ULTRASONIC DETECTION OF DAMAGE IN HETEROGENEOUS
MEDIA
L. Yang and J. A. Turner
Department of Engineering Mechanics, W317.4 NH
University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0526 USA
ABSTRACT. The use of ultrasound for detecting material damage from microcracks within a
heterogeneous medium is discussed. The ability to detect the damage is dependent on the amount of
scattering due to the microcracks relative to the scattering from the heterogeneous background. The
damage is modeled in terms of penny-shaped microcracks that are assumed to be randomly oriented and
uniformly distributed. Scattering from the heterogeneous microstructure is considered in terms of two
types of media: polycrystalline and two-phase. Expressions for attenuation and backscatter coefficient
are derived using stochastic wave theory for both the assemblage of microcracks and for the
microstructure. Incident longitudinal and shear waves are both considered. Wave speed changes are
shown to be weak indicators of damage. However, attenuation and backscatter are shown to change
sufficiently for damage detection. The scattering from the microcracks is assumed independent of the
microstructural scattering. Thus, the results are thought to provide a lower bound on damage
detectability. The results are applicable to detection of microcracking for various applications including
thermal fatigue damage in concrete and mechanical fatigue damage in metals.
INTRODUCTION
The use of ultrasound for detecting material damage from microcracking within a
heterogeneous medium is of considerable importance to quantitative nondestructive
evaluation, materials characterization and related research interests. Early detection of
damage in engineering structures is crucial for maintaining structural integrity. Material
responses, which are often characterized ultrasonically by the decrease in wave velocity or
the increase in ultrasound attenuation, vary with the microstructure and microcracking
changes. The limits of detecting these changes in wave behavior strongly depend on the
amount of increased scattering due to induced damage. Recent theoretical research
[1,2,3,4] has shown how longitudinal and shear attenuations are dependent on either
microstructure or damage from microcracks. In general, the assessment of microscale
damage in complex media is very difficult both experimentally and theoretically. However,
ultrasonic backscattering measurements are often more reliably obtained in comparison
with attenuation. Backscattering measurements have been use to detect and evaluate
damage in steel as discussed by Hirsekorn, et al. [5]. Ultrasonic backscatter for detecting
microstructure is examined by Rose [6], Margetan, et al. [7, 8], Thompson [9] and others.
In this article, attenuation and backscatter are considered in terms of both
microcracks and microstructure of the medium. The damage is modeled as penny-shaped
microcracks that are assumed to be randomly oriented and uniformly distributed. Two
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/$20.00
133
types of microstructure, polycrystalline and two-phase, are the focus of discussion for
media microstructure. General expressions of both attenuation and backscatter coefficient
are derived using stochastic wave theory in the both types of microstructure. The ability to
detect the damage is dependent on the amount of scattering due to the microcracks relative
to the scattering from the heterogeneous background.
THEORETICAL MODEL
The propagation and scattering of elastic waves in heterogeneous media is
described here in terms of the Dyson equation. Such an approach has been discussed by
Frisch [10] and used by others [1,2,3,4]. The equation of motion for the elastodynamic
response of a linear, elastic solid to deformation is given in terms of the Green's dyadic by
(1)
The second order Green's dyadic, Gma(x,x';0, is defined as the displacement response at
location x in the rath direction due to a unit impulsive force acting at the position x' in the
ath direction at time zero. The moduli Cijmn (x) density p(x) are considered to be spatially
heterogeneous and of the form
(2a)
(2b)
Hence, the moduli Cijmn(x) have an average value of C®mn, plus a fluctuation about the
mean SCijmn (x) . As such, the fluctuation has zero mean, (SCijmn (x n = 0 . In Eq. (2b), ~p is
the average density and Sp(x) is a dimensionless measure of the density fluctuations. The
covariance of the moduli is characterized by an eighth rank tensor
(3)
The spatial and tensorial parts of the covariance, W and S, respectively, are assumed to be
independent. The form of the function W implies that the medium is statistically
homogeneous and statistically isotropic. The attenuation and backscatter can be calculated
by inner products on the covariance of the moduli fluctuation. The mean response of
scattering is governed by the Dyson equation [7]
^3z.
(4)
The notation G°[(x,x/) is the bare Green's dyadic, which is defined as the solution to
equation (1) without heterogeneities. The second order tensor M is the self-energy
operator. The above equation can be solved in Fourier transform domain with the
assumption of statistical homogeneity. The spatially Fourier transform of the Dyson
equation, is then solved in the form
134
where M(p) is the spatial transform of the self-energy. The Dyson equation is exact and
describes the mean response of the medium. For more details of the general scattering
theory, the reader is directed to Refs. [1,2,3,4].
ATTENUATION AND BACKSCATTER
Based on the stochastic wave theory, explicit expressions of the dimensionless
longitudinal and shear attenuations are given under the general assumptions: For
microcracked media, the cracks are treated as penny-shaped microcracks that are assumed
to be randomly oriented and uniformly distributed. All cases of heterogeneous media
considered here, including locally isotropic, two-phase, and polycrystalline, the media are
assumed statistically homogeneous and statistically isotropic. In addition, the results for the
polycrystalline medium are restricted to grains with cubic crystalline symmetry. Because
the focus here is on L-L and T-T backscatter, only these components of the attenuation are
needed.
For the case of scattering from a homogeneous medium with damage, the relevant
attenuations are given by [1]:
(6b)
(fib)
where B = cT/cLis the wave speed ratio. Dimensionless longitudinal and transverse
frequencies are defined as XL =0)LlcL and*r =coL/cT. Here, L is the crack length and 6
is defined as the damage density. The coefficients kt, n. (i = 1,2,3) are dependent on
Poisson's ratio and are given in the Ref. [1].
For the case of scattering from a heterogeneous medium which is locally isotropic,
the attenuations are [2]:
LL
h
2 J-i
(l + 2x2L(l-z))2
^(l^Vyi-3^)
*
2
where H is the correlation length and Ay is the magnitude of the fluctuations for each
material parameter. The subscript y denotes the material parameters of /?, A, and ju.
Dimensionless longitudinal and transverse frequencies are defined in this case as
XL = coH I CL and XT = coH I CT .
For the case of scattering from a medium with two uniform phases, the attenuations
are [3]:
135
f
f
x
LL
L r
'
-
+
-
J_,_, ————————
^ + 2+
-2
/
i
r + z + ^
————————dx'
-
where H is the correlation length. Here, fa and /OT are the volume fractions of the two
phases in the medium, respectively.
Finally, for the case of scattering from a polycrystalline medium, the attenuations
are given by [4]:
2 Z
'
where // = (Cn - C12 - 2C44)//9c^ is the dimensionless anisotropy factor applicable to cubic
crystal symmetry. Dimensionless longitudinal and transverse frequencies are defined as
XL = cojS I CL and XT =0)f3 1 CT, where ft is the grain diameter. In the polycrystalline model,
the material density is assumed uniform throughout.
Explicit formulae for longitudinal and transverse backscatter coefficients, &L and
£TT, respectively, are calculated from the attenuation expressions given above. The
scattering attenuation is the integration of the energy that scatters in all directions. Thus,
the attenuations are given as integrals over scattering angles. The backscatter coefficient
represents the amount of energy that scatters only in the backward direction. Thus,
expressions proportional to £LL and %TT may be derived by multiplying the integrands for
aiL and arr by a delta function, S(% + 1) . For example, the longitudinal backscatter from a
polycrystalline material is
2 -1-1 525(1 + 2x2L (1-%}?
'"^
525(1 + 4*;;)
as derived previously by Rose [6] .
Within the Rayleigh-regime, XL and XT are « 1 and further simplification is
achieved. The results here are expressed as the ratio of the backscatter coefficient for the
microcracks to the backscatter coefficient for the heterogeneous background medium.
Under the assumptions outlined above, these ratios are expressed as
j ~t
__
\^ / I
f ft
^M-r
136
-«•'-«- 1
I
'L^3
-1—'
(lie)
VbLL/p
^'1 '3£J\PJ
and
(He)
where
1024 (l-v)2(191v6+896v5 -4722v4 +7532v3 -7345v2 +3564V-756) ,„ ,
MI1 - —————————————————————— -4————2-—————————————— , (llg)
8505
(l-2v) (2-v)
_512.(l-v)'(2v-9)
1215
(2-v) 2
In Eqs. (11), the non-dimensional damage density is defined by £ = NVC, where Vc is the
single crack volume and N is number of cracks per unit volume [11].
Equations (11) give the general expressions for evaluation of the detection limit of
the damage within a heterogeneous medium within the Rayleigh-regime. It is easy to see
that under the assumptions the ratios of backscatter by cracks media and heterogeneous,
two-phase, polycrystalline media, respectively, have a linear dependence on the damage
density and are related to the third power of the ratio of crack length to the correlation
length of considered media. In addition, the results are independent of frequency. The
dependency of the magnitude of the fluctuations of the material properties in heterogeneous
media and the dependency of the volume fractions in two-phase media are also shown.
EXAMPLE RESULTS
In this section, numerical results based on the above derivations are presented for
the special cases of iron and concrete. For iron, the results make use of the material
constants shown in Table 1 . These values yield the ratios of backscattering to be
«/™ = TlNr = 153.882/^1 ,
(12a)
Tl,™ = l^v = 34.303/1 ,
(12b)
($LL)P
\P)
(±TT ) p
\
137
where the notation L is the crack length, and L = 2(a). Here, the result L3 is not same as
that given in the literature [5] due to the different definition of the damage density and the
crack length. But if the difference is considered, the coefficient of 153.882 is replaced with
a value of 9.80, slightly larger than the value of 7.49 from literature [5]. Results for several
values of crack length and crack density are shown in Table 2. The grain diameter used
here is ft = 50 ^urn.
The results show that within the Rayleigh-regime, if the crack length is about less
than the grain length the grain scattering would dominate the total scattering. If the crack
length is larger than the grain length the cracks would cause the main scattering. Therefore,
for most metals it is expected that small cracks, especially ones less than about half the
grain diameter, cannot be detected reliably. The backscatter signals will be buried by the
grain scattering signals. However, in general, two or more small cracks usually compose of
the one large crack, with length that is of the same order of the magnitude as the grain
diameter or even larger [5]. Then the signals of damage scattering increase strongly with
the corresponding increase in crack size. The detection of damage is still thought to be
reasonable in practical applications.
For concrete, the material is considered as a heterogeneous medium with local
isotropy. For the simplified case, the material is assumed as two uniform phases: aggregate
and matrix, respectively. Material constants used for these results are shown in Table 3.
Using this model, the ratios of backscattering reduce to
16.035 £
0.1104A;+A;+0.4459A;
M
-.
con
LlL
___
(13a)
L
(13b)
TZl'n
Some numerical results are shown in Table 4. For concrete, the material
heterogeneities dominate the wave scattering if the cracks length is much smaller than the
material correlation length. The influence observed for different levels of material property
fluctuation is almost the same for the backscatter ratio calculated here. Therefore, the
backscattering contribution from small-scale damage could be considerably difficult to be
TABLE 1. Material Constants of Iron.
v
Iron
0.30
/?(kg/m3)
7860
cL(km/s)
CT (km/s)
Cn(Gpa)
C12(Gpa)
C^CGpa)
5.9
3.23
237
141
116
TABLE 2. Ratio of Cracks to Grain Backscattering (Iron).
Low level
High level
L(jum)
N(l/mm 3 )
20
20
20
40
50
60
250
500
750
100
100
100
138
€(%)
0.104
0.209
0.314
0.334
0.654
1.130
^iron
Klron
0.0102
0.0205
0.0308
0.2632
1.006
3.005
0.0023
0.0046
0.0069
0.0586
0.2243
0.6618
observed in practical experiments. Numerical results using the simplified two-phase model
are shown in Table 5. The results in this case are similar to that observed in Table 4. This
result is not such a surprise given the relative independence to material heterogeneity seen
in Table 4. The majority of damage in concrete structures is initiated at the length scales
that are smaller than the size of the majority of the aggregate. Here again, crack
interlinking would cause a rapid increase in the effective crack length. In general, it is
expected to be sufficient to reach the detection level. As shown in Tables 4 and 5, a few
large cracks are significantly more easily detected than many smaller cracks. Hence, the
results presented here are expected to be useful to the development of experiment
techniques for concrete structures.
TABLE 3. Material Constants of Concrete.
V
0.16
Concrete
p(kg/m3)
2400
cL(kml s)
cT(kml s)
4.24
2.45
TABLE 4. Ratio of cracks to heterogeneous media backscattering (Concrete).
Ml/cm")
L(mm)
H(mm)
1000
1500
1000
1500
1000
1500
100
100
0.2
0.2
0.2
0.2
0.2
0.2
1.0
1.5
10
10
10
10
10
10
10
10
A
P
0.01
0.01
0
0
0
0
0.01
0.01
4l
^
€(%)
Ucon
0
0
0.01
0.01
0
0
0.01
0.01
0
0
0
0
0.01
0.01
0.01
0.01
0.418
0.628
0.418
0.628
0.418
0.628
5.236
17.66
0.0054
0.0080
0.0486
0.0729
0.0120
0.0181
5.388
61.20
TABLE 5. Ratio of backscattering coefficient for cracks in a two-phase medium (Concrete).
Ml/cm3)
500
1000
1500
100
100
100
L(mm)
0.2
0.2
0.2
0.5
1.0
1.5
H(mm)
10
10
10
10
10
10
fa
fm
e
0.25
0.50
0.75
0.50
0.50
0.50
0.75
0.50
0.25
0.50
0.50
0.50
0.004
0.008
0.012
0.013
0.100
0.338
L2con
0
0
0
0
0.0013
0.0143
CONCLUSIONS
Results have been presented for attenuation and backscatter for microcracks and
microstructure using stochastic wave theory. Within the Rayleigh-regime and under the
basic assumptions, the ratio of ultrasonic backscattering from microcracks to
backscattering from microstructure is expressed for incident longitudinal and shear waves.
It is shown that the detectability of the damage is dependent on the scattering due to the
microcracks relative to the scattering from the heterogeneous background. If the
microcrack length is larger than the microstructural length scale, the attenuation and
backscatter might be sufficient for detection of damage. This case is usually true in
practical materials such as applied to fatigue in metals and concrete. Some numerical
results for the special cases for iron and concrete provide an indication of the detection
levels expected. In addition, it was assumed that there is no interaction scattering between
139
the microcracks and the microstructure. Therefore, the detection limit presented here is
considered as a lower bound on damage detectability.
ACKNOWLEDGMENTS
The support of the National Science Foundation (Grant No. CMS-9978707) and the
National Bridge Research Organization (NaBRO) is gratefully acknowledged by authors.
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