Transient and steady-state nanoindentation creep of polymeric materials

International Journal of Plasticity 27 (2011) 1093–1102

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Transient and steady-state nanoindentation creep of polymeric materials Chien-Chao Huang a, Mao-Kuo Wei b, Sanboh Lee a,⇑ a b

Department of Materials Science and Engineering, National Tsing Hua University, Hsinchu, Taiwan Department of Materials Science and Engineering, National Dong Hwa University, Hualien, Taiwan

a r t i c l e

i n f o

Article history: Received 20 December 2009 Received in final revised form 4 November 2010 Available online 4 December 2010 Keywords: Nanoindentation creep Viscoelasticity Power-law creep Polymer

a b s t r a c t The transient and steady-state nanoindentation creep of polymeric materials was investigated. The creep model is used to explain the experimental data of transient and steadystate creep dominated by viscoelastic deformation and power-law creep deformation, respectively. The Burgers viscoelastic model was used to interpret the transient creep in polymers under nano-indentation. Explicit expression for the displacement of transient creep was derived using the correspondence principle of linear viscoelasticity theory. The power law of strain rate-stress relation was used to explain the creep displacement during the steady state. Three polymers of poly(methyl methacrylate), hydroxyethyl methacrylate copolymer, and the fast-cure acrylic resin were used to measure the nanoindentation creep. The transient creep data are in good agreement with the predictions from the Burgers viscoelastic model. The creep displacement is mainly attributed to the viscous flow of the Kelvin element, and the computed values of viscosities (g1,c, g2,c) increase with decreasing preloading rate. By comparing the steady-state creep data with the power law of strain rate-stress relation, the stress exponents for the above polymeric materials were quantitatively determined. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Mechanical properties are of important concern in microelectromechanical systems (MEMS) and the MOS device design, fabrication and application, especially on the micrometer and nanometer scales. The depth-sensing indentation (DSI) technique becomes a standard tool for measurements of the elastic modulus and the hardness of the small sample. Oliver and Pharr (1992) proposed the indentation method to measure the Young’s modulus based on the assumption that the material is purely elastic. However, in many cases, the behavior of the contact area between indenter tip and sample is viscoelastic (exhibiting both viscous flow and elastic deformation) not just elastic (LaFontaine et al., 1990; Syed and Pethica, 1997; Feng and Ngan, 2001a; Li and Ngan, 2004; Tang et al., 2006; Jager et al., 2007; Oyen, 2007; Choi et al., 2008). The extreme case that the load–displacement curve exhibiting a ‘‘nose’’ occurs when the shear viscosity dominates during unloading. When the nose appears, the stiffness at the onset of unloading curve becomes negative, and the Young’s modulus calculated by the Oliver and Pharr method becomes negative. In order to obtain an accurate Young’s modulus from the nano-indentation test, Briscoe et al. (1998) suggested maintaining a sufficient time at the peak load for stress relaxation before unloading. However, such an experiment requires a long time for completion. Based on the linear viscoelasticity, Feng and Ngan (2002) proposed a simple formula to correct the contact stiffness of Berkovich nanoindentation measurement within a short dwell time. In the following year, Tang and Ngan (2003) refined the approach of Feng and Ngan by modifying both the contact stiffness and contact area measurements. They provided a formula to eliminate the effect of creep on modulus measurement for both

⇑ Corresponding author. Tel.: +886 3 5719677. E-mail address: [email protected] (S. Lee). 0749-6419/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2010.11.005

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PP and a-Se viscoelastic materials. A semi-empirical elastic–viscoelastic–viscous model was also proposed by Yang et al. (2004) to describe the creep behavior of polymers under a rapid loading speed (2.5 mN/s) for both Berkovich and flat-ended punch nanoindentors. Then, Ngan et al. (2005) proved the formula proposed by Feng and Ngan (2002) and extended the analysis from linear viscoelasticity to power law viscoelasticity. Subsequently, using the finite element analysis, Tweedie and Vliet (2006) illustrated that the linear viscoelasticity did not hold when the creep compliance was measured by a conical or other sharp nanoindentor performed on the polymeric surface regardless of the loading rate, but was valid when it was measured by a spherical nanoindentor with contact strain less than the elastic limit. However, the above studies were not able to explain the nanoindentation creep composed of both primary and secondary stages attributed to viscoelasticity and power-law creep, respectively. Chu and Li (1977) were first to develop an impression creep test using a cylinderical indenter to characterize the steady state creep. Later, Baker et al. (1992) found that the time-dependent displacement consisted of plastic deformation and experimental drift. Lucas and Oliver (1992) measured the Young’s modulus, hardness, and the stress exponent for sapphire and a 1.9 lm amorphous alumina film on the sapphire substrate by using indentation under an ultra low load. In the same year, Raman and Berriche (1992) used the plastic power law to analyze the indentation creep of polycrystalline tin and sputtered aluminum on silicon substrate. Shortly, Li and Warren (1993) proposed a nano-indentation creep model based on atomic diffusion to explain the creep stage prior to pop-in. Furthermore, O’Connor and Cleveland (1993) applied the strain rate-stress power law to study indentation creep of polystyrene, special poly (styrene-co-divinylbenzene) and poly (divinylbenzene). Chiang and Li (1994) conducted the impression creep experiments on ABS polymers at different temperatures. Lucas and Oliver (1999) studied the indentation power-law creep of high-purity indium. Feng and Ngan (2001b) observed that the indentation creep in aluminum took place only after a strain burst had occurred, but the creep in indium could take place before or after the strain burst. On the other hand, Sakai (2002) presented an analytical solution for the case of an axisymmetric indenter penetrating into a viscoelastic body using the Boltzmann’s hereditary integral. Sakai and Shimizu (2002) also found that a pyramid indenter is an efficient microprobe for viscoelastic behaviors of glass-forming materials at temperatures near the glass transition temperature. For metals and crystalline materials, Li and Ngan (2004) used strain rate vs. stress power law to study the size effect on indentation creep of single-crystal Ni, Al, polycrystalline pure Al, and fused quartz. Tang and Ngan (2004) determined the elastic modulus, hardness, viscosity, activation energy and activation volume for creep of amorphous selenium below the glass transition temperature by using the depth-sensing indentation. Note that the load duration and temperature of their tests are less than 30 min and 40 °C, respectively. Based on the conventional indentation procedures, Goodall and Clyne (2006) proposed a procedure to analyze the steady-state nanoindentation creep. More recently, Seltzer and Mai (2008) made the assumption of the linear viscoelasticity to estimate the viscoelastic properties of PMMA using spherical nanoindentation experiments. They found that plastic and viscoelastic deformation processes occurred simultaneously under a constant load during the spherical indentation. It was Radok (1957) who first proposed a theory of linear viscoelastic indentation under a spherical indenter. The theory was then proved to be valid only if the contact area was not reduced and was extended to more general indentation problems (Lee and Radok, 1960; Ting, 1966; Cheng et al., 2005). Anand and Ames (2006) developed a three-dimensional finite deformation model for the elasto-visco-plasticity of amorphous polymer based on the generalized Kelvin–Voigt model of classical linear viscoelasticity. Furthermore, the strain rate-stress power law was used to analyze the plasticity behavior of crystalline materials (Goodall and Clyne, 2006; Lee and Chen, 2010). Liu et al. (2006) developed both Burgers and Maxwell viscoelastic models for loading and unloading during the Berkovich nanoindentation experiment. By comparing the displacement-load curves, they found that the Burgers model was more effective than the Maxwell model. The indentation creep consists of the primary creep (or transient creep) and secondary (or steady-state creep), representing regions of decreasing creep rate and nearly constant creep rate, respectively. It prompted us to investigate the transient and steady-state nanoindentation creep of polymeric materials. In the next section, an analytical solution of transient creep based on the Burgers viscoelastic model is derived using the corresponding principle between elasticity and Laplace transformation of viscoelasticity; and the power law of strain rate-stress relation is proposed to analyze the steady state creep. The third section describes the experimental creep procedures using the nanoindentation technique. In the fourth section, the Burgers model is used to fit the displacement–time data of transient creep of three polymeric materials tested at different loading rates; the power law of strain rate-stress relation is compared with the steady-state creep data. Finally, the findings of this work are summarized in Section 5.

2. Theoretical analysis The problem deals with the creep behavior of a viscoelastic material subjected to a constant nano-indentation load. A Berkovich indentation tip is pressed into the viscoelastic semi-infinite body under an external load, which is increased from zero to a maximum and then held until the steady state is reached (see Fig. 1(a)). The nanoidentation creep consists of two stages, transient (primary creep) and steady state (secondary creep). In analogy to the loading and unloading periods, the creep displacement consists of viscoelastic displacement and power-law creep displacement. Although the stress in the sample beneath the indenter tip at the transient stage is very large, the power-law creep deformation is small as compared to the viscoelastic displacement because the power-law creep needs time to build-up. Thus the creep displacement at the transient stage is dominated by the viscoelastic displacement. On the other hand,

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a

b 2.5 Seconary Creep

Displacement , ε

2.0

Load, P (mN)

Primary

PR

1.5 1.0

dε = nearly dt constant

0.5 0.0

tR

Creep Time, t

Time (s)

Fig. 1. (a) A typical load-time (P  t) curve for nanoindenation creep test where PR is kept constant. (b) A typical creep curve consists of two stages: transient period and steady state.

the displacement at the steady state is attributed to the power-law creep. These two stages are addressed separately in the following: 2.1. Transient creep The transient creep behavior of the viscoelastic material is simulated by the Burgers model, in which a Maxwell element (the spring and dashpot are in series) is connected in series with a Kelvin element (spring and dashpot are in parallel). The Burgers model was used to solve the displacement during loading and unloading before the reduction of contact area (Liu et al., 2006). Briefly, the governing field equations of linear elasticity in the time domain are analogous to those of viscoelasticity in the Laplace transform coordinate system. That is, if the displacement-load solution of the elastic system is given, the corresponding solution in the viscoelastic system can readily be found in the Laplace transformation coordinate system. The time dependent displacement in the viscoelastic system is then obtained by taking the inverse Laplace transform. Note that the corresponding principle between linear elasticity and Laplace transformation of linear viscoelasticity was examined by Radok (1957) and Lee and Radok (1960) for the analysis of Maxwell viscoelastic model, and was proved valid only if the contact area was not reduced. The solution of Lee and Radok (1960) was later confirmed by Ting (1966). More recently, Cheng et al. (2005) also validated the solution for the loading period before the reduction of the contact area. The Young’s modulus, E, and Poisson’s ratio, m, in the load–displacement equations in the time domain for an elastic system corresponding to the counterparts of Laplace transforms of load–displacement equations for the Burgers viscoelastic system can be written as Liu et al. (2006) and Nowacki (1986)

E!

9K  G1 g1 sðG2 þ g2 sÞ ; 3K G1 G2 þ ½G2 g1 ð3K þ G1 Þ þ 3K  G1 ðg1 þ g2 Þs þ ½g1 g2 ð3K  þ G1 Þs2 











ð1aÞ 2

3K G1 G2 þ ½G2 g1 ð3K  2G1 Þ þ 3K G1 ðg1 þ g2 Þs þ ½g1 g2 ð3K  2G1 Þs ; 6K  G1 G2 þ ½G2 g1 ð6K  þ 2G1 Þ þ 6K  G1 ðg1 þ g2 Þs þ ½g1 g2 ð6K  þ 2G1 Þs2 1 1 1 ¼ þ ; K K1 K2

m!

ð1bÞ ð1cÞ

where K1 and K2 are the bulk moduli of Maxwell and Kelvin elements, respectively, and G1 and G2 are the corresponding shear moduli. g1 and g2 are the Newtonian shear viscosities of Maxwell and Kelvin elements, respectively, and s is the Laplace transform coordinate. For a semi-infinite elastic body indented with a conical tip, the load–displacement relation (P–h equation) derived by Sneddon (1965) is 2

Q e hemc ðtÞ ¼

1 PðtÞ; E

ð2Þ

where P is the load, hevc is the displacement

Qe ¼

2 tan a

;  p 2 1  md 1 ð1  m2 Þ ; ¼ þ Ed E E

ð3aÞ ð3bÞ

2a is the effective included angle of the indenter tip; Ed and md are the Young’s modulus and Poisson’s ratio, respectively, of the indenter tip. Eq. (2) is also valid for a Berkovich indenter tip.

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When the Young’s modulus, E, and Poisson’s ratio, m, of the elastic body in Eq. (3b) are replaced by the values given in Eq. (1), we obtain the Laplace transform of the displacement in a viscoelastic body

8 h i <1  m2 ð3K  þ 4G1 Þ 1 1 3G21 ðg1 þ g2 Þ 2 d  þ Q e L hEmc ðtÞ ¼ þ þ þ  2 G : Ed 4G1 ð3K þ G1 Þ 4g1 s 4g s þ 2 4g1 g2 ð3K  þ G1 Þ 2 g2 9 = s þ g Gþ2g 1 2 h i  L½PðtÞ;    G1 G2 ; 1 Þþ3K G1 ðg1 þg2 Þ s2 þ G2 g1 ð3Kg þG s þ g g3Kð3K  g ð3K  þG1 Þ þG1 Þ 1 2

ð4Þ

1 2

where L is the Laplace transform operator. The inverse Laplace transform of Eq. (4) gives the displacement in the viscoelastic body as     Z t Z t 1  m2d ð3K  þ 4G1 Þ PðsÞ PðsÞ G2 ðs  tÞ 2 Q e hEmc ðtÞ ¼ þ ds þ exp PðtÞ þ ds  4G1 ð3K þ G1 Þ Ed g2 0 4g1 0 4g2   Z t Z t 2 2 3G1 ðg1 þ g2 ÞPðsÞ G2 3G1 ðg1 þ g2 ÞPðsÞ G2 aðstÞ þ  a e d s   b ebðstÞ ds; 2 g þg 2 g þg   0 4g1 g2 ðb  aÞð3K þ G1 Þ 0 4g1 g2 ðb  aÞð3K þ G1 Þ 1 2 1 2

ð5Þ where

a¼ b¼

B Bþ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2  4C ; 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi

ð6aÞ

B2  4C ; 2

ð6bÞ

and

G2 g1 ð3K  þ G1 Þ þ 3K  G1 ðg1 þ g2 Þ ; g1 g2 ð3K  þ G1 Þ 3K  G1 G2 : C¼ g1 g2 ð3K  þ G1 Þ B¼

ð6cÞ ð6dÞ



Substituting a constant rate ðdP=dt ¼ p1 Þ up to the peak load PR at time tR and a constant load PR up to time t into Eq. (5), the time-dependent displacement hevc of creep (t > tR) is

PR ðt  t R Þ PR þ ð1  eG2;c ðtR tÞ=g2;c Þ 4Q e g1;c 4Q e G2;c (" ) # " # 3G21;c ðg1;c þ g2;c ÞPR G2;c G2;c ac ðt R tÞ bc ðtR tÞ Þ Þ ; þ  1 ð1  e  1 ð1  e 2 ac ðg1;c þ g2;c Þ bc ðg1;c þ g2;c Þ 4Q e g1;c g2;c ð3K  þ G1;c Þ ðbc  ac Þ

2

2

hemc ðtÞ ¼ hemc ðt R Þ þ

ð7Þ where 2 hemc ðt R Þ

al ¼ bl ¼

"    #   g2;l g2;l Gg2;l2;ltR 3K l þ 4G1;l P R 1  m2d P t p1    þ R R þ ¼ þ tR  þ e Qe Ed 8Q e ge;l 4Q e G2;l G2;l G2;l 4G1;l 3K l þ G1;l " # (   3G21;l ðg1;l þ g2;l Þ p1 G2;l 1 1 þ  1 tR  þ eal tR   2 al al al ðg1;l þ g2;l Þ 4Q e g1;l g2;l 3K l þ G1;l ðbl  al Þ " # ) G2;l 1 1 ;  1 t R  þ ebl tR  bl bl bl ðg1;l þ g2;l Þ

Bl 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2l  4C l

2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bl þ B2l  4C l

;

; 2   G2;l g1;l 3K l þ G1;l þ 3K l G1;l ðg1;l þ g2;l Þ ; Bl ¼ g1;l g2;l ð3K l þ G1;l Þ

ð8Þ

ð9aÞ ð9bÞ ð9cÞ

C.-C. Huang et al. / International Journal of Plasticity 27 (2011) 1093–1102

3K l G1;l G2;l ; g1;l g2;l 3K l þ G1;l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bc  B2c  4C c ; ac ¼ 2 ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bc þ B2c  4C c ; bc ¼ 2  G2;c g1;c ð3K c þ G1;c Þ þ 3K c G1;c ðg1;c þ g2;c Þ ; Bc ¼ g1;c g2;c 3K c þ G1;c Cl ¼

Cc ¼

3K c G1;c G2;c  ; g1;c g2;c 3K c þ G1;c

1097

ð9dÞ

ð9eÞ ð9fÞ ð9gÞ ð9hÞ

and the subscripts l and c represent the preloading and the creep periods, respectively. Note that the Poisson’s ratio is assumed to be the same in both preloading and creep periods. 2.2. Steady-state creep The formulation of steady-state creep is basically followed the concept of Goodall and Clyne (2006). Stresses and strains vary from very large in the vicinity of indenter tip to nearly zero in the remote regions. The steady-state strain rate of conventional tensile creep satisfies the power law (Goodall and Clyne, 2006), i.e.:

e_ ¼ krn ;

ð10Þ

where r is the applied stress, k is a constant, and n is the stress exponent. Assume that the steady-state creep rate, e_ PC , satisfies the conventional power-law creep. The applied stress used in the conventional creep is substituted by the effective stress used in the nanoindentation creep. In analog to the definition of hardness used in nanoindentation technique, the effective stress, rpc, is defined as the applied load F, divided by the projected contact area Apc

rpc ¼ F=Apc ;

ð11Þ

2 24:5hpc

where Apc ¼ for the Berkovich tip and hpc = hc  hevc. Note that the projected contact area is the cross-sectional area of the indenter at the depth to which it is indented, not the real surface contact area. Similarly, the strain rate is changed from very large in the vicinity of intender tip to infinitesimal small in the remote regions. The strain rate in the conventional creep is replaced by the effective strain rate in the nanoindentation creep. The origins of formulation for the effective strain rate come from the hemispherical geometry of plastically deformed zone produced by a spherical cavity expanding under internal pressure (Marsh, 1964). The effective strain rate during the indentation period was proposed by Mayo and Nix (1988) as

e_ p ¼ ð1=hpc Þðdhpc =dtÞ:

ð12Þ

Substituting Eqs. (11) and (12) into Eq. (10) yields

1 dhpc F ¼k 2 hpc dt 24:5hpc

!n :

ð13Þ

The parameters k and n are assumed to have the same values as those obtained from the conventional tensile creep tests. The nanoindentation creep test involves in heterogeneous stress in the specimen and lasts short period whereas the conventional tensile creep test requires the homogenous stress in the whole specimen and maintains long time. The stresses in material are complicated and related to its microstructure. It implies that the former is reflected the microstructure of materials, but the latter is not. The creep mechanism is also related to the microstructure. Whether Eq. (13) can be used to describe deformation in nanoindentation creep is a key issue. Amorphous polymers with the absence of microstructure are expected to have similar behavior for nanoindentation creep test and conventional tensile creep test. The viscoelastic deformation at the steady-state creep is much smaller than the power-law creep deformation. Thus the steady state creep is dominated by the power-law creep and Eq. (13) is used to analyze the steady-state experimental data in Section 4. 3. Experimental procedure Three polymers used in this study are poly (methyl methacrylate) (PMMA, Lucite), contact lens blanks made of hydroxyethyl methacrylate copolymer (HEMA), and AcryliMet fast-cure acrylic resin. The sample preparation is the same as that reported by Liu et al. (2006). Nano-indentation creep experiments were performed at room temperature (25 °C) using a NanoIndenter XP (MTS Systems, Oak Ridge, TN, USA) equipped with a diamond Berkovich tip. The procedure of the indentation creep, as shown in Fig. 1(a), comprised a loading ramp to a specified peak load and then maintaining at a constant level until the steady state was reached. The preloading rates of 0.2 mN/s (tR = 10 s), 0.04 mN/s (tR = 50 s), 0.02 mN/s (tR = 100 s),

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0.0067 mN/s (tR = 300 s) and 0.002 mN/s (tR = 1000 s) at a peak load of 2.0 mN were used for HEMA. For PMMA and fast-cure acrylic, the preloading rates of 0.25 mN/s (tR = 10 s), 0.05 mN/s (tR = 50 s), 0.025 mN/s (tR = 100 s), 0.0083 mN/s (tR = 300 s) and 0.0025 mN/s (tR = 1000 s) and a peak load of 2.5 mN were applied. Nanoindentation tests conducted at room temperature (25 °C) were performed at six different locations for each specimen, with the distance between two nearest locations being approximately 500 lm. 4. Results and discussion The creep displacement consists of viscoelastic and power-law creep components. A typical indentation depth vs. time curve is similar to the conventional tensile creep curves with a steady-state stage following a transient period as shown in Fig. 1(b). Because of the geometric constraint of nanoindentation, the deformation is always stable, and the tertiary creep (or creep fracture in the tensile creep curve) is not observed (Dieter, 1988). The transient period is assumed to help build up the steady-state power-law creep zone in the sample underneath the indenter tip. Thus the transient creep is presumably dominated by the viscoelastic deformation and the steady-state creep is attributed to the power-law creep deformation. They are discussed in the following: 4.1. Transient creep The mechanical parameters associated with two elements of a typical Burgers model, have no physical meaning (Nowacki, 1986). However, in analogy to equivalent circuit of semiconductor devices, Liu et al. (2006) studied the displacement of polymeric materials underneath the indenter tip during loading and unloading. They compared their Young’s modulus value with the counterpart from the literature and found the Young’s modulus of Kelvin element corresponds to the Young’s modulus of bulk material in the literature. Thus Liu et al. (2006) concluded that the Maxwell and Kelvin elements in Burgers model were contributed by the surface layer and the bulk of polymeric materials, respectively. Here we adopted their approach by using the Burgers model to analyze the nanoindentation creep. The experimental data of nanoindentation creep for PMMA, HEMA and fast-cure acrylic materials are shown in Fig. 2(a)–(c), respectively. The creep displacement at time t in Fig. 2 is the difference between the total displacement at (t + tR) and the total displacement at tR. The total displacement is the sum of the viscoelastic and the power-law creep displacement. The power-law creep displacement is neglected during the transient creep. Hence the transient creep displacement can be given by

hðtÞ ¼ hev c ðt þ t R Þ  hev c ðtR Þ;

a

ð14Þ

b

800

Displacement (nm)

Displacement (nm)

PMMA t R = 10 s t R = 50 s t R = 100 s t R = 300 s Model

600

400

200

0

0

200

400

600

200

0

200

Creep Time (sec)

c

HEMA

400

0

600

tR = 10 s tR = 50 s tR = 100 s tR = 300 s model

800

400

600

800

1000

Creep Time (sec)

1000

Displacement(nm)

fast cure Acrylic tR = 10 s tR = 50 s tR = 100 s tR = 300 s

800 600

model

400 200 0

0

200

400

600

Creep Time (sec) Fig. 2. The transient creep displacement–time curves with varying preloading times, tR: (a) PMMA; (b) HEMA and (c) fast-cure acrylic resin.

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C.-C. Huang et al. / International Journal of Plasticity 27 (2011) 1093–1102 Table 1 The fitting parameters, E1, E2, m1, m2, g1,l, g2,l, g1,c, and g1,c for PMMA, HEMA and fast-cure acrylic resin with different preloading periods, tR. Polymer

PMMA

tR (s)

10

m1 ð¼ m2 Þ

0.33

Loading

g1,l  102 (GPa s)

g2,l  102 (GPa s)

E (GPa)

g1;c1  102 (GPa s)

g2,c  102 (GPa s)

E1 = 7.4 E2 = 4.2

1.27

1.20

E1 = 7.4 E2 = 4.2

10.08

1.21

3.99 4.12 5.07

3.80 4.00 5.00

10.49 12.11 21.23

2.98 3.35 4.28

0.37

0.65

1.99

0.35

0.97 1.42 2.25

0.90 2.40 6.80

3.27 6.22 6.95

1.55 2.01 4.47

0.67

1.10

2.70

0.81

1.79 2.2 2.75

1.90 3.20 7.50

3.26 3.96 5.61

2.03 2.33 3.32

50 100 300 HEMA

10

0.30

E1 = 5.5 E2 = 4.4

50 100 300 Fast-cureacrylic

10

0.35

Creep

E (GPa)

E1 = 5.3 E2 = 3.4

50 100 300

E1 = 5.5 E2 = 4.4

E1 = 5.3 E2 = 3.4

where hevc(t + tR) and hevc(tR) are given by Eqs. (7) and (8), respectively. The solid lines in Fig. 2(a)–(c) are obtained by using Eq. (14) with fitted parameters listed in Table 1. It can be seen from Fig. 2 that the experimental data are in good agreement with the theoretical predictions. As shown in Table 1, for a given tR assigned to a polymer sample, the shear viscosities g2,c and g1,c are, respectively, the smallest and the greatest value among those in the preloading and creep periods. The ratio of g1,c–g2,c decreases with increasing tR (or with decreasing preloading rate). Liu et al. (2006) found that the viscous flow behavior was more pronounced with decreasing shear viscosity, and the Kelvin element corresponded to the bulk material. Thus the nano-indentation transient creep is dominated by the Kelvin element. The transient displacement is attributed mainly to the viscous flow of bulk material. The shear viscosity increases with increasing tR regardless of the Maxwell or Kelvin element for all materials. The preloading period tR is inversely proportional to preloading rate. For a given time, the transient displacement increases with increasing preloading rate, as shown in Fig. 2. This phenomenon is similar to that the loading displacement increases with loading rate (Liu et al., 2006). The transient period, tc, under nano-indentation measured for the above polymers together with the preloading period, tR, are tabulated in Table 2. It is seen that the transient period increases with increasing preloading time (or with decreasing preloading rate) for all polymers. For PMMA, HEMA and the fast-cure acrylic resin, the values of Young’s moduli of Maxwell and Kelvin elements in the transient period are the same as those in the preloading period. It is known that PMMA, HEMA, and the fast-cure acrylic resin are amorphous. The first two polymers are thermoplastic and the last one is thermosetting. Since there are no structural changes under the action of applied load at room temperature, their Young’s moduli are not affected by the preloading rate and creep deformation. 4.2. Steady-state creep According to Eqs. (11) and (12), both the strain rate and the stress decrease monotonically with increasing time, while the stress exponent, n, has the opposite trend. However, the power-law creep displacement hpc is smaller than hevc during the transient period. The power-law creep displacement is more pronounced than the viscoelastic displacement at the Table 2 The transient period of nano-indentation creep for PMMA, HEMA and fast-cure acrylic resin with different preloading periods, tR. Polymer

tR (s)

Transient period (s)

PMMA

10 50 100 300

130 150 190 210

HEMA

10 50 100 300

220 250 300 400

Fast-cureacrylic

10 50 100 300

160 210 240 280

1100

C.-C. Huang et al. / International Journal of Plasticity 27 (2011) 1093–1102

a

b

-1.5

-1.5

tR tR tR tR

-2.0

-2.5

HEMA

= 10 sec = 50 sec = 100 sec = 300 sec

log10 (strain rate)

log10 (strain rate)

PMMA

-3.0

-3.5

-2.0

tR tR tR tR

-2.5

= 10 sec = 50 sec = 100 sec = 300 sec

-3.0 -3.5

7

8

9

10

11

-4.0

12

7

8

c

9

10

11

12

log10 (stress)

log10 (stress) -1.0

log10 (strain rate)

Fast Cure Acrylic -1.5

tR tR tR tR

-2.0

= 10 s = 50 s = 100 s = 300 s

-2.5 -3.0 -3.5

7

8

9

10

11

12

log10 (stress) Fig. 3. The Log10 (strain rate)-Log10 (stress) plot of steady-state creep with varying preloading times, tR: (a) PMMA; (b) HEMA and (c) fast-cure acrylic resin.

steady-state stage. This leads to the investigation of the stress exponent at the steady-state stage. Fig. 3(a)–(c) show the Log10e_ p vs. Log10r lines with slope n for the above three polymeric materials at different preloading rates. For a given stress and polymeric material, the strain rate increases with decreasing preloading time (or increasing preloading rate). It is also found that the range of Log10r function for the straight line (Log10 e_ p vs. Log10r) is wider for a shorter preloading time. The slopes of Log10 e_ p vs. Log10r lines (or the stress exponents) for the above polymers at different preloading rates obtained from Fig. 3 are listed in Table 3. The value of the stress exponent is almost independent of preloading time. The stress exponent ranging from large to small follows the sequence: PMMA, Fast-cure acrylic, and HEMA. As stated before, PMMA, the fast-cure acrylic, and HEMA are amorphous polymers. Further, HEMA is a highly crosslink and hydrophilic polymer. The stress exponents of the above polymeric materials are less than those of inorganic materials (Goodall and Clyne, 2006) by one order of magnitude. This is reasonable because the former is softer than the latter. The thermal drift was found in the indentation creep (Baker et al., 1992; Li and Ngan, 2004). The displacement of silicon wafer underneath indenter tip was found to be time independent for a given load at 25 °C. That is, no thermal drift was observed, and the effect of thermal drift was neglected in this study. The thermal activated process for dynamic loading during nano-indentation was studied by Cordill et al. (2009). They used the dislocation nucleation and motion to explain Table 3 Determination of exponent factor, n, of steady-state creep for PMMA, HEMA and fast-cure acrylic resin with different preloading periods, tR. Polymer

tR (s)

n

PMMA

10 50 100 300

0.479 0.461 0.456 0.460

HEMA

10 50 100 300

0.372 0.355 0.371 0.412

Fast-cureacrylic

10 50 100 300

0.405 0.419 0.425 0.434

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the stair-case yielding of single crystal nickels, which is different from the creep of polymeric materials with no dislocations. Because creep is a thermal activated process, the creep test at elevated temperatures is worthy studying in detail. The current commercial nano-indentation instruments cannot carry out the creep test at elevated temperatures because of thermal drift (Nieh and Lee, 2009). Thus, the high-temperature nanoindentation creep will be studied after the thermal drift problem is overcome. 5. Summary and conclusions Nanoindentation creep of polymeric materials, consisting of the primary and secondary stages, has been investigated. The total creep displacement comprises the viscoelastic and the power-law creep extension. The transient creep and steady-state creep are contributed by the viscoeleatic and the powr-law creep deformation, respectively. The viscoelastic displacement during the transient period is interpreted by the Burgers model, which is constructed by the Maxwell element connected in series with the Kelvin element. The solution of transient creep displacement was derived using the governing field equations of the elastic medium in time domain in analog to those of viscoelastic medium in Laplace transform coordinate system. On the other hand, the steady-state displacement was assumed to follow the power law for the strain rate and stress relation, where the strain rate is equal to the power-law creep displacement rate divided by power-law creep displacement, and the stress is the load over the characteristic cross-sectional area. The creep tests of three polymers, poly (methyl methacrylate), hydroxyethyl methacrylate copolymer, and fast-cure acrylic polymers, were performed using a nanoindentation tester. The salient findings of the present work are: 1. 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