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Research Papers

# Prediction of Asymmetric Yield Strengths of Polymeric Materials at Tension and Compression Using Spherical IndentationOPEN ACCESS

[+] Author and Article Information
Noriyuki Inoue, Yousuke Watanabe

Department of Precision Mechanics,
Chuo University,
1-13-27 Kasuga, Bunkyo,
Tokyo 112-8551, Japan

Akio Yonezu

Department of Precision Mechanics,
Chuo University,
1-13-27 Kasuga, Bunkyo,
Tokyo 112-8551, Japan
e-mail: yonezu@mech.chuo-u.ac.jp

Hiroshi Yamamura

Department of Integrated Science
and Engineering for Sustainable Society,
Chuo University,
1-13-27 Kasuga, Bunkyo,
Tokyo 112-8551, Japan

Baoxing Xu

Department of Mechanical and
Aerospace Engineering,
University of Virginia,
Charlottesville, VA 22904
e-mail: bx4c@virginia.edu

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received May 5, 2016; final manuscript received September 7, 2016; published online February 1, 2017. Assoc. Editor: Xi Chen.

J. Eng. Mater. Technol 139(2), 021002 (Feb 01, 2017) (11 pages) Paper No: MATS-16-1133; doi: 10.1115/1.4035268 History: Received May 05, 2016; Revised September 07, 2016

## Abstract

Engineering polymers generally exhibit asymmetric yield strength in tension and compression due to different arrangements of molecular structures in response to external loadings. For the polymeric materials whose plastic behavior follows the Drucker–Prager yield criterion, the present study proposes a new method to predict both tensile and compressive yield strength utilizing instrumented spherical indentation. Our method is decomposed into two parts based on the depth of indentation, shallow indentation, and deep indentation. The shallow indentation is targeted to study elastic deformation of materials, and is used to estimate Young's modulus and yield strength in compression; the deep indentation is used to achieve full plastic deformation of materials and extract the parameters in Drucker–Prager yield criterion associated with both tensile and compressive yield strength. Extensive numerical computations via finite element method (FEM) are performed to build a dimensionless function that can be employed to describe the quantitative relationship between indentation force-depth curves and material parameters of relevance to yield criterion. A reverse algorithm is developed to determine the material properties and its robustness is verified by performing both numerical and experimental analysis.

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## 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 an appropriate characterization of their mechanical properties down to the nanoscale. Usually, most of polymeric materials exhibit different plastic behavior against a loading mode (e.g., tension versus compression). This difference is due to molecular structural changes subject to loading directions, and usually the compressive yield strength is larger than that of in tension [15]. For materials integrity, it is important to estimate the yield strength, especially the mechanical responses under a tensile loading mode. However, uniaxial tensile test for small polymeric materials is usually difficult, and an alternative mechanical testing is highly needed in the industry.

Indentation test that only requires a small volume of materials becomes a convenient way to probe mechanical properties at a small scale. Many studies have been done on estimation of mechanical properties, such as Young's modulus [6,7], elastoplastic property [8], fracture [9] and their time-dependent behaviors [10] such as creep [11] and fatigue [1215]. In these methods, an indenter with a very high stiffness (e.g., diamond or hard metal) is typically impressed on a target material, and the loading—penetration depth curve (commonly referred to as indentation curve) is recorded continuously to estimate mechanical properties. The loading mode is essentially compressive loading to ensure a continuous penetration of indenter, and such an indentation curve may relate to compressive deformation behavior, and indentation-based measurement for polymeric materials may characterize a compressive deformation behavior, including yield strength and plastic flow stress under compressive loading. As mentioned previously, tensile yield strength may also be required for materials integrity for polymers.

For engineering metals and steels, since the Mises yield criterion is generally used, and the tensile and compressive yield strengths are identical, the determination of yield strength through indentation is used to evaluate mechanical property in tension. On the contrary, for engineering glassy polymers, it is well known that the difference of yield strength between in tension and compression comes from molecular chain arrangement and the microscopic deformation mechanisms that are dependent on the hydrostatic stress level [3,4,16], the Drucker–Prager yield criterion is often considered. So far, there are several studies on indentation mechanics against the Drucker–Prager yield criterion [1721]. In fact, as it is the case of some ceramics [22], bulk metallic glasses [23,24], and polymers [2527], their indentation responses exhibiting hydrostatic pressure-sensitive plastic behavior have been studied. Among these studies, Mai et al. established method to determine the plastic properties of pressure-sensitive plastic materials from the load–displacement curves [27]. However, they conducted three indentation tests with two different indenters (Berkovich indenter and spherical indenter), and it may need some efforts for a testing. In addition, the deformation behavior of polymeric material usually exhibit strain rate sensitivity. If they need several different tests, it may be difficult to set the strain rate (indentation loading rate) constant for those tests. One indentation test (so called single indentation test) as well as fixed indenter shape may be favored to establish a new framework to extract parameters of the Druker–Prager yield criterion, such that we estimate yield strengths in both compression and tension simultaneously.

The present study is establishing a new method to evaluate yield strengths in tension and compression for polymeric material using a spherical indentation with the help of dimensional analysis. The considered material obeys the Druker–Prager yield criterion, exhibiting plastic deformation behavior for general engineering polymers [5,27]. Our estimation process employs a reverse analysis based on the data of indentation loading curve, using a spherical indentation. Our method includes two parts based on the depth of indentation, i.e., shallow indentation and deep indentation. Shallow penetration (shallow indentation) is for elastic deformation, and deep indentation is for full plastic deformation. The former part is to estimate Young's modulus and compressive yield strength. The latter part is to predict the parameters of Drucker–Prager yield criterion, resulting in both yield strengths of compression and tension. The latter method was established by extensive numerical computations with FEM to build the relationship between indentation load-depth curves and material parameters for yield criterion. In other words, a dimensionless function which can correlate the materials properties with the indentation loading curves is established for the reverse analysis. Finally, the proposed method is verified through numerical and experimental results. The plastic properties of PC, and polymethylmethacrylate (PMMA) in bulk form are determined using the proposed method, and are further compared to those directly measurements from the conventional tensile and compression tests. In addition, the strain rate effect in estimation is discussed.

## Materials and Experimental Tests

This study employed two types of engineering glassy polymer materials, PC, and PMMA. At first, uniaxial loading tests were carried for all materials. Tensile tests were performed by using a ball-screw-type universal testing machine (AG-1, Shimadzu Corp.), while compression tests were conducted with electro-hydraulic servo testing equipment (Shimadzu Corp. EHF-EB50KN-10L). The specimen size is a cylinder shape with 10 mm diameter and 10 mm height for the uniaxial compression test; the tensile test was conducted using dumbbell-like plate specimen with rounded junctions, and the width is 10 mm, the gage length is 60 mm, and the thickness is 2 mm. All tests were conducted in the laboratory at room temperature (23 °C).

Figure 1 shows their typical stress–strain curves in tension and compression. Note that their strain rates were set to 10−3 1/s. In Fig. 1, the elastic deformation of all materials shows a linear relationship in the stress–strain curves. Around the yield point, the linear relationship gradually disappears, and it reaches a peak stress. After that, the stress slightly decreases and plastic deformation promotes. In Fig. 1(a), Young's modulus E is measured to be 2.2 GPa for tensile test and 1.7 GPa for compressive test. The averaged value is about 2.0 GPa, and is drawn by the solid line, which is plotted on the experimental data. Note that our measured modulus (E = 2.0 GPa) reasonably agrees with E = 2.4 GPa of previous studies [2830] (in which Poisson's ratio ν is 0.35).

The deviation point from initial linearity may correspond to the yield point where plastic deformation starts. However, it is pretty common in polymers to assimilate the yield stress to the maximum stress [3,27].1 By using this way, the yield strength in tension σYt and compression σYc, was determined. Note that the experimental data of PMMA was referred from the previous study [27]. It is found that all materials show different yield strengths in between tension and compression. Such a difference in yield strength will be introduced for the mechanical model (yield criterion) as discussed later (refer to Sec. 4.1).

It is reported that both yield point and plastic flow stress are dependent on strain rate. A high strain rate will promote the stress, whereas a low strain rate will decrease the flow stress [28,31]. On the contrary, elastic deformation is barely independent of strain rate [31]. Thus, several tests with different strain rate were conducted in order to investigate the strain rate effect on the yield stresses (σYt and σYc) for all materials. Figure 2 shows the changes in both tensile and compressive yield stresses (σYt and σYc) by the applied strain rate. It is found that their yield stresses are strongly dependent on strain rate. In addition, this relationship shows linearity in the present range of strain rate. As discussed later, this relationship will be used when compared with present indentation estimation.

###### Indentation Experiments.

The micro indentation equipment (Dynamic Ultra-Micro Hardness Tester DUH-510 S, Shimadzu Corp.) was used in all the present indentation tests. To realize an elastic indentation deformation, we prepared a large spherical diamond indenter, and its radius is 500 μm. This study set loading rate of $F˙$  = 207 mN/s. Subsequently, to achieve full plastic deformation, small spherical indenter with tip radius of 20 μm is used. The loading rate is set to $F˙$  = 2.28 mN/s. The test number is more than ten, and the averaged data is used for estimations. All tests were conducted at room temperature (23 °C).

The materials used in this study are PC and PMMA. The specimen has a disk shape, and its 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.

## Determination of Elastic Modulus and Compressive Yield Strength

###### Theory.

When a spherical rigid body (spherical indenter) contacts on a half-space with isotropic and homogenous elastic solid, it is well known that small contact deformation can be described by Hertzian contact theory [32], i.e., Display Formula

(1)$F=4E*3Rh32$

F is the applied force (indentation force), R is the radius of spherical indenter, h is the penetration depth, and E* is the reduced modulus that is described by $1/E*(1/E*)=((1−νs2/Es)+(1−νi2/Ei))$ (Note that Young's modulus is E and Poisson's ratio is ν, where the subscript “s” and “i” represent the specimen and indenter, respectively). Equation (1) yields Young's modules from the indentation curve (F–h curve) during the elastic deformation. The contact pressure distribution was also analytically investigated, and the maximum pressure p0 and mean pressure pm are derived from the following equation: Display Formula

(2)$p0=3F2πa2=32pm=(6E*2Fπ3R2)13$

a is contact area between the indenter and the specimen.

Subsequently, we consider the yield point when the plastic deformation starts. With the assumption of Tresca yield criterion, we use the relationship of maximum shear stress τ1 = 0.3 p0 from Hertzian contact theory. By considering τY = τY/2 for yield strength of each stress component, the following equation can be derived: Display Formula

(3)$σY=1.09(E*2FYπ3R2)13$

FY is the indentation force when the plastic deformation occurs. In this study, such a critical point is designated as yield indentation force (or critical indentation force) FY. In fact, several previous studies explore the determination of FY, when the indentation curve deviates from elastic deformation due to the onset of plastic deformation [33]. Regarding this method, the detail investigation including experiment will be discussed later.

This study conducts a shallow indentation, leading to elastically contact deformation (Hertzian contact), which can be simulated by Eq. (1). When we determine the critical point (yield indentation force FY), Eq. (3) leads to the yield strength directly. This estimation process is very simple and the verification will be investigated in Sec. 3.2. As mentioned before, general polymer material has difference in the yield strength between tension and compression. Thus, the above process (Eqs. (1) and (3)) may be used to estimate the compressive yield stress, and this question will be clarified via the following experiment in Sec. 3.2.

###### Experimental Result.

As described earlier, we estimate elastic modulus and yield strength on the basis of Hertzian theory. Figure 3 shows indentation curve of PC and PMMA. These show a very small hysteresis loop in the curve, suggesting that the elastic deformation be dominant (plastic deformation is very small). For the loading curve, Eq. (1) yields the reduced modulus E*. The elastic moduli of PC and PMMA were calculated from the experimental indentation curves of Fig. 3. Note that Poisson's ratios are set to be 0.35 for PC and 0.4 for PMMA [27]. Figure 4 show the calculated Young's moduli from Eq. (1) as a function of indentation depth for PC and PMMA. Although the value of modulus E should become constant theoretically and numerically2, the modulus E in this study is significantly changed at smaller depth < 1 μm. This may be due to the fact of existence of surface roughness and/or hardened surface layer of relevance to residual strain. Subsequently, the modulus decreases and is approaching to the constant value. As shown in the enlarged figures, the variated moduli in the range of 2–4 μm are averaged to be 2.2 GPa for PC and 2.99 for PMMA, which are closed to the experimental data of Fig. 1 (E = 2.0 GPa) and the referenced value (E = 3.71 GPa) [27]. Therefore, we estimated the present elastic modulus of 2.2 GPa for PC and 2.99 GPa for PMMA. Subsequently, these values are employed to Eq. (1) to describe their elastic indentation deformation. The calculated elastic deformation curves are drawn in Fig. 3 as plotted by blue solid lines. For both materials, during shallow indentation depth, the calculated curves agreed well with experimental ones. However, larger indentation force leads to the deviation of the two curves, showing the experimental curve becomes deeper. This indicates the specimen deforms plastically.

The next step is an estimation of yield strength. As shown in Fig. 3, when the specimen undergoes plastic deformation, the experimental curve deviates from elastic deformation curve (calculated by Hertzian equation). This deviation force is the critical yield indentation force FY. From Eq. (3) and FY, the yield strength σY can be estimated easily. In order to determine the critical indentation force FY, the experimental indentation curves is compared with the elastic deformation curve from the calculation of Eq. (1), through which we explore the yield point. Referred with Eq. (1), since the variable F is linear dependency of h3/2, Fig. 5 show the relationship between indentation force F and depth h3/2 (instead of indentation F–h curve) for both materials. It is found that the elastic deformation curves (from Eq. (1)) show a linearity, while the experimental curves slightly deviate. As shown in the enlarged figure, the experimental curve deviates from the calculated one. Here, the deviation point is set to be 1.0% as shown in this figure. For PC and PMMA, the critical indentation forces correspond to 460 mN and 800 mN, respectively. By substituting into Eq. (3), the yield strength is calculated to be 78.4 MPa for PC. This is very close to 70.9 MPa that is obtained from the uniaxial compression test in Fig. 1. Note that as shown in Fig. 2, the yield strength is strongly dependent on strain rate. Our indentation loading rate is set to 207 mN/s, resulting in equivalent strain rate becomes about 10−3 order 1/s (which was simulated from FEM computation). Such a strain rate corresponds to the present condition for compressive yield strength of 70.9 MPa as shown in Fig. 2. Similarly, the deviation point of PMMA (Fig. 5(c)) is 800 mN, and substituted to Eq. (3). This leads to the yield strength of 115 MPa, which is very similar with that of 98 MPa obtained from uniaxial compressive test [27]. Thus, this method based on Hertzian theory can estimate Young's modulus and compressive yield strength σYc readily. Section 4 discusses an estimation of tensile strength.

## Determination of Tensile Yield Strength

###### Material Model.

As mentioned in Sec. 1 and Fig. 1, we observed the difference of yield strength between in tension and in compression, which may come from molecular chain arrangements in response to different loading modes. Such a microscopic deformation mechanism is well known to be dependent on the hydrostatic stress level [3,16]. Thus, the Drucker–Prager yield criterion is often considered for engineering glassy polymers, since it can describe hydrostatic pressure sensitivity for the yield criterion. The Drucker–Prager yield criterion is given by Display Formula

(4)$f=ασii+16σij′σij′−σ¯3=0$

where $σii$ is the normal stress tensor (i.e., hydrostatic stress), $σij′$ is the deviated stress tensor (i.e., von Mises criterion), and $σ¯$ is the equivalent stress. At the yield point, $σ¯$ is set to the yield strength σY of the target material. Note that the σY may be intermediate value in between tension σYt and compression σYc. The parameter α is called the pressure-sensitivity index, and represents the difference of the yield strength in between tension σYt and compression σYc. The difference is often expressed by Display Formula

(5)$m=σYcσYt$

Here, Eq. (4) is specifically investigated at the yield point ($σ¯=σY$) for uniaxial compression and uniaxial tension. At the yield point in uniaxial compression state, Eq. (4) becomes Display Formula

(6)$σY=(1−3α)σYc$

In the contrast, uniaxial tension leads Eq. (4) to Display Formula

(7)$σY=(1+3α)σYt$

Thus, the following relationship can be derived: Display Formula

(8)$σYcσYt=1+3α1−3α$

It is noted that, as mentioned in Fig. 1, the target material behaves like elastic-perfectly plastic deformation (i.e., no work hardening). In addition, the present modeling considers time-independent deformation. Thus, our target model has two independent materials constant, i.e., σY (related to σYc or σYt) and α. Since the above section (based on Hertzian contact) can estimate compressive yield strength σYc, this section will estimate the parameter α. To establish an estimation method for the parameter α, FEM is carried out in Sec. 4.2.

###### Finite Element Model.

This study conducted the FEM of an indentation test with a spherical indenter. The two-dimensional axisymmetric finite element (FE) model was created to compute the response of the indentation test for saving computation cost, as shown in Fig. 6. The model comprises about 28,428 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. In the FEM model, the indenter is assumed rigid and radius is 500 μm. Computations were carried out using the commercially available software, Marc [34].

The FE models were used to conduct a comprehensive parametric study for the range of mechanical properties comprising many polymers. Table 1 shows the mechanical properties for the present parametric FEM study. In total, 36 models were computed. Note that Poisson's ratio is fixed at ν = 0.35 (since engineering polymers including polycarbonate are target materials).

###### Estimation Method.

First, we investigate the effect of material parameter on the indentation curves. The present materials model is described through Eqs. (4)(8) with two independent parameters (σYc or σYt, and α). Note that Eqs. (6) and (7) lead to σY from σYc and σYt, respectively. Thus, the effects of σY and parameter α on the indentation curves are investigated as shown in Fig. 7. Note that both figures include elastic deformation curve as indicated by open mark. Figure 7(a) shows the effect of σY with the fixed α = 0.05, indicating a larger yield stress promotes deformation resistance. Subsequently, Fig. 7(b) shows the effect of parameter α with the fixed σYc = 100 MPa. In fact, since Sec. 3.2 explains that indentation test dominantly measures compressive yield strength σYc, the σYc is fixed Fig. 7(b) in order to show the dependency of the parameter of α. It is found that the indentation curve is strongly dependent on the parameter α, when the material undergoes plastic deformation (after it deviates from elastic deformation curve). According to Eq. (4), an increase in parameter α makes the effect of hydrostatic stress larger. Thus, indentation contact leads to a negative hydrostatic stress (i.e., compression) in Eq. (4), resulting in the resistance of plastic deformation becomes larger. By using such a characteristic, we will establish the method to estimate two independent material parameters, especially the parameter α.

As mentioned previously, many studies have been done on the estimation of elastoplastic properties from the viewpoints of dimensionless function [8,3537]. 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 an estimation method of the parameter α in Eq. (4). In other words, the dimensionless function that correlates the indentation curve with parameter α to be identified will be established through the parametric FEM study. Given the indentation depth at h = 250 μm (h/R = 0.5,3 where R is indenter radius, 500 μm), the indentation force (F(h/R = 0.5)) can be expressed by

$F(h/R=0.5)=f(E*,σY,α,h,R,F˙)$

Dimensionless analysis with Π theory dictates the following relationship, when the σY and h are independent variables. Display Formula

(10)$F(h/R=0.5)σY⋅h2=Π(E*σY,α,Rh,F˙)$

In this study, the indentation depth h is fixed (h/R = 0.5) and the indentation loading rate $F˙$ is also fixed due to the experimental limitation. Thus, Eq. (10) is simplified to the following equation: Display Formula

(11)$F(h/R=0.5)σY⋅h2=Π(E*σY,α)$

Here, E* is the reduced Young's modulus and can be expressed by $(1/E*)=((1−νs2/Es)+(1−νi2/Ei))$, where the subscript “s” and “i” represent the specimen and indenter, respectively.

The data of parametric FE studies are introduced into Eq. (11). The result is shown in Fig. 8. It is found that the $(F(h/R=0.5)/σY⋅h2)$ increases monotonically with $(E*/σY)$. In addition, the $(F(h/R=0.5)/σY⋅h2)$ is strongly dependent on the parameter α. Therefore, the data of $(F(h/R=0.5)/σY⋅h2)$ is dependent on $(E*/σY)$ as well as the parameter α (as described in Eq. (11)). Therefore, the data of $(F(h/R=0.5)/σY⋅h2)$ can be approximated by a simple function:

Display Formula

(12)$F(h/R=0.5)σY⋅h2=A1⋅ln(E*σY)2+A2⋅ln(E*σY)+A3$

The coefficients A1A3 are listed in Table 2, where the coefficients for each α are shown individually with their correlation coefficients $R$. For a given parameter α, we interpolate against the coefficients A1A3 as described in the Appendix. By conducting an indentation experiment up to the depth of h/R = 0.5, the data of $F(h/R=0.5)$ is obtained, and substituted into the dimensionless function (Eq. (12)). The dimensionless functions (Eq. (12) and Table 2) yield one relationship between $(E*/σY)$ and α. When the σY and E* are known, the parameter α can be readily obtained.

We summarized the flowchart of our reverse analysis method. In Sec. 3.2, shallow indentation as elastic contact deformation mode is conducted. Based on Hertz theory, Eq. (1) leads to reduced Young's modulus E* and Eq. (3) provides the compressive yield strength σYc. Next, deep indentation test up to the depth of h/R = 0.5 (as full plastic deformation) is conducted, and Eq. (12) with Table 3 yields one relationship between $(E*/σY)$ and α. Note that Eq. (6) provides relationship between σYc and α theoretically. Therefore, the parameter α can be obtained. In conclusion, the present method directly identifies material parameters (E*, σY (or σYc), α) for mechanical characterization of engineering polymers.

## Validation of the Estimation Method

###### Numerical Experiment.

The present method is now applied to the numerical experiment in order to estimate the accuracy and robustness of our method (i.e., dimensionless function of Eq. (12)). As mentioned, parametric FEM study (Table 1) is carried out to establish the dimensionless functions. Reversibly, those data are employed to estimate the material properties (i.e., σY, and α) by using the proposed method mentioned above. For all cases, we extract the indentation force value at h/R = 0.5 (F(h/R=0.5)), and then estimate the material properties. Note that, for elastic modulus E and σYc, the setting parameters (for parametric FEM computation) were used, in which we assume to know them in prior, since these parameters are determined at the first step (see Sec. 3.2).

We investigated the accuracy of estimated properties (σY and α) by the method as shown in Fig. 9. 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 [3842]. 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 extracted properties to variations in the input data, and to clarify how the input data affects these properties [43,44]. In this study, we conduct sensitivity analysis for four representative materials (in Table 3). The perturbation cases of the indentation response (input data of F(h/R = 0.5)) are set as shown in Table 4. In fact, for the indentation curve, the perturbations for indentation curve are generally set to 4%, since previous studies have used this value for sensitive analysis [31,45]. In addition, the Young's modulus E and the compressive yield strength σYc can be estimated from the first step of estimation. As shown in Fig. 4, the estimation error is about 10%.4 Thus, these perturbation values are set to 10%. Table 4 shows the perturbation cases (designated as Cases 1–3).

The estimations are shown in Fig. 10 and input data is shown by open marks. The estimations show reasonable agreement and robust for all perturbation cases. Several previous studies addressing sensitivity analysis report that their methods have 50% error in maximum [31]. 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 Sec. 5.2.

###### Experimental Investigations.

Experimental validations were carried out. This study used micro indentation equipment as described in Sec. 2.2. For the first estimation step, shallow indentation was conducted to estimate Young's modulus E and compressive yield strength σYc. This was already described in Sec. 3.2 (see Figs. 4 and 5). Next is the second step, in which deep indentation test is conducted to achieve full plastic deformation. The indentation depth is set to h/R = 0.5 (see Fig. 8). This study used the spherical diamond indenter whose radius is 20 μm and the loading rate is 2.28 mN/s.5 Indentation curves of PC and PMMA are shown in Fig. 11. For each test, the number of tests was more than ten times, and the averaged curves are plotted in this figure including the standard deviation. We extracted the value of indentation force F(h/R = 0.5) from each experiment. The value is substituted to Eq. (12), yielding one relationship between σY and the parameter α. Figure 12 show the relationship between σY and the parameter α, in which the estimation is drawn by blue solid line. In this figure, other relationship is also plotted as black line, which is calculated from Eq. (6), where σY can connect to σYc and the parameter α theoretically. This is because the first step already estimates σYc. These two curves are independent, and then unique solution of σY and α can be obtained. The estimations are summarized in Tables 5 and 6, where it compares with the material properties obtained from uniaxial tension and compression test (Fig. 1 for PC and the reference for PMMA6 [27], respectively). It is found that there is very good agreement. Thus, our method is verified in the estimation of tensile and compressive yield strength of polymer materials.

In fact, this method requires two tests with two different indenter tip radii due to the present experimental limitation (e.g., load capacity and displacement accuracy). However, single indentation test at one time is essentially realized for our method, when we capture both small elastic deformation and large plastic deformation during a spherical indentation test.7 If the measurement accuracy and capacity of both load and displacement are sufficient, a single deep indentation is only needed to capture both elastic deformation and plastic deformation, which combines the two-step method. Therefore, our method can reduce to a single indentation test. Such an advantage can be emphasized, compared with other previous methods. Thus, our method is a relatively simple framework. Note that our method (estimation function) does not consider strain rate effect. However, we alternatively compared with uniaxial loading test at the similar rate condition (the strain rate of between indentation and uniaxial loading). The comparison shows good agreement as shown in Tables 5 and 6.

## Conclusions

This study proposes a new method to predict both tensile and compressive yield strength of polymeric materials within the frame of Drucker–Prager yield criterion utilizing spherical indentation. The method will be useful for engineering polymers that generally exhibit different yield strength in between tension and compression due to asymmetric arrangements of molecular structures to loading manners. The proposed method has two steps based on the indentation depth, where the shallow indentation is used to probe the elastic deformation and the deep indentation is described to full plastic deformation. The former part estimates Young's modulus and yield strength in compression. The latter test predicts the parameters embedded in Drucker–Prager yield criterion, thereby determining both yield strength of compression and tension. The latter method is established by extensive numerical computations via FEM so as to build the relationship between indentation curves and material parameters of yield criterion. Dimensionless analysis is employed to establish the relationship with simple formulae, through which, a reverse algorithm is proposed to extract the material properties. Finally, both numerical and experimental investigations (for PC and PMMA) are performed to verify the proposed method. It is concluded that our method provides an easy, quick and precise route for measuring the mechanical properties of polymer materials.

## Acknowledgements

The work of A.Y. is supported by JSPS KAKENHI (Grant No. 26420025) from the Japan Society for the Promotion of Science (JSPS) and Research Grant from The Suga Weathering Technology Foundation (SWTF) (No. 67). The work of H.Y. is supported by the Environment Research and Technology Development Fund (3K153006) of the Ministry of the Environment, Japan.

## Appendices

###### Appendix

As described in Table 2, the coefficients of Eq. (12) are dependent of the parameter α. The dependency is plotted in Fig. 13. In those figures, polynomial approximations are shown by solid lines. The approximation is given by

Display Formula

(A1)$A1,A2,A3=aα2+bα+c$

Three coefficients (a, b and c) are shown in Table 7. With Eq. (A1) and Table 7, the coefficient of Eq. (12) for any parameter α will be obtained.

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Donato, G. H. B. , and Bianchi, M. , 2012, “ Pressure Dependent Yield Criteria Applied for Improving Design Practices and Integrity Assessments Against Yielding of Engineering Polymers,” J. Mater. Res. Technol., 1(1), pp. 2–7.
Oliver, W. C. , and Pharr, G. M. , 2004, “ Measurement of Hardness and Elastic Modulus by Instrumented Indentation: Advances in Understanding and Refinements to Methodology,” J. Mater. Res., 19(01), pp. 3–20.
Oliver, W. C. , and Pharr, G. M. , 1992, “ An Improved Technique for Determining Hardness and Elastic- Modulus Using Load and Displacement Sensing Indentation Experiments,” J. Mater. Res., 7(06), pp. 1564–1583.
Dao, M. , Chollacoop, N. , VanVliet, K. J. , Venkatesh, T. A. , and Suresh, S. , 2001, “ Computational Modeling of the Forward and Reverse Problems in Instrumented Sharp Indentation,” Acta Mater., 49(19), pp. 3899–3918.
Chantikul, P. , Anstis, G. R. , Lawn, B. R. , and Marshall, D. B. , 1981, “ A Critical Evaluation of Indentation Techniques for Measuring Fracture Toughness—II: Strength Method,” J. Am. Ceram. Soc., 64(9), pp. 539–543.
Samadi-Dooki, A. , Malekmotiei, L. , and Voyiadjis, G. Z. , 2016, “ Characterizing Shear Transformation Zones in Polycarbonate Using Nanoindentation,” Polymer, 82, pp. 238–245.
Xu, B. , Yonezu, A. , Yue, Z. F. , and Chen, X. , 2009, “ Indentation Creep Surface Morphology of Nickel-Based Single Crystal Superalloys,” Comput. Mater. Sci., 46(2), pp. 275–285.
Xu, B. , Yue, Z. , and Wang, J. , 2007, “ Indentation Fatigue Behavior of Polycrystalline Copper,” Mech. Mater., 39(12), pp. 1066–1080.
Xu, B. , Yue, Z. , and Chen, X. , 2009, “ An Indentation Fatigue Depth Propagation Law,” Scr. Mater., 60(10), pp. 854–857.
Xu, B. X. , Yonezu, A. , and Chen, X. , 2010, “ An Indentation Fatigue Strength Law,” Philos. Mag. Lett., 90(5), pp. 313–322.
Toyama, H. , Niwa, M. , Xu, J. , and Yonezu, A. , 2015, “ Failure Assessment of a Hard Brittle Coating on a Ductile Substrate Subjected to Cyclic Contact Loading,” Eng. Failure Anal., 57, pp. 118–128.
Drucker, D. C. , 1973, “ Plasticity Theory, Strength-Differential (SD) Phenomenon, and Volume Expansion in Metals and Plastics,” Metall. Trans., 4(3), pp. 667–673.
Drucker, D. C. , and Prager, W. , 1952, “ Soil Mechanics and Plastic Analysis of Limit Design,” Appl. Math., 8, pp. 157–162.
Vena, P. , Gastaldi, D. , and Contro, R. , 2008, “ Determination of the Effective Elastic-Plasticresponse of Metal-Ceramic Composites,” Int. J. Plast., 24(3), pp. 483–508.
Bowden, P. B. , and Jukes, J. A. , 1972, “ The Plastic Flow of Polymers,” J. Mater. Sci., 7, pp. 52–63.
Quinson, R. , Perez, J. , Rink, M. , and Pavan, A. , 1997, “ Yield Criteria for Amorphous Glassy Polymers,” J. Mater. Sci., 32(5), pp. 1371–1379.
Voyiadjis, Z. G. , and Taqieddin, Z. N. , 2009, “ Elastic Plastic and Damage Model for Concrete Materials—Part I: Theoretical Formulation,” Int. J. Struct. Changes Solids – Mech. Appl., 1(1), pp. 31–59.
Giannakopoulos, A. E. , and Larsson, P. L. , 1997, “ Analysis of Pyramid Indentation of Pressure-Sensitive Hard Metals and Ceramics,” Mech. Mater., 25(1), pp. 1–35.
Vaidyanathan, R. , Dao, M. , Ravichandran, G. , and Suresh, S. , 2001, “ Study of Mechanical Deformation in Bulk Metallic Glass Through Instrumented Indentation,” Acta Mater., 49(18), pp. 3781–3789.
Fornell, J. , Concustell, A. , Surinach, S. , Li, W. H. , Cuadrado, N. , Gebert, A. , Baró, M. D. , and Sorte, J. , 2009, “ Yielding and Intrinsic Plasticity of TiZrNiCuBe Bulk Metallic Glass,” Int. J. Plast., 25(8), pp. 1540–1549.
Briscoe, B. J. , Flori, L. , and Pelillo, E. , 1998, “ Nanoindentation of Polymeric Surfaces,” J. Phys. D, 31(19), pp. 2395–2405.
Briscoe, B. J. , and Sebastian, K. S. , 1996, “ The Elastoplastic Response of Poly(Methylmetha-Crylate) to Indentation,” Proc. R. Soc. London A, 452(1946), pp. 439–457.
Seltzer, R. , Adrian Cisilino, P. , Patricia Frontini, M. , and Mai, Y.-W. , 2011, “ Determination of the Drucker–Prager Parameters of Polymers Exhibiting Pressure-Sensitive Plastic Behavior by Depth-Sensing Indentation,” Int. J. Mech. Sci., 53(6), pp. 471–478.
Bucaille, J. L. , Felder, E. , and Hochstetter, G. , 2002, “ Identification of the Viscoplastic Behavior of a Polycarbonate Based on Experiments and Numerical Modeling of the Nano-Indentation Test,” J. Mater. Sci., 37(18), pp. 3999–4011.
Kermouche, G. , Loubet, J. L. , and Bergheau, J. M. , 2008, “ Extraction of Stress–Strain Curves of Elastic–Viscoplastic Solids Using Conical/Pyramidal Indentation Testing With Application to Polymers,” Mech. Mater., 40(4–5), pp. 271–283.
Kwon, H. J. , Jar, P.-Y. B. , and Xia, Z. , 2004, “ Residual Toughness of Poly (Acrylonitrile-Butadiene-Styrene) (ABS) After Fatigue Loading-Effect of Uniaxial Fatigue Loading,” J. Mater. Sci., 39(15), pp. 4821–4828.
Inoue, N. , Yonezu, A. , Watanabe, Y. , Okamura, T. , Yondea, K. , and Xu, B. , 2015, “ Prediction of Viscoplastic Properties of Polymeric Materials Using Sharp Indentation,” Comput. Mater. Sci., 110, pp. 321–330.
Johnson, K. L. , 1985, Contact Mechanics, Cambridge University Press, Cambridge, UK.
Hirakata, H. , Ogiwara, H. , Yonezu, A. , and Minoshima, K. , 2010, “ Evaluation of Incipient Plasticity From Interfaces Between Ultra-Thin Gold Films and Compliant Substrates,” Thin Solid Films, 518(18), pp. 5249–5256.
Marc, 2011, “ Theory and User's Manual A 2011,” MSC Software, Santa Ana, CA.
Cheng, Y. T. , and Cheng, C. M. , 1998, “ Scaling Approach to Conical Indentation in Elastic-Plastic Solids With Work Hardening,” J. Appl. Phys., 84(3), pp. 1284–1291.
Cheng, Y. T. , Cheng, C. , and Scaling, M. , 2004, “ Dimensional Analysis, and Indentation Measurements,” Mater. Sci. Eng., R44(4–5), pp. 91–149.
Chen, X. , Ogasawara, N. , Zhao, M. , and Chiba, N. , 2007, “ On the Uniqueness of Measuring Elastoplastic Properties From Indentation: The Indistinguishable Mystical Materials,” J. Mech. Phys. Solids., 55(8), pp. 1618–1660.
Yonezu, A. , Kusano, R. , and Chen, X. , 2013, “ On the Mechanism of Intergranular Stress Corrosion Cracking of Sensitized Stainless Steel in Tetrathionate Solution,” J. Mater. Sci., 48(6), pp. 2447–2453.
Yonezu, A. , Yoneda, K. , Hirakata, H. , Sakihara, M. , and Minoshima, K. , 2010, “ A Simple Method to Evaluate Anisotropic Plastic Properties Based on Dimensionless Function of Single Spherical Indentation—Application to SiC Whisker Reinforced Aluminum Alloy,” Mater. Sci. Eng. A, 527(29–30), pp. 7646–7657.
Le, M.-Q. , 2009, “ Material Characterization by Dual Sharp Indenters,” Int. J. Solids Struct., 46(16), pp. 2988–2998.
Cao, Y. P. , and Lu, J. , 2004, “ Depth-Sensing Instrumented Indentation With Dual Sharp Indenters: Stability Analysis and Corresponding Regularization Schemes,” Acta Mater., 52(5), pp. 1143–1153.
Xu, B. , and Chen, X. , 2010, “ Determining Engineering Stress–Strain Curve Directly From the Load–Depth Curve of Spherical Indentation Test,” J. Mater. Res., 25(12), pp. 2297–2307.
Phadikar, J. K. , Bogetti, T. A. , and Karlsson, A. M. , 2013, “ On the Uniqueness and Sensitivity of Indentation Testing of Isotropic Materials,” Int. J. Solids Struct., 50(20–21), pp. 3242–3253.
Phadikar, J. K. , Bogetti, T. A. , and Karlsson, A. M. , 2014, “ Aspects of Experimental Errors and Data Reduction Schemes From Spherical Indentation of Isotropic Materials,” ASME J. Eng. Mater. Technol., 136(3), p. 031005.
Zhao, M. , Chen, X. , Yan, J. , and Karlsson, A. M. , 2006, “ Determination of Uniaxial Residual Stress and Mechanical Properties by Instrumented Indentation,” Acta Mater., 54(10), pp. 2823–2832.
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## References

Roesler, J. , Harders, H. , and Baeker, M. , 2007, Mechanical Behavior of Engineering Materials. Metals, Ceramics, Polymers, and Composites, Springer, Berlin.
Bower, D. I. , 2002, An Introduction to Polymer Physics, Cambridge University Press, New York.
Ward, I. M. , and Sweeney, J. , 2013, Mechanical Properties of Solid Polymers, 3rd ed., Wiley, Chichester, UK.
Raghava, R. , and Caddell, R. M. , 1973, “ The Macroscopic Yield Behavior of Polymers,” J. Mater. Sci., 8(2), pp. 225–232.
Donato, G. H. B. , and Bianchi, M. , 2012, “ Pressure Dependent Yield Criteria Applied for Improving Design Practices and Integrity Assessments Against Yielding of Engineering Polymers,” J. Mater. Res. Technol., 1(1), pp. 2–7.
Oliver, W. C. , and Pharr, G. M. , 2004, “ Measurement of Hardness and Elastic Modulus by Instrumented Indentation: Advances in Understanding and Refinements to Methodology,” J. Mater. Res., 19(01), pp. 3–20.
Oliver, W. C. , and Pharr, G. M. , 1992, “ An Improved Technique for Determining Hardness and Elastic- Modulus Using Load and Displacement Sensing Indentation Experiments,” J. Mater. Res., 7(06), pp. 1564–1583.
Dao, M. , Chollacoop, N. , VanVliet, K. J. , Venkatesh, T. A. , and Suresh, S. , 2001, “ Computational Modeling of the Forward and Reverse Problems in Instrumented Sharp Indentation,” Acta Mater., 49(19), pp. 3899–3918.
Chantikul, P. , Anstis, G. R. , Lawn, B. R. , and Marshall, D. B. , 1981, “ A Critical Evaluation of Indentation Techniques for Measuring Fracture Toughness—II: Strength Method,” J. Am. Ceram. Soc., 64(9), pp. 539–543.
Samadi-Dooki, A. , Malekmotiei, L. , and Voyiadjis, G. Z. , 2016, “ Characterizing Shear Transformation Zones in Polycarbonate Using Nanoindentation,” Polymer, 82, pp. 238–245.
Xu, B. , Yonezu, A. , Yue, Z. F. , and Chen, X. , 2009, “ Indentation Creep Surface Morphology of Nickel-Based Single Crystal Superalloys,” Comput. Mater. Sci., 46(2), pp. 275–285.
Xu, B. , Yue, Z. , and Wang, J. , 2007, “ Indentation Fatigue Behavior of Polycrystalline Copper,” Mech. Mater., 39(12), pp. 1066–1080.
Xu, B. , Yue, Z. , and Chen, X. , 2009, “ An Indentation Fatigue Depth Propagation Law,” Scr. Mater., 60(10), pp. 854–857.
Xu, B. X. , Yonezu, A. , and Chen, X. , 2010, “ An Indentation Fatigue Strength Law,” Philos. Mag. Lett., 90(5), pp. 313–322.
Toyama, H. , Niwa, M. , Xu, J. , and Yonezu, A. , 2015, “ Failure Assessment of a Hard Brittle Coating on a Ductile Substrate Subjected to Cyclic Contact Loading,” Eng. Failure Anal., 57, pp. 118–128.
Drucker, D. C. , 1973, “ Plasticity Theory, Strength-Differential (SD) Phenomenon, and Volume Expansion in Metals and Plastics,” Metall. Trans., 4(3), pp. 667–673.
Drucker, D. C. , and Prager, W. , 1952, “ Soil Mechanics and Plastic Analysis of Limit Design,” Appl. Math., 8, pp. 157–162.
Vena, P. , Gastaldi, D. , and Contro, R. , 2008, “ Determination of the Effective Elastic-Plasticresponse of Metal-Ceramic Composites,” Int. J. Plast., 24(3), pp. 483–508.
Bowden, P. B. , and Jukes, J. A. , 1972, “ The Plastic Flow of Polymers,” J. Mater. Sci., 7, pp. 52–63.
Quinson, R. , Perez, J. , Rink, M. , and Pavan, A. , 1997, “ Yield Criteria for Amorphous Glassy Polymers,” J. Mater. Sci., 32(5), pp. 1371–1379.
Voyiadjis, Z. G. , and Taqieddin, Z. N. , 2009, “ Elastic Plastic and Damage Model for Concrete Materials—Part I: Theoretical Formulation,” Int. J. Struct. Changes Solids – Mech. Appl., 1(1), pp. 31–59.
Giannakopoulos, A. E. , and Larsson, P. L. , 1997, “ Analysis of Pyramid Indentation of Pressure-Sensitive Hard Metals and Ceramics,” Mech. Mater., 25(1), pp. 1–35.
Vaidyanathan, R. , Dao, M. , Ravichandran, G. , and Suresh, S. , 2001, “ Study of Mechanical Deformation in Bulk Metallic Glass Through Instrumented Indentation,” Acta Mater., 49(18), pp. 3781–3789.
Fornell, J. , Concustell, A. , Surinach, S. , Li, W. H. , Cuadrado, N. , Gebert, A. , Baró, M. D. , and Sorte, J. , 2009, “ Yielding and Intrinsic Plasticity of TiZrNiCuBe Bulk Metallic Glass,” Int. J. Plast., 25(8), pp. 1540–1549.
Briscoe, B. J. , Flori, L. , and Pelillo, E. , 1998, “ Nanoindentation of Polymeric Surfaces,” J. Phys. D, 31(19), pp. 2395–2405.
Briscoe, B. J. , and Sebastian, K. S. , 1996, “ The Elastoplastic Response of Poly(Methylmetha-Crylate) to Indentation,” Proc. R. Soc. London A, 452(1946), pp. 439–457.
Seltzer, R. , Adrian Cisilino, P. , Patricia Frontini, M. , and Mai, Y.-W. , 2011, “ Determination of the Drucker–Prager Parameters of Polymers Exhibiting Pressure-Sensitive Plastic Behavior by Depth-Sensing Indentation,” Int. J. Mech. Sci., 53(6), pp. 471–478.
Bucaille, J. L. , Felder, E. , and Hochstetter, G. , 2002, “ Identification of the Viscoplastic Behavior of a Polycarbonate Based on Experiments and Numerical Modeling of the Nano-Indentation Test,” J. Mater. Sci., 37(18), pp. 3999–4011.
Kermouche, G. , Loubet, J. L. , and Bergheau, J. M. , 2008, “ Extraction of Stress–Strain Curves of Elastic–Viscoplastic Solids Using Conical/Pyramidal Indentation Testing With Application to Polymers,” Mech. Mater., 40(4–5), pp. 271–283.
Kwon, H. J. , Jar, P.-Y. B. , and Xia, Z. , 2004, “ Residual Toughness of Poly (Acrylonitrile-Butadiene-Styrene) (ABS) After Fatigue Loading-Effect of Uniaxial Fatigue Loading,” J. Mater. Sci., 39(15), pp. 4821–4828.
Inoue, N. , Yonezu, A. , Watanabe, Y. , Okamura, T. , Yondea, K. , and Xu, B. , 2015, “ Prediction of Viscoplastic Properties of Polymeric Materials Using Sharp Indentation,” Comput. Mater. Sci., 110, pp. 321–330.
Johnson, K. L. , 1985, Contact Mechanics, Cambridge University Press, Cambridge, UK.
Hirakata, H. , Ogiwara, H. , Yonezu, A. , and Minoshima, K. , 2010, “ Evaluation of Incipient Plasticity From Interfaces Between Ultra-Thin Gold Films and Compliant Substrates,” Thin Solid Films, 518(18), pp. 5249–5256.
Marc, 2011, “ Theory and User's Manual A 2011,” MSC Software, Santa Ana, CA.
Cheng, Y. T. , and Cheng, C. M. , 1998, “ Scaling Approach to Conical Indentation in Elastic-Plastic Solids With Work Hardening,” J. Appl. Phys., 84(3), pp. 1284–1291.
Cheng, Y. T. , Cheng, C. , and Scaling, M. , 2004, “ Dimensional Analysis, and Indentation Measurements,” Mater. Sci. Eng., R44(4–5), pp. 91–149.
Chen, X. , Ogasawara, N. , Zhao, M. , and Chiba, N. , 2007, “ On the Uniqueness of Measuring Elastoplastic Properties From Indentation: The Indistinguishable Mystical Materials,” J. Mech. Phys. Solids., 55(8), pp. 1618–1660.
Yonezu, A. , Kusano, R. , and Chen, X. , 2013, “ On the Mechanism of Intergranular Stress Corrosion Cracking of Sensitized Stainless Steel in Tetrathionate Solution,” J. Mater. Sci., 48(6), pp. 2447–2453.
Yonezu, A. , Yoneda, K. , Hirakata, H. , Sakihara, M. , and Minoshima, K. , 2010, “ A Simple Method to Evaluate Anisotropic Plastic Properties Based on Dimensionless Function of Single Spherical Indentation—Application to SiC Whisker Reinforced Aluminum Alloy,” Mater. Sci. Eng. A, 527(29–30), pp. 7646–7657.
Le, M.-Q. , 2009, “ Material Characterization by Dual Sharp Indenters,” Int. J. Solids Struct., 46(16), pp. 2988–2998.
Cao, Y. P. , and Lu, J. , 2004, “ Depth-Sensing Instrumented Indentation With Dual Sharp Indenters: Stability Analysis and Corresponding Regularization Schemes,” Acta Mater., 52(5), pp. 1143–1153.
Xu, B. , and Chen, X. , 2010, “ Determining Engineering Stress–Strain Curve Directly From the Load–Depth Curve of Spherical Indentation Test,” J. Mater. Res., 25(12), pp. 2297–2307.
Phadikar, J. K. , Bogetti, T. A. , and Karlsson, A. M. , 2013, “ On the Uniqueness and Sensitivity of Indentation Testing of Isotropic Materials,” Int. J. Solids Struct., 50(20–21), pp. 3242–3253.
Phadikar, J. K. , Bogetti, T. A. , and Karlsson, A. M. , 2014, “ Aspects of Experimental Errors and Data Reduction Schemes From Spherical Indentation of Isotropic Materials,” ASME J. Eng. Mater. Technol., 136(3), p. 031005.
Zhao, M. , Chen, X. , Yan, J. , and Karlsson, A. M. , 2006, “ Determination of Uniaxial Residual Stress and Mechanical Properties by Instrumented Indentation,” Acta Mater., 54(10), pp. 2823–2832.

## Figures

Fig. 1

True stress-true strain curves in uniaxial tension and compression test for polycarbonate (PC) at strain rate of 10−3 s−1

Fig. 2

Relationship between yield strength and strain rate obtained from tension and compression tests

Fig. 3

Indentation curves of shallow indentation using spherical indenter R = 500 μm for PC (a) and PMMA (b)

Fig. 4

Calculated Young's modulus with respected to indentation depth during loading process for PC (a) and PMMA (b)

Fig. 5

Relationships between indentation force and depth h3/2 for PC (a) and PMMA (b)

Fig. 6

Two-dimensional model of FEM computation for spherical indentation

Fig. 7

Examples of indentation curve, in which it investigates the effect of the yield strength σY (a) and the parameter α (b)

Fig. 8

Relationship between (F(h/R=0.5)/σY⋅h2) and (E∗/σr) at each parameter α

Fig. 9

Contour map of error distribution for σY (a) and differences (input value—estimation value) of parameter α (b)

Fig. 10

Comparison between the input value and the estimation from reverse analysis for representative materials for sensitivity analysis

Fig. 11

Indentation curves of deep indentation test using spherical indenter R = 20 μm for PC (a) and PMMA (b)

Fig. 12

Relationships between the yield strength σY and parameter α, which are from theory (Eq. (6)) and experimental estimation from Eq. (12) for PC (a) and PMMA (b).

Fig. 13

Changes in coefficients (A1, A2, and A3) in Table 2 with respect to parameter α

## Tables

Table 1 Mechanical property combination used in the FEM parametric study
Table 2 Coefficients of dimensionless function Eq. (12) with respect to the parameter α
Table 3 Representative materials for sensitivity analysis
Table 4 Case study for sensitivity analysis of representative materials
Table 5 Estimation errors of between the present indentation method and uniaxial loading test for PC
Table 6 Estimation errors of between the present indentation method and uniaxial loading test for PMMA
aReference [27].
Table 7 Coefficients of Eq. (A1)

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