Prediction of viscoplastic properties of polymeric materials using sharp indentation

Computational Materials Science 110 (2015) 321–330

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Prediction of viscoplastic properties of polymeric materials using sharp indentation Noriyuki Inoue a, Akio Yonezu a,⇑, Yousuke Watanabe a, Takeo Okamura b, Kouji Yoneda b, Baoxing Xu c,⇑ a

Department of Precision Mechanics, Chuo University, 1-13-27 Kasuga, Bunkyo, Tokyo 112-8551, Japan Material Analysis Department, NISSAN ARC, Ltd., 1 Natsushima-cho, Yokosuka, Kanagawa 237-0061, Japan c Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904, USA b

a r t i c l e

i n f o

Article history: Received 2 June 2015 Received in revised form 10 August 2015 Accepted 15 August 2015 Available online 8 September 2015 Keywords: Viscoplastic property Strain rate effect Polymer materials Indentation Reverse analysis

a b s t r a c t The present study proposes a new indentation method to predict the viscoplastic properties of polymeric materials utilizing three different indentation experiments. Numerical experiments with finite element method (FEM) are first carried out to simulate response of materials with various elastic and viscoplastic properties. The strain rate is featured by a loading rate of indentation, and the hardening rate can be captured through variations of the indenter shape. Next, a parametric FEM study is conducted to build the relationship between indentation load-depth curves and material parameters. Dimensionless analysis is employed to represent the relationship with simple formulae, through which, a reverse algorithm is proposed to extract the viscoplastic properties. Finally, both numerical and experimental investigations are performed to verify the proposed method. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction The ever-increasing use of polymeric materials with a small volume, such as coatings to achieve a desirable surface properties and thin substrates to achieve a high flexibility of wearable devices, requires appropriate characterization of their mechanical properties down to the nanoscale. Usually, almost all of polymeric materials feature time-dependent mechanical properties and in particular, the plastic properties (often referred to as viscoplasticity) are of interests to the researchers. Given the limits on their volume and scale in applications, the conventional tensile or compressive testing technique is very challenging in the accurate prediction of their mechanical properties. Instrumented indentation technique provides a compelling alternative with the merits of almost no requirements in both volume and scale of samples and accuracy in measurements, and has been widely employed to probe the mechanical properties like Young’s modulus and hardness [1,2]. Elastoplastic properties have been successfully derived from curves of indentation force and penetration depth and impressions (i.e. residual imprints) [3–5]. In these studies, the dimensionless function that correlates the parameters of indentation responses (mostly the indentation curves) with the elastoplas⇑ Corresponding authors. Tel.: +81 3 3817 1829; fax: +81 3 3817 1820 (A. Yonezu). Tel.: +1 434 924 1038; fax: +1 434 982 2037 (B. Xu). E-mail addresses: [email protected] (A. Yonezu), [email protected] (B. Xu). http://dx.doi.org/10.1016/j.commatsci.2015.08.033 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.

tic properties of materials is first established through forward analysis (i.e. the acquirement of indentation responses from serials of assumed mechanical parameters of materials) using computational approaches [6–9]. The elastoplastic properties of unknown materials can then be identified when the indentation responses obtained from practical experiments are assigned to the dimensionless function, which is a reverse analysis process (i.e. identification of material properties from indentation responses). To our knowledge, the reverse analysis may have a higher resolution than that of conventional analytical solution (i.e. hardness measurement and Tabor equation), because it does not require on the accurate estimation of indentation contact areas, and can use a robust representative strain [10,11]. Therefore, the instrumented indentation based on the reverse analysis has been acknowledged in the determination of the elastoplastic properties of materials. The evaluations of time-dependent mechanical properties of materials such as polymeric materials through indentation are mostly focused on viscoelasticities [12,13] and there are only a few of measurements on viscoplastic properties so far. As one of pioneering work, Bucaille et al. employed nanoindentation test to evaluate viscoplastic properties of polycarbonaite (PC) with finite element method (FEM) [14]. By matching the computational and experimental curves with the help of various FEM computations, they could quantitatively predict the stress–strain curve at large plastic deformation. However, their method is a typical forward analysis and relies on intense FEM computations. Kermouche et al. employed an analytical approximate solution by using

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classical indentation parameters (e.g. mean pressure, indentation strain rate and representative stress–strain) and developed a method for evaluating viscoplastic properties of glassy polymer [15]. Their method is relatively simple in comparison of the forward analysis and is expected to be applied to a wide range of materials. For robust estimations of viscoplastic properties of polymeric materials, dimensionless functions (established by parametric FEM study preliminarily) may be more powerful for exploring a simple, convenient and accurate approach and is worth of investigation. More recently, Peng et al. used dimensionless function to estimate the yield stress of polymeric materials (PVC) with an assumption of viscoelastic-perfectly plastic material [16]. However, it did not focus on the flow stress and non-linear work hardening behavior. The present study is establishing a new method to evaluate viscoplastic properties using an indentation method with the help of dimensional analysis. To focus on the study of viscoplasticity in indentation experiments, we employ a polymer material – polycarbonaite (PC) which has little viscoelasticity yet a very strong viscoplasticity with non-linear work hardening behavior. The present method aims to predict a constitutive law of stress–strain relationship up to a large plastic strain with the consideration of the strain rate effect. The process of estimation employs a reverse analysis based on the data of indentation loading curve, and three indentation experiments conducted using two sharp indenters with different angles and loading rates. First, numerical experiments with finite element method (FEM) are carried out in order to simulate indentation loading curves against the materials with various viscoplastic properties. Next, a dimensionless function which can correlate the materials properties with the indentation loading curves is established in the reverse analysis. Finally, the proposed method is verified through numerical and experimental results. 2. Materials and the mechanical constitutive law Three types of polymeric materials, engineering glassy polymer materials, polycarbonate PC, polymethylmethacrylate PMMA, and acrylonitrile butadiene styrene ABS, are investigated in this study. Fig. 1 shows their typical compressive stress–strain curves. These tests were conducted in our laboratory (details please see Section 5.2). In Fig. 1, during the elastic deformation, all of their stress–strain relationships are almost linear. At the yield point, the stress shows a peak, and then slightly decreases when plastic deformation occurs. Plastic flow stress remains almost constant at first, and increases beyond a critical plastic strain. This behavior is well known to be dependent on strain rate. A high strain rate will promote the stress, whereas a low strain rate will decreases the

True stress σt, MPa

150

Strain rate ε 10-4s-1

:PMMA :PC :ABS

120 90

Table 1 Mechanical property of PC, PMMA and ABS obtained with uniaxial compression test. Material

PC PMMA ABS

Young’s modulus

Poisson’s ratio

Material constant

E (GPa)

m

K (MPa sm)

hg

m

2.4 4.2 2.5

0.35 0.35 0.35

70 142 63

1.15 0.70 0.40

0.030 0.055 0.038

flow stress. Fig. 2 shows the stress–strain curves of PC with different strain rates. The duration of plastic deformation strongly depends on the strain rate, yet not for the elastic deformation. This study will focus on plastic deformation with strain rate effect. This non-linear plastic deformation can be approximately described by fitting their corresponding curves of stress–plastic strain, as shown in Fig. 2 and the full plastic deformation can be expressed by the following constitutive law.

r ¼ K  ehg
ep 
e_ m 2

ð1Þ

In Eq. (1), ep is the plastic strain and
e_ is the strain rate. Note that this study ignores stress softening phenomena around the yield point (i.e. peak stress) of relevance to changes of molecular structure. This simplification is developed from the G’sell-Jonas law [17], and has been used by Kermouche et al. [15]. and Bucaille et al. [14]. In addition, the materials constant, m represents the strain rate sensitivity, K is the consistency parameter, and hg is the strain hardening modulus. These three material constants will be identified from indentation experiments in this study. In Fig. 2, three curves with different strain rates ð
e_ ¼ 103 —105 Þ can be fitted well by Eq. (1), and the resultant materials constants are listed in Table 1. Similarly, the other materials PMMA and ABS are investigated and their material parameters are also included in Table 1. Note that elastic property (Young’s modulus E and Poisson’s ratio m) is from the previous studies [14,15,18] and is assumed to be known in advance. 3. Numerical analysis

60

3.1. FE model

30 0

Fig. 2. True stress–true strain behavior in uniaxial compression test for PC at strain rate of 103, 104 and 105 s1.

0

0.15

0.3

0.45

0.6

True strain εt Fig. 1. True stress–true strain behavior in uniaxial compression test for PC, PMMA, and ABS at strain rate of 104 s1.

Fig. 3 shows the FE model of an indentation test with a conical sharp indenter. This study uses two types of conical indenters with different apex angles. The sharper angle of indenter is called ‘‘Indenter A”, whose half apex angle h is 30°, and the other ‘‘Indenter B” has h = 70.3°. These indenters are employed to mimic triangle pyramidal indenters, whose diagonal angle a is 70° for ‘‘Indenter

N. Inoue et al. / Computational Materials Science 110 (2015) 321–330

323

Fig. 3. Axi-symmetric two-dimensional FE model for indentation test.

A”, and is 115° for ‘‘Indenter B” (i.e. Berkovich indenter). In this study, indentation experiment used the triangle pyramidal indenter, while the indentation computation (FEM) used the sharp conical indenter for simplicity of computations (in order to approximate an axisymmetric model). Note that the tip of Indenter A is rounded a little bit and the tip radius is assumed to be 200 nm in this FEM model (Fig. 3). Such values (order of a few hundred nm) may be generally used when a brand new sharp indenter is considered. The two-dimensional axisymmetric FE model was created to compute the response of the indentation test for saving computation cost, as shown in Fig. 3. The model comprises about 30,000 four-node elements, wherein fine meshes were used around the contact region and mesh convergence test was conducted. The model is very large against the indenter contact area, and can be assumed to be semi-infinite model. The bottom of the model is fixed along the direction of indenter penetration. As mentioned above, the conical indenters, whose half apex angle is 30° (‘‘Indenter A”) and 70.3° (‘‘Indenter B”), were employed for the present two-dimensional model. In the FEM model, the indenter is assumed rigid. Computations were carried out using the commercially available software, Marc (MSC, Marc 2011 [19]). Coulomb’s law of friction was assumed, and the coefficient of friction was m = 0.15, which is often employed in indentation analysis [8,20]. The material has stress free and isotropic mechanical properties. Note that as described later, since this material has time dependent mechanical properties, indentation loading rate will affect indentation curve. In the present FEM, the loading rate F_ is set to

Table 2 Mechanical property combination used in the FEM parametric study. E (GPa)

K (MPa sm)

hg

m

1.0 2.4 4.5

50 94 170

0.1 0.38 0.8

0.01 0.047 0.09

1.47 mN/s (fast rate) and 0.0147 mN/s (slow rate), which is the same with experiments (see Section 4.1). The material properties used in this study are shown in Table 2. Since this study addresses typical engineering polymers, Young’s modulus E is set to 1.0, 2.4 and 4.5 GPa, the consistency K is 50, 94, and 170, the strain hardening modulus hg is 0.1, 0.38 and 0.8, the strain rate sensitivity m is 0.01, 0.047 and 0.09. The number of materials combination is 81 as a total. 3.2. Representative computational result This study first examines the effect of viscoplastic properties (K, hg, and m in Eq. (1)) on the indentation curve. As an example, we set the elastic properties of E = 2.4 GPa and m = 0.35, and change viscoplastic properties in FEM computations. This section uses two indenters (‘‘Indenter A” and ‘‘Indenter B”) and two loading _ Fig. 4 shows the dependency of K on the indentation curve, rates F. indicating the indentation force increases with increasing K for both indenters. A larger K (in Eq. (1)) corresponds to a higher yield stress (flow stress), thus leading to a large indentation force. The parameter hg (in Eq. (1)) affects the indentation deformation with the employment of the sharper indenter (‘‘Indenter A”), as shown in Fig. 5(a). On the other hand, ‘‘Indenter B” (Berkovich indenter) does not produce any difference in indentation curves (Fig. 5(b)). Since the strain hardening modulus hg reflects non-linearity of plastic deformation, a larger hg increases flow stress at a large strain. Therefore a sharper indenter associated with a large representative strain may be sensitive to the hg value (as shown in Fig. 5(a)). In addition, Fig. 6 shows the dependency of the strain rate sensitivity m on indentation curve. The sharper ‘‘Indenter A” was used and two tests with different force rates were conducted. The indentation curves are changed, as the indentation loading rate F_ varies. A larger m indicates that the effect of strain rate is large in the stress–strain curve, leading to an increased variation between loading curves at two loading rates (slow rate vs. fast rate), as shown in Fig. 6(c).

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20

K=170

K=170 2 1.5

Indentaon force F ,mN

Indentaon force F ,mN

2.5

K=94 K=50

1 0.5

α=70 0 0

0.5

1

1.5

K=94

15

K=50 10

5

α=115 0 0

2

Penetraon depth h , m

0.5

1

1.5

2

Penetraon depth h , m

(a)

(b)

Fig. 4. Effect of parameter K on indentation curve for Indenter A (a) and Indenter B (b).

2

15

hg=0.8

Indentaon force F ,mN

Indentaon force F ,mN

hg=0.8 hg=0.38

1.5

hg=0.1 1

0.5

0

α=70 0

0.5

1

1.5

2

hg=0.38 10

hg=0.1

5

0

α=115 0

0.5

1

1.5

Penetraon depth h , m

Penetraon depth h , m

(a)

(b)

2

Fig. 5. Effect of parameter hg on indentation curve for Indenter A (a) and Indenter B (b).

4. Reverse algorithm for the determination of viscoplastic parameters

rate. The equivalent strain rate e_ can be expressed by the following equation [21]:

4.1. Definition of strain rate sensitivity

e_ ¼ 0:12 

This study aims to evaluate viscoplastic property, which is strongly dependent on the strain rate e_ , as described in Eq. (1). In other words, the strain rate sensitivity m will be identified via indentation testing. In general, the applied strain rate e_ is constant during uni-axial tensile test, and then it is desirable that the strain rate during indentation test (called indentation strain rate) is constant. Over the past decade, many studies on indentation strain rate have been conducted. The indentation strain rate is usually _ defined as h [21,22], where h is the indentation depth, and h_ is h

the indentation depth rate (dh/dt). We assume that the constancy of

h_ h

makes the indentation strain rate to be constant. Since the sharp conical indenter (e.g. Berkovich indenter) is selfsimilar in geometry, its indentation curve can be described by the Kick’s law (F = Ch2), where F is the indentation force, h is the indentation depth and C is the materials constant. With the differentiation by time, the relationship between force rate F_ and depth rate h_ can be obtained as follows:

h_ F_ ¼2 h F

ð2Þ _

Therefore, it is desirable that FF needs to be set to be constant in indentation experiment to make an constant of indentation strain

h_ h

ð3Þ

However, the present indentation equipment cannot work with the constant of strain rate (i.e.

F_ F

and

h_ h

are not constant). As described later, the present experiment set the loading rate F_ be constant instead. Therefore, we investigate how the strain rate e_ is changed during the present indentation with the constant of _ Fig. 7 shows the changes in strain rate (from loading rate F.

Eq. (3)) during the indentation loading with the loading rate F_ of 1.47 mN/s. The material is assumed to be PC (see Table 1) as an example case. It is found that the strain rate e_ is not constant, i.e. the strain rate exponentially decreases with respect to the indentation depth. The magnified figure shows the range of indentation depth of around 2 lm. As an example case, this study focuses on the indentation force value at h = 2 lm (referred to as F(h=2lm)) to calculate the uniaxial strain rate e_ in the analysis. Therefore, by using Eqs. (2) and (3), the equivalent strain rate e_ can be derived. This equivalent strain rate e_ and the value of F(h=2lm) are used to establish the present estimation method in the next section. 4.2. Dimensionless function and reverse analysis method As mentioned above, many studies have been done on the estimation of elastoplastic properties from the viewpoints of dimen-

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2

F =1.47mN/sec

Indentaon force F ,mN

Indentaon force F ,mN

2

F =0.0147mN/sec

1.5

1

0.5

m=0.01 0

0

0.5

1

1.5

F =1.47mN/sec 1.5

1

0.5

m=0.047 0

2

Penetraon depth h , μm

F =0.0147mN/sec

0

0.5

1

1.5

2

Penetraon depth h ,μm

(a)

(b)

Indentaon force F ,mN

2

F =1.47mN/sec 1.5

F =0.0147mN/sec

1

0.5

m=0.09 0

0

0.5

1

1.5

2

Penetraon depth h ,μm

(c) Fig. 6. Effect of parameter m on indentation curve for Indenter A (m = 0.01 (a), 0.047 (b), 0.09 (c)).

0.5

0.05 0.04

Strain rate s-1

0.4

0.03 0.02

0.3

0.01

0.2

0 1

1.25

1.5

1.75

2

0.1 0

0

0.5

1

1.5

2

2.5

Depth h μm Fig. 7. Example of relationship between strain rate and indentation depth during loading with constant loading rate.

sionless function [3,7,8,23]. Such a simple function may be useful for use in the reverse analysis. Similarly, the present study conducts dimensionless analysis with the aim of developing a new estimation method of viscoplastic properties such as material parameters (K, hg, m) in Eq. (1). In other words, the dimensionless function which correlates the indentation curve with material parameters to be identified (K, hg, m) will be established through the parametric FEM study. Since there are three (K, hg, m) variants to be identified, it requires three independent parameters in an indentation curve. Given the indentation depth at h = 2 lm (F(h=2lm)), the indentation force can be expressed by

_ a; h2lm Þ F h¼2lm ¼ f ðE ; K; hg ; m; F;

ð4Þ

F

lm Fig. 8. Relationship between r h2 2 r h2lm rate = 1.47 mN/s and 0.0147 mN/s.



and rE r

for Indenter A with loading

Dimensionless analysis with P theory dictates the following relationship.

F h¼2lm

rr  h2ð2lmÞ


  E _ ¼P ; F; a

rr

ð5Þ

In general, the representative stress rr and strain er [3] is used to reduce the number of material parameters. In this study, the viscoplastic parameter (K, hg, m) is converted to be rr via Eq. (1). That is, by employing various er, rr can be calculated from Eq. (1) repeatedly, through which we attempt to reduce the valuable dependency of dimensionless function and establish simple dimensionless function as we discuss below. For this calculation,

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the strain rate is required and obtained from Eq. (3). In Eq. (5), since the loading rate F_ is constant and the angle of indenter tip a is fixed, Eq. (5) degenerates to

F h¼2lm

rr  h2ð2lmÞ


 E ¼P

ð6Þ

rr

Here, E* is the reduced Young’s modulus and can be expressed by  2 1m2  1m 1 ¼ Es s þ E i , where the subscript ‘‘s” and ‘‘i” represent the specE i

imen and indenter, respectively.

To obtain three unknown parameters (K, hg, m), three independent functions are established in this study. We use two indenters (‘‘Indenter A” and ‘‘Indenter B”), and two indentation force rates (F_ ¼ 1:47 mN=s and 0.0147 mN/s). Note that the sharper ‘‘Indenter A” is used with two force rates and ‘‘Indenter B” is with only F_ ¼ 1:47 mN=s. Thus, the indentation test is conducted with three types. Since there are 81 cases (in Table 2), the number of parametric FEM study will be 243 (81  3 times). The data of parametric FE studies are introduced into the Eq. (6). The results of ‘‘Indenter A” and ‘‘Indenter B” are separately shown in Figs. 8 and 9. Fig. 8 shows two results under different indentation force rates (indicated by s and 4). For both figures, the r hF2 increases monotonically with rEr . With the representative r

max

stress rr, the data of r hF2 is only dependent on rEr (as described in r max Eq. (6)). Therefore, the data of r hF2 can be approximated by a simr

max

ple function:

F h¼2lm

rr  h2ð2lmÞ

F

lm Fig. 9. Relationship between r h2 2 r h2lm rate = 1.47 mN/s.



and rE r

for Indenter B with loading

Table 3 Coefficient of dimensionless function for Indenter A and Indenter B at loading rate of 1.47 mN/s and 0.0147 mN/s. Loading rate

A1 A2 A3 Representative strain er Correlation coefficients R

Indenter A

Indenter B

1.47 mN/s

0.0147 mN/s

1.47 mN/s

0.0491 1.65 0.768 0.56 0.991

0.0634 1.76 0.974 0.56 0.990

1.83 8.83 12.5 0.07 0.999


¼ A1 ln

E

rr

2


þ A2 ln

E

rr

 þ A3

ð7Þ

The coefficients A1–A3 are listed in Table 3, where the coefficients for each function are shown individually with their correlation coefficients R. In addition, the representative strain (which we selected for the highest R) is also plotted. These dimensionless functions (Eq. (7) and Table 3) yield a serials of representative stress rr and strain er, individually. Thus, three different representative points can be derived once we experimentally obtain three values F(h=2lm) from three different indentation experiments. Note that elastic properties are assumed to be known a priori. Fig. 10 gives the flowchart of our reverse analysis method. Three indentation tests with ‘‘Indenter A” and ‘‘Indenter B” are carried out. With the sharper indenter (‘‘Indenter A”), two tests at indentation force rates of F_ ¼ 1:47 mN=s and 0.0147 mN/s are carried out,

while ‘‘Indenter B” is conducted with the rate of F_ ¼ 1:47 mN=s. We extract the values of F (at 2 lm depth) and substitute to the dimensionless functions (Eq. (7)). As indicated by Step r and Step s in Fig. 10, each data (rr and er) from ‘‘Indenter A” with different loading rates is substituted to Eq. (1), leading to the strain rate

Fig. 10. Flowchart of present method for extracting the viscoplastic properties from three indentation tests.

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N. Inoue et al. / Computational Materials Science 110 (2015) 321–330

Error of K %

(a)

Table 4 Representative materials for sensitivity analysis. Symbol (Fig. 12)

Material

s h } 4

Material Material Material Material Material

r

1 2 3 4 5

E (GPa)

K (MPa sm)

hg

m

1.0 2.4 4.5 1.0 4.5

50 94 170 94 50

0.8 0.38 0.8 0.8 0.1

0.047 0.047 0.09 0.01 0.09

hg Table 5 Case study for sensitivity analysis of representative materials. Case study

F70°1.47 mN/s (%)

F70°0.0147 mN/s (%)

F115°1.47 mN/s (%)

Case 1 Case 2 Case 3

0 +4 4

0 +4 4

0 +4 4

K ⋅ε m E 1.5

Difference of hg

(b)

1.2

hg

hg

1, 2, 3, 4, 5 :Input

0.9

:Case 1

0.6

:Case 2

0.3

:Case 3

0 0

0.03

0.06

0.09

0.12

0.15

K ⋅ε m E Fig. 12. Comparison between the input value and the estimation from reverse analysis for representative materials for sensitivity analysis.

K ⋅ε m E

Error, % 3.46

200

Case 1

(c)

2.595

150

1.73

100

0.865

50

Case 2 Case3

δ

0

0

1

2

3

4

5

6

Material code Fig. 13. Error quantity d of representative materials for sensitivity analysis (Table 4) for Table 5.

5. Validation of the reverse analysis method 5.1. Numerical experiment Fig. 11. Contour map of error distribution for K (a) and differences (input value – estimation value) of hg (b) and m (c).

sensitivity m. In parallel, by using different ‘‘Indenter A” and ‘‘B” (with the same loading rate), Steps s and t give us the hardening parameter of K and hg. In conclusion, the present method directly identifies three parameters (K, hg, m) for characterizing viscoplastic properties of materials.

The present method is now applied to the numerical experiment in order to estimate the accuracy and robustness of our method. As mentioned above, parametric FEM study (Table 2) is carried out to establish the dimensionless functions. Reversibly, those data are employed to estimate the viscoplastic properties (i.e. parameters K, hg, and m) by using the proposed method mentioned above (Fig. 10). For all cases, we extract the indentation

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28

1.47mN/sec 0.0147mN/sec

9

Indentaon force F ,mN

Indentaon force F ,mN

12

α=70

6

3

0

0

2

4

6

α=115

21

14

7

0

8

0

1

2

3

Penetraon depth h , m

Penetraon depth h , m

(a)

(b)

4

Fig. 14. Indentation curves of polycarbonate (PC) from Indenter A (a) and Indenter B (b).

45

1.47mN/sec 0.0147mN/sec

9

Indentaon force F ,mN

Indentaon force F ,mN

12

α=70 6

3

0

0

1.5

3

4.5

6

α=115

36 27 18 9 0

0

0.6

1.2

1.8

2.4

Penetraon depth h , m

Penetraon depth h , m

(a)

(b)

3

3.6

Fig. 15. Indentation curves of polymethyl methacrylate (PMMA) from Indenter A (a) and Indenter B (b).

force value at 2 lm depth (Fh=2lm) from the three different indentation experiments (‘‘Indenter A” with loading rate F_ ¼ 1:47 mN=s and 0.0147 mN/s and ‘‘Indenter B” with F = 1.47 mN/s), and then estimate the material properties. We investigate the accuracy of estimated properties (K, hg, m) by the method as shown in Fig. 11. Note that this figure indicates m the plane of hg  KEe_ to express 81 materials combination (in Table 2) two-dimensionally. On this plane, the error of K is shown in Fig. 11(a), and the difference from the input value for hg and m is shown in Fig. 11(b) and (c). All calculations show a small error, suggesting the relatively good agreement with the input data, which is similar to that of a previously developed estimation methods of elastoplastic properties with reverse analysis [5,6,24–26]. This indicates that the present algorithm is fairly reliable and has satisfactory accuracy in our materials range. Another concern for experimental estimations is the perturbation of indentation response. When we experimentally conduct an indentation test in a laboratory, uncertainties in the experimental indentation responses are usually inevitable due to several factors related to the materials and indentation measurement equipment. A number of previous studies have investigated the robustness of the method, which is a key issue for accuracy. In other words, it is important to investigate the sensitivity of the determined properties to variations in the input data, and to clarify how the input data affects these properties [27,28]. In this study, we conduct sensitivity analysis for five representative materials (in Table 4). The perturbation cases of the indentation response

(input data) are set as shown in Table 5. In fact, for the indentation curve, the perturbations are generally set to 4%, since previous studies have used this value for sensitive analysis [29]. These perturbation cases (designated as Case 1, Case 2, Case 3) are shown in Table 5. The deviations (errors in the estimated values compared to the input values) are shown in Fig. 12. Similar with Fig. 11, the plane of m hg  KEe_ to express our materials combination is shown in Fig. 12. The estimations show reasonable agreement and robust for all perturbation cases. Furthermore, to confirm the robustness of the present sensitivity analysis, the following parameter d is investigated.

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ! u
ts
ts  ts ps 2 u K  K ps 2 ps 2 h  h m  m g g d¼t þ þ ts mts K ts hg

ð8Þ

By referring the previous study [27,28], the quantity d1 reflects a measurement difference between the true and estimated material properties with a perturbation. The present viscoplastic properties are K, hg and m. These parameters are denoted by Kts, hg ts and mts, where subscript ‘‘ts” indicates the true solutions, and by Kps, hg ps and mps, where subscript ‘‘ps” indicates the perturbed estimations. Note that this d represents an ‘‘overall error”, since it includes all 1 Note that for the sensitivity analysis the quantity of d (Eq. (8)) is ‘‘Uniform Factors” scheme [27,28] which gives a reasonable estimate of the reliability of the proposed method. However, a Monte Carlo Analysis would be required for an indepth examination [27,28].

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45

1.47mN/sec 0.0147mN/sec

9

Indentaon force F ,mN

Indentaon force F ,mN

12

α=70 6

3

0 0

2

4

6

8

α=115

36 27 18 9 0 0

0.9

1.8

2.7

3.6

Penetraon depth h , m

Penetraon depth h , m

(a)

(b)

4.5

Fig. 16. Indentation curves of acrylonitrile butadiene styrene (ABS) from Indenter A (a) and Indenter B (b).

Table 6 Error and difference between reference value obtained from compression test and the properties estimated by presented method. Material

Reference value from compression test K (MPa sm)

PC PMMA ABS

Identified value (error% & difference)

hg

m

K (MPa sm)

hg

m

70

1.15

0.030

142

0.70

0.055

63

0.40

0.038

82 (17.1%) 177 (24.6%) 103 (63.5%)

1.09 (0.06) 0.86 (0.16) 0.70 (0.30)

0.027 (0.003) 0.090 (0.035) 0.080 (0.042)

these three parameters (K, hg and m). By using this error parameter d, Fig. 12 can be replotted to Fig. 13, indicating a good robustness of our proposed method. Several previous studies addressing sensitivity analysis report that their methods have 50% error in maximum [29]. This indicates that the present method has relatively better performance for perturbations due to potential experimental errors. Therefore, the actual experimental situation will be addressed in the next section. 5.2. Experimental investigation 5.2.1. Experimental conditions Experimental validations were carried out. This study used micro indentation equipment (Dynamic Ultra-micro Hardness Tester DUH-211 and DUH-510S, Shimadzu Corp.). Similar with the above FE computation, we prepared two sharp diamond indenters. The sharper indenter (‘‘Indenter A”) corresponds to the triangle pyramidal indenter with diagonal angle of 70°. On the other hand, ‘‘Indenter B” is a regular Berkovich indenter with diagonal angle of 115°. This study sets two loading rates, referred to as the fast rate F_ ¼ 1:47 mN=s and slow rate 0.0147 mN/s. The materials used in this study are polycarbonate PC, polymethylmethacrylate PMMA, and acrylonitrile butadiene styrene ABS. The specimen is disk shape, whose diameter is 10 mm and thickness is 2 mm. The specimen surface was mechanically polished before the indentation experiment. Subsequently, heat treatment for annealing (over Tg: glass-transition point) was conducted in order to reduce the residual stress and strain. To verify our estimation from indentation, uni-axial compression test was carried out. This study prepared the polymer materials (PC, PMMA, ABS) with round bar, and the diameter is 10 mm and height is 10 mm. Electro-hydraulic servo testing equipment (Shimadzu corp. EHF-EB50KN-10L) was used for uniaxial compression tests in room temperature (RT). The compressive strain was

measured by two eddy current sensors (EX-201 and EX-305, Keyence) and stroke sensor of this equipment. For all specimens, three different strain rates were used i.e. e_ ¼ 103 ; 104 ; and 105 in order to obtain viscoplastic property through the Eq. (1). The result is summarized in Table 1. 5.2.2. Experimental results Indentation curves of PC, PMMA and ABS are shown in Figs. 14–16, respectively. For each test, the number of tests was more than ten times, and the averaged curves are plotted in this figure. In those figures, the result of ‘‘Indenter A” is shown in (a) and that of ‘‘Indenter B” is shown in (b). These indicated that ‘‘Indenter A” shows a little dependency of loading rate in the indentation curves. We extracted the three values of indentation force F(h=2lm) from each experiment (Indenter A: F_ ¼ 1:47 mN=s

and 0.0147 mN/s, Indenter B: F_ ¼ 1:47 mN=s). These values are substituted to the present method as shown in Fig. 10, yielding the viscoplastic properties (K, hg, m). The estimations are plotted 1.5

:Reference value :Idenfied value

1.2

hg

0.9

:PC :PMMA :ABS

0.6 0.3 0 0

0.009 0.018 0.027 0.036 0.045

K ⋅ε m E Fig. 17. Comparison between reference values obtained from compression test and the property estimated by presented method.

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in Table 6 and Fig. 17. The comparison with the viscoplastic properties obtained from uni-axial compression test shows good agreement. Thus, our method is verified in the estimation of viscoplastic properties of polymer materials. This method requires three tests with two different indenters and loading rates. The elastic property is assumed to be known. By substituting the experimental data into the three dimensionless functions, the viscoplastic properties are directly obtained. Thus, our method is a relatively simple framework when compared with the previous one (e.g. [14]). Note that our method relies on a specific condition such as loading rate, indenter shape and indentation force, which may apply some limits to indentation experiments in practice. However, by using a present concept, we may establish more universal method which will be addressed in the future. 6. Conclusions This study proposes a new indentation method to predict the viscoplastic deformation behavior of polymer materials. This method is a reverse analysis on the basis of simple dimensionless functions. The viscoplastic behavior of the present material is expressed by a non-liner constitutive equation with strain rate coupled, which is similar to G’sell-Jones law. This law has three materials parameter (K, hg, m) to model large strain-hardening behavior up to large strain. In this study, the reverse analysis based on the data of indentation loading curve is established. Indentation experiments are conducted three times using two sharp indenters with different angles and loading rates. Numerical experiments with finite element method (FEM) were first carried out to simulatewrd an indentation loading curve against the material with various material properties, i.e. elastic and viscoplastic properties. It is found that the indentation loading curve depends on the material properties. In particular, the strain rate effect arises by the changing of loading rate of indentation, while the work hardening rate can be captured by a sharp indenter. Next, a parametric FEM study by changing of the material properties which covers various polymer materials is conducted to deduce the relationship between indentation loading curve and material parameters. This relationship can be expressed by dimensionless functions with simple formulae. Finally, our method is verified through numerical and experimental investigations. In the experiments, polycarbonate PC, polymethylmethacrylate PMMA, and acrylonitrile butadiene styrene ABS are employed for indentation test. The estimated viscoplastic

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