There is increasing concern regarding the safety of power plant components used in a high temperature environment for extended periods. For structural materials used at high temperatures, creep rupture is a common failure mode. The creep failure occurs by a process of cumulative damage. This involves the nucleation and growth of cavities at the grain boundaries, their subsequent linkage to form microcracks, and the propagation of microcracks until failure. During this process, the precipitation of the second phase particles such as carbides and intermetallic compounds is accompanied. Nondestructive techniques have long been desired for assessing the creep damage. Ultrasonic methods have a unique potential of detecting internal damage states, while many others (replication, magnetic measurements, and eddy currents) merely inspect the surface or nearsurface region of materials.
Ultrasonics have been studied for estimating the creep damage[1,2]. As the creep progresses, the propagation velocities of ultrasonic waves decrease and the attenuation coefficients increase. Most of these studies have been experimental in nature with little theoretical attempts to relate ultrasonic parameters to the microstructures of crept materials. When ultrasonics are to be used for quantitatively estimating the creep damage, detailed knowledge of mathematical relationships between the microstructural changes and the ultrasonic NDE signatures is required. Ledbetter et al.[3] and Hirao et al.[4] used a composite model to explain the dependence of ultrasonic velocities on the creep voids. Morishita and Hirao[5] considered a double composite model to explain the velocity evolution, including the anisotropy, during the creep life. These are known as the forward problem in NDE data interpretation. In the inverse problem, the sought microstructures are obtained from the measured NDE outputs and forward micromechanics model. Jeong and Hsu[6] used a twophase MoriTanaka method to evaluate the material properties of porous ceramics from measured velocities. Dunn and Ledbetter[7] performed a similar work to estimate the orientation distribution of short fiber composites. More recently, a neural network[8] was applied to the prediction of creep damage based on magnetic properties in power plant piping.
This paper presents a NDE technique to estimate the creep
induced porosity of pure copper samples. The proposed scheme rests on the connection between the measured ultrasonic velocity and the corresponding prediction by composite micromechanics. The approach consists of
 measurement of ultrasonic velocities,
 development of forward model to predict effective velocities, and
 porosity estimation by a neural network. The model can account for void characteristics such as shape, aspect ratio, and orientation distribution.
In order to explain the observed velocity
porosity relationships of the interrupt samples, the voids are modeled as an oblate spheroid, but its aspect ratio can change as the creep progresses. The optimal aspect ratio of voids at a specific creep stage is determined using a least squares sense. A threelayered neural network is trained using the measured velocities and the void aspect ratio as input parameters, and its performance is then tested on another set of creep samples containing 0.0
0.7 % porosity.
2. Experiments
2.1 Creep samples[5]
The test material is a commercial copper of 99.95 mass percent purity. The creep specimens were machined from a rolled plate of 20 mm thick. Specimen geometry and coordinate system are shown in Fig. 1. The x_{1} direction lies along the thickness direction of the original plate, the x_{2} along the rolling direction, and the x_{3} along the loading direction. The nominal stress was 6.0 MPa at the minimum cross section. Shallow notches were introduced to obtain two cube samples, 20 mm on each side, of different damage states (Fig. 1). They underwent slightly different stresses with the equal thermal history. Samples obtained from the notched position and those from the smooth position are illustrated in Fig. 1(a) and (b), respectively. For a comparison purpose, six reference samples of the same dimensions were prepared.
(a) sample from notched region

(b) sample from smooth region

Fig 1: Specimen geometry and sample coordinate system x_{1}, x_{2}, x_{3}.

Eight specimens were crept to failure, four at 500 °C and the other four at 550 °C to know the rupture time. The average rupture time, t_{r}, was 597.8 hours at 500 °C and 270.8 hours at 550 °C. Interrupt tests were then made in the same creep conditions as the rupture test to obtain coupons with different level of creep damage. Eleven interrupting times were chosen relative to t_{r}, i.e., t/t_{r }= 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.75, 0.8, 0.85, 0.9.
2.2 Porosity and velocity measurements
The porosity v_{2} is defined by v_{2} = 1 
r_{*}/r_{1}, where the subscripts "1" and "*" indicate the reference and interrupt samples, respectively. In this study, v_{2} was determined from the weight measurements due to Ratcliffe [9]. This measurement of v_{2} is independent of the sample volume and the fluid density.
A contact, pulseecho technique was used to measure longitudinal (L) and shear (S) ultrasonic velocities in the x_{1}, x_{2}, and x_{3} directions in the reference and creepdamaged cubic samples (20 mm ´
20mm ´
20 mm). Throughout the experiment, 5 MHz broadband transducers of 12.7 mm diameter were used. In the pulseecho testing, velocities are given by 2d/D
t, where d is the sample thickness and D
t is the transit time difference between the first and second echoes. The ultrasonic wavelengths used in the experiment are around 900 mm and 500 mm for longitudinal and shear waves, respectively. These values are one order larger than the diameter of voids induced by creep damage. Thus the frequency dependence of velocities was ignored, and the transit time difference D
t was measured in the time domain using the pulse overlap technique. V_{ij} denotes the ultrasonic velocity propagating in the x_{i} direction and polarized in the x_{j} direction.
2.3 Experimental results
Figure 2 shows a typical photomicrograph of creepdamaged copper sample. The grain shape is nearly equiaxial and the grain size is approximately 0.1 mm. Voids are not randomly positioned. They tend to gather preferentially on the grain boundaries perpendicular to the x_{3} direction. The void diameter is approximately 10 mm at this stage of creep.
The reference sample was found to be isotropic with average velocities V_{L} = 4727 m/s, V_{S }= 2292 m/s. In the micromechanics model described below, a creepdamaged copper will be treated as a twophase composite material composed of the isotropic copper matrix and the voids. The average velocities of the reference sample will be used to calculate the elastic constants of the isotropic matrix.
The velocity measurement results for the Lwave are shown as a function of porosity in Fig. 3. The solid lines represent the leastsquare fitting with the second order polynomials. In general, velocities V_{11}, V_{22 }and V_{33} decrease nonlinearly with increasing porosity or creep damage. In each creepdamaged sample, the velocity in the x_{3} direction is found to be much lower than that in the x_{1} or x_{2} direction. However, the velocities in the x_{1} and x_{2} directions are seen to be very close to each other. The anisotropy of velocities (V_{11} »
V_{22 }> V_{33}) between the loading direction and the other two directions can be correlated with the oriented growth of voids. Similar anisotropic behavior was observed for the Swave velocities, i.e., V_{12}> V_{13} »
V_{23}. Based on these observations, the creep damaged copper samples can be modeled as a transversely isotropic composite with the loading axis, x_{3}, as the symmetrical axis.
Fig 2: Photomicrograph of creep damaged copper:
notch position, t/t_{r}=0.7, 550°C.

Fig 3: Measured longitudinal wave velocities. 
In this section, we intend to establish a micromechanics model that explains the velocity evolution, including the anisotropy, during the creep life. The observed anisotropy can be attributed to the creepinduced voids of nonspherical shape and preferential orientation distribution. The goal of the theoretical model is to incorporate the microstructural characteristics of voids, which will, in turn, be used to evaluate the unknown porosity of the crept copper samples.
3.1 Composites with aligned inclusions
We consider a twophase composite composed of the isotropic matrix and the inclusions. Here the inclusions can be reinforcing particles or voids. Quantities associated with the matrix and the inclusions are denoted by subscripts "1" and "2," respectively. The volume fraction is denoted by n
where,n_{1} + n_{2} = 1, and the density is denoted by r
. The stiffness tensor is represented by C. It is assumed that the inclusions are represented by ellipsoids of identical shape. The local coordinates of an ellipsoidal inclusion in the composite are denoted by the primed axes x_{1}', x_{2}'and x_{3}', while the global (sample) coordinates are denoted by the unprimed ones. The orientation of an inclusion is then specified uniquely by the three standard Euler angles x
=cosq
, y
, and j
. The orientation distribution of the inclusions can be described by the probability density function w(x
=cosq
, y
, j
).
When the inclusions are perfectly aligned in the matrix, the effective elastic stiffness of the composite is given by Hill [10]
 (1) 
Here A is the strain concentration factor tensor, which relates the average strain in the inclusion to the remotely applied uniform strain in the composite, i.e., generally depends in a complex way on the phase moduli, and reinforcement shape and orientation. Numerous approximate methods have been proposed to estimate A. For example, in Mori
Tanaka [11] method, the average strain in the interacting inclusions is approximated by that of a single inclusion in an infinite matrix subjected to the uniform average matrix strain . Using this approach, A is given by [12,13]
 (2) 
where A^{dil} denotes the exact concentration tensor for a single inclusion in an infinite matrix and can be obtained by Eshelby's equivalent inclusion principle [14] as follows:
 (3) 
Here S is Eshelby's tensor and I is the fourthorder identity tensor. For an isotropic matrix, S is a function of the inclusion geometry and Poisson's ratio of the matrix [15].
The effective density of the composite is given by the rule of mixtures
 (4) 
When the inclusions are voids, we set C_{2} =
r_{2}^{ }= 0.
The observed anisotropy can be attributed to the void shape and orientation. To predict the effective velocities of crept samples, we consider the voids to be oblate spheroid. If the relative size of an oblate spheroid is set to be a_{1 }= a_{2} > a_{3}, the aspect ratio is then defined by a
= a_{3} / a_{1} < 1.
3.2 Orientational average
The effective stiffness of a two phase composite containing arbitrarily oriented inclusions can be evaluated from the orientational average of C^{a} weighted by the ODF w(x
, y
, j
) as follows:
 (5) 
where w(x
, y
, j
) denotes the orientation distribution function (ODF).
Here T^{*} is the eighthorder transformation tensor relating a fourthorder tensor in the local (x_{1} ^{'},x_{2} ^{'},x_{3}^{'} )and global (x_{1},x_{2},x_{3}) coordinate systems. Since C^{a} is expressed in the global coordinates, we can take it outside the integrals and write Eq. (5) in a form
 (6) 
Here T is the eighthorder texture tensor defined by
 (7) 
If C_{1} and C_{2} are isotropic and the shape of the inclusion is spheroidal, (C_{2}  C_{1})A is transversely isotropic and thus C^{a} has only five independent components. In this case T can be reduced to a 5´
5 matrix and is given by [16]
For the present study of copper matrix/void inclusion composites, we assume an axisymmetric orientation distribution of voids with respect to the x_{3} axis. The effective stiffness of the composite is then transversely isotropic and can be written as
 (8) 
The ultrasonic velocities in the sample directions are obtained as
 (9) 
3.3 Velocity predictions
In order to predict the velocities of creepdamaged copper based on Eq. (9), the following parameters should be known:
 elastic constants and density of the isotropic matrix, and
 the volume fraction, shape, aspect ratio, and orientation distribution of the voids.
First the isotropic matrix stiffness C_{1} was calculated from the measured quantities of the reference sample: V_{L} = 4727 m/s, V_{S }= 2292 m/s, r
_{1}=8.89 g/cm^{3}. The wellknown relationships for elastic constants, density, and velocities yield the matrix properties: C_{11}=198.3 GPa, and C_{44}=46.7 GPa.
Next we assume that the voids are of an oblate spheroidal shape and its aspect ratio changes as the creep damage increases. Their orientation distribution is assumed to be axisymmetric, thus maintaining transverse isotropic properties. The void aspect ratio for a given porosity was calculated from a least squares method. To this end, an objective function E that is a function of the unknown aspect ratio a
is defined as
 (10) 
In Eq. (10) the superscripts m and p denote measured and calculated quantities, respectively. The measured velocities are the three Lwave velocities shown in Fig. 3 and the predicted velocities are those obtained with Eq. (9). The optimum a
can be obtained by minimizing Eq. (10) and the results are shown in Fig. 4 for the ODF shape parameter s
= 0.5, W_{200 }= 0.02826, W_{400} = 0.02169. As the creep advances, the void aspect ratio continues to decrease, i.e., the oblate spheroid progressively changes to a disk shape.
The agreement between the predictions and measurements was best when s
= 0.5, and their comparisons are shown in Fig. 5. The solid lines show the measured velocities based on the fitting curves of data in Fig. 3. We see a satisfactory agreement throughout the creep life. The progressive damage model is, therefore, appropriate and is capable of explaining the voidvelocity relationships.
Fig 4: Calculated aspect ratios of assumed oblate voids for s= 0.5.

Fig 5: Comparison of predicted and measured spheroidal velocities.
