# Analysis of multiaxial impact behavior of polymers.

A. Saigal (*)Impact performance is a primary concern in many applications of polymers. In this paper, finite element analysis (FEA) and ABAQUS/Explicit are used to simulate the deformation and failure of polymers in the standard ASTM D3763 multiaxial impact test. The specimen geometry and loading mode in this multiaxial impact test provides a close correlation with practical impact conditions. A previously developed constitutive model ("DSGZ" model) for polymers under monotonic compressive loading is generalized and extended for any loading mode and takes into account the different behavior of polymers in uniaxial tensile and compression tests. The phenomenon of thermomechanical coupling during plastic deformation is also included in the analysis. This generalized DSGZ model, along with thermomechanical coupling and a failure criterion based on maximum plastic strain, is incorporated in the FEA model as a coupled-field user material subroutine to produce a unique tool for the prediction of the impact behavior of polymeric materials. Load-displacement curves from FEA simulations are compared with experimental data for two glassy polymers. ABS-1 and ABS-2. The simulations and experimental data are in excellent agreement up to the maximum impact load. It is shown that not accounting for the different behavior of the polymer in uniaxial tensile and compression tests and thermomechanical coupling effects leads to an overestimation of the load and impact energy, especially at large displacements and plastic deformations. Friction also plays an important role in the impact behavior. If one neglects the friction between the striker and polymer disk, the predicted impact loads are lower as compared with experimental data at large displacements.

INTRODUCTION

Polymers have increasingly been applied in applications where impact performance is of primary concern because of their good thermal and electrical insulation properties, low density, high resistance to chemicals and ease of manufacturing (1-4). There are several types of test methods for evaluating the impact strength of polymers (5). The most commonly used are Charpy and Izod. Although the two types of tests provide some information on the relative impact resistance of materials, the particular specimen geometry requirements of both tests make the test results difficult to relate to practical polymer design models. However, the specimen geometry and loading mode for a multiaxial impact test provide a close correlation with practical impact conditions (6). There is continuing interest in the literature regarding experimental results from this type of test, but reports with respect to theoretical characterization are limited. Lack of understanding of the theoretical material characterization requires engineers and scientists to evaluate any impact design failure by experiments performed on finished polymeric products. To reduce experimental trial and error, numerical simulation of impact deformation and failure of polymeric products is proposed.

Considerable progress has been made with respect to the numerical simulation of multiaxial impact tests on polymers during the last two decades (7-l1). Nimmer (7) created an FEA model to simulate a fixed velocity puncture test of BPA-polycarbonate disk, where a constant bilinear stress-strain curve was used to approximate the material behavior. Good agreement was achieved between model prediction and experimental load-deflection data for deflections up to four times the thickness of the test disk. By using the G'Sell-Jonas constitutive model, Billon and Haudin (8) numerically explored the effects of specimen thickness, friction between striker and specimen disk, and quality of the clamping device during a multiaxial impact test on polypropylene, a semicrystalline polymer. They indicated that the material response during impact is difficult to analyze since the deformation is not homogeneous and not isothermal. After validating the application of G'Sell-Jonas constitutive model to a semicrystalline polymer pol yamide 12 for a large range of strain rates during uniaxial tensile test, Schang et al. (9) developed an FEA model to simulate the multiaxial impact test. Model predictions were compared with experimental data for the history of impact load. Even though good agreement was achieved at the beginning of the load-time curves, the maximum impact load was largely overestimated. They pointed out that this result occurred because either the strain rate dependence in the constitutive model was not correctly taken into account or the tensile characterizations were not representative of the impact situation. All the aforementioned simulations are for impact deformation of polymers. Although the focal point of research on the impact on metallic structures has been gradually switched from dynamic deformation to dynamic failure (12), few reports are found in the literature for numerical simulation of polymer failure in multiaxial impact tests.

On the basis of our previous work on numerical simulation of impact failure of polymers and a generalized stress-strain constitutive model for polymers ("DSGZ" model) (10, 11, 13), an FEA model is formulated in this paper using ABAQUS/Explicit to simulate the deformation and failure of two different grades of ABS glassy polymers: ABS-1 with a trade name of Cycolac GPM5600 (melt flow rate: 8 g/10 min; [T.sub.g]: 110[degrees]C; specific gravity: 1.03) and ABS-2 with a trade name of Cycolac GPT3800 (melt viscosity: 2000 poise at 260[degrees]C; [T.sub.g]: 110[degrees]C; specific gravity: 1.02) in the standard ASTM D3763 multiaxial impact test. Both polymers are manufactured by GE Plastics.

MATERIAL CONSTITUTIVE MODELING

Numerical simulation of destructive impact events requires at least two models: material stress-strain constitutive model and failure model. The constitutive model plays a key role (14). Over the past four decades, much effort has been devoted to modeling the stress-strain constitutive relationships for polymers (15-18). Using concepts from the Johnson-Cook model, G'Sell-Jonas model, Brooks model and Matsuoka model, the authors developed a phenomenological constitutive model (called "DSGZ" model) to uniformly describe the entire range of deformation behavior of both glassy and semicrystalline polymers under monotonic compressive loading (13). The DSGZ model explicitly gives the compressive true stress [[sigma].sub.c] dependence on true strain [epsilon], true strain rate [epsilon] and temperature T and is given by:

[[sigma].sub.c]([epsilon],[epsilon],T) = [K.sub.c] {f([epsilon]) + [[epsilon].[e.sup.(1-[epsilon]/[C.sub.3] . h([epsilon],T))]/[C.sub.3] . h([epsilon], T) - f([epsilon])] . [e.sup.[ln(g([epsilon],T))-[C.sub.4]].[epsilon]]} . h([epsilon],T) (1)

where

f([epsilon]) = ([e.sup.[-C.sub.1].[epsilon]] + [[epsilon].sup.[C.sub.2]])(1 - [e.sup.-[alpha].[epsilon]) (2)

h([epsilon],T) = [[epsilon].sup.m] [[epsilon].sup.a/T] (3)

and g([epsilon],T) is the numerical value of h([epsilon],T) and time is measured in seconds. The eight material coefficients in this model are [K.sub.c] (Pa.[s.sup.m]), [C.sub.1], [C.sub.2], [C.sub.3] ([s.sub.m]), [C.sub.4], a (K), m, and [alpha]. In the DSGZ model, the term [epsilon].[e.sup.(1 -[epsilon]/[C.sub.3].h([epsilon], T))]/[C.sub.3] . h([epsilon],T]) describes the initial elastic deformation and the shift behavior of the yield point with strain rate and temperature for glassy polymers, the term f([epsilon]) respresents the strain hardening part of the constitutive relation for polymers, the term ln(g([epsilon],T)) - [C.sub.4] represents the exponential change from initial value [epsilon].[e.sup.(1 - [epsilon]/[C.sub.3].h([epsilon],T)] to stay state value f([epsilon]), and finally the entire constitutive relation is multiplied by [([epsilon]).sup.m][e.sup.a/T], which represents the stress dependence on strain rate and temperature.

It has been well known that the deformation behaviors of polymers are not only sensitive to strain, strain rate and temperature but also relate to the deformation mode (19-22). Unlike metals, polymers often exhibit different behavior in uniaxial tensile and compression tests (21, 22). Suppose [s] is the deviatoric part of a true stress tensor [sigma]. For materials that exhibit the same behavior in uniaxial tension and uniaxial compression, the hydrostatic pressure part of the stress tensor is generally assumed to have no effect on the mechanical behavior of the material. The equivalent stress [sigma] for the stress tensor [sigma] is given as

[sigma] = [square root of (3/2 ([S.sub.ij] . [S.sub.ij])] (4)

However, for materials that exhibit different behaviors in uniaxial tension and uniaxial compression, the effect of hydrostatic pressure has to be included. Using this hydrostatic pressure effect, a generalized DSGZ model is proposed to describe the stress-strain constitutive relationship of polymers under any mode of loading, and is given by

[sigma]([epsilon], [epsilon], T, p) = K{f([epsilon]) + [[epsilon].[e.sup.(1-[epsilon]/[C.sub.3].h([epsilon], T))]/[C.sub.3].h([epsilon], T) - f([epsilon])] . [e.sup.[ln(g([epsilon], T)) - [C.sub.4]].[epsilon]]} . h([epsilon], T) -[gamma]p (5)

where f([epsilon]) and h([epsilon], T) are given by Eqs 2 and 3 by substituting [epsilon] for [epsilon] and [epsilon] for [epsilon] respectively. The equivalent stress [sigma] is defined by Eq 4, the hydrostatic pressure p is defined as 1/3 [[sigma].sub.u] and the equivalent strain [epsilon] is defined as

[epsilon] = [square root of (2/3 ([e.sub.ij].[e.sub.ij]))] (6)

where [e] is the deviatoric part of a true strain tensor [[epsilon]]. The equivalent strain rate [epsilon] is defined as the derivative of the equivalent strain [epsilon] with respect to time t,

[epsilon] = d[epsilon]/dt (7)

and [gamma], called the hydrostatic pressure sensitivity, is a material coefficient accounting for the effect of loading mode. The other eight material coefficients in the generalized DSGZ model are K ([Pa.s.sup.m]), [C.sub.1], [C.sub.2], [C.sub.3] ([s.sup.m]), [C.sub.4], a(K), m and [alpha].

In the generalized DSGZ constitutive model for a uniaxial compression test, the compressive stress [[sigma].sub.c] can be written in the form

[[sigma].sub.c]([epsilon],[epsilon],T) = (K/1 - [gamma]/3){f([epsilon])+[[epsilon].[e.sup.(1-[epsilon]/[C.sup.3].h([e psilon], T))]/[C.sup.3].h([epsilon], T)-f([epsilon])].[e.sup.[ln(g([epsilon], T)) - [C.sub.4]].[epsilon]]}.h([epsilon], T) (8)

Note that Eq 1 for a uniaxial compression test can be obtained from Eq 8 by substituting the term K/1 - [gamma]/3 by [K.sub.c], For a uniaxial tensile test, the tensile stress [[sigma].sub.t] can be written as

[[sigma].sub.t]([epsilon],[epsilon],T) = (K/1 + [gamma]/3){f([epsilon])+[[epsilon].[e.sup.(1-[epsilon]/[C.sub.3].h([e psilon],T)))]/[C.sub.3].h([epsilon], T) - f([epsilon])].[e.sup.[ln(g([epsilon],T)) - [C.sub.4].[epsilon]]}.h([epsilon],T) (9)

The value of [gamma] can be calculated by combining Eqs 8 and 9. For a given strain [epsilon]. strain rate [epsilon] and temperature T, [gamma] has the form

[gamma] = 3 [[sigma].sub.c]([epsilon],[epsilon],T) - [[sigma].sub.t]([epsilon],[epsilon],T)/[[sigma].sub.c]([epsilon],[eps ilon],T) + [[sigma].sub.t]([epsilon],[epsilon],T) (10)

From Eq 10 it can be seen that [gamma] is a function of strain [epsilon], strain rate [epsilon] and temperature T over the entire range of deformation. Because of failure of polymers during uniaxial tensile tests at low strains in comparison to the large strains obtained in compression tests, [gamma] cannot be calculated over a large strain range. Therefore, [gamma] is calculated at the yield stress and assumed to be constant in the form

[gamma] = 3 [[sigma].sub.cy] - [[sigma].sub.ty]/[[sigma].sub.cy] + [[sigma].sub.ty] (11)

where [[sigma].sub.cy] is the yield stress in uniaxial compression test and [[sigma].sub.ty] is the yield stress in uniaxial tensile test. The values of the two yield stresses can be obtained experimentally. The other eight material coefficients in the generalized DSGZ model can be deduced from uniaxial compression stress-strain curves following the procedures given in reference (13).

THERMOMECHANICAL COUPLING AND FAILURE CRITERION

Since the mechanical properties of polymers are sensitive to temperature, an accurate estimate of the temperature rise during plastic deformation is important. Arruda et al. (4) did a series of unlaxial compression tests in which the specimen surface temperatures were monitored using an infrared detector to investigate the relationship between strain rate and temperature rise for the glassy polymer polymethyl-methacrylate (PMMA). It was found that the specimen was nearly isothermal up to a true strain of 0.8 at a strain rate of 0.001/s. but significant temperature rise (around 30[degrees]C) were observed up to the same true strain at strain rates of 0.01/s and 0.1/s. The rise of temperature had a dramatic effect on stress-strain curves. Rittel (23) embedded a small thermocouple in polycarbonate specimen disks to record the transient temperature during impact tests with strain rates ranging from 5000/s to 8000/s. Within a time order of [10.sup.-4] s, a true strain of 0.45 was obtained and the recorded temperature incr eased by nearly 25[degrees]C. The temperature rose significantly in the softening region of the corresponding stress-strain curve. Using a fast response infrared radiometer to monitor the surface temperature of epoxy specimens in a Split Hopkinson Pressure Bar impact test. Trojanowski et al. (24) observed that there was an increase of approximately 50[degrees]C. These experimental results indicate that for polymers, the temperature rise is significant during large plastic deformation at high strain rate.

It has been shown that the highest strain rates encountered in multiaxial impact simulations are of the order of [10.sup.2] ~ [10.sup.3/s] (11). The impact process is completed in less than 0.01 s and for such a short time it is reasonable to assume that the deformation process is essentially adiabatic. The governing equation for the increase of temperature [DELTA]T at each increment of plastic strain is

[DELTA]T = [beta] ([[sigma].sup.old] + [[sigma].sup.new]) [DELTA][[epsilon].sup.pt]/2[rho]c (12)

where [rho] is material density. c is heat capacity, [beta] is the fraction of dissipated plastic energy that converts into thermal energy, [[sigma].sup.old] is the equivalent stress at the beginning of an increment, [[sigma].sup.new] is the equivalent stress at the end of the increment and [[DELTA].sup.pl] [epsilon] is the increment of equivalent plastic strain. At each increment of plastic deformation, the local temperature of the plastic deformation zone will increase by an amount governed by Eq 12. The increase in temperature decreases the equivalent stress [sigma] through the generalized DSGZ model given by Eq 5. This gives a framework to account for the thermomechanical coupling during high strain rate plastic deformation. The issue is how to determine the value of [beta] in Eq 12 for each increment of strain. Rittel (23) found that [beta] is dependent on strain and strain rate during plastic deformation of polymers. The behavior is similar to that exhibited by metals (25). Since there is a lack of avai lable experimental data for [beta] vs. strain [epsilon] and strain [epsilon] rate for the two glassy polymers ABS-1 and ABS-2, a constant value of [beta] = 0.5 is used for all the simulations when the effect of thermomechanical coupling is taken into account. Macdougall and Harding (26) used a similar approximation in their numerical modeling of the high strain rate torsion tests on Ti6A14V bars.

Polymer failure in the multiaxial impact test can be understood as the sudden significant reduction of its load-carrying capability. Failure criteria have to be combined together with the stress-strain constitutive model in order to simulate the impact failure. There are a variety of proposed material failure criteria such as maximum tensile stress, maximum shear strain, and maximum strain energy density (27). As the two polymers investigated in this paper have significant ductility, the maximum plastic strain failure criterion is used. A failure indicator [phi] is created and defined as,

[phi] = [SIGMA][DELTA][[epsilon].sup.pl]/[[epsilon].sub.max.sup.pl] (13)

where [[epsilon].sub.max.sup.P1] is a prescribed maximum equivalent plastic strain, and [[DELTA].sup.p1] [epsilon] is the increment of equivalent plastic strain. Furthermore, the simplifying assumption is made that when the sum of the equivalent plastic strain increments at a material point is equal to or greater than the prescribed value of [[epsilon].sub.max.sup.p1] i.e. when [phi] [greater than or equal to] 1, the material point fails and is permanently removed from future calculations.

FINITE ELEMENT ANALYSIS MODEL AND IMPLEMENTATION

In the ASTM D3763 multiaxial impact test, a cylindrical striker with hemispherical end is dropped from a given height onto the center of a clamped polymer disc. The clamp assembly consists of two circular parallel plates with a 76-mm-diameter hole in the center. Sufficient pressure is applied to prevent slippage of the polymer disk from the clamp assembly during impact. The striker, consisting of a 12.7-mm-diameter steel rod with a hemispherical end of the same dimension, is positioned perpendicular to and centered on the clamp hole. During impact, the striker moves down and the polymer disk is totally penetrated. History of load and displacement of the striker are recorded.

Figure 1 shows the FEA model of the multiaxial impact test created using ABAQUS/Explicit. Nine hundred (900) eight-noded linear brick, reduced integration C3D8R elements with a total of 1519 nodes are used to mesh the polymer disk. An analytical rigid surface is used to model the geometry of the steel striker. The rigid surface is associated with a rigid body reference node that defines the mass and the motion of the striker. The boundary conditions are set as fixed support around the outer edge of the disk and the six degrees of freedom of those nodes located on the circular edge are set to be zero. The striker can only move along the vertical axis, all the other five degrees of freedom of the rigid body reference node are set to be zero. The finite-sliding contact model of ABAQUS/ Explicit is used, and the friction coefficient between the striker and the polymer disk is assumed to be a constant value of 0.3. In order to study the effect of the assumed value of the friction coefficient, additional numerical simulations of the impact behavior were performed using [micro] = 0.

An algorithm of elastic prediction-plastic correction is applied to update the stress tensor of each material point where the first predicted stress tensor is based on generalized Hooke's law and inputted values of elastic modulus and Poisson's ratio. The elastic modulus is 1.7 GPa and the Poisson's ratio is 0.35 for both the polymers, and are assumed to be constant. At the end of each increment of strain, the stress tensor [[[sigma]].sup.new] is corrected using the DSGZ model. The corresponding equivalent stress [[sigma].sup.new] is obtained from Eq 4. Equation 12 is then applied to calculate the increase of the local temperature of polymers. The failure indicator [psi] is updated through Eq 13 and when [psi] [greater than or equal to] 1 the material point fails and is permanently removed from future calculations. Based on this framework of calculation, a user material subroutine is developed and applied in the FEA simulations to implement the generalized DSGZ constitutive model, the thermomechanical coupling model and the failure criterion.

RESULTS AND DISCUSSION

The stress-strain curves under uniaxial tension and compression and the multiaxial impact test data for ABS-1 and ABS-2 were obtained from the literature (22). The nine material coefficients in the generalized DSGZ constitutive model for the two glassy polymers ABS-1 and ABS-2 were calibrated and are shown in Table 1. With larger values of parameters [gamma] and [alpha]. ABS-2 is more sensitive to hydrostatic pressure and to temperature than ABS-1. The compressive stress [[sigma].sub.c] in a uniaxial compression test can be predicted by Eq 8 for different strain, strain rate and temperature. Figure 2 shows comparison of the constitutive model predictions with available uniaxial compression test data (T = 296 K, [epsilon] = 0.0006/s) for the two polymers (22). It can be seen that the DSGZ constitutive model accurately predicts the stress-strain behavior of the two polymers over a wide range of strains.

As previously stated, a constant value of 0.5 is used for the [beta] in Eq 12 in the simulations of the multiaxial impact test. In order to estimate the importance of thermomechanical coupling, simulations were performed in which the effect of the thermomechanical coupling is ignored by setting [beta] = 0. To evaluate the effect of hydrostatic pressure on the behavior of the two polymers, simulations were done in which the constitutive model was calibrated using uniaxial compression data with the parameter [gamma] set to zero. Figure 3 shows comparison of the FEA model predictions with experimental load-displacement data for ABS-1. Figure 4 shows comparison of the FEA model predictions with experimental load-displacement data for ABS-2. From Figs. 3 and 4, with initial temperature of T = 296 K, it can be seen that the FEA model predictions, with [beta] = 0.5 and [gamma] appropriately taken from Table 1, agree well with the load-displacement experimental data of the two polymers up to the maximum impact loadi ng (failure). It is also observed that not accounting for [beta] and/or [gamma] in the simulation models tends to lead to an overestimation of the impact load, especially at large displacements.

From Fig. 3 for a given [gamma], it can be seen that the load-displacement curves for [beta] = 0 and [beta] = 0.5 are about the same. As such, it appears to be reasonable to neglect the effect of thermomechanical coupling for ABS-1. Furthermore, even though a maximum temperature rise of 18[degrees]C was obtained in the simulation, the effect on the predicted load and impact energy is less than 5%. However, as shown in Fig. 4 for ABS-2, the lack of thermomechanical coupling ([beta] = 0) accounts for overestimates of around 15% for the maximum impact load and around 10% for the impact energy. This difference in behavior occurs because ABS-2 is more sensitive to temperature and has a higher temperature rise with a larger plastic deformation and higher stress. Therefore, for ABS-2 the thermomechanical coupling appears to be an important factor for the mechanical behavior at large plastic deformations.

For ABS-1 with [gamma] = 0.25 (Fig. 3), the effect of hydrostatic pressure is not important. For ABS-2 with [gamma] = 0.58 (Fig. 4), the effect of hydrostatic pressure is very important. If this factor is ignored, the impact load is largely overestimated in the region of plastic deformation. These simulation results show that for materials that exhibit different behaviors in uniaxial tension and uniaxial compression, the effect of hydrostatic pressure must be included. If the difference is large, neglecting the hydrostatic pressure effect will lead to a much greater error for the simulations of the multiaxial impact tests. Uniaxial compression or tensile tests taken individually are not representative of the general three-dimensional deformation behavior of polymers. In the application of the generalized DSGZ constitutive model, it is important to experimentally calibrate the parameter [gamma].

Figure 5 shows the simulated equivalent stress distributions on the bottom surface of the ABS-1 disk. In this simulation, as previously mentioned, the friction coefficient between the striker and the polymer disk is set to a constant value of 0.3. Even though the material coefficients in Table 1 are calculated from low strain rates ([10.sup.-4]/s) test data, the calibrated constitutive model can correctly extrapolate to predict the deformation behavior of the two glassy polymers at high strain rates. This has been shown to be true for other polymers as well, such as PMMA, PC and polyamide 12 (13). From the simulation results, it is found that most regions of the disk undergo elastic deformation. The plastic deformation and temperature rise is localized in the impact region, with the maximum values in the center of the disk. The maximum equivalent stress is located in a circular zone around the striker.

Figure 6 shows the effect of friction between the striker and polymer disk on the load-displacement behavior of ABS-2 disk. It can be seen that if one neglects friction, the predicted impact loads are significantly lower at large displacements. Similar experimental observations have been reported for the friction effect in the literature (8).

CONCLUSIONS

A generalized form of the DSGZ stress-strain constitutive model is proposed to describe the stress-strain constitutive relationship of polymers under any mode of loading. The thermomechanical coupling during high strain rate plastic deformation and the failure criteria for polymers are discussed. An FEA model is created using ABAQUS/Explicit to simulate the standard ASTM D3763 multiaxial impact test and a user material subroutine is developed and applied to implement the constitutive model, thermomechanical coupling model and failure criterion in the FEA simulation.

Impact tests on two glassy polymers ABS-1 and ABS-2 were simulated. The predicted load-displacement curves were compared with experimental data. They agree well up to the maximum impact load. The results indicate that the generalized DSGZ constitutive model accurately predicts the stress-strain behavior of the two polymers over a wide range of strains and it correctly extrapolates over a large range of strain rates.

For polymers, the uniaxial compression or tensile tests are generally not representative of the general three-dimensional deformation behavior. The deformation mode should be considered. In the application of the generalized DSGZ constitutive model, it is important to experimentally calibrate the parameter y. The thermomechanical coupling and friction are also important factors. It is shown that not accounting for the different behavior of the polymer in uniaxial tensile and compression tests and thermomechanical coupling effects, tends to overstimate the load and impact energy especially at large displacements and plastic deformations. If one neglects the friction between the striker and polymer disk, the predicted impact loads are significantly lower at large displacements.

ACKNOWLEDGMENT

The support of Lucent Technologies during this research is gratefully acknowledged.

[Figure 2 omitted]

[Figure 3 omitted]

[Figure 4 omitted]

[Figure 6 omitted]

Table 1. Material Coefficients for ABS-1 and ABS-2. Polymer [C.sub.1] [C.sub.2] m a(K) K ([MPa.[s.sup.m]) ABS-1 -0.8 -0.83 0.07 900 0.95 ABS-2 1.64 1.25 0.07 1200 1.39 Polymer [C.sub.3] ([s.sup.m]) [C.sub.4] [alpha] [gamma] ABS-1 0.0028 7 100 0.25 ABS-2 0.002 13 25 0.58

(*.) Corresponding author: Anil Saigal, Professor: Department of Mechanical Engineering: Tufts University: Medford. MA 02155: Email:anil.saigal@tufts.edu

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Author: | Duan, Y.; Saigal, A.; Greif, R.; Zimmerman, M.A. |
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Publication: | Polymer Engineering and Science |

Date: | Feb 1, 2002 |

Words: | 4776 |

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