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Estimation of creep voids using a progressive damage model and neural network Hyunjo Jeong Division of Mechanical Engineering, Wonkwang University, 344-2 Shinyong-dong, Iksan, Jeonbuk 570-749, South Korea Phone: +82-653-850-6690, Fax: +82-653-850-6691 Email : hjjeong@wonkwang.ac.kr Contact

Abstract

In order to develop nondestructive techniques for the quantitative estimation of creep damage, a series of crept copper samples were prepared and their ultrasonic velocities were measured. Velocities measured in three directions with respect to the loading axis decreased nonlinearly and their anisotropy increased as a function of creep-induced porosity. A progressive damage model was described to explain the void-velocity relationship, including the anisotropy. The model study showed that the creep voids evolved from sphere toward flat oblate spheroid with its minor axis symmetrically distributed with respect to the stress direction. This model allowed us to determine the average aspect ratio of voids for a given porosity content. A novel technique, the back propagation neural network (BPNN), was applied for estimating the porosity content. The measured velocities were used to train the BP classifier, and its performance was tested on another set of creep samples containing 0 to 0.7 % porosity. When the void aspect ratio was used as input parameter together with the velocity data, the NN algorithm provided much better estimation of void content.

1. Introduction

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 near-surface 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 two-phase Mori-Tanaka 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 three-layered 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 x₁ direction lies along the thickness direction of the original plate, the x₂ along the rolling direction, and the x₃ 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 x1, x2, x3.

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 v₂ is defined by v₂ = 1 - r_*/r₁, where the subscripts "1" and "*" indicate the reference and interrupt samples, respectively. In this study, v₂ was determined from the weight measurements due to Ratcliffe [9]. This measurement of v₂ is independent of the sample volume and the fluid density.
A contact, pulse-echo technique was used to measure longitudinal (L) and shear (S) ultrasonic velocities in the x₁, x₂, and x₃ directions in the reference and creep-damaged cubic samples (20 mm ´ 20mm ´ 20 mm). Throughout the experiment, 5 MHz broadband transducers of 12.7 mm diameter were used. In the pulse-echo 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 creep-damaged 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 x₃ 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 creep-damaged copper will be treated as a two-phase 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 L-wave are shown as a function of porosity in Fig. 3. The solid lines represent the least-square fitting with the second order polynomials. In general, velocities V₁₁, V₂₂ and V₃₃ decrease nonlinearly with increasing porosity or creep damage. In each creep-damaged sample, the velocity in the x₃ direction is found to be much lower than that in the x₁ or x₂ direction. However, the velocities in the x₁ and x₂ directions are seen to be very close to each other. The anisotropy of velocities (V₁₁ » V₂₂ > V₃₃) 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 S-wave velocities, i.e., V₁₂> V₁₃ » V₂₃. Based on these observations, the creep damaged copper samples can be modeled as a transversely isotropic composite with the loading axis, x₃, as the symmetrical axis.


Fig 2: Photomicrograph of creep damaged copper: notch position, t/tr=0.7, 550°C.

Fig 3: Measured longitudinal wave velocities.

3. Micromechanics Model

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 creep-induced voids of non-spherical 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 two-phase 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,n₁ + n₂ = 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 x₁', x₂'and x₃', 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 fourth-order 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 C₂ = r₂ = 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 a₁ = a₂ > a₃, the aspect ratio is then defined by a = a₃ / a₁ < 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 eighth-order transformation tensor relating a fourth-order tensor in the local (x₁ ^',x₂ ^',x₃^' )and global (x₁,x₂,x₃) 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 eighth-order texture tensor defined by


(7)

If C₁ and C₂ are isotropic and the shape of the inclusion is spheroidal, (C₂ - C₁)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 x₃ 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 creep-damaged 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 C₁ 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/cm³. The well-known relationships for elastic constants, density, and velocities yield the matrix properties: C₁₁=198.3 GPa, and C₄₄=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 L-wave 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, W₂₀₀ = 0.02826, W₄₀₀ = 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 void-velocity relationships.


Fig 4: Calculated aspect ratios of assumed oblate voids for s= 0.5.

Fig 5: Comparison of predicted and measured spheroidal velocities.

4. Neural network

The neural network (NN) adopted in this work is designed in three layers with various numbers of units. The first layer is the input layer, and the second and third layers are the standard hidden and output layers of back-propagation network. The experimental data or a combination of experimental and analyticaldata obtained from the creep samples in the notched position are used to train the NN classifier. The performance of trained NN is tested on the creep samples from the smooth position.


Fig 6: Comparisons between the measured and estimated porosity

Two different methods were employed:

• Method 1. The input layer has three units, and three L-wave velocities, V₁₁, V₂₂ and V₃₃, were used. The second layer contained five units. The third layer had one unit. The single output corresponds to the porosity of creep samples from the smooth position.
  
• Method 2. The input layer has four units. In addition to three L-wave velocities, the void aspect ratio a obtained from the progressive damage model for axisymmetric orientation distribution with s = 0.5 was used. The second layer contained seven units. The third layer had one unit..
Porosity estimation results obtained by implementing three-layered neural networks are presented in Fig. 6. As shown in Fig. 6, both BPNNs were very successful in estimating the porosity in the range 0.0 - 0.7 % by volume. However, the method 2 provides better estimation, and the average difference between the estimated and measured porosity is less than 1.6 %. These results indicate that the BPNN evaluated the small amount of void contents very accurately when the void aspect ratio was used as an input parameter in addition to the velocity data.

5. Conclusions

The damage mechanism of crept copper samples was found to be the voids on the grain boundaries perpendicular to the loading direction. As the creep advances, the porosity increased and the ultrasonic velocities measured in the loading direction were found to be much lower than the velocities in the other two directions. The anisotropy of velocities could be correlated with non-spherical void shapes and their anisotropic orientation distribution. A two-phase composite model that accounts for void characteristics such as shape, aspect ratio, and orientation distribution was described. In order to explain the observed velocity- porosity relationships of the interrupt samples, the voids were modeled as an oblate spheroid, but its aspect ratio could change as the creep progresses. The minor axes of assumed oblate spheroids were assumed to be axisymmetrically distributed with respect to the loading direction, consequently a transversely isotropy was obtained. The progressive damage model together with the measured velocities revealed that the creep voids evolved from sphere to disk. A three-layered neural network using a backpropagation algorithm was employed to quantify the volume fractions of distributed voids. When the void aspect ratio was used as input parameter in addition to the measured velocity data, the BPNN estimated the void content very accurately in the range 0.0 - 0.7 %.

Acknowledgements

The author would like to deeply acknowledge professors T. Morishita (Akashi College of Technology, Japan) and M. Hirao (Osaka University, Japan) for furnishing copper samples. This work was partially supported by Wonkwang University, Korea, in the academic year 2000.

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