An improved correlation between impression and uniaxial creep

JOURNAL OF APPLIED PHYSICS 99, 113513 共2006兲

An improved correlation between impression and uniaxial creep Chun-Hway Hsueha兲 Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6068

Pedro Miranda Departamento Electrónica e Ingeniería Electromecánica, Escuela de Ingenierías Industriales, Universidad de Extremadura, 06071 Badajoz, Spain

Paul F. Becher Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6068

共Received 20 February 2006; accepted 3 April 2006; published online 5 June 2006兲 A semiempirical correlation between impression and uniaxial creep has been established by Hyde et al. 关Int. J. Mech. Sci. 35, 451 共1993兲兴 using finite element results for materials exhibiting general power-law creep with the stress exponent n in the range 2 艋n艋 15. Here, we derive the closed-form solution for a special case of viscoelastic materials, i.e., n = 1, subjected to impression creep and obtain the exact correlation between impression and uniaxial creep. This analytical solution serves as a checkpoint for the finite element results. We then perform finite element analyses for the general case to derive a semiempirical correlation, which agrees well with both analytical viscoelastic results and the existing experimental data. Our improved correlation agrees with the correlation of Hyde et al. for n 艌 4, and the difference increases with decreasing n for n ⬍ 4. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2200727兴 I. INTRODUCTION

Conventional uniaxial creep tests require many specimens to collect a complete creep data set. It is also difficult to perform tests on nanocrystalline materials where typically only small specimens are available or on materials with complex microstructures 共e.g., superalloys and composites兲 where results for the different microstructural components are desired. Hence, it is logical to establish an alternative test for use with small specimens and being able to perform multiple tests on a single specimen. The impression creep test1–9 was introduced by Chu and Li,1 and a comprehensive review of this test was published recently by Li.10 It is a modified indentation test with a flat-ended cylindrical indenter 共i.e., a punch兲 used to characterize the deformation behavior of materials under a constant load and at elevated temperatures. Compared to the conventional uniaxial creep test, the advantages of the impression creep test are the following: 共i兲 creep data can be obtained for small volume of materials, 共ii兲 specimen preparation is relatively easy, 共iii兲 several tests can be performed on the same specimen, 共iv兲 the small specimen size makes environmental control easier, and 共v兲 it is possible to develop portable test equipment. While the uniaxial creep test can be either tensile or compressive, the impression creep test is compressive. To be compatible with impression creep, uniaxial compressive creep is considered in the present study. By comparing the displacement versus time data between the impression creep test and the uniaxial 共compressive兲 creep test under a constant-load condition, the following has been concluded. 共i兲 The displacement versus time curves are similar between these two tests, having pria兲

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mary 共decreasing creep rate兲 and secondary 共steady-state creep rate兲 regions. 共ii兲 The steady-state indenter displacement rate has the same stress exponent and temperature dependences as the uniaxial creep test. 共iii兲 The steady-state indenter displacement rate is proportional to the diameter of the indenter. However, because of the difference in the loading fixtures used in these two tests, the constitutive equation describing uniaxial creep cannot be applied directly to impression creep, i.e., both the strain rate and the stress have different interpretations between these two tests. In order to characterize the creep behavior using impression creep tests, it is essential to establish correlations between the data obtained from impression creep tests and those from uniaxial creep tests. Using the finite element analysis 共FEA兲, a semiempirical correlation was established earlier for materials obeying power-law creep.11 However, we found that this existing correlation is not sufficiently accurate for materials with low stress exponent dependence of creep rates. Hence, the purpose of the present study is to derive accurate semiempirical equations using the FEA to correlate impression creep to uniaxial creep. First, the existing work is summarized, and the reason for questioning its accuracy is stated. Second, an analytical model is developed to derive the exact closed-form solutions for the special case of linear viscoelastic materials subjected to impression creep tests. This serves as a checkpoint for the FEA results. Then, the FEA is used to simulate impression creep tests of materials that obey general power-law creep. Finally, the analytical solution is compared to the FEA result for viscoelastic materials, and the FEA results for the general case are used to derive the semiempirical correlations between impression

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© 2006 American Institute of Physics

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and uniaxial creep tests. The present semiempirical correlations are shown to agree well with the analytical viscoelastic results and the existing experimental data. II. EXISTING SEMIEMPIRICAL CORRELATION AND QUESTIONING

When materials are subjected to a constant uniaxial load at elevated temperatures, they typically exhibit power-law creep, and the steady-state creep rate ␧˙ can be related to the uniaxial stress ␴ by ␧˙ = A␴n ,

共1兲

where A is a proportional constant and n is the stress exponent. For impression creep, the steady-state indenter dis˙ has been found to have the same stress placement rate w exponent dependence and to be proportional to the diameter of the punch indenter d, i.e., ˙ w ⬀ ␴ni , d

共2兲

where ␴i is the average stress imposed by the indenter. Researchers have resorted to trial-and-error methods to correlate ␴i and w˙ / d in impression creep to ␴ and ␧˙ in uniaxial creep. The general forms of these correlations are

␴i = ␣␴ ,

共3a兲

˙ w = ␤␧˙ , d

共3b兲

where ␣ and ␤ are the correlation factors and are generally determined by data fitting; however, multiple pairs of ␣ and ␤ can give similar fittings. In order to find a unique pair of ␣ and ␤, Hyde et al. introduced the reference stress method,11 namely, to determine the correlation factors by requiring that both ␣ and ␤ are constants and independent of material properties. To achieve this, Hyde et al. performed a FEA for impression creep of materials that obeyed power-law creep using the constitutive equation for uniaxial creep as the input. Stress exponent values n ranging from 2 to 15 were simulated. By comparing the steady-state impression creep rates obtained from the FEA with the uniaxial creep rates for various n values, the following semiempirical correlation factors were obtained: ␣ = 3.38 and ␤ = 0.755.11 It should be noted that in order to obtain the empirical values for ␣ and ␤ from experiments, experimental data from both uniaxial and impression creep tests are required. If the reference stress method of Hyde et al. were valid, the impression creep data could be converted directly to uniaxial creep results by using the semiempirical values of ␣ and ␤, and the need of performing uniaxial creep tests is avoided. To examine the accuracy of the reference stress method of Hyde et al., we check the semiempirical correlation factors of Hyde et al. against the existing analytical solution for impression creep of viscous materials,12,13 as shown in the following. Although multiple pairs of ␣ and ␤ can give similar fittings, a relation exists between ␣ and ␤ for each of those pairs. Substitution of Eqs. 共3a兲 and 共3b兲 into Eq. 共1兲 yields

冉冊

w˙ ␴i =A ␤d ␣

n

.

共4兲

Comparison between Eqs. 共2兲 and 共4兲 shows that the different pairs of ␣ and ␤ will give similar fitting as long as the following relation is satisfied:

␣n ⬇ const. ␤

共5兲

The above condition, Eq. 共5兲, can be used to examine the accuracy of the reference stress method of Hyde et al. by comparing ␣n / ␤ obtained from the semiempirical correlation factors of Hyde et al. and the existing analytical solution for impression creep of viscous materials. For viscous materials, the strain rate-stress relation during uniaxial creep is ␧˙ =

␴ , ␮

共6兲

where ␮ is the uniaxial viscosity, and the stress exponent n = 1. For impression creep, the displacement rate-stress relation has been derived, such that12,13 w˙ ␲␴i = , d 16␩

共7兲

where ␩ is the shear viscosity, which is one-third of the uniaxial viscosity for viscous materials, i.e.,

␩ = ␮/3.

共8兲

With n = 1, combination of Eqs. 共5兲–共8兲 yields

␣ 16 = ⬇ 1.7. ␤ 3␲

共9兲

Equation 共9兲 is an exact solution for viscous materials, i.e., n = 1. On the other hand, ␣n / ␤ obtained from the correlation factors of Hyde et al. for n = 1 is 4.48, which is about 2.6 times of the value given by Eq. 共9兲. Hence, we conclude that the correlation factors of Hyde et al. are inaccurate at least for the case of n = 1. In order to accurately correlate the data obtained from impression creep tests to those for uniaxial creep tests, either a modification of the reference stress method of Hyde et al. is required or another approach must be developed. III. ANALYSES

Both analytical modeling for the special case of viscoelastic materials and FEA for general cases are performed to analyze impression creep. While the result obtained from analytical modeling serves as a checkpoint for FEA, the results obtained from the FEA are used to obtain the accurate correlation factors. A. Analytical modeling for a special case: Viscoelastic materials

The constitutive equation describing the stress-strain relation for a viscoelastic material is a time-dependent differential equation whose Laplace transform with respect to time

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t has been demonstrated to be of analogous form to the governing field equation of elasticity.14,15 Hence, knowing the solutions in an elastic system, the Laplace transform of the solutions in a corresponding viscoelastic system can be formulated. Then, with the inverse Laplace transform, the viscoelastic solutions can be obtained. For elastic materials, the displacement-stress relation for impression loading is16 w=

共1 − ␯2兲␲d␴i , 4E

共10兲

where E and ␯ are Young’s modulus and Poisson’s ratio, respectively. Adopting the usual assumptions that the viscoelastic behavior can be described by a Maxwell model, it has been shown that the elastic constants have the following transforms in formulating the Laplace transform of the displacement-stress relation in a corresponding viscoelastic system.14,17,18 E→

9KG␩s , 共3K + G兲␩s + 3KG

共11b兲

where K and G are bulk modulus and shear modulus, respectively, and s is the transform parameter 共when the viscoelastic equation is transformed from the t domain to the s domain兲. Using elastic-viscoelastic analogy, the Laplace ¯ 共s兲, becomes transform of the viscoelastic displacement w

␲d关共3K + 4G兲␩2s2 + 2G共3K + 2G兲␩s + 3KG2兴¯␴i , 16G关共3K + G兲␩s + 3KG兴␩s 共12兲

where ¯␴i is the Laplace transform of ␴i. Inverse Laplace transform of Eq. 共12兲 gives the viscoelastic displacement w共t兲 such that w共t兲 =

再

␲d 共3K + 4G兲␴i 1 + 16 G共3K + G兲 ␩ 3G2 + 共3K + G兲2␩

冕

t

冕

␴idu

0

冋

再

− =

冋

− 3KGt G exp K共3K + G兲 共3K + G兲␩

册

冋

册冎

共15兲

Hence, the steady-state creep rate 共i.e., for t → ⬁兲 of viscoelastic materials is the same as the creep rate of viscous materials, Eq. 共7兲, and it is independent of the elastic constants. Also, when E → ⬁, viscoelastic materials become purely viscous and Eq. 共15兲 is reduced to Eq. 共7兲. Both Eqs. 共14兲 and 共15兲 serve as checkpoints for FEA results. For constant impression loading rate 共i.e., ␴˙ i is a constant兲 and the initial condition of ␴i = 0 at t = 0, the solution of Eq. 共13兲 becomes w共t兲 =




5 − 4␯ 3共1 − 2␯兲2␩ ␲d␴i t − + 16 2␩ E E 2t

冋 冉 冊册

⫻ 1 − exp

− Et 3␩

,

共16兲

where ␴i is linearly proportional to t. It can be obtained from Eq. 共16兲 that w = 0 at t = 0, and Eq. 共16兲 is reduced to the elastic solution when ␩ → ⬁. If the test is under displacement control 共e.g., constant displacement rate兲, Eq. 共12兲 should be reformulated by expressing the impression load as a function of the displacement before taking the inverse Laplace transform. B. Finite element analyses

Materials generally obey power-law creep behavior, and the stress-strain relation for uniaxial loading can be described by ␧˙ =

␴˙ + A␴n . E

冎

␧˙ ij =

− 3KG共t − u兲 ␴i exp du . 共3K + G兲␩ 0

␲d␴i t K + G + 16 ␩ KG

冉 冊




共1 − 2␯兲2 − Et ␲d␴i exp 1+ . 3 16␩ 3␩

共17兲

The tensor form of the above constitutive relation is

t

共13兲

Equation 共13兲 is a general equation describing the timedependent viscoelastic displacement when the test is under load control. Depending upon the loading condition, Eq. 共13兲 can be solved accordingly. For constant impression loading, i.e., ␴i is a constant, the solution is w共t兲 =

w˙共t兲 =

共11a兲

共3K − 2G兲␩s + 3KG , ␯→ 共6K + 2G兲␩s + 6KG

¯ 共s兲 = w

At t = 0 or when ␩ → ⬁ Eq. 共14兲 is reduced to the elastic solution, i.e., Eq. 共10兲. Differentiation of Eq. 共14兲 with respect to time yields

冉 冊册

− Et ␲d␴i t 5 − 4␯ 共1 − 2␯兲2 − exp + 16 ␩ E E 3␩

. 共14兲

1+␯˙ 1 − 2␯ 3A Sij + Sij␴en−1 , ␴˙ kk␦ij + E 3E 2

共18兲

where ␦ij is the Kronecker delta, and the deviatoric stress Sij and von Mises stress ␴e are 1 Sij = ␴ij − ␴kk␦ij , 3

␴e =

冑

3 SijSij . 2

共19a兲

共19b兲

The stress-strain relation described by Eq. 共18兲 is used as the input for the FEA. The impression creep test is simulated using ABAQUS. The algorithm models frictionless contact between a rigid cylindrical punch and a cylindrical specimen. A 1 mm diameter punch is considered, and a fillet with a 2 ␮m radius is assumed at the corner of the punch in order to avoid the singularity problems. It has been shown that a fillet radius of less than 5% of the punch indenter has negligible effects on

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FIG. 2. Comparison between analytical solution, Eq. 共10兲, and FEA results for the elastic displacement of materials subjected to impression loading. The difference between the FEA and the analytical solutions, 关w共FEA兲 − w共analytical兲兴 / w共analytical兲, is also shown.

FIG. 1. The finite element mesh around the edge of the contact region for a material subjected to impression creep tests. The fillet with a 2 ␮m radius at the corner of the rigid cylindrical punch is shown.

the simulated displacement.3 A constant load P ranging from 1 to 5 N is applied on the punch. The specimen size effect has been studied, and the effect is less than 5% if the specimen thickness is five times the indenter diameter.10 Sufficiently large dimensions 共320 mm diameter and 160 mm height兲 of the specimen are used in present simulations to ensure that the simulated results are independent of the dimensions. The mesh is refined in the high stress concentration region around the contact area, and it consists of 28 369 four-node linear axisymmetric quadrilateral elements with reduced integration, accounting for a total of 29 922 nodes. A detail of the mesh around the edge of the contact region is shown in Fig. 1. First, the accuracy of FEA is examined by simulating the elastic and the viscoelastic deformations under a constant impression load, and the results are checked against the analytical solutions. Then, the cases for n = 2 – 8 are simulated to obtain the steady-state creep rates at different loads. This steady-state creep rate is normalized by the proportional constant A as well as the punch diameter d, and it is then related to the average loading stress ␴i 共=4P / ␲d2兲 by ˙ w = C ␴ in , Ad

共20兲

where C is a proportional constant. Combining Eqs. 共4兲 and 共20兲, it can be obtained that C corresponds to ␤ / ␣n. Hence, the relation between ␣n / ␤ and the stress exponent n can be obtained from FEA by using different values of n in Eqs. 共17兲 and 共18兲 as the input for simulations. Then, curve fitting is used to obtain ␣n / ␤ as a function of n. It should be noted that ␣n / ␤ is independent of other material properties. Also,

knowing the n dependence of ␣n / ␤ 共or of C兲 is sufficient for correlating the impression creep data to the uniaxial creep data, and the determination of ␣ and ␤ is not required. However, in order to compare with the correlation factors of Hyde et al., both ␣ and ␤ are determined from the curve fitting routine. IV. RESULTS

First, the FEA results are compared with the closed-form solutions for the elastic case and n = 1 to ensure the accuracy of the simulated results. Then, the FEA results for n = 1 – 8 are used to derive the correlation factors between impression and uniaxial creep tests. The material properties pertinent to polymer materials are considered, such that E = 6 GPa, ␯ = 0.38, and ␮ = 100 GPa s are used in simulations. This uniaxial viscosity datum corresponds to A = 10−5 MPa−1 s−1 for n = 1 in Eqs. 共17兲 and 共18兲. The proportional constant A influences the creep rate and the simulation time in reaching the steady-state creep rate; however, it does not influence C 共and hence ␣n / ␤兲 because the displacement rate is normalized by A when C is determined from Eq. 共20兲. In order to reach the steady-state creep rate at similar simulation times for different n values, a value of A = 10−5 / 2共n−1兲 MPa−n s−1 is selected in present simulations. A. Comparison between FEA and analytical modeling for elastic case and n = 1

For elastic materials, the elastic displacement versus impression load is plotted in Fig. 2. Both the FEA and the analytical 关Eq. 共10兲兴 results are shown, and their difference 关w共FEA兲 − w共analytical兲兴 / w共analytical兲 is less than 0.23%. For n = 1, i.e., viscoelastic materials, the displacement and the displacement rate are plotted in Figs. 3共a兲 and 3共b兲, respectively, as functions of time at different loads. Both the FEA and the analytical results, Eqs. 共14兲 and 共15兲, are shown. At t = 0, the displacement corresponds to the elastic displacement. However, because the elastic displacement is much less than the creep displacement for the time scale shown in Fig. 3共a兲, it can hardly be seen in Fig. 3共a兲. A good agreement between the FEA and the analytical results is ob-

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˙ / Ad FIG. 4. FEA results for the normalized steady-state displacement rate w as a function of the average impression stress ␴i for n = 1. The best fit of the data to Eq. 共20兲 is shown as a solid line.

plying this methodology is plotted as a function of the average impression stress ␴i for n = 1 in Fig. 4. Curve fitting of Eq. 共20兲 with n = 1 to the FEA data in Fig. 4 shows an excellent agreement for C = 0.589 05. This C value corresponds to ␣ / ␤ ⬇ 1.7 that agrees with the analytical solution, Eq. 共9兲. This validates the accuracy of the present simulated results at least for n = 1.

FIG. 3. Comparison between analytical solutions, Eqs. 共14兲 and 共15兲, and ˙ as FEA results for 共a兲 the displacement w and 共b兲 the displacement rate w functions of time at different loads for viscoelastic materials 共i.e., n = 1兲 subjected to impression creep tests.

served in Fig. 3共a兲 although a slight deviation of the FEA results from the analytical solution is observed at the longer times. These differences become more obvious as the load increases and can be more readily observed in Fig. 3共b兲 for the displacement rate. Specifically, while Eq. 共15兲 shows a true steady-state displacement rate when t is sufficiently large, the simulated results show that the displacement rate keeps decreasing with time. This continuously decreasing displacement rate in the simulated results has also been reported elsewhere.11,19 In order to obtain the steady state, it was suggested that the steady state could be deduced by polynomial extrapolation of the displacement rate versus re˙ − 1 / t, data to infinite time.19 However, ciprocal time, i.e., w we find that the result obtained by this method depends considerably on the point at which the simulation is terminated. In any case, the continuous decrease in the simulated creep rate at high displacements is likely an artifact of the excessive mesh distortion in simulations and should be disregarded. Based on the comparison in Figs. 3共a兲 and 3共b兲, the simulated displacement rates are identical to the analytical steady-state displacement rates when the creep displacement is about 50 ␮m 共i.e., 5% of the punch diameter兲 and beyond that point the curve starts to deviate. Hence, in the present work, the displacement rate at the creep displacement of 50 ␮m is considered as the steady-state creep rate. The normalized steady-state displacement rate w˙ / Ad obtained by ap-

B. FEA results for n = 2 – 8

The FEA is then conducted for the cases where n is an integer of 2–8. The steady-state creep rate is obtained at P = 1, 2, 3, 4, and 5 N, respectively, for each stress exponent. When n = 1, primary creep 共i.e., the initial decreasing creep rate兲 is hardly noticeable in Fig. 3共b兲. This primary creep becomes more obvious as n increases, and an example of the ˙ / Ad as a function of time is normalized displacement rate w shown in Fig. 5 for P = 3 N and n = 2, 4, 6, and 8. The normalized steady-state displacement rate is plotted as a function of the average impression stress ␴i for n = 2 – 8 in Fig. 6. The data are then fitted by Eq. 共20兲 for each stress exponent n. Both the fitting curves and the fitting parameter, C, are also shown in Fig. 6, and excellent agreement is observed.

˙ / Ad as a funcFIG. 5. FEA results for the normalized displacement rate w tion of time t for P = 3 N and n = 2, 4, 6, and 8. Primary creep, i.e., the initial decreasing creep rate, becomes more obvious with increasing n.

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J. Appl. Phys. 99, 113513 共2006兲

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FIG. 6. FEA results 共symbols兲 for the normalized steady-state displacement ˙ / Ad as a function of the average impression stress ␴i for n = 2 – 8. The rate w best fits of the data to Eq. 共20兲 are shown as solid lines.

This reaffirms that the steady-state indenter displacement rate has the same stress exponent as uniaxial creep. With the C value obtained for each stress exponent, ␣n / ␤共=1 / C兲 is then plotted as a function of n in Fig. 7. The semiempirical values of ␣n / ␤ based on the reference stress method of Hyde et al. are also included as a dashed line in Fig. 7 for comparison. It can be seen that good agreement in ␣n / ␤ is obtained between the present FEA and the FEA of Hyde et al. when n ⬎ 4; however, the results exhibit an increasing deviation as n decreases below 4. This deviation for small values of n may result from the fact that the FEA was performed by Hyde et al. only for n 艌 2. Chu and Li1 performed both uniaxial and impression creep tests on succinonitrile at 37 ° C to obtain steady-state creep rates. The stress exponent of 4 was obtained, and their experimental results

were fitted by using ␣ = 3.3 and ␤ = 1. Impression creep tests have also been performed on ␤-tin single crystals in different orientations and at different temperatures.20 By comparing to uniaxial creep data from Weertman and Breen21 and Breen22 assuming ␤ = 1, the following results were obtained:20 n = 5, ␣ = 3.5 at 60 ° C and n = 4.5, ␣ = 3.9 at 203 ° C for 关110兴 orientation; n = 4.4, ␣ = 2.8 at 40 ° C and n = 4.3, ␣ = 2.8 at 60 ° C for 关001兴 orientation. Dorner et al.7 performed impression creep tests on TiAl alloy at 750 ° C and compared their results to those obtained from uniaxial creep tests. The stress exponent was found to be ⬃7.5 and the correlation factors of Hyde et al. could be used for fitting. Based on the above empirical values of ␣ and ␤, the corresponding values for ␣n / ␤ are also included in Fig. 7 for comparison, and good agreement is obtained between the semiempirical and existing experiment values. Hence, although the present FEA result is validated by the closed-form solutions only for n = 1, the agreement with existing FEA and experimental results for n 艌 4 共Fig. 7兲 makes us believe that we have determined accurately ␣n / ␤ for the range of stress exponent considered in the present study. To determine ␣ and ␤ from the simulated ␣n / ␤ vs n data, our methodology of curve fitting is described as follows. The semilogarithm plot of the ␣n / ␤ − n relation shows nearly a straight line with a slight deviation from linearity for low n values. While constant ␣ and ␤ yield an exact straight line in the semilogarithm plot of ␣n / ␤ vs n 共e.g., the results of Hyde et al.兲, the slight deviation from linearity for low n values can be accounted for by adding a 1 / n or exp共−n兲 term to a constant ␤. For simplicity, the 1 / n term is adopted in the present study and the fitting equation has the following form:

␣n an = . ␤ b + c/n

共21兲

In this case, the slope of the semilogarithm plot of the ␣n / ␤ − n relation at large n values is described by log共a兲. With curve fitting, the parameter values a = 3.33± 0.04, b = 0.43± 0.06, and c = 1.51± 0.07 yield an excellent agreement, as shown in Fig. 7 by the solid line. Hence, it is straightforward to obtain

␣ = 3.33, ␤ = 0.43 +

共22a兲 1.51 . n

共22b兲

The difference between the result of Hyde et al. and our result, 关␣n / ␤共Hyde兲 − ␣n / ␤共Eq. 21兲兴 / ␣n / ␤共Eq. 21兲, is also shown in Fig. 7, and it is ⬃160% difference at n = 1. V. CONCLUSIONS

FIG. 7. Plot of ␣n / ␤ as a function of n showing the comparison between present FEA results 共solid circles兲, semiempirical values of Hyde et al. 共long dashed line兲, and existing experimental data 共symbols兲. The best fit of present FEA results to Eq. 共21兲 is shown as a solid line, and the difference between the result of Hyde et al. and our result, 关␣n / ␤共Hyde兲 − ␣n / ␤共Eq. 21兲兴 / ␣n / ␤共Eq. 21兲, is shown using the right y axis 共short dashed line兲.

The impression creep test is attractive because it enables both creep data to be obtained from small specimens and several tests to be performed on the same specimen. Also, its steady-state displacement rate has the same stress exponent and temperature dependences as the conventional uniaxial creep test. In order to correlate the impression creep data to the uniaxial creep data, the FEA has been performed to establish a semiempirical correlation for materials obeying

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J. Appl. Phys. 99, 113513 共2006兲

Hsueh, Miranda, and Becher

power-law creep. However, using the existing semiempirical correlation,11 the predicted uniaxial creep rate from the measured impression displacement rate is about 2.6 times of its theoretical value for viscous materials 共i.e., stress exponent n = 1兲. This discrepancy could result from the fact that the existing semiempirical correlation was based on FEA for n 艌 2. An accurate correlation is derived in the present study by 共i兲 deriving the closed-form solution for a special case of viscoelastic materials 共i.e., n = 1兲 subjected to impression creep tests and 共ii兲 simulating impression creep tests for materials obeying power-law creep with n 艌 1. While the result obtained from analytical modeling serves as a checkpoint for FEA, the results obtained from the FEA are used to obtain the accurate correlation. ACKNOWLEDGMENTS

The authors thank Dr. M. K. Ferber and Dr. A. A. Wereszczak for useful comments. This research was jointly sponsored by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory to one of the authors 共C.H.H.兲, Ministerio de Ciencia y Tecnología, Spain and the Fondo Social Europeo 共Grant No. MAT200305584兲 to another author 共P.M.兲, and U.S. Department of Energy, Division of Materials Sciences and Engineering, Office of Basic Energy to another author 共P.F.B.兲 under Contract No. DE-AC05-00OR22725 with UT-Battelle, LLC.

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