NASA Contractor Report 3356 

On the Feasibility of Quanti 

Ultrason .ic Determination of 

.tativ 

Frac 

NASA 

%6 

C.1 

Toughness - A Literature Review 

L. S. Fu 

GRANT NSG-3269 

NOVEMBER 1980 

MSA 

ture 
NASA Contractor Report 3356 

On the Feasibility of Quantitative 

Ultrasonic Determination of Fracture 

Toughness - A Literature Review 

L. S. Fu 

The Ohio State University 

Cohnbtcs, Ohio 

Prepared for 

Lewis Research Center 

under Grant NSG-3269 

National Aeronautics 

and Space Administration 

Scientific and Technical 

Information Branch 

1980 
I. INTRODUCTION 

It is known that the fracture toughness of materials can be improved 

by specific microstructure modifications. For example, the covalent electron 

'bond, low crystal symmetry, and long range order in a solid lead to increasing 

tendency for brittle fracture. The study of stress corrosion gives evidence 

that migration of hydrogen atoms to the crack tip embrittles the material. 

This implies that a change of microstructure tends to change the fracture 

behavior of the material. 

There have recently been a number of studies to relate fracture toughness 

and specific microstructural factors. A survey of the work is presented in 

Section II with emphasis on particular models devised to describe the process 

of fracture. 

Ultrasonic non-destructive methods are commonly used to detect the presence 

and size of flaws. But the criticality of crack-like flaws depends upon the 

microstructural environment in which the flaw resides. For this reason 

considerable attention is currently being given to the ultrasonic characterization of microstructural factors that govern such properties as fracture 

toughness. A review of some recent work in this area, with emphasis on the 

ultrasonic determination of fracture toughness, is given in Section III. 

Finally, Section IV presents a review of mathematical methods available 

for solving boundary value problems related to scattering of ultrasonic waves 

by microstructural factors that govern fracture toughness and ultrasonic wave 

propagation. The purpose of this paper is to delineate a basis for the 

empirically observed correlation between fracture toughness and ultrasonic 

factors. 
II. PlUCTDRE TOUGHNESS AND MICROSTRUCTURE 

2.1 Background 

The fracture toughness is an index that is a measure of material resistance 

to catastrophic crack propagation. This index can be given in terms of the 

critical stress intensity factor K lc' the critical energy release rate G lc' 

the critical J-integral Jlc or the critical crack opening displacement 6 

C’ 

The numeral "1" is referred to the normal separation mode of crack face 

displacement. The fracture toughness is recognized as a material constant 

and is normally evaluated by measuring the onset of crack extension in small 

laboratory specimens for which Kl is defined as a function of specimen 

geometry, crack length and applied load or load-point displacement. Standardizedmethods for conducting the tests, instrumentation, and interpretation are 

described in ASTM specification E-399-74. Due to different methods in computation and measurement, one of the indice of fracture toughness may be easier to 

obtain and once one of them is found the others can be obtained from the 

following relations valid within the scope of linear elastic fracture mechanics: 

Glc = Jlc = [(l - u*)/E]K:~ = 2(1 - v*)~~Y (2.1) 

where the plane strain condition prevails, and v, E and Y are Poisson's 

ration, Young's modulus and 0.2% yield stress in uniaxial tension, respectively. 

Among the indice of fracture toughness, the critical crack opening displacement is the one that is physically meaningful and is directly related to the 

microstructure of the solid. 

Before taking up the mechanics and metallurgical considerations of the 

2 
correlations between the fracture toughness and microstructures, it is of 

interest to point to the different regions of stress-strain behavior near the 

crack tip and the order of magnitude of their sizes, see Fig. 2.1. It is 

noted that there exists a plastic region near the crack tip even for brittle 

materials unless the yield stress is not defined. Nevertheless, a general 

trend is that the more brittle the material, the smaller the plastic region 

at the crack tip. 

2.2 Mechanics and Metallurgical Considerations 

The criterion for crack extension in a solid was originally proposed by 

Griffith [2.1]* for glass and later modified by Orowan [2.2] and Irwin [2.3] 

for metals. The principle states that the elastic energy is released in the 

vicinity of the crack tip when the crack grows, that the energy consumed per 

unit area of crack extension is a material property, and that the total 

energy is conserved during the process of crack extension. The elastic strain 

energy release rate G for crack extension can be defined for a static crack 

system as: 

G=g-g (2.2) 

Where W, and U are work done to the solid and internal energy in solid, 

respectively. When the applied load reaches a critical value, the crack extends, 

and the G-value corresponds to the critical situation is equal to the measured 

value of the fracture toughness, G lc' Assuming the crack extends in such a 

* A bracket indicates number of references at end of the section. 

3 
manner that the work done is equivalent to the plastic work plus the energy 

to create new surfaces, i.e., neglecting kinetic energy carried by the outgoing elastic waves, the fracture toughness can be related to the tensile 

properties of the materials. Depending upon the failure criterion used, a 

few models were proposed. 

(i) The Continuum Plasticity Model (Fig. 2.2) 

Let the crack extend by a distance 6a, the energy release per unit thickness is 

"W2 = 

*au2(1 - v2)6a _ 

rK+ - v2) 

2E 2E 6a (2.3) 

If the plastic work is done within the region of size R, see Fig. 2.2., we 

have 

6w2 = 2RYcfsa + 2ysa (2.4) 

in which y is specific surface energy and cf is true strain at fracture. 

Employing Eqs. (2.3) and (2.4), Krafft and Irwin [2.4] obtained: 

l/2 

v2) 

(RYES + ~)l'~ (2.5) 

(ii) The Ligament Model 

Setting the elastic stress u. 0~ Kl/&equal to EEL for failure of a 1.j 

ligament ahead of a crack and letting r equal to the ligament spacing 11, see 

Fig. 2.3, Krafft [2.5] derived the following relation for Klc 

K lc = CIEsffi (2.6) 
where Cl is a constant and he later replaced cf by n, the hardening exponent 

in the Ludwik type hardening law. 

Hahn and co-works [2.6, 2.71 showed, at temperature just below the 

transition temperature, certain grains will fail to cleave and will abosrb a 

large amount of plasticwork and act as ligaments. They obtained the following 

for Klc 

KIC 

=+q (2.7) 

in which d is the diameter of the cracked particles. 

(iii) Linking-up Model (Fig. 2.4) (Fig. 2.5) 

McClintock, Kaplan, and Berg [2.8] proposed that voids near crack tip 

interact thereby favoring localized intense shear between neighboring voids. 

In addition to promoting void growth, the intense shear can damage small 

precipitate particles causing void nucleation, growth and joining-up on an 

even finer scale. As already shown in Fig. 2.1, this involves more complicated 

stress-strain environment of the crack-tip as the finer scale is approached. 

Using a detailed continuum plasticity analysis, Rice and Johnson 12.91, Fig. 2.5, 

found that crack extension proceeds when the extent of the heavily deformed 

region is comparable to the width of the unbroken ligament separating cracked 

particles: 

II P bc 

The quantity, 6c can be related to Klc by way of Eq. (2.1) 

Klc = v%%% 

(2.8) 

(2.9) 

5 
These relations can then be related in terms of fc and d, the volume fraction 

and diameter of cracked particles, respectively, to fracture toughness, and 

Hahn and Rosenfield [2.10] showed: 

K = 2YE: 

l/2 

lc d 1/2f -l/6 

C 

(2.10) 

(iv) Crack Path Tortuosity Model (Fig. 2.6) 

McClintock [2.11] proposed that if interface decohesion is important, 

then a large aspect ratio of second phase particles would promote crack path 

tortuosity and hence increase K lc' The relation is expressed by 

Klc = AE~ (2.11) 

where A is the particle aspect ratio 12.121 

2.3 Results of Related Metallographical Studies 

In what follows, metallographical studies of certain alloys of interest 

are sampled and viewed in the light of the above mentioned models. 

(i) Aluminum 

In Table I, the mechanical properties of some commercial aluminum and 

corresponding metallographic data for large inclusion particles are given by 

Low, van Stone and Merchant [2.13]. Note the numbers for 6 and the centerC 

to-center spacing of inclusion are roughly equal. Fracture toughness 

essentially increases as 6c increases. The number fc -l/6 is about ten times 

less than a except for 2124-T851 and 7075-T7351. Particle diameter does not 

seem to play an important role in Klc. 

6 
(ii) Titanium. 

Another material of interest to aerospace is titanium. It is corrosion 

resistant as well as having beneficial strength to weight ratio. Large 

amount of work is done for titanium alloys. During the course of three 

separate research programs, Froes and co-workers [2.14, 2.151 and Chesnutt, 

etc. [2.16] measured a large number of fracture toughness values, both valid 

Klc and K 

Q' 

for titanium alloys, and corresponding tensile properties. 

They first describe the variations of microstructure and the second describe 

the effect of microstructure on fracture toughness and topography. Asummary 

of the mechanical properties and corresponding microstructures features for 

three different alloys under specific process and heat treatment is given in 

Table II [2.17]. It is again seen that for constant aspect ratio, larger 

dimple size, implying larger center-to-center spacing of inclusions would give 

larger fracture toughness. On the other hand, for the same dimple size, 

larger aspect ratio of the second phase particle would yield larger fracture 

toughness. 

(iii) Alumina Al203 

For a number of years the influence of microstructure such as grain 

size on the fracture toughness of polycrystalline Al203 has been unclear. 

Although the material is brittle, it is highly anisotropic. Some sources 

suggested an increase of toughness with grain size [2.18] [2.19] and others 

a decrease [2.20]. Furthermore, the results appear to be dependent upon 

the specimen geometry used, for example, the D.C.B. test gives results for 

coarse-grained alumina almost twice those of the S.E.N.B. test [2.21]. 

New specimen geometry is currently under development [2.22, 2.231. It 

appears that there is not enough evidence to draw any conclusion as to 

7 
how the microstructure of the polycrystalline A1203 influences its toughness. 

However, the general trend described in (i) and (ii) is expected to be followed. 

A text given by Lawn and Wilshaw [2.24J is useful for the understanding of 

the fracture of brittle solids. Microstructural aspects and atomic apsects 

and atomic aspects of crack propagation are discussed, however, no indications 

are given in regard to the correlation between fracture toughness and microstructural factors. 

2.4 Deformation and Fracture Processes 

As early as 1928, metallurgists already were aware of the phenomenon 

of "tin cry". During heat treatments of steels, audible sounds of "clicks" 

were noted and were found to be related to the martensitic transformation 

12.25-J. When materials undergo plastic deformation of fracture elastic 

energy is spontaneously released in the form of waves and is detectable 

away from the sources. A study known as "acoustic emission" has been used 

as a means of detection and determination of flaw size in materials [2.26] 

12.271. 

Two types of acoustic emissions were detected: a quasi-continuous 

signal and a burst-type signal. It was found that the first type signal was 

strain-rate sensitive and hence might be related to dislocation pinning and 

cross slip, and the burst type signal was probably related to the rapid 

deformation mechanisms responsible for the formation of stacking faults and 

mechanical twins [2.28]. 

It appears that there should be no doubt that the deformation and fracture 

process during the initiation and subsequent crack propagation is a dynamic 

one. Itappears reasonable to hypothesize that the process involves not only 

local plastic instability but also local sudden unloading. This sudden 

8 
unloading causes waves to be sent out through the material medium. 

2.5 Summary 

The above review suggests that fracture toughness is influenced by the 

microstructure of alloys in such a manner that: 

(a) it increases as the plastic zone size, or equivalently, the 

crack tip opening displacement, increases, 

(b) the crack tip opening displacement should roughly equal both the 

spacing of inclusions or second-phase particles which determine 

hole spacing and the size of the intensely deformed plastic zone 

at the crack tip, and 

(c) the aspect ratio of second-phase particle that causes crack 

tortuosity and crack branching should increase the apparent 

macroscopic K lc' 

The deformation processes associated with crack initiation involves 

local plastic instability and sudden unloading which induce waves to be sent 

through the material. It is suspected that the interaction of these waves 

with neighboring particles or inclusions may have caused crack propagation 

under critical situations. It is indeed this dynamic nature of the deformation 

process associated with crack initiation that serves as the key to the 

quantitative ultrasonic determination of fracture toughness. 

III. QUANTITATIVE ULTRASONICS AND MICROSTRUCTURE 

3.1 Background 

With ultronic measurements it is possible to investigate those properties 

of materials which determine the velocity of propagation and the attenuation 

of ultrasonic waves in them. The intensity of ultrasonic waves is influenced 

9 
not only by the medium of propagation but also by a geometrical factor--the 

divergence of the beam of waves. The ultrasonic attenuation caused by the 

microstructure can be divided into scattering and absorption. Recent progresses 

in quantitative NDE techniques have generated great interest in the field 

of quantitative ultrasonic evaluation of mechanical properties of engineering 

materials [3.1 to 3.31. An extensive and useful review on very recent advances 

is given by Vary [3.4]. Further reference can be found in [3.5]. 

3.2 Tensile Properties 

An elastic medium-that is isotropic possesses only two independent 

elastic constants. Thus if the density has already been measured, all the 

elastic constants for this medium can be determined frbm two different types 

of waves propagated in it, CL the longitudinal wave speed and CT the shear 

wave speed, as follows: 

E = &;(3c; - 4$)/@* - c;> L (Young's modulus) (3.1) 

lJ = PC; (Shear modulus) (3.2) 

Because these relations are not linear, a careful study must be made of the 

effects of errors in the measurements of the velocities on the resulted errors 

in the calculations. These constants that are obtained from wave velocity 

measurements are called dynamic elastic constants. For more references, 

see [3.6-3.121. 

3.3 Yield Stress and Fracture Toughness 

Due to the extremely complex nature of the deformation mechanism 

10 
involved during yield and fracture, there is no theoretical relation between - 

y* 5c and ultrasonic measurements, such as Eqs. (3.1) and (3.2), available 

at this time. 

Vary [3.13] was able to measure the ultrasonic attenuation dependence 

on frequencies and discovered some empirical relations between the fracture 

toughness, yield strength and critical attenuation factors related to grain 

sizes in two maraging steels and titanium alloys. Subsequently, Vary [3.14] 

gave some considerations of the stress wave energy required to create a 

microcrack of "diameter" d to the critical crack tip opening displacement 6c. 

Equating the energy loss by Rayleigh scattering, associated with a proposed 

two-grain microstructure model to Klc * /Y or critical crack tip opening displacement, he found a quantitative relation between the fracture toughness Klc, the 

yield stress and ultrasonic measurements, as 

Ktc/Y = $(v,B,/m)1'2 (3.3) 

where the RHS is directly related to quantitative ultrasonics in which vL is 

longitudinal velocity, B6 is the attenuation slope measured at wave length X = 6 

C 

and m and $ are constants. The attenuation factor is assumed to relate to the 

frequency by power law. The stress wave energy required to create a microcrack 

of size d is equated to the energy loss by Rayleigh scattering of stress wave 

as the wave initiates from the source grain and reaches the scattering grain. 

The model is thus useful in pursuing the understanding of ultrasonic determination of fracture toughness, see Fig. 3.1. 

From the experimental results obtained in [3.13], Vary also provided a 

second equation of the form 

11 
y+AIc lc + BBl = C 

where A, B, C are dimensional constants and Bl is the attenuation slope measured 

at the frequency for which attenuation factor a is unity, for a complete ultrasonic evaluation of the yield stress and fracture toughness. Currently, 

quantitative ultrasonic techniques are also being developed for determining 

strength of composites [3.15-3.181. 

3.4 Ultrasonic Attenuation, Velocity and Microstructure 

The propagation of high-frequency stress waves in solids is determined 

by the attenuation and velocity of the stress wave. Various types of defects 

such as dislocations, and a change in the type or density of defects will 

usually change the propagation behavior of the stress wave. In general, 

dislocation damping is of importance in the case of low-frequency waves and 

the direct scattering by defects and secondphase particles is of importance 

in the case.of high-frequency waves [3.21. 

Scattering of stress waves in a solid is brought about by differences in 

mass density and elastic properties from point to point. These differences 

are related to grain boundaries, precipitates, inhomogeneities in composition, 

and even to smaller groups of defects whose extent may be measured in terms 

of lattice spacings. 

With availability of higher and higher frequencies for examining material 

properties, it is now possible to detect directly fewer defects of smaller 

and smaller sizes [3.15, 3.19, 3.201. 

Various methods are now available for measuring velocity and attenuation 

in solids. A detailed discussion of the methods is given in [3.1], pp. 68-91. 

12 
Vary gave an account for computer signal processing in this respect 13.211. 

The general problem of expressing analytically the attenuation by 

scattering for an ultrasonic beam propagating through a medium containing 

scatterers with any size, shape, distribution of sizes, density of scatterers, 

wave length relative to scatterer size, etc. is a very complex one. A brief 

review is given next in Section IV and some available mathematical methods 

are discussed. 

3.5 Summary 

Experimental measurements using ultrasonic techniques indicate that there 

is a correlation between fracture toughness and ultrasonic factors. It appears 

that ultrasonic measurements at certain frequencies, closely related to material 

microstructure, can provide means for determining fracture toughness. If a 

power law relation can be established between the ultrasonic attenuation factor 

a and the ultrasonic frequency, certain theoretical relations can be derived 

by using a simple two-grain model. It does seem to be feasible to determine 

fracture toughness by the quantitative ultrasonic measurements. 

IV. ELASTIC WAVE SCATTERING AND RELATED MATHEMATICAL METHODS 

4 .l Background 

The model discussed in the previous section suggests that elastic wave 

scattering plays an important role in determining fracture toughness by ultrasonics. This leads to a review of this area. 

The recent thrust in studying the scattering of waves in an elastic 

solid has been highly motivated by its applications in various fields such as 

seismic explorations, nondestructive testing, material properties, and dynamic 

stress concentration. An excellent account of history and fundamentals is 

13 
given in [4.1] and a comprehensive discussion of applications from a theoretical viewpoint is found in [4.2 - 4.41. 

The deviation of the wave from its original path is known as the diffraction, and the radiation of secondary waves from an embedded obstacle is called 

the scattering. In elastic medium the obstacle may be in the form of a crack, 

a rigid body or a second phase particle with different moduli from that of the 

medium. The problem of an elastic scatterer is more difficult than that of a 

vacancy or a rigid inclusion in that both the displacements and tractions at the 

interface are unknowns. 

Historically, many theories of scattering, until a few years ago, dealt 

with scaler waves and simple obstacles. Within that context two regimes were 

apparently distinct from each other, the long wave length, and the short wave 

length or imaging regime. The treatment of vector fields in elastic solids is 

much more complex than classical wave fields. Several useful techniques are 

extended from the classical fields of scalar waves and electromagnetic waves. 

Three such methods useful at high frequency range are briefly sketched. 

It is interesting to note that the bulk of the boundary value problems 

studied are not related to the problem of fracture toughness determination 

but to the application of flaw detection [4.5, 4.61. 

4.2 Geometric Diffraction Method 

At high frequencies the diffraction of elastic waves by obstacles can be 

analyzed on the basis of elastodynamic ray theory [4.7]. For time harmonic 

wave motion, ray theory provides a method to trace the amplitude of a disturbance as it propagates along a ray. The technique of geometrical diffraction 

theory was introduced by Keller [4.8]. The application of ray theory to 

diffraction by smooth obstacles was investigated by Resende [4.9] and to 

14 
diffraction by cracks was studied by Achenbach, Gautesen and McMaken [4.16]. 

Geometrical diffraction theory is based on the use of certain canonical 

exact solutions, for example, the Kirchoff solution for an edge, the diffraction 

of a plane wave by a semi-infinite crack. These canonical solutions are 

appropriately adjusted to account for curvatures of incident wave fronts, 

edges and finite dimensions. The pertinent canonical solution must first be 

obtained. 

Achenbach, et. al. [4.10] showed that the method provides good results 

for normal incident waves on slits and penny-shaped cracks at relatively 

small values of the frequency (Kh a > 1.5) and relatively close to the crack. 

For more complicated geometries, say elliptical cracks, the corrections at 

shadow boundaries and caustics become very cumbersome. 

4.3 Transition Matrix Method 

The general theory of the scattering of acoustic waves is contained in 

the mathematical theory of Huygen's principle [4.11]. The scattered waves 

outside the scatterer are related to the Helmholtz integral over the surface 

of the scatterer. Waterman was first to introduce the transition matrix 

(T-matrix) method for acoustic scattering [4.12] and later [4.13] for electromagnetic waves. He started from the Helmholtz integral formula and expanded 

both the incident and scattered waves in series of the basis functions. 

Using the orthogonality of these basis functions, he showed that the unknown 

coefficients of the scattered waves are related by the transition matrix to 

the coefficients of the incident waves. This approach was applied, in 1976, 

simultaneously by Waterman [4.14] and Varatharajulu and Pao [4.153 to the 

scattering of elastic waves. An excellent summary of how the method can be 

used for acoustic waves and elastic waves is given by Pao 14.161. The transfer 

15 
matrix is derived for waves scattered by an inclusion of arbitrary shape. 

The T-matrix has infinite number of elements. These elements are integrals 

of the basis functions over the bounding surface of the scatterer and depend 

upon the incident wave frequency, geometry of the scatterer and the material 

properties. The integrals are evaluated numerically. Currently, numerical 

results are available for the scattering of compressive waves and shear waves 

by a single infinite cylinder or a spheroid [4.17, 4.181. 

4.4 The Integral Equation Method 

The theory developed by Fredholm [4.19, 4.201 for the solution of certain 

types of linear integral equations has been well-known and widely used in 

mathematical physics and mechanics. The main reason being at least two fold: 

(1) if the kernel is separable, the problem of solving an integral equation of 

the second kind reduces to that of solving an algebraic system of equations. 

Any reasonably well-behaved kernel can be written as an infinite series of 

degenerate kernals [4.21]. (2) The Fredholm theorems provide an assurance of 

the existence of the solution hence approximate methods can be applied with 

confidence when the integral equation cannot be solved in close form [4.20]. 

This method was applied to problems in quantum scattering first by Jost 

and Pais [4.22] in a discussion of the convergence of the Born series. 

Reinhardt and Szabo [4.23] suggested a numerical procedure for elastic scattering by constructing the Fredholm determinant which contains all the scattering 

information. Using a similar approach, Holt and Santoso applied the method first 

to scalar wave scattering [4.24] and then to vector wave scattering [4.25]. 

They were successful in both cases. They further studied the simple model of 

a collinear atom-molecule inelastic collision [4.26]. This results in an 

integral equation in two variables in which an infinite set of coupled channels 

16 
occurs. The method was again successful. 

The integral representation technique is closely related to the Fredholm 

integral equation method. Eyges 14.271 used the integral representation 

technique to solve of Schrodinger and related equations for irregular and 

composite regions. He also calculated modes of two coupled electromagnetic 

dielectric wave guides [4.28], see Fig. 4.1. 

Gubernatis, Domany and Krumhansl [4.29] presented a theoretical study of 

the scattering of ultrasonic waves from a single flaw embedded in an isotropic 

medium through the use of the integral equation formulation. The integral 

equation method permits a systemmatic generation of approximations with which 

the scattering of ultrasonic waves from nonspherical shapes can be treated. 

They subsequently [4.30] obtained the Born approximation, analogous to that in 

quantum mechanics. The Born approximation is of broad utility, however, it 

breaks down for strongly scattering flaws. As a remedy, Gubematis [4.31] 

developed a quasi-static approximation. An interesting conclusion he had is 

that the elastic scattering at long wavelengths is not isotropic. Different 

shapes produce different angular distributions. This feature is distinctive 

to elastic scattering while the scattering for acoustical and quantum mechanical 

problems is isotropic at long wavelengths. 

4.5 Additional Remarks 

Although developments over the past few years have led to promising studies 

of theoretical methods for treating scattering of ultrasound by defects and 

second phase particles in elastic solids, the field is still in its infancy. 

A variety of useful techniques such as partial wave expansions, T-matrix, 

integral equation techniques, variational methods and geometrical diffraction 

theory are being studied. 

17 
Krumhansl [4.32] recently commented on the question "which method should 

you use in which regime?" He argued that, in engineering, a useful theoretical 

framework whould be one which physical intuition or engineering data can be 

entered as conveniently as possible, while, on the other hand, for any 

approximate method, the limits of that approximation must exist. He and his 

group of researchers, therefore, chose to concentrate, at least in the initial 

phase, on the integral equation method. He recognized, however, the partial 

wave expansion, including work by Ying and True11 [4.33] and Varatharajulu and 

Pao [4.15], will furnish important reference for the theoretician and, 

eventually, for the experimentalist. 

A comparison of the Fredholm integral equation method with the T-matrix 

approach in the scattering of electromagnetic waves is given by Holt [4.34]. 

He pointed out that the two methods are entirely different and in many ways 

complementary. They are independent of each other and can serve as independent 

checks of the results obtained. Since the Fredholm integral,equation method 

is a convergent method and is numerically stable, it would seem probable that 

at least for spheroids (of maximum dimension a) the Fredholm integral equation 

can be used at values ka, where k is the wave vector, greater than those the 

T-matrix method can treat. For the purpose of completeness in giving a review, 

his summary of the comparison is given in Table III. 

There are currently many efforts relating the theory and some sophisticated 

interpretation of experimental results on single defects. Not much, however, 

is done for multiple defects. 

V. SUMMARY AND SUGGESTED FUTURE RESEARCH 

From the reviews given in the previous sections, it is seen that the feasibility of determining fracture toughness by quantitative ultrasonics can be 

18 
based upon the findings that fracture toughness is governed by certain material 

microstructure factors, i.e. aspect ratio and spacing of second-phase particles 

or flaws. Since interactions of ultrasonic wave with microstructure are 

measurable in terms of ultrasonic attenuation and velocity, it is possible 

that these measurements obtained at particular frequencies can be correlated 

to determine fracture toughness. Such evidences now exist. Confirmation and 

guidance to experimental set-up is, however, still lacking. 

Most of the theoretical studies on elastic scattering emphasize flaw 

determination, i.e., the reconstruction of flaw geometry, rather than the 

determination of fracture toughness. As a result, most studies are concerned 

with a single flaw of size much larger than that of the second-phase particles 

and their spacings. The reconstruction, or the inverse problem, requires 

measurements at all frequencies. Although not much attention is given to 

the problem of ultrasonic determination of fracture toughness, the mathematical 

methods that have been under development in regard to the flaw determination 

problem can be used for the fracture toughness determination problem. It 

appears that the integral equation method is most promising. 

In searching for a theoretical foundation for ultrasonic determination 

of fracture toughness, it is necessary to consider interactions of ultrasonic 

waves with inclusions at given spacings. Elastic scattering of multiple 

inclusions will provide essential information for the determination, i.e. 

the functional dependence of the ultrasonic attenuation a upon frenquency f 

must be theoretically established. The range of wavelengths of interest must 

correspond to the sizes of grain, subgrain, inclusions and second-phase 

particles that comprise the scattering field. 

To accomplish these, it is recommended that the two-grain model suggested 

by vary and a model where a periodic distribution of grains is present be used 

19 
to study the elastic scattering. These models should provide the functional 

relation between a and f. A comparison of the outcome of the two models 

should provide insight as to how dominant the first neighboring grain next to 

the source grain is to the scattering field. 

The shape of the grains should probably be taken as cylindrical or 

spherical in the beginning for simplicity purpose. To study the effect of 

aspect ratio in the value of fracture toughness, more complicated geometries 

such as ellipses or spheroids, prolate and oblate, will have to be investigated. 

Since the elastic scattering is anisotropic in many cases, the theoretical 

study will eventually provide insight and guidance to the actual experimental 

set-up for an ultrasonic determination of fracture toughness. 

20 
REFERENCES 

L2.11 A.A. Griffith, Phil. Trans. Roy. Sot., A221, 163 (1920). 

12.21 E. Orowan, Trans. Inst. Engrs. Shipbuilders Scotland, 89, 165 (1945). - 

12.31 G.R. Irwin, 9th Inter. Congr. Appl. Mech., VIII, Paper 101 (II), 

University of Brussels, 245 (1957). 

12.41 J.M. Krafft and G.R. Irwin, in Fracture Toughness Testing and Its 

Applications, ASTM STP 381, 84 (1965). 

t2.51 J.M. Krafft, Appl. Mat. Research, Vol. 3. 88 (1963). 

12.61 G.T. Hahn, M.F. Kanninen, and A.R. Rosenfield, Annual Rev. Mat. Sci.. 

2, 381 (1972). 

12.71 R.G. Hoagland, A.R. Rosenfield, and G.T. Hahn, Met. Trans., 2, 123 

(1972). 

12.81 F.A. McClintock, S.M. Kaplan and C.A. Berg, Int. J. Fract. Mech., 2, 

615 (1966). 

L2.91 J.R. Rice and M.A. Johnson, in Inelastic Behavior of Solids, McGrawHill, New York, N.Y., 641 (1970). 

[2.10] G.T. Hahn and A.R. Rosenfield. Proc. 3rd. Int. Conf. on Fract., Munich, 

Germany, L, PL III, 211 (1973). 

[2.11] F.A. McClintock, in Stress Corrosion Cracking and Hydrogen Embrittlement 

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21 
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22 
13.81 W.A. Bessett, E.M. Brody, "Brillouin Scattering: A New Way to Measure 

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of Determining Moduli of Elasticity and the Shear Moduli of Wood," 

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Stress Concentrations, Crane-Russek, N.Y., (1973). 

23 
14.21 

14.31 

r4.41 

t4.51 

14.61 

14.71 

L4.81 

r4.91 

[4.10] 

14.111 

14.121 

14.131 

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~;__ 

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24 
l4.211 K.E. Atkinson, A Survey of Numerical Methods for the Solution of 

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25 
TABLE I. MECHANICAL PROPERTIES OF SOME COMMERCIAL ALlJMINllM ALLOYS AND CORRESPONDING METALLOGRAPHIC 

DATA FOR LARGE INCLUSION PARTICLES AFTER LOW, VAN STONE AND MERCHANT [2.13] AND OTHERS 

Alloy 2014-T6 2024-T851 2124-T851 7075-T7351 7079-T651 

Yield Stress, MNmm2(ksi) 

KIc , MNm -3t2 

6,,w 

Average Spacing of Large 

Dimples, nm 

436 (63.5) 443 (64.4) 435 (63.3) 391 (56.8) 502 (73.0) 

19.5 20.6 26.2 30.9 27.4 

6.3 7.0 11.4 17.7 10.8 

11 9.6 8.6 9.5 

N Identity of Large Q-Phase (A14CuMgSi4) A12Cu(Mn,Fe)3 A120Cu20hfn,Fe)3 (Fe,hfn,CU)fl6 (Fe .h ,cU)+ 

m inclusions A112 (Fe,Md $3 cuA12 %,Si 

Mg2Si 

W2Si 

cufi12 

Volume Fraction of 2.9 4.4 1.0 2.6 0.9 

inclusions, 9: = f 

f-1/6 

C 

8.4 7.8 1.0 0.85 1.02 

Center-to-Center Spacing 

of inclusions, nm 

8.4 7.4 9.2 11.6 9.4 

Average particle diameter,pm 5.8 5.6 4.5 5.7 4.2 

6c is the critical crack-tip opening displacement (6c = 

0.5 KIc2 

EY ) = COD 

C 
- 

TABLE II. MECHANICAL PROPERTIES AND CORRESPONDING MICROSTRUCTURAL FEATURES 

Ultimate ReducAlloy Fracture Tensile Yield Elonga- tion of Primary a Dimple 

Condi- Direc- Toughness (Klc), Strength, Strength tion Area, Aspect Spacing, Size 

tion Alloy tion MPa m(ksi in.) MPa (ksi) MPa (ksi) % % Ratio pm Pm 

A 334 L (LR) 92 (84) 1330 (193) 1260 (183) 7 1.5 6 2 to 5 10 to 15 

T (CR) 63 (57) 1340 (194) 1275 (185) 2 4 

227 ,L (LR) 65 (59) 1240 (180) 1200 (174) 8 24 6 5 to 10 10 to 15 

T (CR) 56 (51) 1230 (178) 1180 (171) 4 12 

B 334 L (LR) 64 (58) 1185 (172) 1150 (167) 4 14 6 2 to 5 5 to 10 

f: T (CL) 50 (45) 1260 (183) 1225 (178) 2 6 

227 L (LR) 63 (57) 1305 (189) 1240 (180) 1 5 6 10 5 to 10 

T (CL) 45 (41) 1360 (197) 1295 (188) 1 6 

C 334 L (LT) 71 (65) 1105 (160) 1095 (159) 13 36 3 5 to 10 10 to 15 

T (TL) 66 (60) 1235 (179) 1180 (171) 5 9 

227 L (LT) 53 (48) 1180 (171) 1130 (164) 13 47 2 5 to 10 10 to 15 

T cm 47 (43) 1275 (185) 1225 (178) 11 32 

D 334 L 940 (136) 890 (129) 23 52 3 2 10 to 15 

T 930 (135) 905 (131) 15 44 

227 L 860 (125) 840 (122) 20 57 5 2 to 5 5 to 10 

T 885 (128) 855 (124) 15 48 
TABLE III. COMPARISON OF BASIC FIM AND T-MATRIX METHODS [4.34] 

..-----2- 

Aspect of Method 

Formulation 

Scattering Parameters 

determined from 

Part of Field 

removed from 

calculation 

Surface enters 

calculation 

Expansion in terms 

of 

Numerical Stability 

Easily Adaptable 

to various shapes 

Fredholm Integral Equation Method 

Volume integral 

equation for 

electric field 

Internal Field 

via integration 

External Field 

Implicitly 

through volume 

integrals 

Fourier transform 

variable 

Theoretically stable 

practically 

instabilities have 

not revealed themselves 

No 

T-Matrix Method 

Surface integral 

equation and Huygens 

principle 

External Field 

via asymptotic form 

Internal Field* 

Explicitly via 

surface integrals 

- implicitly matching 

occurs on surface 

Position space 

variable 

Instability occurs 

when ka is 

increased too far 

(Ref. 6) 

Yes (?) 

28 
I Non1 i near 

I’ 

Linear Elastic Zone 

Elastic Zone Plastic Zone 

Large Plastic 

Stretch Zone 

Fig. 2.1 Schijve's scale of crack dimension. 

29 
tissl Elastically 

Deformed Region 

tza 

Plastically 

Deformed Region 

Fig. 2.2 Model for crack extension where the only work done is 

the plastic work plus energy to create new surfaces. 

30 
0 Holes Ahead of Crack Tip 

Plastic Zone 

Intensely Deformed 

Plastic Zone 

Fig. 2.3 Model demonstrating contained region of intensely deformed material with hole spacing. 

31 
Fig. 2.4 Mechanism of fibrous crack extension: 

schematic. 

32 
a Large Strain Region 

Void Site Enveloped by 

x&-()- Growth 

Further Growth 

Localized Flow Begins 

Final Coalescence 

Fig. 2.5 Model of dutile fracture by the growth and coalescence of 

an initially spherical void with the crack tip. 

33 
LENTICULAR COARSE ALPHA 

Nucleation -Easy 

Growth - Difficult 

DUCTILITY TOUGHNESS 

CT 

Plastic Zo* ~~~ 

CT Crack 

Small Plastic Zone Tortuous Crack Path 

GLOBULAR COARSE ALPHA 

Nucleation-Difficult 

Growth-Easy 

DUCTILITY TOUGHNESS 

i 

v 0 ezz 0 

i Plastic Zone 0 

Large Plastic Zone Non-Tortuous Crack Path 

Fig. 2.6 Schematic representation of effect of coarse alpha 

shape on ductility and toughness. 

34 

I
Fig. 3.1 Diagram of fracture model. "Grains" (S) and (R) are imbedded in a matrix subjected to a local static stress field 

of magnitude 0 . A stress wave of initial amplitude o 

and velocity vfis emitted from (S). The distance betwgen 

(S) and (R) is R and "grain" size is 6. 

35 
/ 

F1 /\ 

4 

Fig. 4.1 Coordinate system for two guides, at a distance R apart. 

36 
~~. .~ 

1. Report No. 2. Government Accession No. 

NASA CR-3356 

4. Title a”” Subtitle ON THE FEASIBILITY OF QUANTITATIVE 

ULTRASONIC DETERMINATION OF FRACTURE TOUGHNESS - 

A LITERATURE REVIEW 

- -- 

7. Author(s) 

L. S. Fu 

3. Recipient’s Catalog No. 

;. Report 0,tl -~ 

November 1980 

6. Performing Organization Code 

8. Performing Organization Report No. 

E-547 

10. Work Unit No. 

9. Performing Organization Name and Address 

The Ohio State University 

Columbus, Ohio 43210 

11. Contract or Grant No. 

NSG-3269 

2. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, D. C. 20546 

13. Type of Report and Period Covered 

Contractor Report 

14. Sponsoring Agency Code 

5. Supplementary Notes 

Final report. Project Manager, Alex Vary, Materials Division, NASA Lewis Research Center, 

Cleveland, Ohio 44135. 

-~-~-.--_ 

6. Abstract 

The report covers three main topics: (a) Fracture Toughness and Microstructure. (b) Quantitative Ultrasonics and Microstructure, and (c) Scattering and Related Mathematical Methods. 

Literature in these areas is reviewed to give insight to the search of a theoretical foundation for 

quantitative ultrasonic measurement of fracture toughness. The literature review shows that 

fracture toughness is inherently related to the microstructure and, in particular, it depends upon 

the spacing of inclusions or second phase particles and the aspect ratio of second phase particles 

There are indications that ultrasonic velocity and attenuation measurements can be used to determine fracture toughness. This leads to a review of the mathematical methods available in solving boundary value problems related to microstructural factors that govern fracture toughness 

and wave motion. A framework towards the theoretical study for the quantitative determination 

of fracture toughness is described and suggestions for future research are proposed. 

7. Key Words (Suggested by Author(s)) 

Fracture toughness; Microstructure; 

Polycrystalline metals; Ultrasonics; 

Ultrasonic measurement; Attenuation; 

Scatter attenuation; Crack extension 

I 

18. Distribution Statement 

Unclassified - unlimited 

STAR Category 38 

I I 

3. Security Classif. (of this report) 20. Security Classif. (ot this page) 21. No. of Pages 

Unclassified Unclassified 38 

For sale by the National Technical Information Service, SpinefIeld. Virginia 22161 

22. Pwx 

A03 

NASA-Langley, 1980