theory of interface and scattered waves in fiber-reinforced MMCs

(cylindrical geometry) needs to be extended to include an interfacial

region of variable characteristics surrounding a transversely isotropic

fiber. The effect of fiber coatings needs also to be investigated and

guided waves of non-radial symmetry should also be included.

We will begin an investigation of the inverse problem for backing out the

thickness and elastic character of the interfacial region. To facilitate

acoustic microscopic studies, we will also begin investigation of the

injection coefficients for determining the efficiency with which a given

external ultrasonic input is converted into distinct guided interface

waves

.

40
3.4 References

[1] D. Graff i, "Mathematical models and waves in linear

viscoelasticity", in Wave propagation in viscoelastic media , F. Mainardi , ed. , Pitman Advanced Publishing Program, (1982),

pp. 1-27.

[2] C.M. Keith and S. Crampin, Geophys. J.R. Astr. Soc, 49 (1977),

pp. 181-208.

[3] J. Lighthill, Waves in Fluids . Cambridge University Press, (1978),

p. 317f f

.

[4] J.L. Synge, Proc. Roy. Irish Acad., 58A (1956), pp. 13-21.

[5] W.L. Pilant, Bull. Seism. Soc. Amer. , 62 (1972), pp. 285-299.

[6] J.F. Nye, Physical Properties of Crystals , Oxford at the Clarendon

Press (1957), pp. 140-143.

[7] D.C. Gazis, J. Acoust. Soc. Amer., 3_1 (1959), pp. 568-573.

[8] D.A. Lee and D.M. Corbly, IEEE Trans, on Sonics and Ultrasonics,

SU-24 (1977), pp. 206-212.

41
4. Experimental Study of Guided Interface Waves

The theoretical studies of guided interface waves presented in section 3

have been complemented with experiments on both planar and cylindrical

model interfaces. The purpose of these experiments has been to test and

verify the theoretical predictions and to evaluate experimental

approaches that might subsequently be applied to actual MMC materials.

The experimental study of planar interfaces has been conducted by

Professor Moshe Rosen and his graduate students at Johns Hopkins

University under subcontract.

4.1 Planar Interfaces

In this section we study the propagation of interface waves along planar

interfaces without an interphase; the incorporation of scattering and

finite interphases will be reserved for later work.

In an earlier series of experiments (see first progress report) interface

waves were generated and detected using a Rayleigh made conversion

technique. Interface bonding was created by applied pressure in several

pairs of materials, including titanium and 4340 steel, titanium and

1040 steel, aluminum and 1040 steel, aluminum and 4340 steel, and nickel

and pyrex glass. In all pairs velocity increased with increasing bond

quality, eventually approaching the theoretical interface wave velocity.

That work has been extended herein to incorporate the effects of

microstructure on interface waves. Several microstructures have been

produced by heat treatment of a 4340 steel. Ultrasonic velocity was then

measured for interface waves along the steel-ti tanium interface at

various frequencies from 1 to 10 MHz, and the effect of the steel

microstructure on interface wave velocity was observed.

42
4. 1 . 1 Background Research

In 1927 Richard Stoneley [1] predicted the existence of an elastic wave

which would propagate along the boundary interface between two adjacent

layers of subterranean rock. By modeling these interfaces as plane

boundaries between two isotropic half spaces and assuming continuity both

of particle displacement and normal tractions across the boundary,

Stoneley extended Lord Rayleigh’s earlier work on surface waves to give a

determinant equation [1,2] which is a special case of equation (19) in

section 3.

The characteristic Stoneley equation for the planar case yields an

2 algebraic equation with 8 roots for v , when v is the complex velocity.

This means that there are 16 possible values for v, but only 8 are

independent, since all signs can be reversed. The equation is

complicated by the presence of four square root functions,

(analogous to those discussed in section 3.2.2, but this time occuring on both sides of the interface), where V can be the shear or longitudinal

velocity in either material. These square root functions are double

valued (+) and introduce branch cuts into the space of solutions.

Pilant has found that, except at isolated branch points, the 16 solutions

always exist, but most of them are usually complex. These complex roots

are often associated to leaky waves. In fact, if one writes down the

displacements for an interface wave (see section 3 or [3,4]) these square

roots appear, together with the velocity, as an exponent in an

exponential. The choice of which branch (+) to choose for the various

square root functions then determines whether that function will

contribute an exponentially increasing or exponentially decreasing part

to the displacement field of the interface wave. Only one contribution

need be exponentially increasing to dominate the entire displacement

43
field, so that only one branch will give rise to a displacement which is

exponentially decaying on both sides of the interface. It is this

branch, when it exists, that produces true Stoneley interface waves. As

noted by Pilant and by Ginsbarg and Strick [5], the Stoneley velocity is

always less than the shear velocity in either material and greater than

the Rayleigh velocity of the linear material.

In cases of complex roots for non-Stoneley branches, as will be

illustrated later, physically meaningful roots are associated to "leaky”

waves, which travel along the interface but radiate characteristic plane

waves into one or both materials, thus leaking energy as they travel.

The amplitude of such waves at the interface decays exponentially with

distance very seriously along the interface [6]. The majority of

Pilant’s computations [2] were restricted, and applied only to the case

where the Poisson ratio for each material is 0.25. Sezawa and Kenai [7]

have graphically determined the range of material constants for Stoneley

wave existence by rederiving the Stoneley equation and investigating the one real root. Scholte [8-10] also analyzed the real Stoneley root and

described the range of existence for numerous theoretical pairs. A

similar project was undertaken by Owen [11] who introduced real material

constants to determine which material couples would support real Stoneley

waves

.

The expressions for Stoneley wave particle displacement derived by

Yamaguchi and Sato [3] suggest that it is somewhat similar to shallow

shear waves in the less dense material. However, the complex dependence

of the displacements on material constants restricts the usefulness of

these theoretical studies to that of valuable guidelines. The computer

codes developed at NBS within this project already go beyond these

studies and permit us the interface wave motion much more clearly in

realistic models of the actual materials being tested.

44
Experimentally, interface waves have been observed in several material

pairs and geometries . Stoneley waves and their leaky counterparts were

generated in plane and cylindrical geometries by Lee and Corbly [4] in

aluminum-steel, aluminum- t i tanium, and steel-ti tanium pairs. Utilizing a surface wave mode conversion technique, they measured interface wave

velocities and suggested possible applications for NDE. Direct

observation of Stoneley waves was made by Claus and Palmer [12] at a nickel-pyrex interface. The transparent material (7070 Pyrex)^ allowed

wave amplitude at the interface to be measured using a differential

interferometer

.

The existence of interface waves guided by a thin film between two bonded

adherents was demonstrated by Rokhlin, Hefets and Rosen [13]. By

measuring the velocity and attenuation of the wave, it was shown that a direct correlation could be made between the strength of the interfacial

bond and the measured shear modulus and general transmission loss factor.

From the dynamics of such interface waves an "effective" shear modulus

was determined which strongly depends on the elastic properties and

integrity of the bond and of the intermediate layer. An analogous

situation may exist at interfaces in composite materials due to the

reaction products between the matrix and the strengthening fibers.

Certain commercial equipment, instruments, or materials are identified

in this paper in order to adequately specify the experimental procedure.

Such identification does not imply recommendation or endorsement by the

National Bureau of Standards, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.

45
Stoneley waves at planar interfaces have also been studied in anisotropic

materials by Lim and Musgrave [14], and Chadwick and Curry [15]. The

connection between planar Stoneley waves and interface waves in cylinders

of isotropic materials has been discussed by Lee and Corbly [4] and by

Thurston [16], who also presented an extensive review of higher mode

(non-radial, polar angle dependent) waves in cylinders. The effect on

Stoneley wave velocities and displacements caused by sliding at the

interface, a realistic possibility with pressure bonded materials, has

been discussed by Meleshko [17] and by Kumar and Murty [18]. Meleshko

contrasts ordinary Stoneley waves with those interface waves in which

the discontinuities in displacement can occur, but the tangential

components of traction are zero across the interface. When this slippage

is allowed, a great effect in displacements can occur at the interface,

but in the examples shown, there is only a small effect on the interface

wave velocity. The work of Kumar and Murty interposes a viscous

layer between the two materials and may be compared with the work of

Rokhlin et al . [13].

4.1.2 Measurement Methodology

Experiments have been conducted to generate interface waves in

titanium-steel pairs, and to study the influences of microstructure

variation in the steel and anisotropy in the titanium.

Different types of heat treated samples of 4340 steel approximately

18 cm x 9 cm x 3 cm were polished to a 30 pm finish. They were

subsequently used as a base plate for Rayleigh wave generation. Samples

of Ti-6A1-4V rolled sheet of size approximately 4 cm x 4 cm x 2 cm were

46
A Time Oscilloscope

Fig. 4.1 Schematic diagram for generation and detection of interface

waves in a compressed duplex block of titanium mounted on steel by using conversion from and to Rayleigh waves on the

steel block.
also polished to a 30 pm finish and used to create the desired interface

by placing it upon the appropriate base plate. The titanium plate

o contained a rolling texture. By rotating the titanium plate 90 on the

steel base plate, it could be oriented so that the rolling direction was either parallel or perpendicular to the direction of interface wave

propagation with a consequent change in shear modulus in the direction of

interface wave propagation.

At the interface the particle motion of interface waves in the linear

material is similar to that of Rayleigh waves. In addition most of the

energy of the wave tends to be carried by the denser material over a

large range of material constant combinations. This suggests a possible

Stoneley wave generation technique by mode conversion from Rayleigh

waves. Indeed, Rayleigh waves were generated and detected on the base

plates using two techniques. The first was a direct surface wave

generation method utilizing a broadbanded conical transducer to obtain

Rayleigh waves over a wide frequency range (DC to 2 MHz). The second

technique involved the use of a mode conversion wedge which produced a

directed Rayleigh wave. A schematic of this device is included in

Figure 4.1.

Two machined aluminum wedges are placed into a sliding mount such that

contact is made with the specimen through the knife-edge surfaces of the

wedges. Longitudinal waves are piezoelectrical ly generated and

propagated down through one wedge. Mode conversion of the longitudinal

wave to a Rayleigh wave occurs at the point of contact between the

knife-edge and specimen surface. Rayleigh wave detection at the second

wedge is accomplished through a similar mode conversion of the Rayleigh

wave to a longitudinal wave. The longitudinal wave is subsequently

detected by the second piezoelectric transducer. Although the Rayleigh

47
3250 t OPTICAL FINISH

3200

3150

VELOCITY 3100--

3050 +

#

METALLOGRAPHIC FINISH

3000

2950

0*0 o

+

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

NORMALIZED STRESS

Fig. 4.2 Interface wave velocity versus applied load at a 4340 steel/Ti-6Al-4V interface. Optical finish 0.5 pm,

metal lographic finish ~4.5 pm.
path length is typically much shorter than the multiply-reflected bulk

waves, actual identification of the surface wave is accomplished by

physically damping the travel surface and detecting the amplitude

decrease on an oscilloscope display. Only the Rayleigh wave will be

affected by the surface damping. Typical Rayleigh velocity measurements

were on the order of 2955 m/s for titanium in the rolling direction,

3055 m/s perpendicular to the rolling direction and 2980 m/s for

4340 steel. Measurement precision was ±5 ms ^ for Rayleigh waves and

±15 ms ^ for the (weaker) interface waves.

In order to force a mode conversion from Rayleigh to interface waves, the

mode conversion wedges were modified to allow a controlled interface to

be placed in the Rayleigh path. Interface quality was controlled as a

function of applied force using a tensile machine in the compressive mode

as shown in Figure 4.1. The Rayleigh wave travels on the base plate from

one wedge to the interface where it is mode converted to an interface

wave. This propagates along the interface to the end of the top plate

whereupon it re-converts to a Rayleigh wave that travels along the base

plate to the second wedge.

4.1.3 Experimental Results

The mode conversion technique, described above, was employed to

demonstrate the effect of surface finish and microstructure at the

interface on interface wave velocity for planar interfaces between

titanium and 4340 steel.

Figure 4.2 shows the effects of varying two parameters, surface finish

and applied pressure, upon interface wave velocity. Specimen preparation

48
VELOCITY

(m/s)

3250 t AS RECEIVED

3200 - 1 oOO

‘°*°

T o°° . 3150 o°’ . o° *

o° 3100 + O

3050 o

o 3000

o 2950

0.00 0.05 0.10 0.15

Y

AUSTENITIZED - ISOTHERMALLY

COOLED

- t H

0.20 0.25

NORMALIZED STRESS

Fig. 4.3 Interface wave velocity versus normalized stress for

pearlitic specimens. The stress was normalized to the yield

stress for the as-received material (480 MPa) and for the

austenitized and isothermal ly cooled material (380 MPa).
VELOCITY(m/s)

Fig. 4.4

3200 T TEMPERED

3150

3100

0°°

o°°

o°° o

O

3050 o 4 - o ,«'

3000

O.M

2950 Q*

?!

\C

2900 -I 1 1

0.00 0.01 0.02

1 1 1 1—

0.03 0.04 0.05 0.06

NORMALIZED STRESS

AIR COOLED

QUENCHED

—i

1 0.07 0.08

Interface wave velocity versus normalized stress for

martensitic specimens. The stress was normalized to the

yield stress for the air-cooled material (1035 MPa) and for

the quenched and tempered materials (both 1550 MPa)

.
involved polishing samples of titanium and steel to two different

finishes. The first was a metallurgical polish which achieved a surface

finish of approximately 3-4 pm. The second polish achieved an optical

finish of less than half a pm. In all cases, wave velocities increased

from the Rayleigh velocity at zero pressure to the theoretical interface

velocity at the required pressure to achieve a coherent interface. The

interfaces with the optically finished surfaces required less force to

reach the interface velocity than the metal lurgical ly polished

interfaces. These results suggest that interface waves may be used to

measure interface coherence and integrity in a nondestructive manner

utilizing only measurements of wave velocity.

The sensitivity of interface waves to microstructural variations was demonstrated by several experiments. Specimens of coarse pearl ite, fine

pearlite, air-cooled martensite, tempered martensite and quenched

martensite were produced and interface velocities measured. It is shown

in Figures 4.3 and 4.4 that the interface waves are sensitive not only to

microstructure, but to the particular heat treatment these

microstructures received. Interface velocity increased asymptotically

from the Rayleigh velocity at zero applied pressure to the limiting

velocity at approximately 0.25 of the yield stress.

In order to ascertain the effects of anisotropy in the titanium on interface wave velocities, limiting interface wave velocities were

prepared on four specimens of 4340 steel of varying microstructure using

the titanium block oriented first in the rolling direction and then in

the transverse direction. The results of these measurements are

summarized in Table 4.1. The measurements taken using the titanium block

oriented in the rolling direction agree very well with the theoretical

predictions discussed below. A fifth specimen with a hard martensitic

structure was also examined, but the amount of pressure available on our

test machine was inadequate to reach a plateau in velocity.

49
Beam

Spread
.
4340 STEEL PROPERTIES STEEL/TITANIUM* DUPLEX PLATE

Microstructure Density

(kg/m3

)

Long

Velocity

(m/s)

Shear

Velocity

(m/s) CaseNo.

Rolling Direction

of Ti Plate

Predicted Wave

Type

Interface Wave Velocity Measured (m/s) Predicted (m/s)

Fine Pearlite (Held at 1200F) 7340 5952 3250

1

A

1 Stoneley 3217 ±15 3247 IB II Leaky

In Ti 3217*15 3232 Tempered

Martensite (Quenched and Temp.)7805 5868 3170 2A 1

Divergent

in Steel 3216*55 3170 2B II Stoneley 3166*15 3162

Bainife

(Air cooled) 7617 5869 3175 3A 1

Divergent

in Steel 3221 * 55 3175 3B II Stoneley 3162*15 3166

Fine Pearlite (As ReceivedHot Rolled) 7839 5934 3235 4A 1 Stoneley 3223*15 3233 4B II Leaky

in Ti 3206*15 3218

•Density of Ti-6AI-4V = 4430 kg/m3

, Longitudinal velocity of Ti-6AI-4V= 6287 m/s in both directions Shear velocity ol TI-6AI-4V: { P«n»"d|cular » al,ec.lon(l| = 3287 m/a,

7

I Parallel to rolling direction ( ) = 31 71 m/s.

Table 4.1 Interface wave velocities for four different microstructures

in 4340 steel and two different orientations of a block of

rolled Ti-6A1-4V.

4.1.4 Discussion

The experimental results presented in the previous section have been

compared, where possible, against theoretical calculations.

Figure 4.5 schematically depicts the situation when a Rayleigh wave

traveling through the steel block passes under the titanium. The first

change that occurs is at the point of contact, C, where a change in

acoustic impedance occurs across the interface. The detailed reactions

are very complex and not treatable using infinite medium models, but one

50
Fig. 4.6 Interface wave velocity versus shear velocity in duplex

block of 4340 steel and Ti-6A1-4V using all measured, bulk

elastic constants except the shear velocity in the steel,

which is allowed to range over values encompassing all

measured cases. In this case the shear velocity is taken as 3171 m/s as is the case for the titanium block oriented with

the interface wave travel parallel to the rolling direction

(B series in Table 4.1).
3.50

o

srf/tuui u; Ajjoo|9a 9abm soejjeiui

Shear

Velocity

of

Steel

in

mm/^s
Fig. 4.7 Interface wave velocity versus shear velocity in duplex

block of 4340 steel and T1-6A1-4V using all measured bulk

elastic constants except the shear velocity in the steel,

which is allowed to range over values encompassing all

measured cases. In this case the titanium shear velocity is

taken as 3287 m/s as is the case for the titanium block

oriented with the interface wave travel perpendicular to the

rolling direction. (A series in Table 4.1)
3.50

o

oasrt/iuui u; A^oo|9a 9Aeyv\ aoejjeiui

Shear

Velocity

of

Steel

in

mm/^s
can see that there will be a transition region lying near the point of

contact where particle motion will be influenced by the geometries of

both the steel block and the combined steel/titanium duplex block.

Outside this transition region, the disturbance can be considered to be a

combination of waves in the combined medium. With the proper choice of

material constants and input surface wave, the "injection” coefficients

will favor a single interface wave in the duplex block. The energy to

make such an interface wave arises from the input Rayleigh wave, and its

energy transport will be determined by its Poynting vector as discussed

in section 3. This energy will be carried in a zone emanating from the

transition region. Outside this zone the material in each medium will be

interacting with similar material which is not moving with the proper

interface mode motion. Accordingly, energy will leak out from the outer

edges of the interface beam, resulting in a beam spread region.

Preliminary theoretical studies have been made of the velocity,

displacement and energy effects associated with a change of steel

vis-a-vis that of titanium at a planar steel/Ti interface. The NBS

computer codes for the cylindrical geometry were found to describe the

planar geometry results with suitable accuracy by choosing a large radius

for the cylinder. A radius of between 30 to 300 wavelengths was generally found to be sufficient, although the code is stable enough that

g one can use a radius of 10 wavelengths at which time the error is much

less than the algorithmic and computer roundoff errors.

In Figures 4.6 and 4.7 we show the calculated variation of interface wave

velocity with varying shear velocity when all other material constants of

both materials are held fixed. This assumption appears to accurately

model the conditions under which the Table 4.1 data were collected where

only the shear modulus of the steel varied significantly and where two

distinct titanium velocities were present (according to the orientation

of the block).

51
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to H

c c

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c

®c ??c Q) a> a) o o > > J

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lumpafli JajnQ in luauodxg qj/wojQ

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and

attenuation

for

the

interface

waves

shown

in

Fig.

4.6

(shear

wave

velocity

in

titanium

block

=

3171

m/s).

For

leaky

and

divergent

waves,

the

exponent

is

only

shown

on

the

side

of

the

interface

where

exponential

growth

occurs.
Fig. 4.9 Displacement field for a Stoneley wave in a

4340 steel/Ti-6Al-4V duplex block. Shear velocity of

steel = 3170 m/s and shear velocity of titanium block is

3171 m/s. Since there is no attenuation for this wave, the

displacements are shown at only one point on the interface.

The ellipses trace the orbit of a particle and the

arrowheads on the ellipse show particle motion. The arrows

show the energy flow (Poynting vector) averaged over one

period. Poynting vector lengths and ellipse sizes are

consistent between each other, but arbitrarily scaled for

graphing.
IN

MM

ATI

MHZ

FOR

5TONELET

WAVE

(CASE

1A)

TI

0.20

<E

- 0.20

-0.60

- 1.00

-1.40

-1.80

- 2.20

-2.60

-3.00

-3.40

-3.80

-4.20

-4.60

-5.00

n t -

Ul O o Uw \' NTERFACE 4.60

4.20

' 0—

O—

3.80 . 8-

lfl)

STEE

CO 4= o {-

LU

cn

cry

LU> 3.00 —

*U

V

>—LU

—1LU5 2.60 O

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4

CO

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i

X

GC

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1 . 40

1

1 . 00 \

r 0.60 r

\

* 0.20 INTERFACE \

•/

-0.20 \y

i i i i 1 H—i 1 1 -1.00 -0.60 -0.20 0.20 0.60 1.00

Z IN MM -1.00 -0.60 -0.20 0.20 0.60 1.00

Z IN MM
Fig. 4.10 Displacement field for an interface wave diverging on the

steel side of a 4340 steel/Ti-6Al-4V duplex block. Shear

velocity of steel = 3050 m/s and shear velocity of titanium

block = 3171 m/s. Since there is no attenuation for this

wave, the displacements are shown at only one point on the

interface. The ellipses trace the orbit of a particle and

the arrowheads on the ellipse show particle motion. The

arrows show the energy flow (Poynting vector) averaged over

one period. Poynting vector lengths and ellipse sizes are

consistent between each other, but arbitrarily scaled for

graphing.
MHZ

FOR

DIVERGENT

WAVE

IN

STEEL

(CASE

2AI

TI

ac

o . uu

4. 60

4.20

3.80

3.40

3.00

2.60

2.20

1.80

1.40

1.00

0.60

i

1

0.20

y

INTERFACE

. \

0.20

-1 00 - 0.60 - 0.20 0.20 0.60 1.00

Z IN MM
Fig. 4.11 Displacement field for an interface wave diverging on the

titanium side of a 4340 steel/Ti-6Al-4V duplex block. Shear

velocity of steel = 3170 m/s and shear velocity of titanium

block = 3171 m/s. Since there is no attenuation for this

wave, the displacements are shown at only one point on the

interface. Poynting vectors that are too large are not

plotted. The ellipses trace the orbit of a particle and the

arrowheads on the ellipse show particle motion. The arrows

show the energy flow (Poynting vector) averaged over one

period. Poynting vector lengths and ellipse sizes are

consistent between each other, but arbitrarily scaled for

graphing.

Fig. 4.12 Displacement field for a leaky wave in a

4340 steel/Ti-6Al-4V duplex block. Shear velocity of

steel = 3250 m/s and shear velocity of titanium

block = 3171 m/s. Displacements that are too large are not

plotted. The ellipses trace the orbit of a particle and the

arrowheads on the ellipse show particle motion. The arrows

show the energy flow (Poynting vector) averaged over one

period. Poynting vector lengths and ellipse sizes are

consistent between each other, but arbitrarily scaled for

graphing.
10.00 oo©o©dcooo

c\i coo©

GO

CMo

CMoooCModo

CMoqdo

rrooo

!1 \TM\S -

QQ

0>

V)

CO O

i—e

<D>

co £

coEE

cc

LULU u; 1
Four wave types are found to be present in each of the two cases. By

varying the orientation of the titanium block, a Stoneley wave mode can

be obtained over a shear modulus range in steel from 3070 to 3300 m/s. A

greater range could be obtained by using titanium with more varied shear

wave velocity. Outside the Stoneley region two different types of leaky

2

waves can occur. Some of the characteristics of each of these four

waves are summarized, for the case of a titanium block oriented in the

rolling direction, in Figures 4.8-4.12.

Figure 4.8 gives the growth exponent and the attenuation for the four

modes. One sees that in the Stoneley modes the growth exponent is

negative indicating exponential decay away from the interface. This

displacement field is shown in Figure 4.9. However, the degree of this

decay is not uniform over the "Stoneley" part of the shear velocity

diagram; near the low steel shear-velocity end of the curve, the exponent

approaches zero, indicating that the displacement field extends

progressively farther into the steel as the shear modulus decreases. The

energy of the wave, and the proportion contained in the steel, increase

accordingly. At the "high" end of the Stoneley region the reverse

situation holds with the displacement field extending infinitely far into

the titanium. Also, although Rayleigh wave plots are not shown here, it

is easy to see that the Stoneley waves extend much further into the steel

than do Rayleigh waves. For Stoneley type waves there is good agreement

between theory and experiment in Table 4.1 except for case 1A, which is

being reevaluated.

The computer codes for the cylindrical model are for radial-axial motion

only and, by necessity, do not permit divergence of displacement inside

the cylinder. Accordingly, waves modes with transverse motion and those

leaking simultaneously on both sides of the interface are excluded from

these calculations. Both of these restrictions will be removed in future

work.

52
Near the upper end of the Stoneley region, Fig. 4.8, one sees three wave

modes coalescing, corresponding to a branch point on the complex

interface surface. Two other possible interface waves are obtained from

our calculations. Each of these waves, designated divergent waves,

corresponds to a leaky wave whose leakage angle is zero. Their

displacement fields are illustrated in Figures 4.10 and 4.11. Thus they

do not attenuate along the interface, but correspond to an interface mode

whose amplitude increases exponentially away from the interface. Unless

the growth exponent is relatively small, we expect these divergent waves

will have a low injection coefficient and pick up only a small amount of

energy from the input Rayleigh wave. Since the energy content of such a

wave increases exponentially as one moves away from the interface, these

waves can also be expected to be confined to a narrow zone near the

interface.

However, in the case of samples 2A and 3A of Table 4.1, which lie just

below the Stoneley region, the exponent is found to be' quite small, so that it may be possible to detect such a wave. The large error spread

associated with those two sample points requires reinvestigation. The

divergent wave coexistent with the Stoneley wave generally has far too

large a growth exponent to be detectable, except where the two wave

velocities are almost equal.

Beyond the Stoneley region, the interface wave becomes a leaky wave,

leaking a shear wave into the titanium as shown in Figures 4.8 and 4.12.

One sees from Figure 4.12 that case IB (and case 2B) are weakly- leaking

waves. The two leaky wave samples, IB and 2B in Table 4.1, show good

agreement between theory and experiment. Radiation from one of the

o samples tested (IB) is predicted at 11 and may be measurable.

53
4*1.5 Summary of Results for Planar Interface

The results presented in this report demonstrate the following:

o Feasibility of generating interface waves at planar interfaces of

dissimilar materials by mode conversion of Rayleigh waves

propagated in the more dense medium.

o Effect of interface integrity upon the measured interface wave

velocity. This type of measurement yields information on the

degree of perfection of interface contact.

o Effect of microstructure near the interface on interface wave

velocity. Correlation of this data is not yet complete, but

appears to indicate that over a range of frequencies where

wavelength is substantially greater than bulk microstructure

characteristics, the interface wave type and velocity can be

predicted from bulk elastic moduli near the interface. When the

bulk moduli vary with depth, the velocity will become dispersive,

but should still be predictable from theory. This points to the

possibility of carrying out inverse modelling to obtain

information on the behavior of bulk moduli near the interface from

interface wave measurements.

4,2 Cylindrical Interfaces

In this section we will discuss our work pointing to the application of

leaky interface waves to the measurement of elastic properties in the

interface region of fiber reinforced metal matrix composites. This work

promises to lead to a technique for direct ultrasonic imaging of

cylindrical interfaces.

54
As a leaky wave travels along an interface it radiates acoustic energy

out of the interface. The leaked ultrasonic energy forms a displacement

field at the surface of the matrix that can be ultrasonically detected.

This wave field contains direct information on interface properties such

as the quality of bonding the distribution of natural defects and the

elastic constants. It is our ultimate aim to use the leaky interface

wave in a novel form of acoustic microscopy so that local elastic

constants can be measured and the three-dimensional structure of the

interface reconstructed.

The viability of the approach depends first on the existence of suitable

interface waves for the material combinations and geometries of import in

MMCs. This report addresses that first question in depth. Subsequent

work will address methods for the microscopic measurement of interface

waves in actual MMCs and the analysis of this data to obtain the

quantities of importance.

4.2.1 Wave Propagation of Embedded Cylinders

The results reported in section 4.1 demonstrate that Stoneley and leaky

waves can be propagated along planar interfaces in certain combinations

of materials. However, as pointed by Owen [11], and now extended in our recent work to the case of cylindrical fiber geometries, true Stoneley

type waves exist in only a limited range of materials (shear moduli of

matrix and reinforcement approximately equal) which are generally not of

importance for MMCs.

On the other hand it appears that leaky waves can be usefully propagated

along the interface types — high modulus reinforcements in a relatively

low modulus matrix — needed for metal matrix composite applications. We

55
have investigated this in detail, particularly in the fiber-reinforced

cylindrical geometry. Detailed calculations and numerical experiments

were conducted for this geometry on several model systems. We considered

the behavior of model aluminum matrix composites embedded with various

fibers, including stainless steel, SiC, boron and graphite.

The cylindrical interfaces in MMCs may be thought of as occuring between

a rod-shaped fiber of fixed radius and a tunnel-like matrix of the same

radius. Not only do these cylindrical interfaces exhibit dispersive wave

behavior, but both empty tunnels and bare cylinders do also. This

dispersive behavior can be conveniently summarized by a dispersion curve

relating the phase velocity of a particular ultrasonic wave mode to

normalized frequency. The velocity at a particular frequency (j is found

by solution of a secular equation, (3.33). The root of this equation

corresponds to a complex velocity

v(w) = vR

(u) + i Vjfa)

where vR

(w) corresponds to the real part of the velocity and Vj(u) is the

imaginary part that leads to leakage from the interface and thus

amplitude attenuation along the interface. The waves thus characterized

are radially symmetric and have no displacements in the tangential

direction. Since they propagate in the direction of the cylinder axis

and have only radial and axial displacement components, we refer to them

as radial-axial modes.

To determine the amount of attenuation along the interface of a radial-axial mode, we express the equation for the mode as:

<f>(z,t) = 4

> (r)exp [i(wt-kz)]

56
where <#>( z,t) is the wave amplitude, (r) is the initial amplitude

o dependence on r, and k = u/v is the wavenumber. It is useful, as in

section 3.2.4, to introduce the factor k. = gj r = 2irr f, where r is the

o o o o interface radius and f is the frequency. The velocity, v, may then be

thought of as a function of k . We can write:

o

<£(z,t) = <t> (r) exp[i«t] exp[-i(x /v) (z/r )]

o o o

Multiplying both top and bottom of the exponent of the first term by the

complex conjugate of v gives

<K z, t) = <p (r)exp

o

(Jt - ( Z/r

o )( V

R

'C 0/ ( V

R

+ vj))

•exp -(z/r

o

)( v

I V(vR

+ V

I })

= <p (r) x phase factor x attenuation,

o

We, then, define the quantity

-20 log

1Q exp /( 2 2 . v

i

k

o/(vr

+ V

I

} /(r

o

. n

= 4Cnrv

J log10 (e)/(Vg + Vj) ( 1 )

as the attenuation coefficient. It represents the exponent governing the

decrease in wave amplitude per unit frequency and per unit distance along

the interface. Since the attenuation coefficient depends only on r *f.

and since fibers in MMCs are of very small radius, this suggests that

57
.
4.80 o N E o 3c

>> •rH w E

•rH 3 o ^“H o cd

0) c > 3

0) u X o

•u> t*H Vm <u o -oo

<U E 0c +j . 0) c w

<u 3 c E •rH o 0) T3 a 0 3

<u 3 u

T3 a X

TJ

w

CJ •rH >>

•u T3 O

cd c

r—H

<u 3 cd 3 O •rH o* TJ 0)

_cd 3 t- 0 fr- <HH

cr>

T*

bi) •rH [-U
3.00

n

i

i

~i

i

i

i

i

i

i

i

1

1

1

1

1

—

1

18.8

(ZHIAI x ujuj)/gp ui uojienuduv

srt/uiuj uj Ajjoo|0a :|auuni |\/

Radius

x

Frequency

in

mm

x

MHz
5.40

o

o

CD

O

00

o o o CNj CD O

^ CO CO

s^/iuiu u; A)po|9/v :poy |eajs

N

X

E

E

c

>

o

c

a)3

O’

Q)

i. LL

X

CO3M

T3

(U cc

Fig.

4

15

Calculated

velocity

vs

radius

x

frequency

fora

nonattenuated

radial

displacement

mode

ina

steel

rod.
9.00

>%

o

c

a)3

crow

LL

X

</) T3

(0

CC

sri/unu Uj A)!30|9A

Fig.

4.

16

Calculated

velocity

vs

radius

x

frequency

fora

nonattenuated

surface

radial

displacement

mode

ina

silicon

carbide

rod.
guided interface waves may exist in fiber-matrix material combinations

not previously considered, over frequency ranges and distance scales

suitable for acoustic microscopy.

Knowledge of the attenuation factor is important for experimental

purposes. For instance, if the calculated attenuation of an

interfacial mode does not exceed the value of 2 dB/mm*MHz up to

r *f = k. /2ir = 1 mm*MHz for radius r =1, then the attenuation along a

o o o typical interface of 30 mm length at a frequency 1 MHz will be 60 dB.

This level of attenuation would not preclude detection of the arrival.

When the attenuation for a chosen frequency is too high, e.g.

5 dB/(mm , MHz) , this mode will be totally attenuated at the far end of an

interface of 30 mm length [150 dB] This permits us to determine which

modes are detectable with our present techniques and equipment.

In the following, we plot both the attenuation coefficient and the phase

velocity versus tor . The dispersion curve for an aluminum tunnel

o containing no fiber was calculated and found to contain several wave

modes. In Fig. 4.13 a radial displacement mode propagating axially along

the tunnel is shown. This mode has a purely real root and thus suffers

no attenuation. In Fig. 4.14 we show a second radial-axial mode that has

been found to exhibit frequency-dependent attenuation. As noted, this

attenuation, when high, can for practical purposes make this second mode

unobservable in experiments. If, for a particular frequency, we were to

measure a velocity on the curve of Fig. 4.14 in a fibrous composite, we would know immediately that the bonding was very weak or nonexistent at

the interface, because the wave would have propagated along the tunnel.

Figures 4.15 and 4.16 show the dispersion curves for a radial

displacement mode propagating axially along bare fibers of stainless

steel and silicon carbide, respectively.

58
5.40

i

1

|

j

|

|

1

1

s

1

1

1

1

1

1

1

1

18.8

(zhi/\| x uuu)/gp u; uojienuauv

CO

to c© q t- CO © o

o

co co cm

o

cdu

0) u

C

cVw

cr-H

'T'

hJD U.

srf/iuui u; A}i30|3A

Radius

x

Frequency

in

mm

x

MHz
4.20

I

n

1

1

1

1

1

I

1

1

1

1

1

1

1

1

1

18.8

(zhw x ww)/gp u; uojienuduv

Fig.

4.18

Attenuated

radial

displacement

mode

calculated

for

an

aluminum-steel

interface.
The dispersion behavior for two interface waves at cylindrical

aluminum-steel interfaces are shown in Figs. 4.17 and 4.18. The curve in

Fig. 4.17 can be crudely thought of as a combination of the tunnel mode

of Fig. 4.13 and the rod mode of Fig. 4.15. Similarly, the curve in

Fig. 4.18 corresponds to the tunnel mode (Fig. 4.14) and the rod mode

(Fig. 4.15).

We find that leaky interface waves in the cylindrical geometry are always

connected to leakage of acoustic energy out of the interface into the

metal of the matrix rather than vice versa. This leakage can be remotely

detected and its measurement is an important feature of our planned

approach to ultrasonic characterization of interfaces in MMCs by means of

acoustic microscopy. Table 4.2 lists the calculated velocities for the

three weakly attenuated radial-axial modes in aluminum-steel.

ALUMINUM - STEEL

FREQUENCY X RADIUS [MHz x mm ] 8.0 16 32

MODE V A V A V A

1 3.18 1.04X10"2 3.180 9.6x1 0

-3 3.18 9.55x1 0"3

II 4.855 0.21 4.438 5.3x1 0"2

4.31 2.3x10-2

III 5.05 0.21 — cut off at 8.64 [MHz x mm] >

Density of Aluminum p = 2.77 x 10 -3 [g/mm3], Longitudinal Velocity of Al = 6.323 [mm/ps], Shear Velocity of Al = 3.10 [mm/ps]

Density of Steel p = 7.9 x lO-Sfg/mmS], Longitudinal Velocity of Steel =

5.92 [mm/ps], Shear Velocity of Steel = 3.25 [mm/ps]

V= velocity of interface wave [mm/ps]

A= attenuation dB per unit radius

Table 4.2 Three prominent leaky modes for steel fibers in an aluminum

matrix. The modes are given for 3 values of frequency times

radius, and the attenuation in the axial direction is given

by dB per radius unit (i.e. distance along axis/fiber

radius)

.

59
We have extended these theoretical calculations to several material

combinations found in MMCs. Table 4.3 shows the velocity dependence as a

function of frequency for five similar modes in aluminum-silicon carbide,

while Table 4.4 displays such modes in aluminum-boron. Our preliminary

experiments could only be designed after such calculations were

available. We would like to emphasize that the presence of cylindrical

curvature introduces new modes, and, therefore, new opportunities for the

ultrasonic measurements of MMC interfaces. The use of leakage from the

interface for interface characterization is one of the unexpected outputs

of this program.

ALUMINUM-SILICON CARBIDE

FREQUENCY x RADIUS [MHz x mm]

MODE

2 5 10

V A V A V A V A

1 6.1 4.00 5.6 3.8 4.8 3.2 4.7 5.5

II 5.8 2.00 7:8 2.9 8.6 8.0 8.25 1 8.0

III 7.6 2.50 8.1 3.2 9.0 5.0 5.5 2.2

IV 0.8 52.00 2.8 50.00 4.4 30.0 10.2 2.5

V 0.1 53.00 0.22 50.00 2.2 30.0 6.6 4.2

VI 0 23.2 1.30 20.0 7.5 5.5 5.6 1.25

Density of Aluminum p= 2.77 x 10 *3 [g/mm 3

], Longitudinal Velocity of Al = 6.323 [mm/ps], Shear Velocity of Al = 3.1 [mm/ps]

Density of Silicon Carbide = 3.2 x 10 -3[g/mm 3

], Longitudinal Velocity of SiC = 9.649 [mm/ps], Shear Velocity of SiC = 5.193 [mm/ps]

V= velocity of interface wave [mm/ps]

A= attenuation dB per unit radius

Table 4.3 Six prominent leaky modes for silicon carbide fibers in an

aluminum matrix.
Fig. 4.19 Model metal matrix composite sample used to test

theoretically predicted dispersion. Aluminum-steel

interface composed of 3.2 mm radius 316 steel rod shrink

fitted into a 2024 aluminum alloy cylinder.

ALUMINUM - BORON

FREQUENCY x RADIUS [MHz x mml

MODE 2 4 6 10

V A V A V A V A

1 7.05 0.01 6.12 0.43 6.00 0.605 5.96 1.07

II '2.00 20.0 8.5 3.2 10.3 1.5 10.7 0.4

III 12.30 0.36 12.80 0.37 13.2 0.4 13.4 0.38

IV cut off at 4.45 [MHz x mm ] 23.4 0.5 12.7 0.4

V 2.50 9.5 5.50 3.7 7.7 2.6 8.21 3.05

Density of Aluminum = 2.77 x 10 3 [g/mm3], Longitudinal Velocity of Al = 6.323 [mm/fxs], Shear Velocity of Al = 3.1 [mm/ps]

Density of Boron * 2.352 x 10-3[g/mm3], Longitudinal velocity of Boron *13.228 [mm/^s], Shear Velocity of Boron = 8.657 [mm/ps]

V= velocity of interface wave [mm/(is]

A= attenuation dB per unit radius

Table 4.4 Five prominent leaky modes for boron fibers in an aluminum

matrix.

4.2.2 Aluminum-Steel Interfaces

To systematically confirm the theoretical predictions we have intensively

studied model specimens containing an aluminum matrix-steel rod interface.

Two fabrication methods were used to prepare model samples, shrink

fitting and casting. The first model was composed of a 3.2 mm radius

316 stainless steel rod, shrink fitted into a 2024 aluminum alloy

cylinder (Fig. 4.19). Shrink fitting created a very good cylindrical

interface between these two materials because of the large difference in

their thermal expansion coefficients.

61
2021

Al

Alloy

Matrix

316

Stainless

Steel

Rod

oo

oo

(zhw x uiui)/gp ui uouenuauv

srf/uim ui. Aj!00|SA

O

CM

bi) u.

Radius

x

Frequency

in

mm

x

MHz

The

interface

wave

velocity

measured

asa

function

of

frequency

for

the

model

shrink

fit

sample

(points)

and

the

theory

(continous

line).
/
5.40 tt

*

A

—

-

A

—

1

1

—

—

*

t

18.8

(zHW x iuui)/gp uj uojienuauv

bD

s^/luiu u; A;joo|9A

4.22

Velocity

of

leaky

interface

mode

measured

as

a

function

ol

frequency

(triangles).

Comparison

with

the

theory

lor

the

aluminum-steel

interface

(continous

line).
Fig.

4.23

Microstructure

of

aluminum-steel

interface.

Top

side

is

2024

aluminum,

the

bottom

side

is

316

stainless

steel.

Visible

flaws

and

separations

along

the

interface

cause

the

decrease

of

interface

velocity

comparing

to

the

calculated

values.

The

vertical

marker

corresponds

to

250

pm.
It is possible to measure the leakage energy for such modes as shown in

Fig. 4.24 using an ultrasonic transducer or an interferometer to register

the displacements at the outer surface of the matrix. We used surface

wave wedges driven by 2.5 MHz transducers to generate interface waves by

mode conversion. A broadband point longitudinal transducer of 1.5 mm

diameter was used to detect normal displacements at the outer surface.

The tangential components of the same leaky arrivals were detected using

a shear wave transducer placed on a cone-shaped waveguide. All

2

measurements were done using a MATEC 5100 ultrasonic system.

Figure 4.25 shows the experimental results obtained on the surface

parallel to the interface of the sample shown in Fig. 4.19. There is a

pronounced maximum visible for both polarizations along the direction

parallel to the interface. This maximum is connected to an angle of

maximal energy flow associated to the Poynting vector integrated from the

point of origin of the mode at the interface. This maximum identifies

the maximum flow of acoustic energy due to leakage. After this point the

energy attenuates as the square of the displacement attenuation.

Figure 4.25 illustrates this angle for two of the modes listed in

Table 4.2. This angle of maximum energy leakage is a valuable way to

separate modes and thus contributes to a more detailed interface

characterization.

Figure 4.25 shows a comparison of theory and experiment for the three

modes listed in Table 4.2. Mode II of that table exhibits a maximum

Certain commercial equipment, instruments, or materials are identified

in this paper in order to adequately specify the experimental procedure.

Such identification does not imply recommendation or endorsement by the

National Bureau of Standards, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.

64
4 (13.63°) Mode I (Aluminum-Steel)

Fig. 4.26 Schematic diagram showing two possible arrivals (Modes

II from Table 4.2) on the cylindrical sample.
R

(mm)

R

(mm)

R

(mm)

Amplitude

(mV)

(a)

7

6

5

4

3

2

10

19.40

13.00

6.60

0.20

o R . 2.5 MHz S<|> 1 mm

a R . 2.5 MHz • U 1.5 mm

T • 2.5 MHz • Interface Wave

Experiment

Theory

-2.00 10.80 23.60 36.40 49.20 62.00

(b) Z (mm)

25.80

19.40

13.00

6.60

0.20

-2.00 10.80 23.50 36.40 49.20 62.00 1 2 3

|

* Theory | |

$ | |

l 5 T~1 Experiment

, , ,, ! ! hi i Liu

i \\uUrftT\ 1 * -M-ntT i t i i i ! ! ! i' ! ; . : :

oo R . 2.5 MHz L a R. 2.5 MHz S

T -2.5 MHzInterface

Wave

(c) Z (mm) Amplitude (mV)

Fig. 4.27 Experimental identification of radial displacement modes

created in the cylindrical sample by mode conversion.
Fig, 4.28 The calculated displacement field of the leaky radial axial

mode I (Table 4.3) for the Al-SiC interface superimposed on

the geometry of the sample. Each diagram represents

calculations for different frequencies.
R

(mm)

X \ X X X X x X X V

v v v \ X X X X X X \ X X X x X X X X X X x X X x X X x VOX Xx X X X V X x > > f * X X X V C K - . -

x X X x X

X X X X x x X X x x x x X X X X X X X X X X x x X X x X x X X X X X X x X X X X X x

X X X X X x X X X X X X x X X X X X X x X X X X X X x x X X X X X x X X X X X x x x X X X x x x X ^ "X. X xxx

Xy X X X xxx

X X X X X x x X X X X X x x X X X X X v x

’X ^ ^ ^ ^

0.00

10.00

3

„ 1 } * ~

9 (i $ < < Q 0 « « •« e e M|||§

0.00

>c ^r' xx *x x

X \\ V *\V4 vA "» \A^ ' A

"* V\

O C£> <K> 0.00 4.00 8.00 12.00 16.00 20.00

© o « «

24.00 28.00

(a)

3.125

MHz

(b)

2.5

MHz

(c)

2.0

MHz

(d)

1.0

MHz

Z (mm)
(ZHN x uiui)/ap u| uojjenuauv :xMV*W IV/OIS

ou

cdo

u)c0•HCO

1

cd

srf/iuui uj AjjQO|0A :xujew IV/OIS

Radius

x

Frequency

in

mm

x

MHz
angle of about 50 , corresponding to the maximum measured in Fig. 4.25

and reproduced here. The high leakage angle is associated, in this case,

to a high attenuation along the interface; this explains why the

interface velocity of this mode could not be measured at the end of the

sample 60 mm down the interface.

Measurements carried out at the end of the sample on the face

perpendicular to the interface are also shown in Fig. 4.27. At 2.5 MHz,

the maximum energy flow for Mode I of Table 4.2, which was also detected

o by velocity measurements, is 13*14 . The accuracy of measurement is

still not very high, but an energy maximum in approximately the right

place was found in all experiments. The small maximum occuring near the

o interface at about 3 could be due to the third mode listed in Table 4.1

or to another predicted mode with a similar low angle of leakage. We

expect to increase measurement accuracy both of leakage angle and mode

velocity to further explore this possibility.

4.2.3 Aluminum-SiC Interfaces

In this section we confirm the existence of leaky interface modes in a

typical metal matrix composite system, that of aluminum-silicon carbide

The theoretical predictions for the Al-SiC systems are summarized in

Table 4.3 (and a similar summary for Al-boron systems is given in

Table 4.4). Figure 4.28 presents the calculated displacement field for

Mode I in Table 4.3 (that mode which was experimentally measured) for a

variety of frequencies. The displacement field shows the main features

of leaky interface waves in the Al-SiC system: strong dispersion;

substantial attenuation along the interface, and a high characteristic

angle of leakage. This figure also clearly shows the horizontal energy

shift mentioned earlier. Figure 4.29 illustrates the dispersion and

65
Fig. 4.24 The displacement field of mode I (Table 4.2) for

aluminum-steel interface calculated for each measured

frequency a) 5 MHz, b) 2.5 MHz, c) 2 MHz, d) 1 MHz. The . characteristic single of leakage and the attenuation are

frequency dependent. The ellipses trace the orbit of a

particle and the arrowheads on the ellipse show particle

motion. The arrows show the energy flow (Poynting vector)

averaged over one period. Poynting vector lengths and

ellipse sizes are consistent between each other, but

arbitrarily scaled for graphing. The thick continuous

integrated Poynting vector showing the energy flow of a wave

packet starting at the edge of the interface. Flow lines

starting from other interface points are parallel, and

obtained by shifting this curve.
R

(mm)

Matrix

(a) 5 MHz

Fiber

Matrix

(b) 2.5 MHz

Fiber

Matrix

(c) 2 MHz

Fiber

Matrix

(d) 1 MHz

Fiber

-2.00 10.80 23.60 36.40 49.20 62.00

Z (mm)

m-h « , t U
Following exploration of other possible methods, interface waves were

found to be best generated by the conversion of rod surface acoustic

waves (SAWs). Polymethyl -methacrylate wedges were used to generate and

detect these ultrasonic surface waves over a range of frequencies.

Figure 4.20 presents the interface wave velocities (as separate points)

measured as a function of frequency for the sample shown in Fig. 4.19.

These can be compared with the curve of theoretically predicted

velocities. The experimental results were obtained for frequencies of

0.5, 1, 2, 3.5, and 5 MHz. The accuracy of the measurements was

+ 15 m/s. In this case a very good agreement between the theory and the

experiment was obtained.

Casting was also used to fabricate test samples. Figure 4.21 shows a metal matrix composite model of aluminum cast around a number of aligned

steel rods of radius 1.195 mm spaced 4 mm apart. The velocities of leaky

interface waves were measured along the interfaces as a function of

frequency and the results are plotted in Fig. 4.22. For this sample, the

experimental results (triangles) deviate from the theoretical predictions

(continuous line). Small flaws, and separation at the interface

(Fig. 4.23) were present in this sample, and are thought to be the origin

of the slight deviation of the experimental results from the theory at

higher frequencies. A much larger deviation from the predicted behavior

occurs at low frequency for fr <2.0 mm*MHz, when the wave length

o becomes comparable with the spacing between the rods and rod interactions

occur

.

The displacement field of the weakly-leaky radial displacement mode

(Mode I in Table 4.2 and Fig. 4.17) is shown in Figs. 4.24a-d for

different frequencies. The length of the interface is 60 mm, the radius

of the rod is 3.2 mm, and the outer radius of the aluminum matrix is

25 mm. The elliptical particle trajectories in these figures show the

62
relative amplitudes of the z (horizontal) and r (vertical) displacement

components at points uniformly distributed in the sample. The direction

of energy flow, the Poynting vector, is depicted by the arrow stemming

from the center of each ellipse. The length of the arrows corresponds to

the magnitude of the Poynting vector. These figures show the

characteristic angle of leakage (given by the Poynting vector) as well as

the decay of wave amplitude. Note that for higher frequencies,

(e.g. 5 MHz), a clear boundary, defined by the Poynting vector, separates

a field of evident exponential amplitude growth. Interestingly, this

region’s displacements are not associated with the transport of energy,

which cannot occur perpendicular to the Poynting vector. The continuous

thick lines show the energy path followed by a "wave-packet” starting at

the edge of the interface as found by integrating the Poynting vector.

For Stone ley type and other non-attenuating waves (such as Love type

waves), the Poynting vector is parallel to the interface. For leaky

waves, this non-zero characteristic angle of leakage depends on the

imaginary part of the velocity and on the attenuation. The angle is, in

fact, determined as that which balances the exponential growth in the

r direction with the exponential decay in the z-direction. For the

cylindrical geometry, the attenuation as well as the characteristic angle

of leakage are frequency dependent. Both increase with increasing

wavelength at constant r in Figs. 4.24a-d. Figure 4.24c shows that this

o mode propagates at an angle of about 14 at 2.5 MHz. Exact measurements

of the angle of the Poynting vector are affected by the shift in energy

flow in the matrix close to the interface. Rays close to the interface

do not travel in straight lines. This effect is newly discovered, and

has not been previously reported. It is of a different nature from the

Goos-Hanchen shift [19,20] observed for a bounded beam at optical

interfaces, or the Schoch effect [21,22] observed for planar interfaces

in the vicinity of the critical angle of a reflected acoustic beam.

Here, the observed changes due to curvature in energy flow are frequency

dependent and are seen, for this mode, to be larger at lower frequencies.

63
Fig. 4.25 The upper diagram presents the idea of the characteristic

angle of leaking. The lower diagram shows the experimental

results obtained through an aluminum matrix parallel to the

interface. Results were obtained for both in-plane (S) and

normal (L) displacements along the sample length. The

measured leakage maximum allows evaluation of the

characteristic leakage angle.
///A JE|

00 /

Surface-wave

regime

Leaky-wave

regime
7.00 XXX XXX XXX XXXX x 5 x X X 5 x 5 x X X 'X' XXX

xx>' xXXXx, XXXX

XX

xx.

NV' ;' X* VXN VX ' X X.v \ x

3.00

0.33

Al

SiC

0.13 0.07 0.27 0.47 0.67 Z (mm x 10-2)

•w c

<+* •**

<y Aow

<0

COo

GOI A

COVAcV£

01o

rta

CO0Aw

w.OV

"03

gs - u

4)C

<y -ao3O

cfl u

<y

E-> O

CO

bO

•r* UvAH o

co

(0

kj£

CO

cfl 3a

T5Caa

o> "O

>>Oc

a>3

O'

<u

>>

b£) uoav«

JC

0^

eg

bO O fcu
CJ 3

•rH 3

CO TJ3

c/3 •rH 3 f“3

•rH >> -3 o

33 >>o

E

E ^“H

cd CM E

CO 3

3

<u •rH A E H 3

f-H3

a TT

/-H CM a O

E CM 3

cn 3

O O

•rH M> CO 3

1 •rH r-H < -3

<D U) -3 •w

U) •MCm O X

3

>> •rH 3

-3 0) C/3 E

O c/3 3 3

bfl >

3 -o JC o H 3

ro

hi

•rH U-.
Fig. 4.32 Generation and detection of the leaky radial displacement

interface waves in the Al-SiC sample. The transmitting

transducer (T) and the receiving transducer (R) are each

attached to aluminum waveguides- The surface wave generated

on the SiC rod converts into the interface wave at the

interface. A second conversion takes place at the end of

the interface. The receiver on the aluminum waveguide then

detects the surface wave from the aluminum tunnel.
30

mm
(zhin x tuui)/ap ui uoiienuauv :xujbw iv/OIS

O

CM

O©e ©

©©

CM

©©

od

© ©©

©

ib

©

©

© • CM

r- N

O©

©©

cb

©

ib

o

©

oo

CM

o

o

X

E

E

c

>»o

c

0) 3

craw. LL

X

(0

3

'S

CT3 QT

P0

po

hJD

stf/iuiu u; A^jooieA :xuie^ |V/0!S

Measured

velocities

at

the

Al—

SiC

interface

as

a

function

of

frequency

(points)

compared

with

the

theory

(continous
attenuation respectively, for this mode. The displacements and energy

flow close to the interface are shown in Fig. 4.30.

An Al-SiC sample, composed of a 3.2 mm radius SiC rod shrink fitted into

a 2024 aluminum cylinder, was used to test the theoretical velocity

predictions listed in Table 4.3. The sample’s geometry is shown in

Fig. 4.31. The ultrasonic shear and longitudinal velocities were

measured on each of the components separately, and the data was used in

the theoretical computations. The measurement technique for generation

and detection of leaking radial-axial waves is schematically presented in

Fig. 4.32. Three transducers, each of different frequency, were used as transmitters. These were placed on a concave aluminum waveguide. Only

the circular edge of the silicon carbide was in contact with the

transmitting waveguide. The interface wave was generated by mode

conversion from a surface acoustic wave on the rod, which projected out

from the matrix tunnel. At the other end of the interface, the interface

wave reconverted into a surface acoustic wave and travelled along the

aluminum tunnel, where it was detected using another aluminum waveguide

of convex curvature.

Knowing (from theory) the surface wave velocities on the silicon carbide

rod and aluminum tunnel separately, and measuring the delay time in the

two waveguides, the velocity of the interface wave along the Al-SiC

interface was obtained. Figure 4.33 presents the measured velocities

along with the theoretical predictions. The accuracy of the velocity

measurements was 60 m/s. We expect to improve upon this accuracy by

modifying the sample shape and by using more efficient methods for

generating surface waves. These results indicate, as suggested earlier,

that leaky interface waves do exist in a form suitable for interface

characterization using acoustic microscopy methods in fiber-matrix

combinations of technological importance.

66
4.3 Future Directions

Weakly leaky modes in the cylindrical geometry provide a potential new

tool for interface characterization in fiber reinforced MMCs. The

leakage from these modes offers the opportunity for application of

acoustic microscopy to image interfaces from the surface of a sample.

Preliminary experiments at low frequencies appear very promising. These

experiments will be continued and extended to high frequencies using

fiber sizes typical in MMCs. The potential of microfocused laser

generated ultrasound for such applications will also be studied.

The general theory discussed in section 3 offers the opportunity to

characterize interphase regions of finite extent. Techniques for

preparing controlled interphases are being developed in both cylindrical

and planar geometries, and work will begin on correlating the theory with

experiments. Work is also underway to study the effect of frequency on

the generation depth of interface waves on determining the elastic

properties near interface material.

These measurement methods will be applied to the study of the elastic

properties near the interface in specially designed samples of MMCs

produced by the SDIO/ONR Advanced Composites Consortium. Success in this

project will greatly advance the ability to model and, therefore, design

composites. Ultimately a combination of these techniques with acoustic

microscopy methods may permit in situ diagnosis of metal matrix, and

perhaps, ceramic composites.

67
4.4 References

[1] R.N. Stone ley, Proc. Roy. Soc. (London), 106

(1924), pp. 416-429.

[2] W.L. Pilant, Bull. Seism. Soc. Amer., 62 (1972), pp. 285-299.

[3] R. Yamaguchi and K. Sato, Bull. Earthquake Res. Inst, of Tokyo

Univ. , 33 (1955), pp. 549-559.

[4] D.A. Lee and D.M. Corbly, IEEE Trans, on Sonics and Ultrasonics,

SU-24 (1977), pp. 206-212.

[5] A.S. Ginzbarg and E. Strick, Bull. Seism. Soc. Amer., 48 (1958),

pp. 51-63.

[6] R.A. Phinney, Bull. Seism. Soc. Amer., 51 (1961), pp. 527-555.

[7] K. Sezawa and K. Kenai , Bull. Earthquake Res. Inst, of Tokyo

Univ.. 17 (1939), pp. 1-8.

[8] J.G. Scholte, Proc. Ned. Akad. V. Wetensch. Amst., 45 (1942),

pp . 20-25

[9] J.G. Scholte, Proc. Ned. Akad. V. Wetensch. Amst., 45 (1942),

pp. 159-164.

[10] J.G. Scholte, R. Astron. Soc. London Monthly Notices (Geophys.

Suppl.), 5 (1947), pp. 120-126.

[11] T.A. Owen, Prog. Appl . Mater. Res., 6 (1964), pp. 69-87.

68
[ 12]

[13]

[14]

•[15]

[16]

[17]

[18]

[19]

[ 20]

[21 ]

[ 22]

R. O. Claus and C.H. Palmer, Appl . Phys. Lett,, 31 (1977),

pp. 547-548.

S. I. Rokhlin, ML Hefets and M. Rosen, J. Appl. Phys., 52 (1981),

pp. 2847-2851.

T. C. Lim and J.P. Musgrave, Nature, 225 (1970), p. 372.

P. Chadwick and P.K. Currie, Quart. J. Appl. Math., 27 (1974),

pp. 497-503.

R.N. Thurston, J. Acoust. Soc. Amer. , 64 (1978), pp. 1-37.

V.V. Meleshko, Soviet App. Mech. , 16 (1980), pp. 382-385.

V. Kumar and G.S. Murty, IEEE Trans, on Sonics and Ultrasonics,

SU-29 (1982), pp. 138-142.

F. Goos and H. Hanchen, Ann. Physik, 1 (1947), pp. 333-346.

F. Goos and H. Hanchen, Ann. Physik, 5 (1949), pp. 251-252.

A. Schoch, Acustica, 2 (1952), pp. 1-17.

A. Schoch, Acustica, 2 (1952), pp. 18-19.

69
5. Acoustic Emission Assisted Measurement of Fiber and Interface

Strength

Our objective in this sub task is the in situ measurement of fiber and

interface strength in MMCs. The need for strength measurements arises

from the increasing awareness of the crucial role that interfaces play in

the mechanical performance of composites. At the same time,

sophisticated models of interfacial microfracture are now available

(Argon et al , [1]) which can be used to predict the failure of bulk

composites on the basis of interfacial strength. Such models require

experimental corroboration and of course can not be used quantitatively

without accurate data on interfacial strength.

In one type of method for determining interface strength, that of

Broutman [2], a single fiber is embedded in a transparent polymer matrix,

which is then loaded in compression. The load at which the interface

fails is detected optically and a stress analysis used to determine the

local stress at the interface at the point of failure. Since optical

techniques are not applicable in opaque MMCs, we have proposed using

acoustic emission (AE) as a means of acoustically detecting the failure.

Moreover, since acoustic emissions are also emitted by other deformation

processes, they may provide information on other aspects of MMC

deformation micromechanisms. It is thus useful to begin with a very

brief overview of the microscopic origins of acoustic emission.

5.1 Microscopic Origins of Acoustic Emission

Acoustic emission can be thought of as the naturally generated ultrasound

created by local, rapid load changes within a body. If an ultrasonic

transducer with sensitivity in the 0. 1-1.0 MHz frequency range is

attached to almost any type of composite, acoustic pulses are detected

70
during the application of load. To intuitively appreciate how such a

natural phenomenon arises, consider the following "thought’' experiment.

Imagine that a body has been placed under load and attained mechanical

equilibrium throughout : the internal force field then exactly balances

the tractions applied to the surface. Now suppose that a crack grows

within the body: the crack surfaces move in order to make them stress

free. Elastic waves radiate from the crack to restore equilibrium with

the surface tractions. Behind the elastic wavefronts, the internal

forces are relaxed. When the elastic wavefronts eventually reach the

surface, they cause surface displacements. These displacements begin the

process of restoring equilibrium between the cracked interior surface and

the surface tractions. The subsequent arrival of elastic waves multiply

scattered within the body is detected ul trasonical ly and called acoustic

emission.

Obviously, any dynamic mechanical instability is capable of generating

these displacements. Dislocation motion, shear fracture, fiber fracture

and interface decohesions are all potential sources of acoustic emission.

It would therefore be gratuitous to assume that only interface decohesion

is detected acoustically in the metal matrix composite analogue of a

Broutman- type test. Previous research however leads us to believe that

we may be able to unambiguously attribute an acoustic emission signal to

a particular micromechanism from the attributes of the emitted signals.

The background to this is presented in the more extensive review of

microscopic origins of acoustic emission included in Appendix I.

Our approach has been accordingly to begin with a review of the

elementary micromechanisms of composite deformation and fracture, to then

design model experiments where acoustic emission may be used to clarify

the mechanisms, and finally to analyze these tests to deduce the strength

of fiber-matrix interfaces produced by various processing routes. Here,

solidification processing pathways are examined. Future work will extend

71
Fig.

• •V..‘

• •/.**•*. •,

•*•••*• • #*• .* • •

* •••*•** •

* *••**••« v

•*/ *• * *• "• • V * •*.' • I

* • |

*

• \ Interphase

; : i:

Matrix!®

tensile

Decohesion

5.1 Mechanisms of interface failure during longitudinal loading

of a fiber reinforced composite.
a+do i

Shear

lag

model

of

load

transfer

from

matrix

to

fiber (1

omg i

lud i

na 1i

oad i

ng ).
the approach to materials processed by other groups in the SDIO-IST/ONR

Advanced Composites Consortium.

5.2 Fibrous Composite Fracture Micromechanisms

The mechanical properties of a fiber-reinforced metal matrix composite

are in large part influenced by the transfer of load from the matrix to

the fiber. This in turn depends upon the fiber’s intrinsic strength and

ductility, its orientation to the matrix stresses, and the mechanical

properties of the interface and matrix. Let us begin by considering only

longitudinally-oriented fiber composites under tensile load. Depending

upon the relative strengths of fiber and interface, several failure

sequences are possible and these are discussed below.

5.2.1 Load Transfer from the Matrix to the Fiber

If a fiber-reinforced composite is loaded along the fiber axis,

deformation will at first be elastic throughout. If the interface does'

not fracture, with continued loading the matrix will eventually begin to

flow plastically, work hardening until the tensile stress in the fiber is

sufficient to cause fiber fracture.

The stress distribution along the length of such a fiber segment with an

ideal interface can be estimated by the shear lag (or fiber loading)

model [3], Figure 5.2. Consider the fiber element shown in Figure 5.2.

If o is the tensile stress in the fiber, t is the interface shear stress

and r is the fiber radius, then the force balance across an element of

length d£ is

2 2

(cr + da)77T = arrr + 27rrTd£

72
Thus, the shear stress Is proportional to the gradient in the tensile

stress in the fiber:

da/dg = 2t/t

so that everywhere at which there is a change in fiber tensile stress,

such as at a pre-cut notch, or a crack, at the fiber ends, interfacial

shear stresses are set up proportional to the stress gradient. Since the

shear stresses are independent of the stress itself, they can exceed the

interface strength even at relatively low fiber stresses, say, on

reloading a previously strained specimen or during fatigue testing. 1

The stress distributions for various fiber lengths are shown in

Figure 5.2. After integration, we find that the tensile stress increases

from 0 at its free end (where it has fractured) up to some limiting value

CT f

^ the tenslle strength of the fiber) at the "ineffective length" ee (over which reinforcement is minimal) from the free end. For simplicity

we have treated the fiber strength as if it were a constant (cf. in

Fig. 5.2), whereas it is well known that, owing to intrinsic defects,

there is a statistical size-strength effect [3], However, the

experimental technique utilized here measures each fracture individually,

SO that a

f

is a floating constant and this variation is in practice taken

into account.

For example, the shear stresses can be estimated from the above as

T = (r/2) (d<j/df ) . At constant load, since da/a = -dA/A = -2dr/r, only a

5% variation in radius will produce a 10% change in tensile stress. This

has obvious implications for quality control. If there is a 10% change

in stress over a distance of r/10, then da = 0.1a and r/df = 10, so that T= CT/2 ’ the flow stress ln ^e matrix. Thus, even small changes in

fiber diameter can act as a mechanism for generating large interfacial

shear stresses capable of nucleating interfacial cracks.

73
If the fiber is shorter than £ = 2£ , referred to as the "critical

c e length," the fracture stress is not reached (nor is reinforcement very

efficient), and the fiber can be pulled out of the matrix endwise

without breaking it. If fibers of lengths greater than £^ are present,

they will repeatedly fracture under continued loading until this critical

length is reached at a "critical aspect ratio"

£ /d = cr r/2t.

c f i

where d = 2r is the fiber diameter and t\ is the shear strength of the

interface, or matrix, whichever is lower. This ratio, attained after a

stress large enough to fracture the strongest fiber segment is reached,

will be useful in measurements of interface strength.

5.2.2 Micromechanics of Fiber and Interface Fracture

When a fiber fractures, the fiber stress is only relaxed along the end

gradient length £ , so that the elastic energy released as AE ought to

depend only on the load, not the total fiber segment length. For shear

fracture of the interface, on the other hand, the crack may extend

multiple values of this length £^ and so in principle relax a larger

volume, creating much larger emissions than fiber fracture, but this may

be limited as discussed below. To understand these effects, as well as estimate interfacial energy from interfacial strength, it is necessary to

briefly analyze the mechanics of fiber and interface fracture.

The total energy change due to fiber fracture is given by the sum of the

changes in elastic and surface energy,

74
AE

f = -2(A

f

/2E

f

)

cr2

(x) dx - + 2-y^A^ < 0

eea

2

( x) dx

o

where ct ( x) = o

^ is the stress distribution before fracture and

a’(x) = cr^x/^ is the distribution after. Note that fiber unloading only

occurs over the end lengths 2£

g

and not in the central portion, where the

stress is still o

(Fig. 5.2). Hence, acoustic emission from fiber

fracture ought to be independent of fiber length. Here and A^. are the

surface energy and cross sectional area of the fiber. The factor of 2

arises in both terms since there are two ends and two fracture surfaces.

After integration,

AE, = -2(Ar /3Er ) i a

2

+ 2~r r A, < 0

f

v f f'ef f f ~

Using values for SiC fibers of A

^ = 1.53 x 10 ^ m

2

, = 10

2

m,

E^, = 480 GPa and = 3 GPa, we obtain the elastic energy release of

-4 2

1.9 x 10 J. The surface energy 2nr^A^., using = 20 J/m [4], is

-7

6.1 x 10 J. Thus the surface energy term is negligible and the

remainder of elastic energy is radiated as kinetic energy, i.e., acoustic

emission. This is orders of magnitude larger than the kinetic energies

produced by thermoelastic sources [5] which although inefficient are

detectable, so that the acoustic emission from fiber fracture ought to be

easily detectable.

An interfacial shear failure at the end of a fiber may be in the form of a circumferential crack that moves in from the end, advancing the tensile

stress gradient ahead of it (Fig. 5.2). Once the stress gradient

impinges on that at the other end, the available stored elastic energy

will drop sharply (Fig. 5.2),. and the crack should arrest. Thus

75
interfacial shear cracks should be less likely to form once the fiber has

broken down into its critical lengths. The change in total energy by the

advance of such a crack over distance 2-2^

is given by

AE = -(A./2Er ) s

v f f

2-2

a (x)dx - a ^(x)dx

2-2

+ 2ttt 2-r.dx (5.1)

where

a(x) = ar , for 2 < x < 2-2 v J f e — ~ e a(x) = Grx/2 , 0 < x < 2 v J f e - - e

and

cr’(x) = ar x/2 , for 0 < x < 2

and where is the interfacial adhesive energy (the factor of 2 coming

from the two surfaces of the crack).

The adhesive energy nr. is defined by2nr.=nr + nr - nr +nr. , where nr l l r m rm irr r is the reinforcement and nr the matrix surface energy, nr is the m rm interface bond energy, and is a dissipative term which includes

frictional and plasticity phenomena. In ceramic-reinforced metals, the

reinforcement surface energy nr generally is >> than nr or nr so that

the preferential path for ’’interfacial” fracture is often through the

matrix if the interface bonding is good and matrix ductility limited (so

that ~i^ rr

is small); this has been observed by Evans, et al . [6],

Crowe, et al . [7], Arsenault and Pande [8], and others.

76
This gives

AE = -

s Af

(«-«

c )/2Ef

2

oc + 2~r. f i 2ttt( 2-2

c

)

Notice that the total energy change vanishes as ^“^

c

» hence shear crack

extension becomes thermodynamically less favorable as the fiber segments

2 approach critical length. On the other hand, in large segments, where

(2-2 ) is large, large interface cracks could produce large energy

changes (acoustic emission) compared to fiber fractures, simply because

of the potentially much larger volume (2-2^) of stressed fiber released.

This can only occur, however, if the total energy change is large and

negative, i.e., if the adhesive energy term is small relative to the

elastic energy (i.e., brittle fracture). But this may not happen in MMCs

because: (a) plastic deformation greatly increases the apparent surface

energy 2'k of cracks, and (b) when the fiber is loaded in tension, the

crack tends to close because of the greater Poisson contraction of the

matrix, and frictional effects could become substantial, similarly

increasing 2nr^. The reverse is true if the load is reversed and we hope

to examine this latter effect through compressive loading tests.

There is a critical stress for such debonding, given by the Griffith

condition that the incremental increase in surface energy of the crack be

less than the incremental decrease in stored strain energy for a change

in length dx:

This phenomenon is similar to that of inclusion breakup in the working

of steels, wherein fracture takes place at the largest inclusions first

[9].

77
( 2ttt ) { 2t\ ) dx < (3E/3x)dx (5.2)

where, from Eqn. 5.1 we obtain the stored energy density

(3E/3x) = (7rr

2/2E

f

) a

2

1 - —

provided that 0 < x < (€-2^). This stored energy density decreases

continuously during straining until it is zero at the ineffective length

halting further fiber breakup. Initiation is most likely at the fiber

end (x = 0) where

so that interfacial fracture will occur at a debonding tensile stress in

the fiber, combining (5.2) and (5.3),

This is precisely equal to the expression derived independently by

Outwater and Murphy [10], if we take = 2t^, where is the mode II

energy release rate. It is also proportional to that derived by Gurland

and Plateau [9] for failure at inclusions in metals. It was later

rederived and verified experimentally by Wells and Beaumont [llj using

compressively loaded filaments of steel of varying radii embedded in

epoxy. The curve fit agreement was much better than that of critical

stress theories, they noted. If we take 2-7 = G , the energy release

2

rate, and G = K /E„, where K is the critical stress intensity, we

c c f c obtain

(3E/3x) = (3E/3x)

max = Trr2

a

2/2E

f (5.3)

(5.4)

78
K = 2^Te7

c if (5.5)

It predicts that the interface fracture toughness should be independent

of the fiber radius and crack size.

5.2.3 Interphase Effects on Fiber Fracture

A reaction zone (interphase) between the matrix and fiber may fail by a

tensile microfracture (Fig. 5.1), subsequently serving as a notch on the

fiber and possibly reducing its effective strength. The microfracture

mechanics of such brittle layer failures has been studied by Ochiai and

Murakami [4,12]. They employ a two-component model of a brittle layer

and fiber (no matrix). The layer exhibits a size effect because thinner

layers have statistically fewer defects and statistically higher

strengths. If thin enough, the brittle layer may not fracture before the

fiber. Above this critical thickness, the strength is reduced as l/4"c,

where c is the layer thickness.

Prolonged exposure of SiC to an aluminum melt degrades fiber strength, a

possible contributing factor being such notch-induced fiber fractures.

Since these can be prevented by shear fracture at the interphase/fiber

boundary, one is interested in establishing the relative component

strength levels for optimum strength. To do this, as suggested by the

work of Ochiai and Murakami [12], it is necessary to define at least

three critical stress levels:

= stress for initiating a notch

E

o = stress for extending the notch into the fiber, and

cP = stress for debonding of interphase.

These give the possible failure sequences shown in Table 5.1:

79
TABLE 5.1

FAILURE SEQUENCES FOR WEAK AND STRONG INTERFACES

Relative Strength Failure Mechanism Interface Type

E

o

I o

> d

D

> a

D

Debonding occurs; no notching Weak

E

o

D N

I o > o Notches form interface debonds;

no extension into fiber

Weak

a

DEI > o > o Notches form and later extend

to cause fracture fiber; no debond.

D

o >

I o > E

o Notches immediately fracture

fiber; no debond.

Strong

Strong

An important conclusion to be drawn is that the fiber fracture can be

prevented by shear fracture of the interface at an interphase notch.

Ochiai and Murakami define the maximum strength that an interface can

have to permit extension of the notch into the fiber [4,12], but do not

give an experimental method for measuring the actual (not this maximum)

interface strength. In practice [4] they used a pull-out test which

established the critical aspect ratio from which the interface strength

was calculated using the shear lag model. They experimentally

demonstrated the interphase thickness-strength reduction effect with data

on B/Ti alloys [4]. Here the interphase thickness was systematically

increased by high temperature anneals, and the strength accordingly

dropped. The effect did not occur in the W/Al system, and they

attributed this to interfacial cracking. The present tests indicate that

80
acoustic emission could be used to check for interfacial cracking and

thus clarify such a situation.

5.3 Model Experimental Approach

Studies of MMC fracture micromechanisms usually must contend with both

very complex stress distributions and slip mechanisms within a

heterogeneous deforming body, and an absence of non-invasive

methodologies for in-si tu observation of the fracture processes at the

microscale. The complex internal stress distributions arise, during

loading, from elastic incompatibility between matrix and reinforcement

and from spatially varying residual stresses due to differential thermal

contraction of the constituents during cooling from process temperatures.

Often the reinforcement is sufficiently randomly distributed so that the

local stress is not directly predictable.

For these basic studies of MMC fracture micromechanisms, we have

attempted to eliminate some of the problems associated with complex

stress states in multiply-reinforced composites through the design of a model composite. This composite still retains the essential features of

import, but is in a configuation more amenable to analytic work with

interfaces more accessible to direct NDE characterization. The model

composite consists of a single ceramic fiber around which is grown a

single crystal aluminum tensile sample. Using acoustic emission,

interrupted test metallography and digitally recorded stress-strain

behavior, we are exploring methodologies for quantitative determination

of key microstructure parameters such as the fiber fracture strength, the

fiber-matrix interface cohesion and the matrix slip behavior.

Dumbbell geometry single crystal tensile samples were used for this first

study (later we will look at comparison tests). They were prepared from

99.99% aluminum and 140 pm diameter silicon carbide fibers with

81
Fig. 5.3

Cap

Al Charge

SiC Fiber

Bridgman apparatus for growing directionally solidified

monocrystal Al/SiC monofilament composites.
ACOUSTIC

EMISSION

INSTRUMENTATION

FOR

ADHESION

MEASUREMENT

Fig. 5.4 Acoustic emission instrumentation for monitoring composite

tensile tests.
unmodified surfaces using a Bridgman technique shown schematically in

Figure 5.3. A high density graphite mold was internally machined to

accommodate the growth of an aluminum single crystal with a 57 mm gauge

length and 4 mm gauge diameter. The fiber was centrally located down the

longitudinal axis of the mold, retained at the top by means of a screw

and at the bottom by graphite glue attachment to a plug. The plug acted

as a weight on the fiber, maintaining it centrally during subsequent

crystal growth.

The mold was attached to a water-cooled pedestal, the system evacuated

and backfilled with argon and then heated using a radio frequency

o induction furnace. The mold was heated to approximately 900 C and

solidification achieved by lowering through a water-cooled copper chill

at constant velocity. A combination of steep temperature gradient and

slow growth speed was used to ensure a single crystal solidification

[13]. Two growth velocities, with drastically different resulting

reinforcement-matrix interfaces, were used. Samples with shallow

reaction zones were grown at 0.41 mms ^ (140 s solidification time) while

samples with extensive reaction zones were grown at a velocity of

0.0083 mms ^ (1.9 h solidification time). Samples with no fibers were

also grown so that fiber effects could be separated from matrix behavior.

The oxide normally present on an aluminum single crystal is a potential

extraneous acoustic emission source during tensile testing. It was removed by electropolishing the entire sample in a 6% perchloric

acid/methanol solution at -30°C.

The specimens were mechanically loaded in tension using a screw- type

machine driven at constant crosshead velocity (0.5 mm/min) , (Fig. 5.4).

The effective machine stiffness was measured to be 0.96 MNm ^ using the

maximum load rate method [14]. The load cell and strain gauge voltages

were continuously digitized throughout the tests, the data stored on a

minicomputer for later analysis.

82
Fig. 5.5

(a)

(b)

SEM micrographs of transverse sections of SiC filament in A1

showing development of Al^C^ interface with increasing

solidification times: (a) Rapidly solidified materials

(140 s), (b) More slowly solidified (1.9 h)

.
Stress/MPa

Strain

Fig. 5.6 Stress-strain and AE-strain curves for single crystal A1

specimen with no SiC fiber.

AE/Rms

Volts
Acoustic emission was continuously measured during testing using a

specially designed system shown schematically in Figure 5.4. Conical

PZT-5 piezoelectric elements were used as transducers attached to either

sample end. The signals from these were amplified, bandpass filtered and

passed through a high speed digital rms-to-dc converter with a 17 ms averaging time. The rms voltage was digitized at 14 bits per 1.5 ms interval and recorded on a minicomputer. The largest transient bursts

exceeded the amplifier linear range (200 mV rms) and are therefore

nonlinear in amplitude.

A dual waveform recording system was also used in parallel with the rms measurements system so that signals could be later evaluated in detail.

This system featured 8-bit digitization with a 50 ns sampling interval.

SEM micrographs showing the transverse sections of undeformed single

crystal composites grown at two velocities are shown in Figure 5.5. It

can be seen that extensive reaction occurred at the fiber-matrix

interface in the slowly grown sample. This resulted in the formation of

a layer approximately 10 pm thick Al^C^ at the interface in 1.9 h. Less

reaction was found in the more rapidly solidified material. Assuming the

parabolic growth law, the reaction zone present after 139 s of growth was ~1.4 pm in thickness. Radial cracks in the fiber are polishing

artifacts, aggravated by fiber degradation, since they were not observed

in the fibers extracted by dissolving the matrix.

Stress-strain and acoustic emission-strain behaviors for the two

composites and an aluminum single crystal with no fiber show that:

o The aluminum single crystal containing no fiber had a smooth

stress-strain and acoustic emission-strain behavior (Fig. 5.6).

The data was in agreement with that reported for similar material

in the past [16].

83
Strain

Fig. 5.7 Stress-strain and AE-strain curves for monofilament SiC/Al

single crystal composites grown at slower rate (1.9 h

solidification time).

AE/Rms

Volts
Stress/

MPa

Fig. 5.8 Stress-strain and AE-strain curves for monofilament SiC/Al

rapidly solidified single crystal composites (140 s solidification time).

AE/Rms

Volts

AE/Rms

Volts
Fig. 5.9 Optical micrograph. Longitudinal section of slowly grown

composite pulled ~5% showing multiple transverse fiber

fractures.
240

s}U9Ag jo jaqiunisj aAijeimuno
o The more slowly grown composite exhibited a smooth stress-strain

behavior but large "burst” acoustic emissions were observed

superimposed on a smooth background behavior (Fig. 5.7).

o The more rapidly grown composite exhibited a discontinuous

stress-strain curve with 23 large load drops (Fig. 5.8). Strong

acoustic emission signals accompanied each load drop.

The samples were pulled to a plastic strain of approximately 6%.

Following testing, samples were longitudinally sectioned and polished to

reveal the state of the SiC fibers. Figure 5.9 shows a typical result

indicating multiple transverse fractures of the fiber. To determine the

number of fractures, samples were immersed in a 10% NaOH solution at 50°C

to dissolve the aluminum matrix, leaving behind the fragmented fiber.

The cumulative distributions of amplitudes of the AE events in

Figures 5. 6-5. 8 is shown in Figure 5.10, which summarizes the results of

five tests. For the more rapidly grown materials, there is a close

correlation between the number of events and number of fiber fractures as indicated by the arrows. The number of events (30 and 38) are also close

to the number of load drops (23 and 27), and this is improved if we

consider that some of these drops resulted from multiple fractures. On

the other hand, fiber fractures can only account for about half of the AE

events observed during testing of the more slowly grown material. Since

plasticity sources are probably unaffected by the solidification times

used, we suggest that these extra signals are associated with interface

cracking

84
5.4 Discussion

It is possible to measure the fiber and interfacial strength in situ

through the use of tensile tests, acoustic emission monitoring, and

post-test examination of the specimen and fractured fibers. First, the

tensile test is used to estimate the fiber strength o^. Then, post-test

examination of the lengths of the broken fibers provides the critical

aspect ratio ^ c/d. The interface strength can then be directlycalculated from the shear lag formula.

The size of the load drops in the tensile tests can be used to estimate

the fiber strength, since each load drop corresponds to a fiber fracture.

This is a function of machine stiffness, so that it is important to

examine the machine effects on load drop magnitude.

Consider a tensile specimen with a single fiber oriented along the

tensile axis. If the fiber breaks, there will be a localized plastic

strain, which causes a load drop whose magnitude depends on the machine

stiffness. For example, if a dead weight ("soft") load mechanism is

used, extensions occur instead of load drops. Our analysis assumes use

of a "hard” or constant crosshead velocity machine and follows that of

Clough [14]. The total extension of the crosshead moving at velocity V

over time duration At is equal to the sum of the extensions u of the

P

machine length (pull rods, grips, load cell) and u

g of the specimen:

VAt = u + u = P/k + u p s s

where P is the load and k is the machine stiffness (N/m) . If a fiber

fractures, during time increment dt there will be a sudden drop in load

with the corresponding extensions:

85
d(VAt) = Vdt = du

g + dP/k

Crosshead motion is negligibly small during the load drop so that we can

3 omit the term Vdt and express the load drop as

AP = kAu

s

The fracture of a fiber of length 22 into lengths 2

will cause a

localized increase in cross-sectional stress Ao - A r o r /A at the

f f s

3The duration of the load drop is At = Ae^/e^, where Ae^ is the load

drop plastic strain at the fracture and e^ is the local strain rate at

the fracture. We can estimate the strain increment from

Ae^ = Aa/(da/de^), where Act is the stress increment (A^/A^ct^ and

(dcr/de^) is the local rate of strain hardening. Taking typical values of

these we obtain a strain of 0.001. Assuming the deformation to be

entirely localized during the load drop, the local strain rate will be

increased by the rate sensitivity of the stress. Rate sensitivity data

on pure A1 (Clough [15]) gives an increase in strain rate by a factor of

3.5 for a 2% stress change (calculated from the loss in fiber

reinforcement). Since the local strain rate before fracture was

V/^

c = (8.3 x 10 ^)/(0.001) = 0.0083s ^

, the local strain rate during the

load drop will be ~3.5 x 0.0083 = 0.029s ^

. Thus the duration of the

load drop will be e^/e^ = 0.0001/0.029 = 33.2 ms. During this time the

cross-head will extend Vdt = (8.3 x 10 ^) (0.0332) = 0.28 pm. The sample

itself will extend an amount dP/k, where dP is the load drop (typically

0

7 N) and k is the machine stiffness (10 N/m) , or 7 pm. Thus the

cross-head displacement is negligible in comparison to sample extension

during a load drop and we can omit the term Vdt.

86
fracture, where and A

g are the fiber and specimen cross-sectional

areas and ais the fiber strength. The plastic strain extends

approximately 4^/2 along each fiber, over the length of ineffective fiber

reinforcement, where the matrix tensile stress is higher. This will

cause an increase in length of the specimen of approximately

Au = 4 be = 4 Ao/ (do/de) = 4 a (A /A )/ {do/de

)

S C C C C x X s

where Ae^ is the localized plastic strain and ( do/de ) is the localized

rate of work-hardening (dcr/de^)

.

When 4 = 4 , this is equal to the

macroscopic rate of work-hardening. The fiber strength (combining the

above two equations) is then

a

f = AP(As/Af )(acr/de)/k£

c

. (5.6)

The last three load drops of the rapidly solidified composite averaged

4.5 N (0. 31MPa) , A /A

f = 943, da/de = 0.18 GPa, k = 0.91 MNm"

1 and

4^ ~2.4 mm. Thus, the fiber strength o

is ~0.3 GPa, only 10% of that of

a virgin fiber prior to composite processing.

Fiber loading theory indicates that the shear strength for fiber fracture

is given by the relation [3]

t. = a, d/24 (5.7)

i f c

v '

where d is the fiber diameter. As a sample calculation, for a rapidly

grown sample, substituting 4 ^ = 2.4 mm, ct^. = 300 MPa and d = 0.14 mm gives a value for 2t^ ~17.5 MPa. Equating 2t^ with the tensile stress

indicates that the interfacial shear strength in the rapidly solidified

87
Fig. 5.11 SEM micrograph of interfacial region in slowly-grown

composite (non-deformed) showing presence of several

interface cracks prior to testing (courtesy C. Handwerker).
sample is approximately equal to the matrix shear strength at 5/ strain.

Thus fiber-matrix failure occurs here by matrix shear rather than

interfacial fracture.

Turning now to the slowly grown material, we see that though many fiber

fractures occurred, no detectable load drops were observed on the

stress-strain curve. The noise-limited smallest detectable load drop is

~0.4N (0.03 MPa). Thus < 30 MPa, an order of magnitude less than that

of the rapidly grown sample. This degradation of strength is likely

associated with notch cracking of the 10 pm thick (brittle) Al^C^ layer

at the interface. Results quoted below also indicate low fracture

toughness of the near-interface matrix, so that the notches could

initiate in the matrix and be much larger than 10 pm, greatly lowering

the fiber strength.

Metallography provides additional information on failure mechanisms. The

plastic strain appears to be, as assumed in the above model, largely

localized to the sections between fiber fracture. Evidence that the

deformation is localized to the fiber ends is obtained by examining the

distance between fiber ends after testing. If the total elongation,

after 5% strain, is (0.057)(0.05) = 0.00285 m, and this is divided

between the separations between 51 pieces, the average displacement

between fibers will be 0.00285/50 = 57 pm, or about 1 radius. This is

comparable to the separations between fiber ends in Figure 5.9, so that

the strain appears to be localized as assumed.

Interface cracking occurred in this material and this provides important

information on the properties of the matrix just external to the

interface. An SEM micrograph (courtesy C. Handwerker) of the interface

region in a slowly solidified sample (Fig. 5.11) shows the presence of

several small (~10 pm) interfacial cracks present prior to pulling the

samp 1 e

88
Radial cracks are also present in the Si-rich matrix, which, although

possibly caused by polishing, nevertheless indicate severe

embrittlement of the matrix near the interphase. If we conservatively

assume the stress limited by the matrix flow stress (~20 MPa) and take

r-~ 1/2

the crack half length to be c = 5 urn, then K ~cr ^ttc = 0.08 MPam

c net

indicating severe embrittlement. This is close to the value of

1/2 K

c

< 0.1 MPam obtained from load drop magnitudes (Table 5.2).

2 2

Further, we estimate that nr.ic= G /2 ~ K /2Er = 0.04 J/m , or about 4% of

c f the surface energy of A1 , indicating poor bonding. These observations

indicate that the model of Ochiai and Murakami [4, 12] ought to be

modified if there is any embrittlement of the matrix material just

outside of the interphase. The low fracture toughness of the matrix

surrounding the fiber can effectively act as an additional brittle layer

surrounding the fiber and interphase, roughly doubling the notch crack

size.

We can calculate the interface strength from the shear lag

equation (5.7). The critical length 2^ was reduced in the slowly

solidified material. Typically 2^ ~1.4 mm so that from the shear lag

equation 2t\ < 3 MPa. This is much less than the matrix shear strength

and is consistent with brittle shear failure in the locally embrittled

matrix or at the matrix-interphase boundary. Tensile and shear fractures

in or just outside the layer may be the source of the additional acoustic

emission signals in this material (Fig. 5.7). Such a thick layer and

Si-enriched matrix are absent in the more rapidly grown material, which

also does not emit these extra signals.

As preliminary examples of the use of the measurement technique, the

following table summarizes measured fiber and interface strengths as well

as interface fracture toughness and interface adhesive energy for two

rapidly cooled, and two slowly cooled SiC fiber/Al single crystal

composites grown from the melt.

89
TABLE 5.2

MEASURED FIBER AND INTERFACE STRENGTHS

Property

Rapidly

Solidif ied

(139 s) Comments

Slowly

Solidif ied

(1.9 h) Comments

Fiber Tensile Strength 300 Fiber <30 Fiber

(MPa) (Eqn. 5.6) 288 Fracture <40 Fracture

Matrix Shear Strength (MPa) 9.5 9.0

(Figs. 5.7 and 5.8) 7.7 6.4

Interface Shear Strength 8.7 <3.0

(MPa) (Eqn. 5.7) 9.8 <2.8

Interface Critical Stress >1.2 No <0.12 Interfacial

Intensity K

c (MPam^^) >1.2 Interfacial <0.16 Fracture

(Eqn. 5.7) Fracture

Interface Adhesive Energy >1.4 <0.014

nr. (J/m^) (Eqn. 5.4) >1.4 <0.026

As can be seen, the time of exposure to the melt greatly affects the

fiber and interface strengths. The interfacial fracture toughnesses and

adhesive energies of the slowly cooled composites are extremely low,

again consistent with the interpretation of brittle shear fracture of the

interface as monitored by acoustic emission. The adhesive energy of the

rapidly cooled composite is found to be greater than that of pure

2

A1 (~1.0 J/m [1]), in agreement with acoustic emission results that

there was little or no interfacial fracture in that material.

5.5 Future Directions

By using specially designed specimens, combined with analyses of tensile

test and acoustic emission results, direct, in situ measurements have

90
been made of fiber and interface strengths, adhesive energy and fracture

toughness. By changing the sample configuration and type of loading

(e.g. , compressive vs. tensile), determination can be made of other

parameters, such as the relative effect of interface friction on cohesive

strength. Compressive longitudinal tests appear attractive as opposed to

transverse loading because the radial symmetry under compressive loading

imposes a constant radial stress around the interface.

Since measurements are made of individual fracture events, rather than

statistical ensembles, the data from the individual events can be

combined to unambiguously evaluate the accuracy of statistical theories

of strength. This is the inverse of the conventional practice of

inferring individual microfracture properties from bulk properties. As a

simplified example, the average length of fiber can be determined during

testing from the number of large AE events (i.e., the number of fiber

fractures). Since the individual fiber fracture strengths are known from

the load drops, a Weibal 1-type statistical distribution of fiber strength

versus length can be constructed. Such a result awaits further analysis

of data already at hand.

While these experiments have been carried out on a model system, the

technique can be extended to other, more complex advanced structural

composites. We are at present in the process of making arrangements to

test multiple fiber MMCs produced at the MIT facilities using techniques

similar to those described here.

91
5.6 References

[1] A. Argon, J. Im, and R. Safoglu, Met. Trans. 6A (1975), p. 825.

[2] K. Gutfreund, L.J. Broutman and E.H. Jaffe, "International

Investigations of Boron Fiber-Reinforced Plastics," Advanced

Fibrous Reinforced Composites . 10 . pp. E-25 to E-40, Society of

Aerospace Material and Process Engineers (SAMPE) , 1966.

[3] A. Kelly, Strong Solids . Oxford, Clarendon Press (1977), p. 131.

[4] S. Ochiai and Y. Murakami, J. Materials Sci., 14 (1979),

pp. 831-840.

[5] R.B. Clough, "Energetics of AE Source Characterization," to be

published (March-May 1987) in Materials Evaluation .

[6] A.C. Evans, M.C. Lu, S. Schmauder, and M. Ruhle, Acta. Met. 34

(1986), 1643.

[7] C.R. Crowe, R.A. Gray, and D.F. Hasson, "Microstructure Controlled

Fracture Toughness," Fifth Int. Conf . on Composite Materials

ICCM-V . W. Harrigan, ed. , AIME (1985), p. 843.

[8] R.J. Arsenault and C.S. Pande, Scripta Met. 18 (1984), p. 1131.

[9] J. Gurland and J. Plateau, Trans. ASM, 36 (1963), p. 442.

[10] J.D. Outwater and M.C. Murphy, Paper 11c, 24th Annual Technical

Conf. on Composites , (The Society of the Plastics Industry,

New York, NY, 1969).

92
[11] J.K. Wells and P. Beaumont, J. Materials Sci., 20 (1985),

pp. 1275-1284.

[12] S. Ochiai and Y. Murakami, Z. Metal Ikunde, 72 (1981), pp. 827-831.

[13] M.C. Flemings, Solidification Processing . McGraw-Hill, NY (1974),

p. 54.

[14] R.B. Clough, "Rational Basis and New Methods for Proportional

Limit, Machine Stiffness, Critical Stress Intensity, and Crack

Velocity Measurements," in Recent Developments in Mechanical

Testing , STP 608, Philadelphia: American Society for Testing and

Materials (1976), pp. 20-44., see Equation (10).

[15] R.B. Clough, "Effects of Composition and Strain Rate on the Tensile

Behavior of Selected High-Purity Aluminum Alloys," M.S. Thesis,

Univ. of Arizona, Tucson (1965), p. 62.

[16] C. Scruby, H. Wadley, and J.E. Sinclair, Phil. Mag., 44 (1981),

pp. 249-274.

93
Appendix I

.

.
2. MICROSCOPIC ORIGINS

by

H. N. G. WADLEY and J. A. SIMMONS

CONTENTS

2.1 Introduction

2.2 Micromechanical Modeling

2.2.1 Formulation

2.2.2 Transducer spectral sensitivity

2.2.3 Detectability criteria

2.2.4 Source directivity

2.3 Microscopic Dislocation Sources

2.3.1 Detectability

2.3.2 Single crystal deformation

2.3.3 Grain size effects

2.3.4 Precipitation effects

2.4 Microscopic Fracture Sources

2.4.1 Detectability

2.4.2 Cleavage microfracture

„ 2.4.3 Intergranular microfracture

2.4.4 Particle microfracture

2.4.5 Microvoid coalescence

2.4.6 Environmental factors

2.4.7 Amplification factors

2.5 Microscopic Phase Changes

2.5.1 Solid state phase transformations

2.5.2 Detectability of martensitic transformations

2.5.3 Liquid-solid transformations

1
2.1 Introduction

Acoustic emission can be thought of as the naturally generated

ultrasound created by local mechanical instabilities within a body.

Imagine, for example, a body which had been placed under load

sometime in the past and is now in elastic equilibrium throughout.

Suppose a small crack appears within the body at a point distant to

that where the loads were applied. The surfaces of the crack are

able to move in such a manner that they become stress free. In so

doing, they release some of the stored elastic energy in the body.

This release of elastic energy is in the form of elastic waves

that propagate freely throughout the body suffering

reflections/mode conversions at its boundaries. In fact, the

propagation of these elastic waves is the mechanism for

communication of the changed elastic state in the immediate

vicinity of the crack to the rest of the body. The waves enable the

entire body to change its shape and accommodate the crack, and each

propagating wavefront carries a component of this shape change. In the perfectly elastic body used for mathematical modeling, these

waves would propagate indefinitely and mechanical equilibrium

would never be established. However, in practical materials,

absorption (conversion to heat) occurs, and the waves eventually

dissipate allowing the body to assume a new (cracked) shape and

establish a new equilibrium with the loads throughout.

A transducer attached to a body is capable of detecting the

motion of the surface with which it is in contact. Its response,

following a local crack extension, is the acoustic emission we

2
observe in experiments and tests. Since the transducers are usually

constructed from piezoelectric slabs and have a resonant behavior,

their sensitivity varies with frequency and is usually greatest in the

range 0.1 - 2.0 MHz. It is only these frequency components of the

emitted waves that are sensed. Neither the static surface strains

(the difference in shapes of the two equilibrium states discussed

above) or the very high frequency components are sensed.

Observed acoustic emission signals, at least close to the

source are dominated by surface displacements associated with

wavefront arrivals at the transducer location. If the wavefront

contains appreciable frequency components in the transducer

bandwidth, then a voltage will be created across the faces of the

transducer, and provided this exceeds the background noise, the

emission will have been detected.

The detectability of acoustic emission, then, depends upon the

temporal nature of the source because this determines the amplitude

of each spectral component in each wavefront. If the source

operates so slowly that there is sufficient time for the body to

return to quasi-equilibrium before the crack has appreciably

extended, it is possible that no detected signal will be emitted

(however a (static) strain gauge might register the change of shape

if the crack grows sufficiently large). Conversely, if the crack

extends rapidly and then stops so that its growth time was around

I-*- transducer bandwidth, then the emitted wavefronts would be

dominated by frequency components in the detectable range.

Clearly, it is very important to gain a deeper understanding of

3
the microscopic origins of acoustic emissions, since this will allow

us to determine the likelihood of detecting various events of

potential interest and of distinguishing between them.

The purpose of this part is, therefore, to develop an

understanding of the relationship between local mechanical

instabilities and the ensuing acoustic emission. In particular, we

will develop expressions for the surface motion produced by such

microscopic acoustic emission sources as dislocations, micro- • cracks and phase transformations, particularly those involving the

formation/annihilation of martensite. Using simple results from

tensor wave propagation theory, we shall then use these

micromechanical models to develop detectability criteria for

microscopic sources. These criteria are finally used to identify the

origin of acoustic emission in materials undergoing deformation,

fracture and phase changes.

In this part, we employ the usual assumption of linear

isotropic elasticity. All metals only approximate these

assumptions. In practice, metals exhibit weak nonlinear behavior

due to internal friction mechanisms, are anisotropic to a greater or

lesser degree, and if preferred grain orientation (texture) is present,

will not exhibit spherical wave spreading. In addition, the

polycrystalline nature of engineering materials results in grain

scattering which can be particularly strong for spectral components

whose wavelengths approach or exceed the grain size. In what

follows, all these effects have been ignored, not because they cannot

be treated, but rather because the relative contribution they make is

4
Schematic

diagram

of

the

acoustic

emission

process.
very sensitive to detailed aspects of the microstructure and varies

from case to case. The treatment does not extend to situations

where sample normal modes dominate the acoustic emission. This is

still an emerging area of research.

2.2 Micromechanical Modeling

In this section, the principles for developing detectability

criteria are presented for acoustic emission occurring following a

general mechanical instability. These principles will be applied to

acoustic emission from dislocation motion, crack growth and phase

transformations in subsequent sections.

2.2.1 Formulation

The sequence of events giving rise to an acoustic emission

signal are summarized in Figure 1. Elastic waves are generated due

to a local change of stress in a region of Volume V. These propagate

(spherically in an isotropic body) as a mechanical disturbance

through the structure causing a time varying surface displacement

ii(t). JJ.(t) varies with source and receiver positions due to (i) inverse

square law decay (because of wavefront area increase with distance

between source and receiver), (ii) source directivity and (iii) wavefront reflection/mode conversion at a free surface. A sensor

located on the body detects the surface disturbance and generates a

voltage waveform which, provided it exceeds the background noise,

is observed as an acoustic emission signal.

Viewed from the perspective of the surrounding elastic

structure, the acoustic emission source appears as an apparent

change of stress (stress drop) and changes in tractions at the

5
surface of the structure. Using a Green's function approach we can

write 1 the displacement at position r as arising from both internal

(volume stress) and surface (traction) sources:

Ui(r,t) = J dr'J G jj, k’(r,r' ,t-t')Acrjk( , t')dt'

- JdS'k'lG i j(r,r',t-t)Atjk(r',t

, )dt' (1)

where Gy (r,r',t) are components of the dynamic elastic Green's

tensor representing displacement in the xj-direction at r as a

function of time, t, due to a unit strength force impulse applied at r' and t=0 in the xj-direction. Thus the Green's tensor is the solution

to the wave equation for a unit force impulse source. The notation

",

H is used to denote partial differentiation so that Gy^' is the

corresponding wave equation solution for a unit force derivative

(dipole) impulse. Acjk and Axjk are the volume stress and surface

traction changes associated with the source and S.' a vector normal

to the surface of the structure.

The time integral in equation (1) comprises a convolution.

This convolution provides the basis, in principle, for predicting

surface displacement waveforms from stress change sources if the

Green's tensor is known (either calculated or measured). In subsequent sections these stress changes will be deduced for

internal microscopic defect sources. Because the source and sensor

are of finite size the representation requires a Green's tensor to be

evaluated between every source and every receiver point. This is a

numerically exhausting task beyond normal computing capabilities

6
(b)

DIPOLE

WAVEFORMS

c\ig>

Ll

Epicenter

displacement

waveforms

for

various

infinetisimal

multipole

sources

in

an

isotropic

elastic

half-space,

a)

Monopole

sources,

b)

Dipole

sources.
(c)

QUADRUPOLE

WAVEFORMS

eex.riufr.i.z‘zig Fig.

2

(c).

Epicenter

displacement

waveforms

for

various

infinetisimaS

quadrupole

sources

in

an

isotropic

elastic

half-space.
and is further compounded by the possibility that each stress

component might have a different temporal character. To simplify

the formulation we note that the stress change is greatest at the

centroid of the source, and that defect sources are often small in

comparison with I r-£ I . Also for sources contained within the body,

the surface traction source term, Atjk, can be considered zero.

For infinitesimal sources, the usual approach 2 > 3 is to expand

the Green's tensor as a Taylor's series about the centroid source

*

point iq:

Gjj.k’d.l'.t) = Gjj

; k'(r,r 0

',t) + Gjj.kM'd.lo'.tJAri' + .... where Arf = f\ - r'g I -

Substituting into equation (1), gives:

Uj(Lt) = |Gij i k'(Ll' 0 .t-t

, )Aajk(t, )cit'

+ lGij,kT(r,ro,t-t')Arjki(t’)dt' +

where the quantity:

Aojk(t) = /A<7jk(r',t)dr'

is the dipole (or seismic moment) tensor and the quadrupole tensor

AT jkl(t) = /Act

j k (Jl’.t) A£|'dr' . Examples of epicenter displacement waveforms for monopole, dipole

and quadrupole sources are shown in Fig. 2. At practical frequencies

only small error is introduced for microscopic sources by truncating

the Taylor's series at the dipole (first) term provided the source to

receiver distance is much greater than the source size. The source

quantity Aojk(t) is the volume integral of the stress change, i.e., it

is the average stress drop considered distributed on the point r'g - It

is also called the source dipole tensor

4,5 and is also equivalent to

7
the quantity referred to as the seismic moment in seismology6 . Since the stress drop can be thought of as a dipole density, equation

(1) amounts to a summation of convolutions between the dipole

density at each source point r' and the Green's tensor between rl and

r. The final step of the formulation is to include the effect of the

tranduction process upon the signal. To date, only "nondisturbing"

transducers of finite area Sj have been considered 1. By

"nondisturbing" one means that the change in waveform caused by

the presence of the transducer can be neglected because it is small

compared to the waveform itself. This approximation is excellent

for optical interferrometric detection schemes7 * electromagnetic

acoustic and capacitance transducers8 . Its validity for piezoelectric

transducers is, however, yet to be determined, for they undoubtedly

load the surface.

The transducer response will be considered in terms of the

surface displacement at the location of the transducer, Sj, without

the transducer attached. The point impulse response will be denoted

Tj(r,t), re Sj, and is defined as the voltage at time t produced from

the transducer by a displacement which, without the transducer

present, would have been a 5-function at point r in the i-direction

at time zero. Under this definition, the voltage at time t due to an

infinitesimal source is:

V(t) = JjTi (r,t-t

, )G ij>k

, (i,ji

, o

,t'-t")Ao’j k (t")drdt" (2)

St

8
This equation for a point transducer simply has the form of a

convolution between the source function, the impulse response of

the body, and that of the transducer. In the frequency domain the

convolutions become products and so we can write that the complex

(but scalar) voltage as a function of frequency co :

V(co) = Tj^Gjj^coJAojkfG)). (3)

It is important to note that under this definition of transducer

sensitivity, the impulse response Tj(t) is a vector quantity. The

output depends both on the magnitude and direction of the surface

displacement. Normally it is assumed that sensitivity exists only

for one displacement direction reducing the formulation to a scalar

one. Schemes for vector transducer calibration are emerging which

can determine the full vector impulse response

9

. 2.2.2 Transducer Spectral Sensitivity.

Examination of equation (3) indicates that information is

transmitted independently, frequency by frequency, from the source

to the observed voltage signal. Thus, the detectability of an

acoustic emission signal is a frequency dependent concept and is

determined by the signahnoise ratio at each frequency, or band of

frequencies. It is obviously very dependent upon the frequency

dependence of the transducer. In Table 1 we list the estimated

smallest detectable signals (sensitivities) in a typical laboratory

background noise environment.

9
TABLE 1 BANDWIDTHS AND MINIMUM DETECTABLE SIGNALS FOR A.E. TRANSDUCERS

Transducer type Passband (MHz) Sensitivity

Air gap capacitor DC - 50 1 CH 2 m

Electromagnetic Acoustic 0.5 -15 1-2 x 10* 4 ms' 1 Damped PZT 0.1 - 2 1 0‘ 13 m

Resonant PZT 0.1 - 0.3 10-14 m

Common types of transducers used for quantitative acoustic

emission measurements are damped PZT or air gap capacitors. Both

are principally displacement sensitive with sensitivities of ICH 3 and

10 _12 m respectively. Other, more sensitive, but narrower bandwidth

transducers are frequently used for source detection and location

work in the field. Some piezoelectric transducers and all electromagnetic acoustic transducers (EMAT's) respond to surface

velocity. For an EMAT, the smallest detectable velocity is

-0.1 -0.2 mm. s' 1

* and a piezoelectric device might be 1-2 orders of

magnitude more sensitive.

2.2.3 Detectability Criteria.

With these sensitivities in hand, we can now attempt to

delimit the concept of detectable acoustic emission sources. For the

purpose of the present discussion, we shall assume point-like

transducers located in the far-field of the source on the surface of a

half-space. Examination of the Green's tensor components, Gjk,| , for

a linear elastic isotropic half-space show that for a source at a

distance x3 perpendicular to the surface at which the transducer is

attached, the waveform is dominated by longitudinal and transverse

1 0
wavefront arrivals with 5(t) form (. here implies time derivative).

It can be shown that:

_ S

|

3 / 5 i3 5k 3 5(t-x 3 /a) (5jk = 5j35k3) 5(t-x 3 /c) x G J k ' l(t) P

+

C

3

)

(4)

where a and c are the longitudinal and transverse wave speeds

respectively and 5jk is Kronecker’s delta (1 for j = k and 0

otherwise).

Substituting equation (4) into equation (2) and assuming the

source is of infinitesimal size with 5(t) time dependence allows us

to simply deduce the epicenter displacement waveform. In order to

achieve detectability the maximum value of this signal must exceed

the minimum displacement sensitivity, i.e.,

djS? 2*px3

k(t) max I" ji

t >o L q

3^^13 (^" x 3/c) + —3^A<J23(t“X3/c)

8 13

• 1

+

a

3 ^ a 33 (t-X3/a)J (5a)

or

v;<

^(x)

27ipx 3

max

t >o

(5b)

where dj represents the smallest detectable displacement and vj the

smallest detectable velocity, both in direction xj.

Since no transducer is of broader band than all microscopic

acoustic emission sources, we have introduced factors K(t) and k'

(

t)
which account for transducer filtering. Here, x is the source

duration, and if the transducer is of constant sensitivity from DC1/t, k and k' are unity. They will be less than unity for transducers

with bandwidths smaller than that of the source.

To estimate the value of K(t) for a real transducer let us

assume the displacement pulse to be detected is of Gaussian form to

which we assign a width x and magnitude M, i.e.,

u(t) = Mexp[-t2

/x 2

] (6a)

with Fourier Transform

u(co) = MxVrcexp[-co 2 x

2

/4] . (6b)

Let us assume that the response spectrum of the displacement

transducer is Gaussian with bandwidth B by which we mean:

R(co) = kexp[-co2/4B2

] (7)

where k is a normalization constant and R(co) is the voltage produced

from a c-w excitation of frequency co. To determine k we assume

that the transducer will produce a signal of magnitude 1,

independent of B, if the input is a constant in time of magnitude 1.

Thus, if u(t)=1, so that u (co)=2t:5 (co), then the response is 2j:k5(co),

whence k=1. This is equivalent to R=1 at co = o or jR(t)dt=1.

The output from such a transducer has the frequency spectrum

MxV7iexp[-co 2 /4B 2 ]exp[-co 2 x

2

/4] whose time response is

MtB

Vi + b 2 t

2

exp[-t2 B 2

/(1 + B2 t

2

)]
a Gaussian whose peak at t = 0 has been reduced by the factor of

xB/y 1 + B2 x

2 . Therefore, we define

k(x) = xB/^j (1 + B 2 x

2

) (8)

To find k'(x) for a velocity sensitive transducer, we shall use a

differentiating filter of the form kcoexp[co 2 /4B 2

] where, as above, it can be shown that normalizing using a time function of constant

slope 1 gives k=-i. Again we use the time function given in (6)

whose maximum derivative, V (2/e) M/x occurs at t = ± x/V2. After

filtering, with the differentiation filter, we have a frequency

spectrum -iMxaW7texp[-co 2 (l + B2 x

2 )/4B 2

] whose inverse transform is

MxB/*V (1 +B2 x

2 )d/dt (exp[-t2 B 2/(1 + B2 x2

)]). The maximum derivative

for this time response is V (2/e) Mx2 B2

(1 + B2 x

2

) occurring at

t - ± V B/2(1 + B 2 x

2

)

.

Thus,

K '(x)

xB2

1 + B 2 x

2

(9)

2.2.4 Source Directivity

The final factor affecting detectability is the directionality

(or directivity) of the source. In linear isotropic elastic materials

the stress change tensor is symmetric i.e., Aajj = Aajj We can

therefore, use a representation quadric to describe the directivity of

the source.

Let the stress drop tensor be represented by its principal

values A, , A. and X {\X l<IX l<IA. I) along principal directions e , and Then the equation of the directivity surface is:

^i e i 2

+ ^2 e

2

2 + ^3 e3

2 = 1 • (*19)

In terms of the regular axes, xj, £j = EjjXj where E is an orthogonal
matrix with Et = E- 1 whose components Ey are the cosine of the

angle between ej and xj. Thus, the three stress change components,

9 § Aoj3 (or Aoj3 ) are given by:

A a i3 = Ejj Xj Ej 3 . (11)

First let us consider the stress change component, Ag33> that causes

out of plane motion at epicenter.

Ag33 = X-j E 13

2 + ^2^23^ + A,3 E332 ~ A + (ki-A) E 13

2

+ (X2 -A)E23

2 + (^.3-A) E332

, (1 2)

where A = (X^ + X2 +X3)/3 gives the dilatational component of the

stress drop tensor. Thus, the directivity pattern for Ad33 contains

an omnidirectional term (A) and terms which are proportional to the

square of the cosine of the angle between the £j and x^ axes. The

maximum value in this pattern is X3 and occurs when the axis

points in the x^ direction. Should two of the principal stresses be of

opposite sign, there will exist orientations for which there is no

vertical motion at the epicenter surface due to Aa33 . Provided all

three principal stresses are of the same sign, there will then always

be a non-zero minimum vertical motion whose magnitude will be

determined by the smallest principal stress, X<\.

The analysis of directivity for either of the stress change

components Ag13 or AG23, that produce in-plane epicenter motion is
similar. Thus, we only consider Aa13 :

^ E E + X. E E 4-X.E E

(13)

The latter three equalities following from the orthogonality of x 1 and X 3

. The maximum in-plane motion at epicenter is determined

by the maximum of the three quantities I A. -A. I ,

I X -X I and

I A. 2

-A. I • Suppose is largest, and the direction orthogonal to

both and g.j is &k- Then, the maximum epicenter motion will be

obtained when x^ek, and &\ and ^

are in the = 0 plane at ±45° to

the axis. The magnitude of the maximum radiation for equation

(13) is ±\X\-X\\!2 and is polarized in the &i direction. Should two of

the principal stresses be identical, then there will exist an

orientation in which there is no horizontally polarized motion

anywhere in the x 1f x2 plane. If all three principal stress values

differ, then there is always a horizontally polarized component in

the x 1f x2 plane, and this component is greater than min \X j-A.jl/2.
2.3 Microscopic Dislocation Sources

2.3.1 Detectability.

The stress change (Aoki) or dipole tensor (Dki) created by the

formation of a dislocation loop in a linear elastic body has been

evaluated by Eshelby10 * Kroner11

*, Kroupa12 and Burridge and

Knopoff 13 . The latter developed a simple relation between the dipole

tensor and the dislocation Burgers vector together with the

dislocation loop normal:

Here Cjjki are the elastic stiffness constants, bj are the components

of the dislocation Burgers vector and Aj(t) the projected swept out

area on the plane whose normal is in the xj direction.

For the case of isotropic elasticity:

where p

is the density.

To determine detectability criteria for a dislocation source, conside

the case of the formation of a planar dislocation loop on a plane with

normal £ at a depth r in a half space, Fig. 3a. Using expression 5a for the

displacement detectability at epicenter we obtain:

Dkl(t) = Cjjki bjAj(t) (14)

Cyki = ^SjjSki +ji(5jk5j| + 5j|5jk)

where X and p, are Lame constants, which are related to the

longitudinal and transverse wavespeeds:

(15)

pa2 = X+ 2p,

PC2 = |4

(16a)

(16b)

27TX3dj< K

t
Similarly, the velocity detectability at epicenter is given by:

where ic(t) and K'(t), as given by equations ( 8) and (9) , account for

filtering by the transducer. If a, and £3 are coplanar, these

expressions can be more simply expressed in terms of twice the

angle between a and the x 1f x2 plane.

The first term on the right hand side in (17) and (18)

corresponds to the out-of-plane (normal) motion at epicenter

associated with the arrival of a longitudinal wave, while the second

term corresponds to in-plane (tangential) motion (parallel to the

projection of the Burgers vector) at epicenter associated to the

arrival of a shear wave. It is of interest to note that while acoustic

emission due to the latter, in-plane, motion is sensitive to the

Burgers vector and slip plane orientation, it is not, to first order,

sensitive to the direction of dislocation motion; that is, it cannot

distinguish between edge dislocations slipping in one direction and

screw dislocations slipping in the perpendicular direction.

The greatest sensitivity to dislocation motion is obtained

from an in-plane sensitive transducer with its poling axis aligned

with the Burgers vector direction from dislocations slipping in the

horizontal plane, or from dislocations slipping in a vertical plane

with vertical Burgers vector and with the poling axis of the

"] maxA(t)

t (18)

1 7
X3

Fig. 3. The strongest horizontal displacement from an

infinitesimal dislocation occurs either when (a) b is

coplanar and n perpendicular to the surface or (b) vice

versa.

Epicenter

Position

Fig. 4. An inclined dislocation loop buried in an isotropic elastic

half-space.
transducer aligned normal to the slip plane, Fig 3. In either case, if we set a=6.4 mmps"1

, c=3.2 mmps-1

, b=2.9 x10- 10 m (typical

aluminum values) and r = 40 mm, the detectability criteria for

horizontally sensitive transducers become (from eqs (17) and (18)):

dj ^3.6 x 10* 10 k(t) max[A(t)] mm j = 1,2 (19)

t

\)j .<3.6 x 10‘ 10 k'(x) max [A(t)] mmps-1

j = 1 ,2 (20)

t

If, for instance, the horizontal (d-|, d2 ) displacement detection

threshold is 10- 14 m and B is 5 MHz, then k(t) -0.035 for a

dislocation that propagated for a time x= 15 ns. Then the

dislocation loop would have to have a radial velocity of 3 mmps- 1 expanding to a 45 pm radius in order to be detectable. Such a

calculation highlights the often neglected importance of transducer

bandwidth (as well as sensitivity) in determining detectability. For

instance, it is possible to show that detection could be achieved for

a 1.5 pm radius loop if the bandwidth were infinite provided

dislocation speeds are close to the limiting shear wavespeed. In

fact, if xB «1, k(t) ~tB and x~1/rVl8rcB.

While the dislocation configurations shown in Fig. 3 produce

the maximum horizontal epicenter displacements, they produce no

out-of-plane motion. If, however, the source is rotated so that the

first term in equation (17) becomes non zero for j = 3, then finite

vertical displacements are produced at epicenter (Fig. 4). For the
situation shown in Fig. 4;

b3 = bcoscj)

n 3 = cos0

where b is the magnitude of the Burgers vector and n the unit normal

to the plane of slip. For a glissile dislocation, b is perpendicular to

n and the maximum displacement occurs at 6 * $ = 45°. It is 1/8 the

maximum horizontal amplitude for a horizontal or vertical loop of

the same radius and velocity. Thus, the detectability criterion is:

d3 ^0.45 x 10- 10 k(t) max [A(t)] mm (21)

t

d3 ^0.45 x 10* 10 k'(t) max [A(t)] mmiis" 1 (22)

t

If a transducer is used with 10 -14 m displacement sensitivity and

bandwidth such that ic(t) is always unity, then the detectability

criteria becomes:

rr :> 0.03 (23)

Detectability criteria for velocity sensitive transducers depend upon

the transducers sensitivity i.e., its smallest detectable velocity, and

upon the acceleration of the source. At present, only very crude

estimates exist for the former (even though it appears some

conventional AE transducers are at least partially velocity

sensitive). The acceleration history of a dislocation is even less

clear, and will be extremely mechanism dependent. Small changes of

temperature, composition, heat treatment, cold work, grain size,

etc. can all have a potentially large effect upon acceleration. While

this shows promise for those interested in fundamental studies of

the dynamics of dislocations (as well as cracks and phase

1 9
transformations), it raises serious questions about reliability of

such transducers for source detection and characterization in NDE

applications.

Restricting further discussion to displacement detectability

criteria only, we see that, from equation (23), rf 0.03 m2 s- 1 for x 3 =40 mm and aluminum physical property values. Unstable dislocation

motion in polycrystalline metals is likely to proceed at velocities ? of

0.1 to 1 times the shear wavespeed (i.e.~300-3,000 ms* 1 ). Thus a single

dislocation propagating 10-100 pm is potentially detectable. This is

indicative of the extraordinary sensitivity of acoustic emission

instrumentation. The distance of dislocation propagation is, however,

strongly influenced by microstructure, as will be discussed below, and

not all dislocation motion is detectable.

One further factor can have a very important effect upon

detectability, and this is the possibility for cooperative slip

involving simultaneous motion of large numbers of dislocations.

These "microyield" phenomena typically involve thousands of

dislocations simultaneously moving on the same slip system. If the

dislocations are closely spaced (c.f. the shortest wavelength of

observation), and if they move synchronously, then their emitted

wavefields add. The detectability criterion then becomes

nrr ^ 0.03 m2 s* 1 (24)

where n is the number of moving dislocations. This criterion is very

easy to satisfy for many metals, especially near the onset of general

yield.

20
2.3=2 Single Crystal Deformation

The most well understood case of plastic deformation is that

of a single crystal oriented for initially single slip. This has been

the starting point for studies of acoustic emission by a number of

groups 14-16 . A typical experimental result is shown in Fig. 5 for high

purity aluminum 14 . In the experiment, the sample was deformed at a

relatively slow constant strain rate of ~10-4 s

-1 and acoustic

emission in the 0.1 -1.0 MHz frequency range recorded with a

principally vertical displacement sensitive transducer

(~10 -14 m sensitivity) located on the tensile axis. The acoustic

emission in this experiment was first detected during nominal

elastic loading and reached a maximum intensity after straining

about 2% beyond general yield. Beyond this point, acoustic emission

decreased, even though the slip per unit time (strain rate) remained

essentially constant. Other studies by Fleischman et al, 1indicate

that the strain at which the maximum occurs is bandwidth sensitive,

and occurs at higher strains when larger bandwidths are used.

For an explanation of this behavior, it is necessary to consider

the microscopic nature of slip. In an fee single crystal slip tends to

occur in <11 0> directions on the closed packed {111} planes. There

are many possible slip systems that could be activated by the

application of stress -- typically, the system with highest resolved

shear stress in the slip direction will be activated. If in Fig. 4 the

stress is normal to the surface (c 33 ), then the system with cose cos({>

closest to 0.5 will be the first activated; that is the system for

which e~<|)~45 0

. Thus, we expect these dislocations to generate

21
Fig. 5. The variation of acoustic emission with stress and strain

for a high purity aluminum single crystal undergoing

constant strain rate uniaxial tensile deformation. The

measurements were made over a frequency band of 0.1 - 1.0

MHz with an amplification of 98 dB.

Fig. 6. Schematic diagrams of (a) the hypothetical slip distance

distribution, (b) the velocity distribution and (c) the

distribution of the r? product. Only events for which

rr^0.03 m2 S

’

1 generate detectable acoustic emission. In pure metals this occurs more frequently before

workhardening begins.

A.E.

POWER

/mW
1000

500

0

Fig. 7. The integrated acoustic emission power (energy) for

polycrystalline aluminum decreases with decreasing grain

size. Also shown is the necessary dislocation velocity for a

single dislocation to generate a detectable acoustic

emission signal.

Fig. 8. A comparison of the grain size dependence of acoustic

emission for high purity aluminum and an Al-Mg alloy solid

solution. The effects of Mg are to enhance the emission at

small grain size and supress it at large grain size.

VELOCITY

OF

SINGLE

DISLOCATION

TO

SATISFY

DETECTABILITY

CRITERION

/ms' 1
finite vertical displacements at epicenter and the detectability

criterion (24) to be valid provided K(t) = 1.

For example, a typical event might be the motion of a dislocation to

form a 150|im radius loop at an average radial velocity of 300ms- 1

. Then

rf = 0.045 m2 s- 1 and xB~ 0.5 x 10* 6 sx 106 Hz~1.0 so that the event should

create a signal approximately 1.5 times the background noise and,

therefore, be just detectable.

Let us suppose that a is a large fraction of the crystal radius

(a reasonable supposition at the initiation of plastic deformation of

a perfect single crystal), say about 1mm, then for a detectable

signal from a single loop, ? must be~35 ms*1 which is not an

unreasonable value to expect. In practice, both r and r for slip

events are statistically distributed, Fig. 6, and only those events

whose nrr products are at the extreme right of the distribution

generate detectable events. At the initiation of deformation a

substantial fraction of slip events may propagate sufficiently far

that even moderately slow dislocation groups would give detectable

signals. Work hardening, however, reduces the mean free path

between sessile dislocations, restricting r and resulting in few

events that radiate detectable elastic waves; thus the emission

power decreases. At a strain of ~2% and a bandwidth of ~2 MFIz, a

maximum emission is observed which indicates this is the strain at

which the greatest fraction of deformation events satisfy the

criteria nr?>0.03 ms- 1 and K(t) = 1. However, if B is increased, ic(t)

can tend to unity for shorter duration events. Shorter duration

events tend to become more numerous as work hardening develops

22
(because r decreases and r remains constant or increases), and so

the maximum in emission shifts to higher strain 16 .

2.3.3 Grain Size Effects

It is well known that grain boundaries are effective barriers

to moving dislocations. Thus in a polycrystalline metal the distance

that a dislocation loop can expand will, at most, be that of the

radius of the largest grain in the sample. The distribution of slip

distances ever some range of plastic strains will now be a

complicated function of the grain size distribution and grain interior

dislocation-dislocation interactions. It seems reasonable to

suppose that, at the initiation of slip, a substantial fraction of

moving dislocations will propagate over a distance equal to about

the mean grain diameter.

As the grain size is reduced, the critical velocity of a single

dislocation required to give a detectable signal increases. For

example, for a mean grain diameter of 1mm, r * 35 ms* 1 while, for a

diameter of about 35 }im, r =1000 ms-1

. In fee pure metals we

expect dislocation velocities of several hundred meters per second

so that emission from small-grain material must come from groups

of cooperatively moving dislocations. The probability that enough

dislocations move together to give a detectable signal decreases

with grain size. A disappearance of detectable acoustic emission

has been observed, Fig. 7, by several groups in agreement with the

above model.

A note of caution should be given, however. Some workers 17

have observed an apparent increase, at very small grain sizes, in the

acoustic emission. Up to the present time this controversial

23
observation has not been fully resolved. If the observations are

ultimately confirmed, they may be linked to an increase in the yield

stress with decreasing grain size which increases the average

dislocation velocity.

In alloys of aluminum, substitutional impurity segregation to

dislocations can occur, even when the alloy element concentration is

below the solid solubility limit. This drastically alters the

dynamics of dislocations and leads to effects such as yield points,

Luders' band propagation and dynamic strain aging during mechanical

deformation. The occurrence of these phenomena is almost

invariably accompanied by intense acoustic emission signals. Both

the incidence of the effects themselves and their associated

acoustic emission signals vary greatly with grain size; see for

example Fig. 8 where the acoustic emission energy is plotted as a

function of grain size for an AI-1.3wt.%Mg alloy solid solution and

compared with the result for aluminum.-

The peak in the acoustic emission of the solid solution at a

grain size of around 80 |im arises from a competition between two

effects; as the grain size increases, the duration, x, increases but r decreases. The velocity decrease occurs because of the fall in flow

stress with increasing grain size and, for grain sizes greater than

about 500 jim, the yield stress is insufficient to separate mobile

dislocations from the cloud of impurity atoms (Cottrell atmosphere)

segregated to their core 14 . This results in the drift-controlled

motion of dislocations at velocities too low to give acoustic

emission, even though propagation distances may be greater than

24
QUENCHED 10 100 1000 10,000

AGEING TIME AT I20°C (min)

100,000

Fig. 9. Ageing supersaturated solid solutions (in this case Al5.5wt%Zn-2.5wt%Mg) may produce "shearable" precipitates

which cause an "intense shear band" mechanism of slip with

enhanced acoustic emission.

2000

1500

1000

500

100 200 300 400 500 600 650

TEMPERING TEMPERATURE / °C

Fig 10. The acoustic emission due to plastic deformation in Fe-base

(in this case Fe-3.25wt%Ni-0.21 wt%C) alloys is strongly

influenced by tempering.

0.2%

PROOF

STRESS

/MPa
that required for the radiation of detectable signals in pure

aluminum.

2.3.4 Precipitation Effects

There are two extreme classes of dislocation-precipitate

interaction.

(1) When the precipitates are strong, widely spaced and

incoherent with the matrix, dislocations bow between

particles, leaving behind a dislocation loop around the

particle.

(2) When the precipitates are weak, closely spaced and

coherent with the matrix, dislocations may penetrate the

precipitates and shear them.

Process (1) encourages uniformly distributed slip while

process (2) leads to the formation of intense slip bands. These

bands form because the successive passage of dislocations through

shearable precipitates reduces their strength, creating a local weak

spot in the crystal and thus an ensuing local strain instability.

Studies in which the precipitate strength, distribution and

coherency have been systematically varied in aluminum alloys

indicate that the uniform slip process of dispersion-hardened alloys

(strong relatively widely spaced incoherent precipitates) gives only

moderate levels of detectable acoustic emission. This occurs

because r is limited to the interparticle spacing (a few

micrometers). In contrast, when precipitate shear occurs, intense

acoustic emission signals are observed 18 ' 20 Fig. 9. Similar results

are also obtained in quenched and lightly tempered ferritic steels21

Fig. 10.

25
These studies serve to illustrate the complementary role that

acoustic emission plays in the experimental study of plastic flow.

In contrast with stress-strain measurements and hardness testing

where the integrated static response of the material to an applied

load is measured, acoustic emission measures that part of the slip

distribution for which the criterion nrr > 0.03 m2 s* 1 is satisfied (the

high speed events), and this is very sensitive to metallurgical

variables, often much more so than the stress-strain curve.

26
2.4 Microscopic Fracture Sources

2.4.1 Detectability

We are here concerned with the acoustic emission generated by

the formation of isolated microcracks and the extension of

macrocracks by the coalescence of microcracks at a cracktip. A

crack can be viewed as a material inhomogeneity; thus the

combination of dipole components that are equivalent to the crack

depend upon the ambient stress in addition to the radius of the •

crack.

For infinitesimal cracks circular in shape (penny-shaped) the

dipole tensor is given by:

where x> is Poisson's ratio, p the shear modulus, Ck| mn is the elastic

stiffness tensor, a is a unit vector normal to the crack face, opq the

ambient stress, r the crack radius and Wm p

are the components of

the matrix:

Substituting equation (25) into equation (5) gives the

expression for detectability of a microcrack at a depth x3 beneath

the epicenter of a half space:

(25a)

W

-vn i

n

3

-un^ -un i n 3

2 °n 2^t) -un 2 n 3

-un 2 n 3 2-n 3

2 u

(25b)

27
di <

4(1-u)k(t) max

7cp(2-'o)x 3

naxr.

t ^LvSj 1 ( W1 n 3 + w3 n 1

)

o . r 2 (t-x^/c

>l.r(

r t-xq/c^ t>(n..w)

5j 2 ( w2 n 3 +w3 n

2)) c

3 + 25j3( ^_2 V

+ w3 n 3 )

r 2 (t-x?/a).r(t-X3/a)

a

3 ] (26a)

4(1 -\))k'(x) maxr . . , v\ ^ 7ip(2-a))x 3

t >o L

' ^

j

1 ( wi n 3 + w3 n i ) + Sj2(w2 n 3 + w3 n 2 ))

2r(t-xa/c)r 2 (t-x^/c) + r 2 (X-x^/c) r (t-x^/c)

c

3 + 25j3(

u(n.w)

1 -2d

2r(t-x,/arr2 (t-xs/a) + r2(t-x^/a)'rVt-xq/a)

+ w3 n 3 ) a3 J (26b)

where ^is a vector in the direction of maximum crack face

displacement; i.e.,

(27)

wm = 2(2-\) )Wmp apq nq.

If the force apq nqis parallel to a, the crack is termed mode I

while if the force is in the crack plane it is mode II /III type. For

these particular cases, w is parallel to apq nq (although this would

not be so for mixed mode cracks since has differing eigenvalues of

2 and 2-v in the crack plane and perpendicular to the crack plane

respectively).

The directivity pattern of the acoustic emission from a mode

II/III crack is the same as that for slip (glissile) dislocations with w

assuming the role of b.. Indeed, it can be shown that the coefficients

of the two horizontal (j = 1,2) terms in equation (26) correspond to

28
the projection of the vector w3 n + n3 w into the x 1f x2 plane using one

projection factor, while the vertical (j = 3) term involves the

projection of the same vector onto the x3 axis using a different

projection factor plus a vector which is independent of the position

of x3. The relative epicenter displacement magnitude as a function of

angle for both cracks and slip dislocations is summarized in Table 2

for the case that the x3 axis, w (or £) and n are coplanar.

TABLE 2

DIRECTIONALITY FACTORS FOR 5-FUNCTION WAVEFRONT COMPONENTS FOR

MODE I AND MODE II /III MICROCRACKS

SOURCE TYPE LONGITUDINAL WAVEFRONT

(out of plane displacement)

SHEAR WAVEFRONT

(in plane displacement)

MODE I MICROCRACK a(1-cx2 )+ a3 cos20 sin20

(or prismatic dislocation)

MODE Il/m MICROCRACK a3 sin4> cos2<}>

(or slip dislocation)

where a= c/a = ratio of shear to longitudinal wavespeeds

9= angle between crack face (dislocation plane) normal and the x3 axis

<t>= angle between the x

3 axis and w (or £).

Using similar methods to those shown for dislocation sources,

it is possible to determine detectability criteria for microcrack

sources. As an example, suppose a vertical displacement sensitive

transducer is used to detect signals from a mode I horizontal

microcrack a depth x3 below the surface. Then, w = (0,0, w3 ), a = (0,

29
0 , ri 3 ), and from equation (26a)

ds <[ jtp(1-2'u)a3 ] » 3

( 28 )

Using physical properties of steel 0o= 0.29, p= 7.8 x 103 Kgrrr3 and

a = 5.9 x 103 ms- 1

, the detectability criteria is:

(J 33 r2 r>1.6 x 10 14 K(x)x3 d 3 (mks units).

Source factors favoring detectability are high ambient stress,

large crack radius and fast crack speed. The criterion is affected by

elastic properties so that for aluminum (d = 0.34, p = 2.7 x 103 Kgm-3

, a = 6.4 xIO3 ms- 1 ): 033 r2 f^ 0.6 x10 14 k(x) x3 d 3 (mks units)

It can be seen that in aluminum, cracks of only a third the area of

those in steel are detectable, other conditions being equal.

In metals, the dimensions of microcracks induced by

deformation are closely coupled to the dimensions of microstructure

constituents such as grain size, inclusion or precipitate diameter,

inter-particle spacing etc. For example, in a ferritic steel at or below its ductile to brittle transition, cleavage cracks that are

typically a grain diameter in size form during deformation. Thus,

within the statistical variations of the microstructure, it is

possible to obtain estimates of r in equation (28). It is well known

that brittle microcracks such as those associated with the cleavage

described above propagate at velocities close to the limiting value

of the shear wave speed while at the other extreme, ductile

microvoid coalescence occurs under essentially quasi-static

conditions. Thus, it is possible to estimate crude values for r in

equation (28) for various fracture micromechanisms. If we assume

30
Shear

Detectability

Wavespeed

Limit

AijjiqepdidG

<£>© I

(Ao

CJo

CMo

j..stu/Aiioo|aA )|oejo 0

jre

C\J H fe E Oc 0

<—0

COc

re

fe- CO 0 re CO o

03

CO

re COE C

fe- reE

re o

'c > ~o

re

_c; **— 0 o0E o

CO

fereE o re X wo o

re ofe_ CL E *D CL

CO

re 0fe- C

re 0

re

fe— £ o

re o 0fe- **- re

re "OC o

03 ^ re -C CL O 3 c ^ ® o ~ in O CO

*«« ~0 CO C CO re g

CD to >O 0

_Q ~

re coc

CO 0 E ~

•- re c *

03 * o0 0 re SI fe- re o

‘0o

Q. >N

reE B > reo0

.0

re 0 o

"O

re o 0

*o O C ' c o o

CO0

Q.00 E

fe_

fe— 0 Oo o

ere CO 0 0

re c oo

re g

re "5 < co 0

CO0 00

CO0a

.1 E § a> E° 0 ^ ^ Eow_O N E 0

_q

re « c

rec

re

CO O ^ ® ® » 0 0 C O ^ 2 2 0 ~

*— !=

©

re £ 0C 0 . _ +* ^ 0 = c o ifl 2

re to $ O 0

re -•->

m

Q. 0 r 0 *0 i

CO

reo

v-re>

g>

Ll
(a) Carbide Crack Mechanism

t t t t t t t t t t t t t t t t t t

i I I I \ I \ I * I I I M M I 1 (b) Edge Dislocation Piieup Mechanism

t t t t t t t t t t t t t t t t t t

H I l M I l.t_ I 1 I I I i W i (c) Slip Plane Intersection Mechanism

t t f t t t t t t t t t t t t t t t

i i i I i i I I I i I I i I i I I 1

Fig. 12. Schematic diagram of three micromechanisms of cleavage

crack nucleation in iron-carbon alloys, (a) nucleation by a

grain boundary Fe3 C film, (b) grain boundary edge

dislocation pile-up nucleation and (c) nucleation by

intersection of edge dislocations on {011} slip planes.
all events to occur at constant stress (a 33 ) and source-receiver

distance (x3 ) } then we can map the fields of each micromechanism on

a plot of r verses nr2

, and superimpose the detectability criterion

(equation 28) for various d3 and ic(t) values. This is shown for steel

in Fig. 11 where o33 = 500 MPa and x3 = 0.04 m.

Only micromechanisms to the right of the detectability

criterion are detectable. Thus we see that intergranular and

cleavage brittle fracture are probably very reliably detected even

with insensitive transducers provided their bandwidth extends up to

~1.0 MHz. However there is clearly no possibility of detecting

microvoid coalescence with the current generation of sensors. Thus,

the reliability of acoustic emission for detecting crackgrowth

depends upon the operative fracture micromechanisms.

2.4.2 Cleavage Microfracture

Cleavage microfracture is a common fracture micromechanism

in materials below their ductile to brittle transition temperature.

Thus, many ceramics, refractory metals and some ferritic steels

fail by this mode at ambient temperature. In composite materials

where ceramic fibers, whiskers or particles are incorporated in

order to reinforce a soft, ductile matrix, reinforcement failure also

often occurs by cleavage.

Cleavage microcracks occur on specific crystallographic

planes, {001} for instance in Fe. They are believed to be nucleated

either by pre-existing flaws (e.g. iron carbide cracks in steels),

dislocation pile-ups at internal boundaries or at the intersection of

slip planes, Fig 12. Very limited plasticity may accompany

microcrack propagation which may thus proceed at velocities close

31
to that of the shear wavespeed (the theoretical upper limit). At low

stresses, the cracks are often arrested at grain boundaries due to

the absence of optimally oriented {100} planes in the adjacent grain

at the site of crack impingement.

A horizontal microcrack in steel with r * 10 ^im, r = 1000 ms-1

, g33 = 500 MPa and x3 = 0.04 m gives rise to a vertical displacement

epicenter signal whose peak amplitude (equation 28) is u 3 ~2.5 x 10 =11 m and has a time scale of ~10”5 m * 103 ms-1

, i.e. 10 ns. Such a

displacement signal is readily detectable under laboratory conditions

even with narrow band transducers, Fig. 11.

2.4.3 Intergranular Microfracture

In many engineering alloys, grain boundaries are a site of

weakness and may prematurely fail under load. This weakness is

often associated with segregation of embrittling chemical species

to the interface or with the formation of brittle phases at the

boundary. For example, in some low alloy steels the co-segregation

of Ni with P, As, Sb and Sn to grain boundaries will cause a

transition from cleavage to intergranular fracture and appreciable

upward shifts in the ductile to brittle transition temperature.

Glassy grain boundary phases in ceramics such as Al 2 0 3 are also

linked to the formation of intergranular fracture in ceramics. In aluminum alloys, liquid metals such as indium and gallium promote

intergranular fracture.

In these systems an intergranular crack, once nucleated, is

capable of rapid propagation over considerable distances due to the

absence of crack arresting features in the microstructure. Only

when the crack reaches a grain boundary triple point and must

32
branch radically from the maximum tensile stress plane is arrest

possible in materials with uniformly weak grain boundaries. If the

distribution of embrittling agent is not uniform, as for example

occurs in some ceramic systems, then the dimensions of the

embrittled region act to control the crack advance distance.

Systems exhibiting brittle intergranular fracture are often

copious emitters of detectable acoustic emission. Crack radii are

often 5-10 times those of cleavage microcracks and crack

velocities are at least as high (and sometimes greater) than those

during cleavage. Thus, acoustic emission strengths of 20-100

times those of cleavage are possible.

In composites, fiber delamination can be likened to

intergranular fracture since an interface debond is the basic crack

advance mechanism, and the crack advance distance can be many

fiber' diameters. Since fiber cleavage and debonding are two of the

main damage accumulation processes in composite materials, it is

clear that their damage evolution has a high acoustic emission

detection probability and is the reason for the success of many

acoustic emission monitoring activities in these materials.

A 2.4.4 Particle Microfracture

Some inclusions and precipitates in engineering alloys are

brittle at ambient temperature and will undergo brittle (cleavage)

fracture under tensile loading. The interface at these

inhomogeneities is also often weak and frequently fails at low

stress. The generation of acoustic emission from these

microstructure constituents, then, depends upon the intrinsic

properties of the particles: the strength of their interfaces and the

33
particle dimensions.

In steels, the fracture of manganese sulphide particles appears

to be an important emission source, particularly in the absence of

cleavage or intergranular failure modes. Ono et al 22 have examined

this source of emission in detail. They observed a strong orientation

dependence of the emission from MnS inclusions. During hot rolling

of steel, the inclusions are elongated in the rolling direction and

flattened in the rolling plane forming an ellipsoidal shape. When

stress was applied in the rolling plane along the prior rolling

direction (L orientation), or perpendicular to the rolling direction (T

orientation), very weak signals were observed. These signals

apparently originated from cracks/disbonds over the minor axis of

the ellipsoid. However, when the steel was tested so that stress

was applied normal to the rolling plane (ST orientation), copious

acoustic emission was observed, presumably associated with

.fractures extending over the major axis.

The sulfur content (i.e. number of inclusions), size and aspect

ratio of the inclusions, together with the stress state, control the

acoustic emission from inclusions. The inclusion size and shape is

determined by the solidification pathway and by post solidification

thermomechanical processing. In heavily rolled steels, with sulphur

contents greater than about 0.06 wt%, manganese sulfide fractures

associated with short transverse loading can be a significant

emission source.

In many steels, crack growth occurs predominantly near welds

where MnS has been melted and then reformed at interdendritic

interstices during resolidification. If the sulphur content is high,

34
Disbond

Inclusion

Fig. 13. The sequence of micromechanisms involved in the growth of

a ductile crack. The process begins (i) with debonding at

inclusion-matrix interfaces, (ii) Voids grow at the

interface and plastic deformation localizes between

adjacent voids, (iii) The intense deformation nucleates

further microvoids at precipitate interfaces, (iv)

Coalescence of the microvoids results in a ductile crack.
large elongated inclusions are deposited. These can subsequently

become the site of defects such as lamellar tears. Since defect

formation involves crack growth distances of >10 jim, they are

substantial acoustic emitters.

Numerous types of inclusions of varying ductility are found in

aluminum alloys. McK. Cousland and Scala23 and others have found

that only those that are rich in iron ore fracture at ambient

temperature. These are found to give detectable signals and like the

signals from MnS in steels, the amplitude distribution of the

acoustic emission scales with size distribution of fracturing

particles.

Precipitates, due to their smaller size, and (usually) reduced

aspect ratio are less prone to fracture until very high stresses have

been attained. Because of the small size (often ~<1 jim) they are

undetectable emitters. The exception has been in some specially

heat treated steels containing large (>5 jim) spheroidal carbides,

but these are not usually present in engineering alloys.

2.4.5 Microvoid Coalescence

Ductile fracture occurs by a sequence of processes, depicted

schematically in Fig. 13.

(i) Early during loading cracks/disbonds occur at

inclusions, causing a stress concentration in the

matrix between inclusions.

(ii) Plastic deformation of inter-inclusion ligaments

occurs resulting in the growth of the inclusion

nucleated voids and an intensification of the local

stress.

35
(iii) Secondary voids are nucleated at finely distributed

precipitates within the deforming ligaments.

(iv) The secondary voids grow and link up resulting in a

crack whose dimension is the inter-inclusion

spacing.

The link-up of voids may occur by one of two modes. In materials with low yield strength and high work hardening capacity,

the decrease in net load supporting area (due to void growth) is

'

balanced by the increased flow stress (due to work hardening) of the

deforming intervoid ligament. Under this stable condition, hole

growth continues almost until the voids overlap. This quasi-static

mode of coalescence is likely to be undetectable due to the very low

crack growth velocity and small intercarbide ligament thickness.

However, in materials with high yield strength and limited work

hardening capacity, hole growth causes a loss of load supporting

area that cannot be compensated by work hardening, and unstable

strain ensues resulting in a premature shear coalescence. This

latter process may occur at intermediate velocity over distances

determined by the inter-inclusion separation (10-100 |im). This

alternating shear is thus a potentially detectable source of acoustic

emission.

The design of structures is usually based upon the premise that

crack growth occurs by stable microvoid coalescence, and,

therefore, efforts are made during materials selection and design to

ensure that insufficient stress develop during service for this mode of

fracture to exist. Should undetected defects exist in construction

materials, or if a manufactured component is subjected to greater than

36
anticipated stress, ductile crack advance may occur. In tough low

strength steels, only intermittent inclusion fractures will indicate

this, and, if these inclusions are absent or the region ahead of the crack

was appreciable prestressed in the past (e.g. during proof testing) so

that inclusions are already fractured, the possibility exists for silent

crackgrowth - a disconcerting phenomena to those interested in the use

of acoustic emission for NDE purposes.

Today, given the improved quality of materials and fracture

mechanics analyses, the likelihood of a ductile fracture failure is

becoming rare. Rather, the emerging concern is that some kind of

embrittling phenomenon occurs due to environmental effects. This

reduces the materials resistance to crackgrowth and changes the

mechanism of fracture from a ductile to brittle one; one that is often

detectable. Thus, acoustic emission monitoring for these situations

may well be worthy of further study.

2.4.6 Environmental Factors

The generation of detectable acoustic emission signals during

environmentally assisted fracture such as hydrogen embrittlement,

stress corrosion cracking and corrosion fatigue will depend upon the

mechanism of crack extension favored. As an example of this, McIntyre

and Green24 have measured the acoustic emission per unit area of crack

extension (dE/dA) in three ferritic steels under various environmental

conditions. Under vacuum the ambient temperature fracture mode

involves microvoid coalescence and, therefore, is not a major emission

source. However, as shown in Table 3, considerable differences in

emission/unit area were obtained through variation of grain size and

environment.

37
TABLE 3

EFFECT OF METALLURGICAL AND ENVIRONMENTAL VARIABLES ON dE/dA

FOR THREE FERRITIC STEELS

Steel Grain Size

(Microns)

Fracture Mode Environment dE/dA(1 0"2 V2 smm'2

) 817 M40 1 7 Intergranular 3.5% NaCI 9

M

1 00 •• vi 108

ii 1 7

IV H2 at 26600 Pa 31

897 M39 1 0 Transgranular 3.5% NaCI 5

'

•• 1 0

•• H2 at 39000 Pa 2

h

1 0

M H 2 at 101000 Pa 1 .5

AISI 4340 1 1 Intergranular 3.5% NaCI 31

200 •I ii 210

•• 1 1II H2 at 26600 Pa 37

During intergranular fracture the acoustic emission activity

unit area was approximately proportional to grain size. For a fixed

grain size, transgranular cleavage generated an order of magnitude

less acoustic emission than the intergranular mode.

A great deal remains to be done to fully understand the role of

environmental factors upon fracture micromechanisms and the

ensuing acoustic emission. Nevertheless, it is evident that there

exists considerable potential for acoustic emission, particularly

when the favored crack advance mechanism involves intergranular

failure, and the microstructure contains coarse grains.

2.4.7 Amplification Factors

In Section 2.4.1 it is shown that the acoustic emission

amplitude is proportional to the crack face displacement w. In determining a detectability criterion it was assumed that w

occurred only by elastic strain and that no plastic deformation

38
occurred. In practice, even brittle fractures have some associated

dislocation emission at the crack tip. This plastic deformation is

capable of allowing w to increase beyond the value attained purely

elastically. Indeed, it is quite common for w to increase 1-3 orders

of magnitude in low yield strength materials by crack tip flow

processes. If this plastic deformation occurs entirely during the

period of crackgrowth, then the acoustic signal emitted potentially

could be amplified by the ratio of elastic to plastic crackface

displacements.

A second source of signal enhancement stems from the

relaxation of the faces of a precrack when microcrack extension

occurs at its tip25 - 26 . Once again, we note that to first order the

acoustic emission amplitude is proportional to the change in crack

volume. If a microcrack occurs at the tip of a pre-existing crack,

the volume in question is now the sum of the microcrack volume and

the change in macrocrack volume facilitated by its crack tip

extension. This factor may amplify the microcrack signals by 1-3

orders of magnitude, and also seriously affects the spectrum of the

emitted wavefield.

39
2.5 Microscopic Phase Changes

2.5.1 Solid State Phase Transformations

Solid state phase transformations almost invariably cause the

development of an internal stress due to density, modulus and thermal

contraction differences between the different phases of an alloy.

Consequently, we expect that the development of the stress field itself

together with microfracture, microplasticity or other mechanisms by

which the stress field is relaxed (e.g., loss of coherency) are potential

sources of acoustic emission.

Speich and Schwoeble27 have systematically investigated the effect

of cooling rate on the acoustic emission accompanying the austeniteferrite transformation in plain carbon steels. As the cooling rate

increased, the reaction products changed from pearlite through bainite to

martensite. Their observations are summarized in Table 4.

TABLE 4

ACOUSTIC EMISSION DURING CONTINUOUS COOLING OF PLAIN CARBON STEEL

Transformation

product

Phase transformations Acoustic emission

activity

Pearlite Diffusion-controlled simultaneous

growth of lamellar ferrite and

cementite

None detectable

Bainite Diffusion-controlled growth of small

carbides (<1 pm) and lath ferrite

None detectable

Martensite Diffusionless transformation in which

laths or plates typically 20 pm in diameter and several micrometers thick

transform from f.c.c. to body-centered

tetragonal structure at 10-80%

of the shear wave speed.

Very energetic

signals detected

From their work it is clear that the diffusion-controlled

nucleation and growth of ferrite and carbides, while probably

40
causing the development of appreciable stresses, fail to generate

detectable elastic waves in the acoustic emission range of

frequencies. However, the diffusionless transformation of austenite

to martensite, is easily detectable and provides the basis for a

simple non-destructive method of deducing, Ms> the temperature at

which martensite starts to form on cooling and, Mf, (the

temperature at which martensite formation on cooling is essentially

completed).

2.5.2 Detectability of Martensitic Transformations

Martensitic transformations are diffusionless changes of

phase involving local shear and dilatations that cause changes of

shape. In some respects they are similar to the formation of

deformation twins in that both involve invariant planes of strain,

i.e., the region that transforms has a characteristic habit plane, h., with components hj defined by:

hjXj * 0 and h^ + h|+h3

2 = 1 (29)

The plane of a twin is usually a low index rational plane; that

of martensite, however, is usually not. In addition to the habit

plane, there is also a characteristic deformation direction d, defined

with IdJ = 1 and a deformation magnitude m proportional to the

distance from the habit plane. The invariant plane strain then takes

the matrix form: I + mdM where d. and h. are each column vectors,

and t stands for transpose.

For the deformation twinning case the habit plane is a

twinning plane (e.g. {112}) and the deformation direction a twinning

direction (e.g. < 1 1 1 >). The subsequent transformed region, thus, has

41
the same lattice as the parent phase, and is only transformed by one

of the point group symmetry properties of the parent phase (e.g. it may be a mirror image of the surrounding phase).

In a martensitic transformation, the lattice structure of the

transformation product is different to that of the parent phase. The

habit plane is usually irrational and the deformation direction may

not be coplanar with the habit plane which results in a nonzero

volume change (dilatation). Thus, the region that transforms

undergoes a change of shape. This change of shape is a mechanism

for the generation of acoustic emission, and given mi, h. and the

initial shape, we could evaluate the source function as discussed

above.

However, this would fail to incorporate important other

effects. First the change of shape occurs in a constraining medium

so that (residual) stresses form which interact with the change in

elastic constants of transformed region to generate further

emission. If the value of rn is small, the residual stress can be

accommodated elastically leading to a thermoelastic transformation.

However, often the stress is such that considerable plastic

deformation and twinning occur. These can be additional emission

sources. Also, the change of shape is different to that of the

unconstrained case. Thus, in attempting to predict acoustic emission

from martensite, care must be taken in choosing the appropriate

shape change.

The expression for the stress change associated with the

martensitic transformation of an ellipsoidal region of volume v has

been derived by Simmons and Wadley28 :

42
Ao(t) = [I + ACD] -i [(C + AC)p* - ACp°] v(t) (30)

where C is the stiffness matrix (using Voigt notation) of the parent

phase, C + AC that of martensite, p* the unconstrained shape change,

P° any pre-existing elastic strain (from an imposed stress or nearby

source of residual strain) and D is a shape matrix.

Examination of equation (30) reveals that six factors

associated to the transformation affect the acoustic emission:

e Volume of region transformed.

• Dilation strain.

• Shear/rotational strain.

• Habit plane.

• Residual stresses (through interaction with AC).

• Time dependence of the transformation.

Equation (30) can be considerably simplified if we suppose AC

is very small (it is actually unknown for most transformations):

Ac(t) - Cp*v(t) (31)

This is an expression very similar to that deduced for a plastic

deformation.

To get some idea of the type and magnitude of acoustic emission

signals from actual martensitic transformations we need data

concerning the change of shape. Dunne and Bowles29 have made careful

measurements of this at the surface of an Fe-21 .89°Ni-0.82°C alloy.

(Caution: it may however be different internally.)

43
They find for one case studied that

/0.1752\ A0„2006\

Jl= 0.5550 ] d = "0.6550 ] (32)

\0.81 31 / \0.7284/

and the magnitude of the plane strain vector, m = 0.19.

We shall assume isotropic elasticity and estimate values for X

and jli from Ledbetter and Reed30 : X = 10.5 x 10-2 MNmm*2

, and |i= 7.3 x 10"2 MNmm-2 . If p = 8.09 x 103 Kgm-3

, then a= 5.57 mmiis- 1 and

c = 3 mmiis- 1

. To fully predict the source function we need to know the

velocity surface of the transformation. The mechanism of growth of

a region of martensite is very poorly understood, but it is generally

thought that the velocity in the habit plane is much greater than that

in the perpendicular direction. Here, we shall consider an ellipsoidal

shaped region with a circular habit plane in which the radial

velocity is independent of direction and has a value of 2 mmjis- 1

while the rate of semi-axis growth perpendicular to the habit plane

is assumed 10% of this.

Thus:

v(t) = (3.2 )tc t3 mm3

v(t) = 10t2 mm3 |is- 1 From equation (31) we can write that the stress change rate is:

Ao(t) = t2

-0.0287

-0.0159

_-0.0244

-0.0157

-0.0623

-0.0890

-0.0244"

-0.0089

0.2032.

MNmm^S"1

( 33 )

Using the method described in 2.2, we can identify the

principal values of the stress change:

44
Aa(t) =t2 0

0

0.176 0

0.085

0

0

0

0.550

MNmrrifiS"

1

( 34 ) where the associated directions are such that the third principle

direction is parallel to the x3 axis (the [001] crystallographic axis),

while the others lie in the x 1? x2 plane, but rotated 80.46° in the

clockwise sense.

Rotation about the [001] axis enables achievement of maximum

vertical and horizontal displacements at epicenter. Both

displacement components have a parabolic time dependence, like

that of the microcrack case considered earlier.

Assuming the source to be 40 mm beneath epicenter, we can

estimate, as before, that for a vertical displacement transducer

with 5 MHz bandwidth, 10’ 11 mm sensitivity and x = 0.3t, the

smallest detectable - if optimally oriented - lath has a 3.25 jim

diameter and a growth time of 1.62 ns. Of course, since in this case

one principle stress change value is negative, a direction will exist

in which no vertical (out of plane) signal is generated.

The maximum horizontal motion occurs in a plane containing

the [001] direction at 80.46° from [100] in a clockwise sense. Using

identical bandwidth and sensitivity, the smallest detectable lath of

optimal orientation would have a 1.6 jim diameter and a 0.8 ns

growth period. Since all the principle stress change values differ, a

horizontal displacement occurs for all orientations. It can be shown

that a 2.25 jam diameter lath can produce a detectable horizontal

displacement even when oriented such that the weakest signal

propagates to the receiver.

45
TOTAL

COUNTS

/

cm

TEMPERATURE (°C)

14. On cooling a high carbon steel (MS ~200°C) acoustic

emission signal (counts) are detected.

100 150 200 300 350

TEMPERATURE (°C)

15. The total number of martensite generated by acoustic

emission signals per unit volume is dependent upon carbon

content in Fe-base (here 4300 series) alloys. The Ms temperature also decreases with increasing carbon

concentration.
Acoustic emission techniques are very useful for following the

kinetics and measurement of the dynamics of martensitic

transformations. For example, in Fig. 14 the acoustic emission

emitted during the continuous cooling of a high carbon steel is

shown. Ms

for this steel was about 200°C; only a few acoustic

emissions were generated above this temperature, possibly

associated with a local stress-assisted transformation. As the

temperature continued to decrease below Ms , increasing rates of

emission were observed, each emission presumably associated with

the rapid growth of a martensitic plate across an austenite grain.

The maximum transformation rate appeared to be about 60°C below

M s and emission ceased entirely at about 100°C below Ms. The observation that intense acoustic emission signals are

emitted at the beginning of martensite transformations has been

used by Ono et al 31 to measure Ms values as part of an alloy

development program. They reported Ms values for numerous steel

alloys with different compositions and microstructures and report

the technique to be a simple accurate method for deducing

martensite transformation temperatures. They also reported that

the acoustic emission accompanying the transformation was

extremely sensitive to the microscopic processes involved in the

transformation.

The carbon concentration of low alloy steels27 has a very

strong effect on both the temperature dependence of the emission

during continuous cooling and the number of detectable signals per

unit volume, Fig. 15. The temperature dependence of the emission is

consistent with the decrease in Ms with increasing carbon

46
concentration. The effect of carbon concentration on the number of

detectable signals, however, is more likely to be a manifestation of

changes in the dynamics and morphology of the martensite

transformation.

Cold work also influences the acoustic emission during the

martensitic transformation of steel. The nature of the cold work

effect is very sensitive to the carbon and nickel concentrations

which control martensite morphology and the transformation

kinetics. The cold work effect, like the carbon concentration effect

resides in changes to the morphology and dynamics of individual

martensitic transformations. Further measurements of the

individual signals may provide a much improved understanding of

this aspect of the martensite transformation. Measurements of the

rate at which emission is detected, which measures the rate at

which laths form, may throw light on the kinetics of the

transformation, and particularly on auto-catalytic phenomena where

they occur.

While almost all solid state phase transformations result in

the development of internal stresses, only those associated with

rapid martensitic transformations apparently generate detectable

elastic waves directly. However, plastic deformation and even

fracture are sometimes induced by internal stresses, providing

indirect detection of their development.

For example, rapidly cooled high carbon steels have a very high

internal stress which is normally relieved by a tempering treatment

that allows dislocation and impurity atom migration. Shea and

Harvey32 have observed, using acoustic emission methods, that
isothermal tempering at too low a temperature will result in microcracking, believed to be due to impurity atoms locking of

dislocations.

Similarly, while the development of stress in and around a

second-phase precipitate has so far not proven directly detectable

by acoustic emission methods Cannelli and Cantelli 33 have observed

signals from hydride precipitates of niobium, tantalum and vanadium

as they relax this localized stress by fracture and adjacent lattice

plastic deformation, Fig. 16. These results suggest that there may

well be some merit in researching the use of acoustic emission

techniques to detect the point during aging when the coherency

stresses of small precipitates are relaxed by the formation of

interfacial dislocation structures around semicoherent precipitates

(e.g. when 0"-> 0' in the Al-Cu system). The prismatic punching of

dislocation loops during precipitation and cooling in other alloy

systems (e.g. Mo-C) might also be worthy of investigation.

Magnetic domain walls in ferromagnetic materials can be

induced to move by the application of magnetic fields. This is

accompanied by electromagnetic emission (the Barkhausen effect).

The motion of these walls under the action of monotonic or

alternating magnetic fields has also been found to generate elastic

radiation (acoustic emission), the details of which have been found

to be dependent on microstructure variables such as dislocation

density and carbide distribution. It has been suggested that

measurement of this emission would provide a non-destructive

method of microstructure characterization. Ono and Shibata34 have

also been able to estimate the residual stress for a given

48
microstructure state since the emission from domain wall motion is

sensitive to the stress state. The potential also exists to determine

other microstructure information if residual stresses are absent.

2.5.3 Liquid-Solid Transformations

There have been few studies of acoustic emission during

solidification. Work on Pb-Sn and Sn-Bi35 indicate the generation of

acoustic emission signals when solidification conditions are such

that interdendritic porosity (solidification shrinkage of the final

interdendrite liquid) occurs. The emission associated with this

process is very intense, although the precise physical mechanism is

unclear. This work indicates that the plastic deformation of

primary dendrites could also generate low intensity acoustic

emissions.

TABLE 5

Summary of results for Solidification emission of

Al_-4„5Wt. %CU-0.2Wt.%Tl ALLOY

H2 contents per Total acoustic Pore fraction

100 g at STP (cnrr3

) emission count %

0.05 1.05 x 104 0.19

0.17 2.75 x 104 0.46

0.23 6.35 x 104 0.63

Feurer and Wunderlin36 measured the acoustic emission during

solidification of an AI-4.5wt.%Cu-0.2wt.%Ti alloy and observed

acoustic emission generated, they believed, by the formation of

porosity. They varied the volume fraction of porosity by adjusting

the hydrogen content of the melt and found that the solidification

4 9
acoustic emission was proportional to the volume fraction of

porosity (Table 5).

They considered the emission to be generated by the unstable

formation of hydrogen "bubbles" in the melt close to the liquid-solid

interface. The lower hydrogen solubility of solid aluminum results

in a hydrogen supersaturation in the melt close to the liquid-solid

interface. The relief of this supersaturation acts as the driving

force for hydrogen bubble formation in an analogous role to that of

strain energy reduction in the formation of cracks. This dilation

source then radiates longitudinal elastic waves that are transmitted

through the liquid-solid interface and are ultimately detected by a

transducer as acoustic emission.

Rapid solidification, where solidification occurs at speeds up

to 1 or 2 ms- 1

, has become a vigorous area of research because of

the advantageous properties that may be achieved through refined

microstructure and homogeneous distributions of alloy elements.

There is a pressing need for process control sensors that could

characterize microstructure and measure process variables during

solidification.

Clough et al 37 have made acoustic emission measurements on

aluminum alloys during pulsed electron beam melting-solidification.

It has been found that acoustic emission signals are emitted during

solid state electron beam heating (because of thermoelastic effects

similar to those well known to be responsible for laser generation

of elastic waves), melting and resolidification. The acoustic

emission emitted during resolidification, following beam cut-off

after attainment of a steady state temperature field, increases in

50
this situation with the electron flux, probably as a result of the

increased volume of resolidifying metal. For a given melt depth the

acoustic emission from an aluminum alloy (2219) is up to 100 times

more energetic than that from nominally pure aluminum (alloy 1100)

Fig. 17.

Metallographic studies on the copper containing 2219 alloy

indicated the occurrence of solidification cracking and course slip

bands. Both phenomena were absent in the commercial purity

aluminum (alloy 1100). These results indicate that the large

solidification and thermal contraction stresses set up during rapid

solidification are responsible for plastic deformation in both

materials. The acoustic emission from dislocations is very weak in

the fine-grained aluminum alloy 1100 whereas in alloy 2219 it is

much stronger. Work by Hsu and Ono38 indicates that dislocation

motion in aluminum alloys at high temperatures generates more

energetic emission than dislocation motion at room temperature.

This, together with the additional emission of hot tearing, is

considered to result in much greater levels of detectable emission.

51
References

1. J.A. Simmons and R.B. Clough "Theory of Acoustic Emission" in

Proc. Int. Conf. Dislocation Modelling of Physical Systems, J. Hirth and M. Ashby Eds, Pergamon Press 1981.

2. J.E. Sinclair, J. Phy. D., 12, p. 1309, 1979.

3. J. A. Simmons and H.N. G. Wadley, Unpublished Work.

4. C.B. Scruby, H.N.G. Wadley, J.E. Sinclair, Phil. Mag. A44 . p. 240,

1981.

5. C.B. Scruby, C. Jones, J.M. Titchmarsh and H.N.G. Wadley, Met.

Sci. 15, p. 241, 1981.

6. K. Aki and P.G. Richards, Quantitative Seismology, W.H.

Freeman, San Francisco, 1980.

7. L.E. Drain, J.H. Speake and B.C. Moss, SPIE, 136 . p. 52, 1977.

8. C.B. Scruby and H.N.G. Wadley, J. Phys. D, H, p. 1487, 1976.

9. J.A. Simmons .Christian D. Turner and H.N.G. Wadley, J. Acoust.

Soc. Am. 82 (4), 1987.

10. J.D. Eshelby, "The Continuum Theory of Lattice Defects", in

Solid State Physics, F. Seitz and D. Turnbull, Eds, 3., p. 79,

1956, Academic Press.

11. E. Kroner, Erg. Angew. Math, 5, p. 1, 1958.

12. N. Haskell, Bull. Seism. Soc. Am. 54, P- 1811, 1964.

13. R. Burridge and L. Knopoff, Bull. Seism. Soc. Am., 54, p. 1875,

1964.

14. H.N.G. Wadley, C.B. Scruby and J. H. Speake, Int. Met. Rev., 25,

(2), p. 41, 1980.

15. N. Kiesewetter and P. Schiller, Phys. Status Solidi A, 48, p.

439, 1978.

52
16 . P. Fleischman, F. Lakestani, J.C. Baboux and D. Rouby, Mat. Sci.

Eng. 2£, 205, 1977.

17. R C. Bill, J.R. Frederick and O. K. Feibeck, J. Mat. Sci, 14, p. 25,

1977.

18. K. L. Rusbridge, C.B. Scruby and H.N.G. Wadley, Mat. Sci. Eng., ££,

151, 1983.

19. C.B. Scruby, H.N.G. Wadley and K.L. Rusbridge, Mat. Sci. Eng., 59,

169, 1983.

20. S. Hsu and K. Ono, "Acoustic Emission of Plastic Flow - II Alloys", in Proc. 5th Int. Acoustic Emission Symp., Tokyo, p.

294, 1980.

21 . H.N.G. Wadley, C.B. Scruby, P. Lane and J.A. Hudson, Met. Sci., 15,

p. 514, 1981.

22. K. Ono, M. Shibata and M. A. Hamstad, Met. Trans., 1QA . p. 761,

1979.

23. S. McK. Cousland and C.M. Scala, Met. Sci., 15, p. 609, 1981.

24. P. McIntyre and G. Green, Brit. J. NDT, p. 135, 1978.

25. Achenbach, J. D. and J. G. Harris, Jour. App. Mech. (46), p. 107,

1979.

26. H.N.G. Wadley and C. B. Scruby, Int. J. Fract., p. 117, 1983.

27. G.R. Speich and A. J. Schwoeble, ASTM Spec. Tech. Publ. 571, p.

40, 1975.

28. J.A. Simmons and H.N.G. Wadley, Theory of Acoustic Emission

from Inhomogeneous Inclusions in Wavepropagation in

Homogeneous Media and Nondestructive Evaluation, G.C.

Johnson, Ed., AMD-62, p. 51, ASME, New York, 1984.

29. D. P. Dunne and J.S. Bowles, Acta. Met. 17, p. 201, 1969.

30. H.M. Ledbetter and R. P. Reed, Mat. Sci. and Eng. 5 (6), p. 341,

1970.

53
31. K. Takashima, Y. Higo and S. Nunomura, Acoustic. Emission

during the Martensitic Transformation of 304 Stainless Steel

in Proc. 5th Int. Acoustic Emission Symposium, Tokyo, p. 261,

1980.

32. M.M. Shea and D. J. Harvey, Scr. Met. IS, p. 135, 1982.

33. G. Cannelli and R. Cantelli, Acoustic Emission Stimulated by

Hydride Formation in Niobium, Tantalum and Vanadium, in

Advances in Acoustic Emission, H.L. Dunegan and W.F. Hartman

(Eds), p. 330, 1981, Dunhart, Knoxville.

34. K. Ono and M. Shibata, NDT International, p. 227, Oct. 1981.

35. H.M. Tensi and W. Radtke, Metall. 32 (7), p. 681, 1978.

36. J. Feurer and R. Wunderlin, Proc. Int. Conf. on Solidification and

Casting, 2., Metals Society, p. 18, 1977, London.

37. R.B. Clough, H.N.G. Wadley and R. Mehrabian, Heat Flow-Acoustic

Emission-Microstructure Correlations in Rapid Surface

Solidification, in Proc. 2nd Int. Conf. on Use of Electron and

Laser Beams for Metals Proc., E.A. Metzbower (Ed), p. 37, 1983,

ASM, Metals Park.

38. S.Y.S. Hsu and K. Ono, Acoustic Emission of Plastic Flow - I. Metals. Proc. 5th Int. Acoustic Emission Symposium, Tokyo,

p. 283, 1980.

54
U.S. DEPT. OF COMM. 1. PUBLICATION OR 2. Performing Organ. Report NoJ 3. Publication Date BIBLIOGRAPHIC DATA

REPORT NO.

March 1988 SHEET (See instructions) NBSIR 87-3630

4. TITLE AND SUBTITLE

Composite Materials Interface Characterization

§. AUTHOR(S)

Wadley, Simmons, Clough, Krasicka, Biancaniella, Rosen, Hsieh

So PERFORMING ORGANIZATION (If joint or other than NBS. see in struction s) NATIONAL BUREAU OF STANDARDS

U.S. DEPARTMENT OF COMMERCE

GAITHERSBURG, MD 20899

7. Contract/Grant No.

So Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

10. SUPPLEMENTARY NOTES

I"") 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 bliography or literature survey, mention it here)

Interfaces and interface zones in composites are studied using ultrasonic and

acoustic emission techniques on models of single reinforcing elements in a matrixc

The theory of ultrasonic scattering and guided interface waves is developed for a model composite with a planar, cylindrical or spherical interface zone with viscoelastic properties varying in a direction perpendicular to the interface zone. Experimental confirmation was carried out for both the planar and cylindrical

geometries with a perfect interface. Both Stoneley and leaky waves were detected

on the planar geometry for a Steel-Ti system. Leaky waves were quantitatively

measured in the cylindrical geometry for steel fibers in aluminum and were detected

for SiC fibers in aluminum. By using specially designed specimens of SiC fibers in

aluminum, acoustic emission and tensile testing permitted determination of fiber and

interface strengths as well as adhesive energy and fracture toughness. The results

show that interface mechanical properties are sensitive to processing variables.

Acoustic emission provides an effective technique for discriminating

mode types such as fiber fracture or delamination 0 between failure

12. KEY WORDS (Six to twelve entries ; alphabetical order; capitalize only proper names; and separate key words by semicolons)

acoustic emission; Al-SiC; Al-Steel; composites; fiber reinforced composites;

guided interface waves; interface; interphase; leaky waves; shear lag model;

Stoneley waves; ultrasonics

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