Nanoindentation of polymers with a sharp indenter

Nanoindentation of polymers with a sharp indenter C.Y. Zhang and Y.W. Zhanga) Department of Materials Science and Engineering, National University of Singapore, Singapore 119260, Singapore

K.Y. Zeng Department of Mechanical Engineering, National University of Singapore, Singapore 119260, Singapore

L. Shen Institute of Materials Research and Engineering, Singapore 117602, Singapore (Received 27 January 2005; accepted 25 March 2005)

A five-step indentation scheme is proposed to extract the elastic and viscoelastic properties of polymeric materials using a sharp indenter. In the formulation, analytical solutions to the elastic-viscoelastic deformation based on the concept of “effective indenters” proposed by both Pharr and Bolshakov [Understanding nanoindentation unloading curves. J. Mater. Res. 17, 2660 (2002)] and Sakai [Elastic recovery in the unloading process of pyramidal microindentation. J. Mater. Res. 18, 1631 (2003)] were derived. Indentation experiments on polymethylmethacrylate following the five-step scheme were performed. The elastic-viscoelastic parameters were extracted by fitting the solution based on Sakai’s effective indenter to the experimental results using a genetic algorithm. It was found that the solution based on Sakai’s effective indenter was able to correctly extract the elastic properties. Based on this prediction and the experimental results, Pharr and Bolshakov’s effective indenter profile could be determined. The extracted elastic-viscoelastic parameters using the solution based on Pharr and Bolshakov’s effective indenter were independent of the reloading levels. I. INTRODUCTION

The nanoindentation technique has drawn much interest recently for both its efficiency and versatility in measuring the mechanical properties of small volumes of materials and thin films. However, the application of nanoindentation to polymeric materials is still a challenging issue. Since the unloading curve of polymers depends not only on the holding time but also on the unloading rate, the widely used Oliver & Pharr method1 is greatly limited. It is well known that when the unloading rate is low, the initial slope of the unloading curve, which is defined as the contact stiffness, may even become negative, causing the so-called “nose” effect. It is believed that the viscoelasticity and/or viscoplasticity of polymeric materials are responsible for this phenomenon. Several methods have been proposed to study the indentation of polymers. Cheng et al.2 derived an analytical solution to the indentation on polymers with a flatended punch based on the standard viscoelastic solid model. This model is able to predict short-time creep behavior of polymers. Recently, Zhang et al.3 derived a

a)

Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/JMR.2005.0200 J. Mater. Res., Vol. 20, No. 6, Jun 2005

semi-analytical solution to the indentation creep of a polymeric film/substrate system with a flat-ended punch based on a more complicated generalized Kelvin model. This model is able to predict relatively long-time creep behavior of polymers. For these flat-ended punch indentation studies, the instantaneous plastic deformation of polymers is usually assumed to be small and thus neglected. Unlike the flat-ended punch indentation in which instantaneous plastic deformation is negligible, the indentation with a sharp conical/pyramidal indenter often gives rise to large permanent deformation. Rikards et al.4 proposed a numerical method based on finite element simulation to determine the plastic properties of polymers with a Vickers indenter. Assuming that the total indentation depth is decomposed into the elastic part, the plastic part, and the viscoelastic part, and further assuming that the viscoelastic part is negligible, they designed a refined experimental method to identify the yield stress and the tangent modulus through an optimization process with Young’s modulus and Poisson’s ratio as inputs. Shimizu et al.5 applied the hereditary integral to derive the analytical viscoelastic solution to the indentation using a conical/pyramidal indenter. Since they did not explicitly consider plastic deformation, a geometrical factor was used to alleviate the influence of plasticity. Using a © 2005 Materials Research Society

1597

C.Y. Zhang et al.: Nanoindentation of polymers with a sharp indenter

first-order approximation, Bucaille et al.6 applied a model proposed by G’Sell and Jonas7 to describe the mechanical behavior of polymers. Since the G’Sell– Jonas model was unable to capture the viscoelasticity of polymers accurately, this method cannot be extended to model general indentation tests of polymers where the viscoelastic deformation is predominant. To describe a full range of viscous-elastic-plastic response of a polymer during the indentation test, Oyen and Cook8 constructed a one-dimensional model in directly analogous to the Maxwell model (i.e., a Newtonian dashpot in series with a Hookean spring) from a series of mechanical elements, such as Newtonian dashpots, Hookean springs, and plastic sliders. The model could fit the loading– unloading curve very well. However, the extracted modulus was much higher than literature values. In a subsequent paper,9 they extended the model to study the indentation response of time-dependent films on stiff substrates by directly incorporating the effect of the substrate-induced stiffness. The extracted Young’s modulus of the films agreed very well with the bulk counterpart obtained by compressive tests. Very recently, Ovaert et al. 10 adopted a threedimensional visco-elastic-plastic constitutive model11 to describe the mechanical response of polymers and applied an iterative algorithm based on finite element analysis to extract the mechanical parameters. The material model was based on the assumption of the total stress decomposition rather than the commonly accepted total strain decomposition. Lu et al.12 derived the analytical solutions to the indentations of solid polymers with both Berkovich and spherical indenters. A complete recovery of indentation impression after unloading was necessary so that the deformation of materials was in the regime of linear viscoelasticity. By scaling and dimensional analysis, Cheng and Cheng13 also derived an analytical solution to the indentation of linear viscoelastic material with a conical indenter. This paper focuses on extracting the elastic and viscoelastic properties of polymers using nanoindentation with sharp conical/pyramidal indenters. It is assumed that the total indentation deformation can be decomposed into a plastic component and an elastic-viscoelastic component. Thus a five-step test scheme is designed to separate the two components so that they can be studied separately. Nanoindentation experiments on polymethylmethacrylate (PMMA) using the five-step test scheme have been performed. For the elastic-viscoelastic part, analytical solutions based on the concept of “effective indenters” proposed by Pharr and Bolshakov14 and Sakai15 are derived using the “correspondence principle” between elasticity and viscoelasticity. Fitted with the experimental results, the two solutions are combined to determine the elastic and viscoelastic parameters of PMMA. 1598

II. THEORY AND EXPERIMENT A. Separation of the viscoelastic deformation

It is known that the elastic deformation and the plastic deformation are time-independent, whereas the viscoelastic deformation and/or the viscoplastic deformation are time-dependent. This suggests that the timeindependent part and the time-dependent part should be studied separately.16,17 It is known that if an indentation with a sharp indenter is carried out in a sufficiently short period of time, the deformation of the time-dependent part is negligible during the short-period test. Besides, considering the strain-hardening, i.e., an increasing stress is required to produce further plastic deformation after the material is strained beyond the yielding point, it is possible to separate the plastic deformation and the viscoelastic deformation and study them individually. Here a new indentation scheme is designed which consists of five steps (Fig. 1): (1) a fast loading step to the maximum load Pmax, (2) a fast unloading step to a very small load Pmin, (3) a holding step under Pmin for a period of thold, (4) a fast reloading step to the creep test load Pcreep, and (5) a final holding step for a period of tcreep. In the fast loading/unloading steps (Steps 1 and 2), the elasticplastic deformation is dominant and only negligible viscoelastic deformation (relaxed in Step 3) is induced. The choice of Pcreep < Pmax is to prevent further timeindependent plastic deformation and ensure the dominance of the viscoelastic deformation during the reloading and creeping steps. Considering much work has been done to extract plastic properties,18–22 the paper focuses only on the extraction of elastic-viscoelastic parameters by indentation creep tests (Steps 4 and 5). B. Analytical solutions to the viscoelastic deformation

To solve the viscoelastic problem whose boundary conditions change with time, Radok23 put forward a

FIG. 1. Schematic illustration of the five-step test scheme.

J. Mater. Res., Vol. 20, No. 6, Jun 2005


method of functional equations to replace the elastic parameters in the corresponding elastic solution by the equivalent viscoelastic operators in the viscoelastic constitutive equations. This method has been used to derive the analytical solutions to the indentation on purely viscoelastic materials with spherical and other axisymmetric indenters.12,24 However, in the present study, the material surface is no longer flat during the reloading process and the deformed configuration will inevitably affect the indentation creep test. Thus, Sneddon’s solutions25 to the indentation on an elastic half-space using an axisymmetric indenter cannot be applied directly to derive the analytical viscoelastic solutions. Recent studies showed that indentation on a deformed elasto-plastic sample can be well reproduced by that on an undeformed elastic half-space with an “effective indenter.”14,15 Pharr and Bolshakov found that the unloaded surface profile of an impression is not exactly conical or pyramidal in shape, and the profile of the “effective indenter” can be conveniently represented by the simple power-law relation [Fig. 2(a)]14 z ⳱ Br

n

.

(1)

On the basis of extensive experimental results for various ceramic, metallic, and even organic materials, Sakai15 found that the unloading/reloading indentation processes for a locally deformed conical/pyramidal impression can be well approximated by the equivalent mechanical process of a conical/pyramidal indenter with the effective face angle of ␤eff on a flat elastic half-space [Fig. 2(b)]

␤eff = ␤ − ␤r , tan ␤r =

hr tan ␤ hm

(2)

,

where ␤ is the face angle of the original conical/ pyramidal indenter, hr is the residual indentation depth, and hm is the maximum indentation depth. Obviously, Sakai’s approximation uses the assumption that the deformed profile is a straight line and there is no convex curvature in the impression surface. Hence, once the “effective indenter” profiles are known, it is possible to obtain the analytical solutions to the indentation on a plastically deformed visco-elastic-plastic material by combining the Sneddon’s solution with the Radok’s method. Sneddon’s method25 yields the following elastic solution to the indentation with Pharr and Bolshakov’s effective indenter P=

冋

册

n ⌫共n Ⲑ 2 + 1 Ⲑ 2兲 4␮ 1 − ␯ 共n + 1兲共公␲B兲1/n ⌫共n Ⲑ 2 + 1兲

1Ⲑn

h1+1 Ⲑ n , (3)

and the following elastic solution with Sakai’s effective indenter P=

4␮cot␤eff 2 h , ␲共1 − ␯兲

(4)

where P is the applied load; ␮ and ␯ are the shear modulus and the Poisson’s ratio, respectively; h is the indentation depth, and ⌫ is the gamma function. It is convenient to decompose the (Cauchy) stress and strain tensors into the deviatoric and volumetric parts

FIG. 2. Schematic illustration of the replacement of an indentation on a deformed surface with an effective indentation on a flat surface; (a) Pharr and Bolshakov’s effective indenter and (b) Sakai’s effective indenter. J. Mater. Res., Vol. 20, No. 6, Jun 2005

1599


␴ij = sij + ␦ij ␴kk Ⲑ 3 , ⑀ij = eij + ␦ij ⑀kk Ⲑ 3 ,

(5)

where ␦ij is the Keronecker symbol, sij is the deviatory part of the stress tensor; and eij is the deviatory part of the strain tensor. Here the Einstein summation convention is used. For viscoelastic materials, the governing equations can be expressed in the general form Pdsij = Qdeij , Pv␴kk = Qv⑀kk ,

(6)

where Pd, Qd, Pv, and Qvare polynomials of differential operators as shown below: N1

P = d

冱

pdi

i=0 N2

Q = d

冱

qdi

i=0 N3

P = v

冱

pvi

i=0 N4

Q = v

冱

qvi

i=0

⭸n ⭸t n ⭸n ⭸t

n

⭸n ⭸t n ⭸n ⭸t n

Fig. 3). With the assumption that the volumetric part responds elastically under a moderate hydrostatic pressure,2,3,11 the constitutive equations of such a model can be written as

兵1 + p1⭸t + p2⭸2t其sij = 兵q0 + q1⭸t + q2⭸2t其eij , ␴kk = 3K0⑀kk

,

where G0␩1 + G0␩2 + G1␩2 + G2␩1 G0G1 + G0G2 + G1G2 ␩1␩2 p2 = G0G1 + G0G2 + G1G2 p1 =

2G0G1G2 G0G1 + G0G2 + G1G2 2G0共G1␩2 + G2␩1兲 and q1 = G0G1 + G0G2 + G1G2 2G0␩1␩2 q2 = G0G1 + G0G2 + G1G2

,

(10)

q0 =

,

,

,

(11)

(7)

,

where the coefficients pdi , qdi , p␷i , q␷i (i ⳱ 0,1,2, . . .) can be calculated from material constants; Ni(i ⳱ 1,2,3,4) are the numbers of the components in the polynomials and they are not necessarily equal to each other; and t is time. From Radok’s method,23 i.e., replacing the elastic parameters by the equivalent viscoelastic operators, the corresponding function equations for the indentation with Pharr and Bolshakov’s effective indenter14 and Sakai’s effective indenter15 can be derived respectively as 关Pd共PdQv + 2PvQd兲兴关P共t兲兴 =

冋

⌫共n Ⲑ 2 + 1 Ⲑ 2兲 ⌫共n Ⲑ 2 + 1兲

册

1Ⲑn

2n 共n + 1兲共公␲B兲1 Ⲑ n

关Qd共PdQv + 2PvQd兲兴关h1+1 Ⲑ n共t兲兴

, (8)

and 2cot␤eff ␲ 关Qd共PdQv + 2PvQd兲兴关h2共t兲兴 .

关Pd共PdQv + 2PvQd兲兴关P共t兲兴 =

(9)

Following the widely accepted method to describe the viscoelastic response of polymeric materials, i.e., combining Hookean springs and Newtonian dashpots, we add one Kelvin–Voigt unit (i.e., a spring in parallel with a dashpot) to the standard viscoelastic solid model to capture the long-time creep response of polymers3 (see 1600

FIG. 3. Constitutive model used to describe the elastic-viscoelastic behavior of polymeric materials.



where ␩1 and ␩2 are the viscosity coefficients of the dashpots, Gi (I ⳱ 0, 1, 2) are the shear moduli of the spring elements, and K0 is the bulk modulus of the first spring. The shear moduli Gi and the bulk modulus K0 can be related to their Young’s moduli by

4

冱a s

i

i

i=0 4

s

冱b s

h0 − = s

i

4

冱s+t hi

(18)

,

i

i=1

i

i=0

Ei 共i = 0, 1, 2兲 , 2共1 + ␯兲 E0 . K0 = 3共1 − 2␯兲 Gi =

4

where −ti are the roots of the equation (12)

Here it is assumed that all the three springs share the same Poisson’s ratio ␯ [2, 24]. In a creep test, a load Pcreep is suddenly applied and held for a period of time. Thus the load P can be expressed as a step function, P共t兲 = Pcreep H共t兲 ,

(13)

where H(t) is the Heaviside unit step function. Substituting Eqs. (10), (11), and (13) into Eqs. (8) and (9), we obtain the corresponding function equations for Pharr and Bolshakov’s and Sakai’s effective indenters respectively as

冉冱冊冋册冉冱冊 4

⭸

i

2n

关H共t兲兴 = ⭸ti 共n + 1兲共公␲B兲1 Ⲑ n 4 1Ⲑn ⌫共n Ⲑ 2 + 1 Ⲑ 2兲 ⭸i bi i 关h1+1 Ⲑ n共t兲兴 , ⌫共n Ⲑ 2 + 1兲 ⭸t i=0

ai

Pcreep

冱b s = 0, and the i

i

i=0

rest of the parameters hi(i ⳱ 0, 1, …, 4) can be obtained by Eq. (18). Therefore the viscoelastic solutions to the indentation creep test by Pharr and Bolshakov’s effective indenter and by Sakai’s effective indenter can be found by conducting an inverse Laplace transform

冉

册冊

冋

共n + 1兲共公␲B兲1 Ⲑ n ⌫共n Ⲑ 2 + 1兲 h共t兲 = Pcreep 2n ⌫共n Ⲑ 2 + 1 Ⲑ 2兲

and

冉

h(t) = Pcreep

冉

4

h0 −

冱h e

冊

−t Ⲑ ti n Ⲑ n+1

i

i=1

2cot␤eff ␲

冊冉 1Ⲑ2

冱h e

−t Ⲑ ti

i

i=1

(19)

,

4

h0 −

1 Ⲑ n n Ⲑ n+1

冊

1Ⲑ2

i=0

and

冉冱冊 4

Pcreep

⭸i

2cot␤eff ai i 关H共t兲兴 = ␲ ⭸t i=0

冉冱冊 4

bi

i=0

⭸i

⭸ti

(14)

关h2共t兲兴

,

(15)

共n + 1兲共公␲B兲1 Ⲑ n 2n 4

冋

⌫共n Ⲑ 2 + 1兲 ⌫共n Ⲑ 2 + 1 Ⲑ 2兲

册

1Ⲑn

冱a s

i

i

i=0

s

(16)

,

4

冱b s

(20)

C. Experiments based on the five-step scheme

where ai,bi (i ⳱ 0, 1, , . . . , 4) can be calculated from p1, p2, q0, q1, and q2. Equations (14) and (15) can be solved by conducting a Laplace transformation h1+1 Ⲑ n共s兲 = Pcreep

.

i

i

i=0

A series of indentation tests following the five-step scheme has been conducted on PMMA (E ⳱ 2.6 to 3.2 GPa, Goodfellow Ltd., Huntingdon, UK). The tests were performed at room temperature using the MTS Nano Indenter XP (MTS Corporation, Nano Instruments Innovation Center, Oak Ridge, TN) under the loadcontrol mode with the Berkovich indenter, which can be modeled as an equivalent cone with a surface angle of ␤ ⳱19.68°. The maximum load of 8.000 mN was applied in 2 s and then immediately reduced to 0.005 mN in 2 s. The choice of such a small load is to avoid the overshoot in the unloading process. After a 500 s holding, several creep loading levels (5.0) for polymeric materials,14 the extracted parameters are in fact insensitive to B [see Eqs. (19) and (22)]. Once the effective indenter profile is determined, the elastic parameters, E0 and ␯ and the viscoelastic parameters E1, ␩1, E2, and ␩2 can be uniquely extracted by fitting the solution based on Pharr and Bolshakov’s effective indenter with the five-step experimental results. Table II lists the extracted values of these parameters. It is found that the values of elastic modulus and Poisson’s ratio are nearly the same as those obtained using Sakai’s effective indenter, validating the method used to determine the profile of Pharr and Bolshakov’s effective indenter. It is interesting to notice that the values of viscoelastic parameters are nearly independent of the creep loading levels and are not too far from the ones extracted using Sakai’s effective indenter. C. Discussion

The fitting curves at different creep loading levels using the solutions based on the two effective indenters are


1603


shown in Fig. 5, together with the experimental results. It is seen that a good fitting for both solutions can be achieved, indicating that the current material model (i.e., one spring and two Kelvin units in series) is adequate to describe the elastic-viscoelastic behavior of PMMA. It is known that two Kevin units are able to describe a creep process with two characteristic time scales, i.e., ␶1 and ␶2. It is believed that these time scales are related to the motion modes of polymeric chains, the elastic restoring force arising from the potential barriers to the motions and the slippage of the molecular chains and the restoring force arising from the entropy-related force.27 Therefore, two Kevin units in the constitutive model seem necessary to describe these two these deformation mechanisms. Although a good fitting is achieved for the solution based on Sakai’s effective indenter, the extracted values of the viscoelastic parameters are dependent on the creep loading levels, which is in contrast to the extracted ones using Pharr and Bolshakov’s effective indenter. This is possibly caused by the profile difference between Sakai’s effective indenter and Pharr and Bolshakov’s effective indenter as illustrated in Fig. 6. Below a critical contact radius ar, Sakai’s effective indenter has a face angle smaller than that of Pharr and Bolshakov’s effective indenter at the same indentation depth, thus making it “sharper” than Pharr and Bolshakov’s effective indenter. Hence, the extracted values of the viscoelastic parameters are overestimated. However, above the critical contact radius, Sakai’s effective indenter behaves more “bluntly” than Pharr and Bolshakov’s effective indenter. Hence, the extracted values of the viscoelastic parameters are underestimated. This explains why the values of the extracted viscoelastic parameters E1, ␩1, E2, and ␩2 using Sakai’s effective indenter decrease with increasing Pcreep/Pmax. To use Sakai’s effective indenter to extract more accurate values of the viscoelastic parameters, one needs to choose a proper value of Pcreep/Pmax so the contact radius should be as close to ar as possible. It appears that a good agreement of the extracted parameters between the effective indenters can be achieved when the Pcreep/Pmax ⳱ 0.5 for the present experimental results. The extracted elastic parameters E0 and ␯ by both Pharr and Bolshakov’s and Sakai’s effective indenters agree well with the product sheet provided by the manufacturer, i.e., E ⳱ 2.6 to 3.2 GPa and ␯ ⳱ 0.38 to 0.42. Furthermore, the values of E0 and ␯ are nearly the same with those extracted by indentation on both bulk PMMA and PMMA film using a flat-ended punch where E ⳱ 2.94 to 3.00 GPa and ␯ ⳱ 0.4.3 Therefore, it appears that the elastic properties are independent of indenter’s shape and the stress distribution beneath the indenter. Previous studies often indicate that a separate Newtonian dashpot is required to describe long-time indentation creep of bulk PMMA.3 In the present study, the 1604

model with a separate Newtonian dashpot in series with two Kelvin–Voigt units3 is also used to describe the creep behavior in the five-step loading scheme. It is found that the separate dashpot is redundant since the extracted viscosity coefficient for the separate dashpot is extremely large, indicating a negligible viscoplastic deformation during the creep deformation. This finding is also supported by the fact that the model without the separate dashpot is able to fit the creep curves very well and the elastic-viscoelastic parameters in the model can be uniquely extracted. It is known that a separate dashpot is able to describe viscoplastic deformation. In the flatended punch indentations, it was often assumed that the plastic deformation was negligible, and therefore the indentation behavior was totally viscoelastic. However, irreversible deformation, such as plastic pileups and residual grooves are often observed during flat-ended punch indentations.28 Hence, a viscoplastic unit—that is, the separate Newtonian dashpot—is required to describe the irreversible deformation. Since the current study adopts a five-step scheme, the negligible irreversible deformation during steps 4 and 5 renders the separate dashpot redundant.

IV. CONCLUSIONS

A five-step test scheme is proposed to separate the plastic and the elastic-viscoelastic deformations to extract the elastic-viscoelastic parameters of polymeric materials. The analytical solutions to the viscoelastic part are derived based on the concept of “effective indenters.” A genetic algorithm is used to extract viscoelastic parameters in combination with the analytical solutions. By combining the effective indenter proposed by Sakai and the one proposed by Pharr and Bolshakov, the elasticviscoelastic parameters can be uniquely determined. It is found that the values of extracted viscoelastic parameters using Pharr and Bolshakov’s effective indenter are independent of reloading levels, and the extracted elastic parameters are in good agreement with literature values and consistent with the ones obtained by indentation tests with a flat-ended punch.

REFERENCES 1. W.C. Oliver and G.M. Pharr: An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7, 1564 (1992). 2. L. Cheng, X. Xia, W. Yu, L.E. Scriven, and W.W. Gerberich: Flat-punch indentation of viscoelastic material. J. Polym. Sci. B 38, 10 (2000). 3. C.Y. Zhang, Y.W. Zhang, and K.Y. Zeng: Extracting the mechanical properties of a viscoelastic polymeric film on a hard elastic substrate. J. Mater. Res. 19, 3053 (2004). 4. R. Rikards, A. Flores, F. Ania, V. Kushnevski, and



5.

6.

7.

8.

9.

10.

11.

12.

13. 14. 15. 16.

F.J. Baltá Calleja: Numerical- experimental method for the identification of plastic properties of polymers from microhardness tests. Comput. Mater. Sci. 11, 233 (1998). S. Shimizu, T. Yanagimoto, and M. Sakai: The pyramidal indentation load-depth curve of viscoelastic materials. J. Mater. Res. 14, 4075 (1999). J.L. Bucaille, E. Felder, and G. Hochstetter: Identification of the viscoplastic behavior of a polycarbonate based on experiments and numerical modeling of the nano-indentation test. J. Mater. Sci. 37, 3999 (2002). C. G’Sell and J.J. Jonas: Determination of the plastic behavior of solid polymers at constant true strain rate. J. Mater. Sci. 14, 583 (1979). M.L. Oyen and R.F. Cook: Load-displacement behavior during sharp indentation of viscous-elastic-plastic materials. J. Mater. Res. 18, 139 (2003). M.L. Oyen, R.F. Cook, J.A. Emerson, and N.R. Moody: Indentation response of time-dependent films on stiff substrates. J. Mater. Res. 19, 2487 (2004). T.C. Ovaert, B.R. Kim, and J.J. Wang: Multi-parameter models of the viscoelastic/plastic mechanical properties of coatings via combined nanoindentation and non-linear finite element modeling. Prog. Org. Coat. 47, 312 (2003). S.G. Bardenhagen, M.G. Stout, and G.T. Gray: Threedimensional finite deformation viscoplastic constitutive models for polymeric materials. Mech. Mater. 25, 235 (1997). H. Lu, B. Wang, J. Ma, G. Huang, and H. Viswanathan: Measurement of creep compliance of solid polymers by nanoindentation. Mech. Time-Depend. Mater. 7, 189 (2003). Y.T. Cheng and C.M. Cheng: Scaling, dimensional analysis, and indentation measurements. Mater. Sci. Eng. R 44, 91 (2004). G.M. Pharr and A. Bolshakov: Understanding nanoindentation unloading curves. J. Mater. Res. 17, 2660 (2002). M. Sakai: Elastic recovery in the unloading process of pyramidal microindentation. J. Mater. Res. 18, 1631 (2003). C. Marano and M. Rink: Shear yielding threshold and viscoelasticity in an amorphous glassy polymer: A study on a styrene– acrylonitrile copolymer. Polymer 42, 2113 (2001).

17. L. Cheng, L.E. Scriven, and W.W. Gerberich: Viscoelastic analysis of micro- and nano-indentation, in Fundamentals of Nanoindentation and Nantribology, edited by N.R. Moody, W.W. Gerberich, N. Burnham, and S.P. Baker (Mater. Res. Soc. Symp. Proc. 522, Warrendale, PA, 1998), p. 193. 18. K. Tunvisut, E.P. Busso, N.P. O’Dowd, and H.P. Brantner: Determination of the mechanical properties of metallic thin films and substrates from indentation tests. Philos. Mag. A 82, 2013 (2002). 19. S. Suresh, A.E. Giannakopoulos, and J. Alcala: Spherical indentation of compositionally graded materials: Theory and experiments. Acta Mater. 45, 1307 (1997). 20. Y.T. Cheng and C.M. Cheng: Scaling relationships in conical indentation of elastic- perfectly plastic solids. Int. J. Solids Struct. 36, 1231 (1999). 21. T.A. Venkatesh, K.J. Van Vliet, A.E. Giannakopoulos, and S. Suresh: Determination of elasto-plastic properties by instrumented sharp indentation: guidelines for property extraction. Scripta Mater. 42, 833 (2000). 22. A.E. Giannakopoulos and S. Suresh: Determination of elastoplastic properties by instrumented sharp indentation. Scripta Mater. 40, 1191 (1999). 23. J.R.M. Radok: Visco-elastic stress analysis. Q. Appl. Math. 15, 198 (1957). 24. L. Cheng, X. Xia, L.E. Scriven, and W.W. Gerberich: Sphericaltip indentation of viscoelastic material. Mech. Mater. 37, 213 (2005). 25. I.N. Sneddon: The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int. J. Eng. Sci. 3, 47 (1965). 26. M. Sakai and Y. Nakano: Elastoplastic load-depth hysteresis in pyramidal indentation. J. Mater. Res. 17, 2161 (2002). 27. L.E. Nielsen and R.F. Landel: Mechanical Properties of Polymers and Composites, 2nd ed. (Marcel Dekker, New York, NY, 1994), p. 92. 28. A. Krupic˘ ka, M. Johansson, and A. Hult: Viscoelasticity in polymer films on rigid substrates. Macromol. Mater. Eng. 288, 108 (2003).


1605

Nanoindentation of polymers with a sharp indenter

Nanoindentation of polymers with a sharp indenter

Suggest Documents